Chapter 1 Photonic Crystals: Properties and Applications

Chapter 1

Photonic crystals: properties and applications

This chapter is intended as a brief overview of the history, concepts, characteristics and applications of photonic crystals (PhCs). A single chapter is insufficient to review a field that continues to grow almost exponentially in the literature almost 20 years after it began, so attention is given to aspects most relevant to the work presented in the remainder of this thesis. Sections 1.1–1.4 introduce the concepts of one-, two-, and three-dimensional photonic crystals and bandgaps and the motivation that led to their development. The theory of band structures and Bloch modes of uniform two-dimensional photonic crystals and photonic crystal slabs is then considered in Sec- tions 1.5 and 1.6. Section 1.7 is concerned with the properties and applications of defects in PhCs, and specifically those designed for resonant behaviour and waveguiding. Inte- gration of these two functions into photonic devices is discussed in the context of two basic operations: coupling and add/drop filtering. Finally, Section 1.8 focuses on a different class of applications, namely those that exploit the unusual dispersion properties of the Bloch modes that exist in uniform photonic crystals.

1.1 Introduction

The optical properties of periodic structures can be observed throughout the natural world, from the changing colours of an opal held up to the light to the patterns on a butterfly’s wings. Nature has been exploiting photonic crystals for millions of years [1], but humans have only recently started to realise their potential. One-dimensional periodic structures in the form of thin film stacks have been studied for many years [2], but three-dimensional photonic crystal were first proposed by Yablonovitch [3] and John [4] in 1987. Yablonovitch proposed that three-dimensional periodic dielectric structures could exhibit an electromagnetic bandgap - a range of frequencies at which light cannot propagate through the structure in any direction. He also predicted that unwanted spontaneous emission within a semiconductor can be prevented by structuring the ma-

5 6 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS terial so that the frequencies of these emissions fall within a photonic bandgap; since no propagating states exist at that frequency, emission is effectively forbidden. John [4] showed that many of the properties of PhCs survive even when the periodic lattice becomes disordered. In such structures, if the index contrast is sufficiently large, strong light localisation can still occur, in analogy to the electronic bandgaps of amorphous semiconductors. Perhaps more relevant to much of the PhC research that has followed, and to the topic of this thesis, was Yablonovitch’s [3] interpretation of the cavity modes that can be introduced into a periodic structure by creating a defect or “phase slip”. While resonant cavities in distributed feedback lasers had had already been demonstrated using this approach [5], Yablonovitch showed that modes could be localised in three-dimensions and explained the effect in terms of defect states in the photonic bandgap. From this observation, and the initial proposals for limiting spontaneous emission, the concept of controlling light with periodic structures has developed rapidly into a topic of worldwide research. Bandgaps in periodic materials were already well understood from solid-state physics, where the presence of electronic bandgaps in semiconductors has revolutionised electronics. Many of the concepts from solid-state research have been carried over to photonic crystals including the notation and nomenclature, and perhaps this is what has allowed the field to make such rapid progress in less than twenty years.

1.2 One-dimensional photonic crystals

Although the term photonic crystal (PhC) is relatively recent, simple one-dimensional (1D) PhCs in the form of periodic dielectric stacks have been used for considerably longer [2]. Their wavelength-selective reflection properties see them used in a wide range of applications including high-efficiency mirrors, Fabry-Pérot cavities, optical filters and distributed feedback lasers. As illustrated in Fig. 1.1, the simplest PhC is an alternating stack of two different dielectric materials. When light is incident on such a stack, each interface reflects some of the field. If the thickness of each layer is chosen appropriately, the reflected fields can combine in phase, resulting in constructive interference and strong reflectance, also known as Bragg reflection. In contrast to two- and three-dimensional PhCs, 1D Bragg reflection occurs regardless of the index contrast, although a large number of periods is required to achieve a high reflectance if the contrast is small. Since the absorption in dielectric optical materials is very low, mirrors made from dielectric stacks are extremely efficient, and can be designed to reflect almost 100% of the incident light within a small range of frequencies. The main limitation of these dielectric mirrors is that they only operate for a limited range of angles close to normal incidence. Another, more recent application of 1D PhCs is the fibre Bragg grating (FBG), in which the refractive index of the fibre core is varied periodically along its axis, typically approximating a sinusoidal profile. This case is somewhat more complex because the 1.3. TWO-DIMENSIONAL PHOTONIC CRYSTALS 7

d z y x Figure 1.1: Schematic of a one-dimensional photonic crystal consisting of a periodic stack of dielectric layers with period d. refractive index varies continuously, rather than discretely, as in the previous example, but the properties are essentially the same. The main difference is that the refractive index contrast in the FBG is so small (∆n ≤ 0.5%) that the operational bandwidth is very narrow and thousands of periods are typically required to obtain the desired reflectance properties. FBGs are now an integral part of fibre optic systems, being used in dispersion compensation, filters, and a wide range of other applications [6].

1.3 Two-dimensional photonic crystals

Both two-dimensional (2D) and three-dimensional (3D) PhCs can be thought of as generalisations to the 1D case where a full 2D or 3D bandgap appears only if the 1D Bragg reflection condition is satisfied simultaneously for all propagation directions in which the structure is periodic. For most 2D periodic lattices this occurs providing the index contrast is sufficiently large, but for 3D structures only certain lattice geometries display the necessary properties, and only then for large enough index contrasts (Ref. [7], Chapter 6). The 3D case is discussed in more detail in Section 1.4. Instead of a stack of uniform dielectric layers, 2D PhCs typically consist of an array of dielectric cylinders in a homogeneous dielectric background material, as illustrated in Fig. 1.2, although there are many other possible geometries. If the refractive index contrast between the cylinders and the background is sufficiently large, 2D bandgaps can occur for propagation in the plane of periodicity — perpendicular to the rods. Light at a frequency within the bandgap experiences Bragg reflection in all directions due to the periodic array of cylinders. However, as in the 1D case where light could still propagate in two-dimensions, in a 2D PhC propagation can still occur in the non-periodic direction, parallel to the cylinders. Thus, an alternative means of confinement is required in the third dimension to avoid excessive losses due to diffraction and scattering. This issue is discussed further in Section 1.6. As in semiconductor devices, much of the interest in photonic crystals arises not 8 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

(e)

(a) (b)

Figure 1.2: (a) Schematic of a 2D hole-type PhC slab consisting of low refractive index cylinders in a high-index slab. (b) Scanning electron microscope (SEM) image of a fabricated hole-type PhC in a silicon slab (from Ref. [8]). (c) Schematic of a 2D rod-type PhC consisting of high refractive index cylinders in a low-index background. (d) SEM image of a fabricated rod-type PhC formed from GaAs rods on a low-index aluminium oxide layer (from Ref. [9]). from the presence of a bandgap alone, but rather from the ability to create localised defect states within the bandgap by introducing a structural defect into an otherwise regular lattice. For example, the removal of a single cylinder from a 2D PhC creates a point-like defect or resonant cavity, and the removal of a line of cylinders can create a waveguide that supports propagating modes. Many potential applications based on this concept have been proposed and demonstrated, a number of which are discussed in more detail in Section 1.7. A second class of 2D PhC applications exploits the unique properties of the propagating modes that exist outside the bandgaps in defect-free PhCs. The discrete translational symmetry of PhCs imposes strict phase conditions on the ﬁeld distributions that they support. As a result, only a discrete number of modes are supported for any given frequency and light propagating in these Bloch modes can have very diﬀerent properties to light in a homogeneous medium. We review some of the potential applications related to this in Section 1.8. 1.4. THREE-DIMENSIONAL PHOTONIC CRYSTALS 9 1.4 Three-dimensional photonic crystals

Three-dimensional PhCs have proved to be the most challenging PhC structures to fabricate. Whereas 2D PhC research has gained significant benefit from well-established 1D PhC thin-film and semiconductor processing technology such as plasma deposition and electron-beam lithography, fabrication of 3D PhCs has required the development of entirely new techniques. For this reason, it was more than three years after the initial proposal for 3D band gap materials [3, 4] before a structure was calculated to exhibit a bandgap for all directions and all polarizations [10]. The design consisted of dielectric spheres positioned at the vertices of a diamond lattice. This followed experimental reports the previous year in which a partial bandgap in a face-centred- cubic (FCC) lattice of spheres was mistakenly identified as a complete bandgap [11]. This latter result highlighted the requirement for rigorous theoretical and computational tools capable of dealing with high-index contrast dielectrics.

Since these early studies, a wide range of 3D PhC geometries exhibiting complete bandgaps have been demonstrated both in theory and experiment. As an example, a 3D “woodpile” PhC is shown in Fig. 1.3. Due to the challenges involved in fabricating high-quality structures with features on the scale of optical wavelengths, early photonic crystal experiments were performed at microwave and mid-infrared frequencies [12– 14]. With the improvement of fabrication and materials processing methods, smaller structures have become feasible, and in 1999 the ﬁrst 3D PhC with a bandgap at telecommunications frequencies was reported [15]. Since then, various lattice geometries have been reported for operation at similar frequencies [16–18].

Waveguiding and the introduction of intentional defects in 3D PhCs has not progressed as rapidly as in 2D PhCs, largely due to the fabrication diﬃculties and the more complex geometry required to achieve 3D bandgaps. Theoretical studies have demonstrated the potential for novel photonic circuit designs [19, 20], but to date only a few experimental results have been reported [21, 22]. Although much of the recent interest in PhCs has focused on telecommunication related applications, the original concept of controlling spontaneous emission has not been forgotten. Recent experiments have demonstrated both inhibition and enhancement of spontaneous emission from quantum dots embedded in both in 2D [23] and 3D [21, 24] PhCs. The presence of a photonic bandgap at black-body radiation frequencies has also been shown to modify the thermal emission properties of heated tungsten 3D PhCs [25,26].

3D PhCs formed in low-index contrast materials such as silica or polymer are also potentially useful for applications such as those described in Section 1.8 where a complete bandgap is not required. Superprism eﬀects have been calculated in 3D polymer PhCs [27], and tunable bandgap eﬀects have also been demonstrated using both nonlinear [28] and liquid crystal tuning [29,30]. 10 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

(a) (b)

Figure 1.3: An example of a 3D PhC woodpile structure known to exhibit a full photonic bandgap. (a) Schematic of an ideal woodpile PhC. (b) SEM image of a real 3D woodpile structure fabricated in silicon (from Ref. [14]).

1.5 Band structure and Bloch modes of 2D photonic crystals

The remainder of this thesis is concerned with 2D PhCs and their application to optical processing and photonic integrated circuits. In this section we review the properties of the 2D band structure and associated Bloch modes of uniform PhC lattices. A truly 2D PhC is invariant in the direction parallel to the cylinder axis, and thus has cylinders of infinite length. For a rigorous analysis of such a structure, the out-of-plane wavevector component must be included in a full 3D calculation, but this can be im- practical for large structures given the computational demands of 3D calculations. In a 2D calculation only in-plane propagation is considered, but there are some simple modifications that can be made to correct partially for out-of-plane wavevector components. Although real PhC structures have finite-length cylinders and usually rely on a slab waveguide geometry to prevent out-of-plane losses, the underlying physics is the same, and significant insight can be obtained by considering the ideal case. Unless otherwise stated, the structures studied throughout this thesis are treated as 2D and the light is taken to be propagating in the plane of the PhC. The implications of this approximation are discussed in general terms in Section 1.6.4, and more specifically in the chapters relating to each of the devices.

1.5.1 Band structure calculation Figures 1.4(a) and (d) show two typical 2D PhC geometries consisting of a square and a triangular array of cylinders of radius r and refractive index ncyl embedded in a background dielectric of refractive index nb. The choice of index contrast (ncyl > nb or ncyl < nb) has a number of important consequences, not only from the point-of-view of fabrication, but also in terms of fundamental properties. To distinguish between these 1.5. BAND STRUCTURE AND BLOCH MODES OF 2D PHOTONIC CRYSTALS11 two geometries, we will refer to the former as rod-type and the latter as hole-type PhCs. Examples of both are shown in Fig. 1.2. Many of the techniques for solving quantum mechanical problems in solid state physics can be used for electromagnetic ﬁelds by casting Maxwell’s equations in the form of an eigenvalue equation in terms of either the electric (E) or magnetic (H) ﬁeld

2 ω 2 ∇ E = − c εE, (1.1) 1 ω 2 ∇ × ε ∇× H = c H, (1.2) where ε = ε(x, y, z) is the dielectric constant, which may depend also on frequency, and c is the speed of light. A time dependence of exp(−iωt) has been included implicitly and we assume that ε is everywhere real and positive, and that µ = µ0 corresponding to a lossless, non-magnetic material. Whereas plane wave based numerical methods (see Section 1.9) typically solve Eq. (1.2), the Bloch mode matrix method described in Chap- ter 2 involves the solution of Eq. (1.1). We discuss here the band structure calculations in terms of Eq. (1.1), although the conclusions are equally valid for Eq. (1.2). In Eq. (1.1) the eigenvalue term (ω/c)2ε is analogous to the energy eigenvalue in Schrödinger’s equation describing the propagation of electrons in a potential. For a 2D PhC such as those illustrated in Fig. 1.2, the periodic modulation of the dielectric constant can be written expressed as ε(r) = ε(r+lpq). Here, lpq = pe1 +qe2 is a general lattice vector defined in terms of the basis vectors e1 and e2, and r = r(x, y) is a vector in the plane perpendicular to the cylinders. The propagation of a wave in such a periodic medium is governed by the Floquet-Bloch theorem which states that the solutions to Eq. (1.1) correspond to plane wave fields modulated by a periodic function. These Bloch modes have the form

ik0.r Emk(r) = e umk(r), (1.3) where umk(r) has the periodicity of the PhC, i.e. umk(r) = umk(r + lpq). Thus, for a given Bloch vector k0 = (kx, ky), the eigenvalue equation (1.1) can be solved to yield a collection of Bloch mode eigenfunctions Emk and corresponding eigenvalues ωm(k0). When plotted over reciprocal space, the ωm(k0) form discrete bands, where each band is a continuous function of k0, indexed by m = 1, 2, 3 ... with increasing frequency as seen in Fig. 1.5. A 2D bandgap occurs at frequencies where no solutions exist for all possible (real) k0-vectors. Note that an equivalent calculation for the modes of a uniform dielectric of refractive index n yields a conical band surface defined by 2 2 (ω/c) = (k0/n) , where k0 is the free-space wavevector. The eigensolutions are also periodic functions of k0 on the reciprocal lattice defined by the basis vectors b1 and b2, where ei.bj = 2πδij, as shown in Figs. 1.4(b) and (e). Thus, to characterise the band structure completely, it is necessary only to find solutions for k0 vectors within the first Brillouin zone, defined as the region of reciprocal space centred on k0 = 0 in which any two wavevectors are separated by less than a reciprocal lattice vector. The first Brillouin zones of a square and a triangular lattice are illustrated in Figs. 1.4(c) and (f) respectively. 12 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

For lattices with rotational and/or mirror symmetry in addition to translational symmetry, the complete band structure can be obtained by applying these symmetry operations to the solutions in an even smaller region of reciprocal space - the irreducible Brillouin zone, indicated by the white triangles in Figs. 1.4(c) and (f). The origin of the Brillouin zone is labelled as the Γ-point in both cases, while the other two vertices are labelled M and X for the square lattice and M and K for the triangular lattice. For many applications it is suﬃcient to calculate only the modes corresponding to k0 vectors lying on the edge of the irreducible Brillouin zone, since the local minima and maxima of the band structure tend to lie on the symmetry axes and at the high- symmetry points of the Brillouin zone. Thus, a typical band diagram for a 2D PhC shows only the band structure along the edges of the irreducible Brillouin zone between each pair of labelled symmetry points, as can be seen in Figs. 1.5(c) and (d). From such a diagram, the position and width of all bandgaps can be determined, along with many other characteristic features of the band structure. In Section 1.5.3 we consider the two most common lattice types used for 2D PhCs – square and triangular – and give examples of the band structure for these lattices.

1.5.2 TE and TM modes In the analysis described in Section 1.5.1, no assumptions are made about the polarisation of the electromagnetic fields. However, in the case of a 2D PhC, where propagation is restricted to a plane perpendicular to the cylinders, the solutions decouple into two distinct polarisation states - transverse electric (TE) and transverse magnetic (TM). For TE (TM) modes, the electric (magnetic) field lies in the xy-plane, while the magnetic (electric) field is aligned with the z-axis, along the cylinders. In general the band structures of the TE and TM states are quite different due to the boundary conditions imposed at the dielectric interfaces. Therefore most 2D PhCs are designed for operation in a single polarisation state. The choice depends largely on the geometry of the PhC and the desired properties; in rod-type geometries, the largest bandgaps tend to occur for TM polarisation, so for bandgap applications this is usually the best choice; the reverse is the case in hole-type PhCs where TE polarised states tend to have the largest bandgaps (Ref. [7], Chapter 5).

1.5.3 Band structures of square and triangular lattices The ﬁrst 2D PhC structures found to exhibit photonic bandgaps were square and triangular lattices of cylinders [31,32]. These geometries are still the most common, and the ones considered in this thesis. Systematic studies of both rod- and hole-type geometries have been undertaken to identify the optimal choice of cylinder radius, index contrast and lattice type in order to maximise the frequency range of the bandgap for either one or both polarisations [33, 34]. In all cases, there is a minimum refractive index contrast below which gaps do not exist for either polarisation, and not all geometries 1.5. BAND STRUCTURE AND BLOCH MODES OF 2D PHOTONIC CRYSTALS13

Real lattice Reciprocal lattice Brillouin zone (a) (b) (c) b1 e2 M G X e1 Square b2 d

y ky

x kx (d) (e) (f)

d b1 M K e1 2p G e2 d Triangular b2

Figure 1.4: (a) and (d): Diagrams of square and triangular PhC lattices in real space generated by the basis vectors e2 and e2. (b) and (e): The corresponding reciprocal lattices and basis vectors b1 and b2. The boundaries of first Brillouin zones are indicated by the dotted lines. (c) and (f): Representation of the first Brillouin zones and the irreducible region (white triangles) of the two lattices. have overlapping TE and TM bandgaps, regardless of the index contrast. Figure 1.4 shows a square and a triangular lattice represented in both real and reciprocal space along with the basis vectors used to generate any lattice vector. The first Brillouin zone of each lattice type is shown in parts (b), (c), (e) and (f) of the figure. In (c) and (f), the irreducible Brillouin zones are defined by the lines of symmetry joining the high symmetry points, which are labelled according to convention. As an example, we consider here the band structures of two typical PhC geometries: the first is a rod-type PhC consisting of a square lattice of silicon cylinders of index ncyl = 3.4 and radius r = 0.25 d embedded in silica with index nb = 1.46, and the second is a hole-type PhC formed by a triangular lattice of air holes (ncyl = 1) of radius r = 0.32 d in silicon (nb = 3.4). When calculating the properties of PhCs, it is standard practice to express all spatial dimensions relative to the lattice period d as this allows structures to be scaled for operation at any desired frequency, although material dispersion must be taken into account for large changes of scale. Thus, the cylinder radius is typically written as r/d and the wavelength as λ/d. The dimensionless frequency d/λ is equivalent to the parameter ωd/2πc used in some texts. 14 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

Figure 1.5(a) shows the first four TM polarised bands (see Section 1.5.2) of the square lattice PhC plotted as a function of frequency over first Brillouin zone. Since the eigenvalues ωm(k0) are continuous functions of k0 within each band, they form band surfaces. Observe that there is a bandgap between the first and second bands. Figure 1.5(b) is a contour plot of the second band showing the equifrequency, or isofre- quency contours over the Brillouin zone. Visualising the bands in this manner can be informative for some applications, but as mentioned in Section 1.5.1, it is often sufficient to calculate the bands along the perimeter of the irreducible Brillouin zone, indicated by the dashed line. Figure 1.5(c) shows such a band diagram for the same PhC, for both the TE and TM polarised modes. The TM bandgap is clearly seen over the normalised frequency range of 0.249 < d/λ < 0.298, but there is no equivalent gap for TE polarisation. Figure 1.5(d) shows the band diagram for the triangular lattice PhC with the parameters given earlier in this section. In this case, there is a large bandgap for

TE polarisation between 0.216 < d/λ < 0.298 but none for TM.

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Figure 1.5: (a) TM band surfaces for a 2D square rod-type photonic crystal with the parameters deﬁned in the text. (b) Equifrequency contour plot for the second band in (a). (c) Band diagram for the same structure showing TE and TM modes on the edge of the irreducible Brillouin zone. (d) Band diagram for the 2D triangular hole-type photonic crystal with the parameters given in the text.

For bandgap applications, the position and size of the gap are usually the critical 1.6. PHOTONIC CRYSTAL SLABS 15 parameters, while the details of the band structure above and below the gap are much less important. Recently, however, the unusual propagation properties of the Bloch modes have been attracting renewed attention and a number of novel applications have been proposed. These applications require the band surfaces to be engineered to have very speciﬁc properties, so a full calculation of the bands over the Brillouin zone is required. We discuss some of these in-band applications in Section 1.8.

1.6 Photonic crystal slabs

We have so far discussed only the properties of purely 2D PhC geometries which are invariant along the z-axis, and thus have cylinders of infinite length. This approximation is valid if the cylinders are many wavelengths long, but for practical purposes this is not always desirable or possible to achieve. Even in geometries that can be considered as purely 2D, diffraction losses are high since light is free to propagate out of the xy- plane even when there is a bandgap for in-plane propagation. An exception to this are photonic bandgap fibres in which light propagates almost parallel to the cylinders and is confined to a central core by the periodic cladding [35]. But for most other applications, a method is required to confine the light in the out-of-plane direction within a 2D PhC of finite height. One way to achieve this is to cap the slab above and below with another PhC structure that has a bandgap for propagation in the z- direction. Suggestions have been made for 1D [36, 37], 2D [38] and 3D [39] capping layers, but to date these approaches have not been demonstrated experimentally. A more common technique is to confine the light in the slab using total internal reflection (TIR) at the slab/cladding interface as in a planar dielectric waveguide. This approach was proposed in 1994 [40], and demonstrated experimentally in 1996 [41], but was not studied rigorously until 1999 [42] for uniform PhC slabs and later for PhC waveguides [38, 43, 44]. To achieve TIR, the refractive index of the cladding must be lower than that of the slab, or in the case of rod-type PhCs, the high-index rods. A bulk dielectric such as air or silica is typically used for the low index cladding material, as illustrated in the examples of Fig. 1.2. A more detailed discussion of slab design in both rod- and hole-type PhCs is provided in Sections 1.6.2 and 1.6.3.

1.6.1 The lightline

As with any index-guiding structure, for light to be conﬁned within the slab, it must satisfy a total internal reﬂection condition to ensure that light in the core does not couple to the radiation modes of the cladding. The dispersion relationship for a mode in a homogeneous dielectric cladding of refractive index ncl is given by

ω 2 n = k2 + k2, (1.4) cl c 0 z 16 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

where k0 = (kx, ky) is the wavevector component in the plane of the slab. Thus, 2 2 if k0 > (nclω/c) , then kz is imaginary and the ﬁeld in the cladding is evanescent, 2 2 corresponding to TIR. If k0 < (nclω/c) however, kz is real and light can propagate 2 in the cladding. When plotted in k − ω-space, the relation k0 = (nclω/c) deﬁnes the light cone, as illustrated by the white cone in Fig. 1.6. In a conventional band diagram this appears as a lightline, such as those in Fig. 1.7. Above the lightline a continuum of propagating modes exists in the cladding, whereas no propagation is allowed below. Therefore, modes of the PhC that lie inside the cone (above the lightline) can couple to the radiation modes and leak into the cladding but modes that lie below the lightline decay exponentially in the cladding and are guided within the slab.

0.8 0.7 0.6 0.5

d/ 0.4 0.3 0.2 M K 0.1 2p 3 0 G 0 k -4p -2p y 3 0 4p 3 kx 3

Figure 1.6: Band surface plot showing the ﬁrst four bands of the hole-type PhC considered in Section 1.5.3, but with a ﬁnite slab of height 0.4 d surrounded by air, as in Fig. 1.5(d). The white surface centred at Γ represents the light cone. Modes outside the cone are guided in the slab, while those inside the cone can radiate into the cladding.

The band properties of a PhC slab are in many ways similar to those of the 2D PhCs discussed in Section 1.5.3, but a number of additional effects must also be considered including the slab thickness, refractive index contrast and symmetry properties of the slab and the cladding. One of the most important differences is the effect of out-of-plane propagation on the mode structure. In the 2D analysis, the TE and TM polarised modes are decoupled and can be treated separately, but this is not the case for the modes of a PhC slab. However, if the slab and cladding are symmetric about the z = 0 plane, 1.6. PHOTONIC CRYSTAL SLABS 17

these modes can be classified according to the symmetry of the Hz field component with respect to the reflection plane. In this case, the TE and TM modes of a purely 2D structure map onto even and odd slab modes respectively and bandgaps can exist for one or other of these symmetries. These are not true bandgaps because light can always propagate above the lightline in cladding modes. However they correspond to gaps in the spectrum of guided modes, and thus share many of the properties of 2D photonic bandgaps.

1.6.2 Hole-type photonic crystal slabs The fabrication and experimental demonstration of PhC-based devices has progressed much more rapidly for hole-type PhCs than for rod-type structures, despite both geometries receiving almost equal attention in theoretical studies. This difference is in part due to the fabrication challenges involved in forming circular rods with smooth, vertical sides, but is also related to the issue of vertical confinement. In a hole-type PhC, the high-index material surrounding the holes can also serve as a high-index slab for TIR guidance, but in rod-type PhCs, where the high-index regions are separated from one another, alternative methods are required to achieve TIR while maintaining structural integrity. This issue is discussed further in Section 1.6.3. There are two cladding designs that are used most for hole-type PhC slabs. The first of these consists of a high refractive index PhC slab on top of a lower refractive index substrate with an air cladding above. Such structures are relatively easy to fabricate using existing thin film deposition techniques to lay down the multiple layers [41, 45], but the resulting asymmetric slabs do not allow for the simple splitting of the odd and even modes. More recently, methods have been developed to fabricate thin membranes of PhC material suspended in air, producing symmetric air-clad PhC slabs, also known as membrane-type or air-bridge PhCs. High-quality low-loss PhCs have been fabricated in both Si and GaAs slabs using this method [8,46]. Figure 1.7(a) shows the band diagram calculated for the same photonic crystal as in Fig. 1.5(d), but now for a finite slab of height 0.4 d with air above and below. The corresponding band surfaces are shown in Fig. 1.6. Only the modes below the lightline are plotted, while the region above the lightline is shaded light grey and contains a continuum of radiation modes. The even mode bandgap in the range 0.292 ≤ d/λ ≤ 0.388 is shaded dark grey and corresponds to the TE bandgap in the 2D calculation, although it has shifted to higher frequencies.

1.6.3 Rod-type photonic crystal slabs Rod-type PhC slab geometries are inherently diﬀerent from hole-type structures because the regions of high refractive index are not connected, and thus the rods must be attached to a substrate material. Index guidance at the bottom of the rods is achieved by choosing a substrate of lower refractive index, or even etching the rods into the 18 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

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0.6 0.4 0.5 0.3 0.4

0.3 d d 0.2 0.2 odd 0.1 odd 0.1 even even 0.0 0.0 Γ M K Γ Γ X M Γ (a) (b)

Figure 1.7: (a) Band diagram of a hole-type PhC slab with the same parameters as in Fig. 1.5(d) and a slab height of 0.4 d. The cladding above and below the slab is air. (b) Band diagram of a rod-type PhC with the same parameters as in Figs. 1.5(a)–(c), but for cylinders of height 1.5 d surrounded above and below by a silica cladding. Only the guided modes below the cladding lightline are plotted. The regions shaded light grey lie above the lightline and contain a continuum of cladding modes.

substrate so that the index of the rods is lower at the bottom than at the top. If the rods are left surrounded by air, then index confinement at the top of the rods is provided by the rod-air interface [47]. The main drawback of this geometry is the strong asymmetry due to the different cladding materials above and below the rods. One solution is to fill the region between and above the rods with a low-index material that matches the substrate index, thus creating an up-down symmetric array of finite rods analogous to the hole-type PhC membrane structure [48, 49]. A similar design was proposed by Martinez et al., [50] in which two different materials were used – one to fill the spaces between the rods, and another with a slightly lower index for the cladding above and below. However it is not clear whether such a small index contrast contributes significantly to the guidance, given the already strong contrast between the rods and the cladding. In these last two approaches, it is important that a sufficiently large index contrast is maintained between the rods and the surrounding material to ensure that the in-plane bandgap is not destroyed.

Figure 1.7(b) shows the band diagram calculated for the same photonic crystal as in Figs. 1.5(a)–(c), but now for ﬁnite rods of height 1.5 d with the SiO2 background extending above and below. The bandgap for odd-symmetry modes in the range 0.293 ≤ d/λ ≤ 0.314 corresponds to the TM bandgap in the 2D calculation, again shifted to higher frequencies. 1.6. PHOTONIC CRYSTAL SLABS 19

1.6.4 2D vs 3D modelling of PhC slabs

Rigorous theoretical analysis of PhC slabs requires full 3D modelling tools, but these involve intensive numerical calculations that can take days or weeks to run on a desktop computer, or may be impossible due to memory requirements. Although parallel computing methods and access to supercomputers may be an option, significant results and insight can be obtained using 2D methods. When operating below the lightline, the main difference between 2D and 3D calculations is a frequency shift in the band structure. This occurs because the band structure is largely determined by the wavevector component k0 that lies in the plane of the PhC, rather than the total k. Since k0 = k − kz, out-of-plane propagation has the effect of reducing k0 for a given frequency. Thus, band features associated with a particular k0 are shifted to higher frequencies, as observed in Sections 1.6.2 and 1.6.3. In many cases this effect can be corrected to some extent by replacing the refractive index of the background material in the 2D calculation with an effective refractive index. For hole-type PhCs, an appropriate effective index is found by first calculating the guided modes of the slab without any holes. The effective indices of the slab modes are proportional to their in-plane wavevector. Since most PhC slabs are designed to support only a single slab mode, calculating the 2D band structure with the effective index of the fundamental slab mode approximates the effect of out-of-plane propagation [51]. For relatively low slab-cladding index contrasts in both symmetric and asymmetric hole-type PhC slabs, the effective index method generally agrees well with the full 3D band structure and waveguide mode calculations [51]. Experimental results have also shown good agreement with 2D calculations [52, 53]. The approximation deteriorates for high index contrasts, such as in membrane-type slabs, as the tighter slab mode confinement introduces significant waveguide dispersion and other effects not easily accounted for in 2D calculations. The method is also unsuitable for studying strongly resonant behaviour in cavities as the operation of these structures can be severely degraded by out-of-plane losses. In these cases it is essential to understand and control the radiation properties [54] and hence 3D calculations are generally required. Another case where 3D methods are needed is the interaction region between modes of different symmetry [55]. Applying the effective index approximation to rod-type PhCs is somewhat less straightforward since the material between the rods is often the same as the cladding. In this case, or for hole-type PhCs with large holes, it may be appropriate to calculate first an average refractive index of the PhC, taking into account the refractive indices and the relative fill-fractions of the cylinders and background material (Ref. [56], Chap- ter 10). The average index value could then be used as the slab index when calculating the guided slab modes. Although applying an average index to a PhC is strictly valid only when the wavelength is significantly larger than the lattice period, this approach may at least allow the effective index method to be applied to rod-type PhCs as an intermediate approximation between 2D and full 3D calculations. However there are 20 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS no reports in the literature of this method being tested. In Chapters 3 and 4 we present several novel PhC waveguide-based devices, all of which are studied using the purely 2D techniques described in Chapter 2. Although the devices have not been explicitly designed for implementation in a PhC slab geometry, their operation relies on generic PhC properties such as waveguiding, mode coupling and omnidirectional reflection. The exception to this is Chapter 5, where 3D numerical calculations are used to verify the efficient coupling characteristics of rod-type PhCs. Rather than presenting devices optimised for fabrication, this thesis is intended to highlight some of the unique properties of 2D photonic crystals and demonstrate novel ways to utilise these properties.

1.7 Defect states in PhCs: cavities and waveguides

Guiding and trapping light using waveguides and resonant cavities are two fundamental optical functions that enable a range of all-optical devices to be created. Waveguides not only perform the tasks of their electrical analogues, wires, by transferring light from one part of a circuit to another, but are used in many other devices such as couplers, junctions and interferometers. Resonant cavities have many potential applications that make use of the sharp spectral response and very strong field intensities that occur at resonance. There are various methods for achieving efficient waveguiding and many others for producing high quality optical cavities, but very few single technologies allow both to be engineered in a single integrated structure. Two dimensional photonic crystals can provide just such a combination due to their versatile geometry that allows simultaneous fine tuning of a number of parameters. 3D PhCs could potentially provide even greater control over light, but they are considerably more challenging to fabricate. In addition, the complex lattice structures required to achieve full 3D bandgaps can make it very difficult to create high quality defects. The possibility of producing cavities and waveguides in PhCs by introducing defects was recognised not long after the first 2D bandgap structures were proposed. In 1994, Meade et al. suggested that both cavities and waveguides could be created in 2D PhCs by simply changing or removing a single cylinder or a whole line of cylinders, respectively [40]. A defect created in this way is surrounded by the uniform PhC that acts as an omnidirectional mirror for light at frequencies in the bandgap, thus trapping light in the defect region. In a cavity defect, this can result in sharp resonant behaviour, whereas a linear defect allows light to propagate along the defect while being confined in the transverse direction by perfectly reflecting PhC walls. In a PhC slab, this is only true for light confined vertically by TIR, but the behaviour is essentially the same. 1.7. DEFECT STATES IN PHCS: CAVITIES AND WAVEGUIDES 21

1.7.1 Cavities

Resonant cavities provide both sharp spectral responses and large field enhancement within the cavity when the resonance condition is satisfied. The former can be used for narrow bandwidth filters and wavelength selective couplers, both of which are required in wavelength-division-multiplexing (WDM) optical systems for operating on individual frequency channels. High field intensities due to light being trapped in a small cavity can enhance light-matter interaction, making them ideal for photonics applications such as lasers and nonlinear optics. They are also useful in sensing applications and for more fundamental research into cavity quantum electrodynamics and control of spontaneous emission. A review of many of these applications is given in Ref. [57]. An ideal optical cavity would trap light indefinitely and have a single resonant frequency, but real cavities are limited by losses, both due to radiation and absorption. A measure of how close a cavity approximates an ideal resonator is given by the quality factor Q, which is proportional to the lifetime of light in the cavity,

stored energy U Q = ω = −ω , (1.5) 0 power loss per cycle 0 dU/dt where ω0 is the resonant frequency, and U is the energy stored in the cavity (Ref. [58], chapter 8). Therefore, an ideal cavity would have an infinite Q. A more practical definition for the purposes of measuring Q in experiment or simulation is given by the relative width of the resonance, ω Q = 0 , (1.6) ∆ω where ∆ω is the full-width at half-maximum (FWHM) of the resonant power spectrum. Field intensities are also determined by the volume of the cavity V , so to achieve the optimum field enhancement, the ratio Q/V should be maximised. Silica microspheres hold the record for the highest Q values, with experimental demonstrations of Q > 109 [59], but the geometry is not easily integrated with other photonic devices and the modal volumes are relatively large. Toroid microcavities are perhaps more promising as they have been demonstrated with Q > 108 in an on-chip silicon-on-insulator (SOI) geometry [60]. However, despite being fabricated on-chip, a tapered optical fibre was still required for coupling to the toroid. While this method allows fine-tuning, it may not be suitable for dense integration as the condition for coupling between the fibre and the resonator is likely to be delicate and hence sensitive to perturbations. In contrast, PhC cavities can be coupled into directly from nearby waveguides in the same structure, requiring no additional alignment or coupling control. Whereas a cavity in a 3D PhC could theoretically provide perfect confinement, cavities in PhC slabs rely on TIR for confinement in the vertical direction, and so their performance is largely dependent on out-of-plane losses. For this reason, it is informative to separate the contributions due to in-plane effects Qk and out-of-plane 22 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

eﬀects Q⊥. These are related to the total Q of the cavity by 1 1 1 = + . (1.7) Q Qk Q⊥

From this equation it can be seen that the cavity Q is limited by the smaller of Qk and Q⊥. Qk depends on a number of factors including the number of PhC layers surrounding the cavity and the scattering losses due to imperfections in the lattice. Q⊥ depends on the out-of-plane radiation, most of which occurs when light is reflected at the interfaces surrounding the cavity [61]. Recall from Section 1.5.1 that Bloch modes of a 2D PhC extend throughout the crystal lattice and can thus be characterised by a single frequency eigenvalue ωm(k0) for every in-plane wavevector k0. When light is confined to a small cavity, it is no longer represented by a single wavevector, but the modal field instead contains a superposition of spatial frequencies in the same way that a finite beam can be expressed as a 2 2 superposition of plane waves. Any components with k0 < (nclω/c) lie inside the light cone, and can couple to radiation modes leading to losses that reduce Q⊥ and hence result in significant reductions in the total Q of the cavity. The first PhC cavities to be fabricated exhibited Q values of only a few hundred [62,63]. These low values were due not only to out-of-plane losses inherent in the cavity design, but also to scattering losses introduced by structural variations and surface roughness. In 2001, Vuˇcković et al. reported a systematic theoretical study of various cavity designs to establish methods for reducing radiation losses [64]. Cavities with Q > 104 were calculated by varying the radius and shape of the cylinders in the area surrounding the defect. Structures based on some of these designs were also fabricated and measured to have Q = 2800 [65]. Many of these early PhC cavities were designed for lasing and included quantum wells in the slab structure that produce photoluminescence when optically pumped from outside the slab but the first experimental report of direct coupling between a PhC cavity and a waveguide was in 2001 [66], where a cavity was created in a section of PhC between the ends of two waveguides. When light was coupled into the end of one waveguide, a sharp resonant transmission of Q = 816 was observed Although radiation losses were considered in the various cavity designs discussed above, the direct relationship between Q and the wavevector components of the mode was not explored. The wavevector distribution of a mode at the resonant frequency is given by the 2D spatial Fourier transform of the field. In this k-space representation, the light cone forms a circle of radius (nclω/c). Any component of the Fourier transform that lies within the circle contributes to radiation losses and hence reduces Q⊥ and the total Q of the cavity. Theoretical studies have shown that by careful adjustment of the size and shape of the cavity, it is possible to redistribute almost all of the mode energy outside the light cone [67, 68]. The first experimental demonstration of this design approach produced Q values of 1.3 × 104 [69]. An alternative approach to reducing radiation losses was proposed by Johnson et al. whereby the cavity is modified to 1.7. DEFECT STATES IN PHCS: CAVITIES AND WAVEGUIDES 23 achieve cancellation of specific radiated field components in the far-field [70]. There are many possible ways to optimise the geometry of a cavity, and it is not immediately obvious which one should be taken. In Ref [69], the defect was formed by reducing the size of many holes in a large defect region, with the holes near the centre of the region being changed more than the ones closer to the outside. A number of other ‘graded lattice’ designs have been proposed with predicted Q values of more than 105 [71, 72]. The feature of all these designs is that there are no abrupt changes in the lattice structure at the edges of the defect. Using a simple 1D model, Akahane et al. [54] showed that this feature is the key to high-Q cavity design; the modal field should vary smoothly across the edges of the defect. By shifting the position of two holes, a Q of 4.5 × 104 was measured in a cavity formed by the removal of only three holes in a triangular lattice. The small cavity size resulted in Q/V = 1.2×105/λ3, at least an order of magnitude higher than any other optical resonator at the time ∗. Further refinements in the cavity design have more than doubled these values [74]. The same technique was also used to design a different cavity geometry with an experimentally measured Q = 6 × 105 [75]. In the last three examples, light was coupled into the cavity from a PhC waveguide separated by a number of lattice periods - a simple, yet convincing demonstration of the possibilities for integrated optical circuitry. Bistable switching for powers as low as 40 µW was demonstrated recently using a PhC cavity coupled between two waveguides that exhibited a linear transmission resonance of Q = 3.3 × 104 [76]. Although the study of high-Q cavities is not a major theme of this thesis, resonance effects are considered in detail and some of the design rules used to reduce radiation losses in cavities could be applied to the resonant coupling devices presented here.

1.7.2 Waveguides

The concept of creating a linear defect in a PhC slab for waveguiding was already introduced at the start of Section 1.7, but we now consider the properties in more detail. In many ways the design of efficient waveguides is considerably easier than the design of resonant cavities because the confinement of light by the PhC is required only in one direction within the slab. There are two conditions for propagating modes to exist in any waveguide, whether metal, dielectric or PhC: first, the walls must be reflective, and second a phase condition must be satisfied. The latter requirement results in a discrete spectrum of guided modes. Hollow metal waveguides are used for guiding microwaves as metals have relatively low absorption at those frequencies, and they reflect light for all incident angles. Dielectric waveguides, on the other hand, rely on total internal reflection, so they only guide modes that satisfy the TIR condition on all the walls. PhC slab waveguides use the in-plane photonic bandgap for lateral mode confinement and TIR for vertical confinement. Since the PhC walls reflect light for all incident angles, PhC waveguides have many similarities to metal waveguides and

∗Higher values have since been reported in toroid microcavities [73]. 24 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS some of the concepts of microwave circuit design such as impedance matching have been applied to PhCs [77,78]. There is justifiably some debate as to whether photonic crystal waveguides provide any benefit over high-index contrast dielectric waveguides, such as those based on the well-established SOI technology [79]. For simply transferring light from one part of an optical circuit to another, SOI waveguides not only tend to have lower losses, but can also be positioned anywhere, rather than having to conform to a PhC lattice. PhC waveguides, on the other hand, can be directly integrated with other PhC structures such as cavities to produce functional devices. In addition, light guided by a photonic bandgap can exhibit very different behaviour to TIR guided light, including very strong dispersion and strong reflections that we exploit in Chapters 3 and 4.

1.7.3 Linear waveguide modes A linear defect is typically created in a PhC by removing or changing nearest-neighbour cylinders along one of the symmetry directions of the lattice. In a square lattice this usually corresponds to the Γ − X direction, while in a triangular lattice the waveguide lies along Γ − K, as illustrated in Fig. 1.8(b) and (c). Since the waveguide only breaks the PhC symmetry in one direction, the structure remains periodic along the waveguide axis. Thus the modes are characterised by a wavevector k directed along the guide and a corresponding frequency. As for any 1D periodic lattice, the waveguide dispersion curves can be represented on a band-diagram showing the mode frequency as a function ˜ ˜ of the Bloch vector k0 over the 1D Brillouin zone defined by −π/d ≤ k0 ≤ π/d, where d˜ is the periodicity in the waveguide direction. For the common waveguide directions described above for the square and triangular lattices, the period along the waveguide is just the lattice period d˜= d, but for waveguides in other directions, generally d˜6= d. The bands of the PhC surrounding the waveguide can be represented on the same plot by projecting the 2D band diagram onto the k−axis of the waveguide. In a square lattice for example, the Γ − X band structure is plotted as before, but the modes lying in the rest of the Brillouin zone are plotted as a function of their kx component only. Such a combined waveguide dispersion and projected PhC band diagram clearly shows where the waveguide modes lie in relation to the bandgap and the Bloch modes of the PhC. For 3D calculations, the lightline can be projected onto the waveguide axis in the same way. More details on the construction of these diagrams are given by Johnson et al. [38]. As an example, Fig. 1.8 shows the dispersion curves and modal field profiles for a waveguide formed by leaving out a single line of holes in the Γ − K direction of the hole-type PhC considered in Section 1.5.3. For TE polarisation there are two modes: one with even symmetry, extending down from the top of the bandgap to the cutoff frequency d/λ = 0.221, and one with odd symmetry that lies in the frequency range 0.245 ≤ d/λ ≤ 0.256. The properties of these modes are discussed in more detail in Section 1.7.4. Waveguide modes lying below the lightline are lossless in an ideal PhC slab. In prac- 1.7. DEFECT STATES IN PHCS: CAVITIES AND WAVEGUIDES 25

0.32 (a) PhC bands 0.3 0.28 0.26

l (b)

d/ 0.24 0.22 0.2 PhC bands 0.18 0 0.25 0.5 (c) k (2p/d)

Figure 1.8: (a) Dispersion curves for a waveguide formed in the triangular PhC lattice of Section 1.5.3 by removing a line of cylinders along the Γ − K direction. Two waveguide modes exist in the bandgap: an even symmetry mode (solid curve) and an odd symmetry

mode (dashed curve). Hz ﬁeld distributions of the odd and even modes are shown in (b) and (c) respectively for k = π/d, clearly illustrating the diﬀerence in symmetry.

tice however, propagation losses are significant. The first experimental demonstration of PhC waveguides was performed using millimetre waves in a lattice of alumina rods with a period of 1.27 mm, in which propagation losses of 0.3 dB/cm were measured [80]. The lattice in this case was essentially 2D, since the rods were considerably longer than the wavelength, and there was no guidance in the vertical direction. Waveguiding at telecommunications wavelengths in PhC slabs was first observed in 1999 but transmission spectra and loss measurements were not reported until 2001 [81, 82]. Much effort has gone into reducing loss since then with recent reports of 15 dB/cm propagation loss

in SiO2 clad structures [83] and < 5 dB/cm in air-clad Si slabs [84,85]. These losses are within an order of magnitude of high-index-contrast SOI waveguides, for which losses of 0.8 dB/cm have been reported [86]. Apart from the initial millimetre-wave experiments, the great majority of waveguiding demonstrations in PhC slabs have involved hole-type PhCs. Although a few experimental demonstrations of waveguiding in rod-type PhCs have been reported, the propagation losses are at least an order of magnitude greater the lowest-loss waveguides in hole-type structures [47,48]. Below the lightline, the main contribution to waveguide loss is thought to be structural disorder. The length scales of this disorder can range from several tens of nm, in the case of hole position and radius, down to only a few nm, in the case of surface roughness. Positional and size disorder has been shown to reduce the size of the bandgap in bulk PhC, and simultaneously decrease the transmission bandwidth and total transmission of PhC waveguides [87, 88]. Calculations have also shown that the first 26 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS row of cylinders on the waveguide edges is the most important for minimizing in-plane scattering losses; as long as these cylinders are in the correct places, disorder in the rest of the lattice has a minimal effect on the waveguide transmission properties [89]. Surface roughness has only recently been included rigorously in scattering loss calculations for PhC waveguides using a Green function theory. Very good agreement with experimental loss measurements was demonstrated, suggesting that the dominant loss mechanisms were identified [84]. The prospect of low-loss bends in PhC waveguides has perhaps attracted more attention than any other aspect of PhC applications [90]. High-efficiency bends have been demonstrated in many different photonic crystal geometries [48, 80, 82, 91]. Since the PhC waveguide walls reflect light for all incident angles, sharp bends do not suffer from the same bend losses that occur in dielectric waveguides [92], but this also means that without careful design, they can reflect a considerable amount of light as well. In contrast, if a dielectric waveguide is bent too sharply, the TIR condition is violated and light radiates into the cladding. Although this is major drawback of low-index- contrast dielectric waveguides originally proposed for integrated optics applications, the stronger index confinement in SOI waveguides allows low-loss curved waveguide bends of less than 5 µm in radius [93]. Alternative methods have also been proposed to achieve even sharper bends [94].

1.7.4 Dispersion One aspect where PhC waveguides differ significantly from TIR-based waveguides is in their dispersion properties. Large and variable dispersive effects are a characteristic feature of light propagation in all periodic media, originating from the wavelength dependence of coherent scattering in these materials. Dispersion control is particularly important when transmitting short pulses through waveguides because of the corre- spondingly broad frequency spectrum. If the dispersion properties change significantly over the spectral range of the pulse it can lead to distortion or, in the worst case de- struction of the signal. This effect can also be used to great advantage, however if the properties can be controlled. By tuning the dispersion appropriately, dispersive effects of other components in the system can be compensated for, or alternatively pulses can be slowed down, compressed or stretched. Nonlinear interaction lengths can also be significantly reduced by slowing the light down. The dispersion properties of a waveguide mode are characterised in part by the group velocity, vg = dω/dk0 and the group velocity dispersion GVD = dvg/dk0; the slope and the curvature of the mode dispersion curve, respectively. Figure 1.8(a) shows a typical pair of dispersion curves for a waveguide formed in a triangular hole-type lattice. Consider first the even mode represented by the solid curve. The curve is almost straight over most of its frequency range, indicating a constant group velocity and a near-zero GVD. As the mode approaches cutoff however, it flattens out and vg → 0 at the edge of the Brillouin zone, as expected for the modes of most periodic 1.7. DEFECT STATES IN PHCS: CAVITIES AND WAVEGUIDES 27 structures. The sharp change in the slope just before cutoff also results in a large GVD at these frequencies, a property first measured experimentally with an indirect interferometric technique [95]. In contrast, the odd mode, represented by the dashed curve in Fig. 1.8(a) has zero vg in the middle of the Brillouin zone, as well as at the edges. Group velocities as low as 0.02c, where c is the speed of light in a vacuum, have been demonstrated experimentally via group delay measurements [83]. Even smaller values of vg < 0.001c have been inferred from direct time-resolved observations of pulse propagation in PhC waveguides, although these have not been confirmed rigorously using other measurement techniques [96]. One of the advantages of the geometrical complexity of PhC waveguides is that there are a vast number of possibilities for tailoring the dispersion properties to suit the desired application. Some of these include changing the waveguide width [97], changing the radius of some of the cylinders [98, 99], or even placing additional cylinders along the waveguide to create a coupled cavity waveguide (CCW) [100, 101]. Other, more complicated coupled resonator designs have been proposed for stopping light altogether, raising the possibility of optical data storage or buffering [102].

1.7.5 Integrated waveguide and cavity devices Waveguides and cavities form the building blocks for a vast range of optical components and devices. In this section we consider two classes of device that have been successfully demonstrated in photonic crystals by integrating waveguides and resonant cavities.

Waveguide couplers

Directional couplers perform various functions in conventional optical circuits, including power splitting and combining and wavelength-selective coupling. Switching and wavelength tuning can also be achieved by incorporating nonlinear or other tunable elements into the structure (See, for example, Ref. [103], Chapters 17, 18). In its most basic form a coupler consists of two parallel identical waveguides running together over some distance L. When placed closely together, the modes in each guide interact to produce supermodes — modes of the coupled waveguide system. If each waveguide in isolation supports only a single mode, then the coupled system generally splits these into two supermodes, one at a lower frequency and one at a higher frequency than the original. For coupled dielectric waveguides, the fundamental, lowest frequency mode is always symmetric (even) with respect to the symmetry axis of the waveguide pair, while the second mode is antisymmetric (odd), with a nodal line running along the axis. Two identical coupled PhC waveguides can also support a symmetric and an antisymmetric mode, but the lowest frequency mode is not always the symmetric one. This issue is studied and explained in more detail in Chapter 3. Although this property may introduce some additional design considerations for PhC waveguide couplers, the coupling mechanism is unchanged as it depends only on the magnitude of the mode 28 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS splitting, and not on the order of the two modes. The mode splitting is characterised by the frequency-dependent parameter ∆β = |β+−β−|/2, where the β± are the propagation constants (or wavevectors) of the even and odd supermodes respectively. If monochromatic light of unit power is injected into one of the waveguides and allowed to propagate, it couples into the other waveguide. After a propagation distance L, the power remaining in the input guide is given by cos2(∆βL), while the power transferred to the second guide is sin2(∆βL). The coupling length is the distance over which a π phase difference accumulates between the two modes π π Lc = = , (1.8) 2∆β |β+ − β−| corresponding to the propagation distance required for all of the light in one guide to be transferred into the other. Couplers of length Lc/2 transfer 50% of the light into the other guide and are widely used as 3 dB power splitters and combiners. The shortest coupling lengths achievable in dielectric waveguides are several tens of microns, and even this requires novel coupler designs [93, 104]. Coupled PhC waveguides however, exhibit very strong mode splitting, and coupling lengths of only a few wavelengths have been predicted [105,106]. Various PhC directional coupler designs have been proposed and demonstrated experimentally [107, 108]. Modified coupler designs incorporating a short waveguide cavity between the two guides have also been suggested to reduce the coupling length further and improve coupling efficiency [109]. The sensitivity of PhC waveguide dispersion properties to small variations in geometry or refractive index makes them ideal for coupler-based switching applications. Optical switches based on both electro-optic and nonlinear refractive index modulation have been proposed as methods of tuning the coupling strength to allow or block coupling between the guides [110, 111]. A basic add/drop filter can also be formed by arranging multiple couplers along a single waveguide, each designed to couple out a single wavelength channel of a WDM signal [112]. One limitation of using a standard directional coupler response for wavelength switching or filtering is that the transmission spectrum is almost sinusoidal. To achieve a transmission band narrow enough to filter a single WDM or DWDM channel, the coupler must be many coupling lengths long, and even then there are transmission peaks close by on either side of the desired frequency. Although it has been suggested that this property could be used for a channel interleaving device, it is by no means ideal for add/drop functions. Some alternative designs for improved add/drop performance will now be considered.

Channel add/drop filters The ability to add or remove one or more wavelength channels from a multiplexed signal is an essential function in optical networks, and as such, all-optical add/drop filters are one of the components required for future all-optical processing. Almost all proposed PhC add/drop filters utilise a resonant cavity coupled to one or more waveguides to improve the wavelength selectivity of the device. Light propagating along a waveguide 1.7. DEFECT STATES IN PHCS: CAVITIES AND WAVEGUIDES 29 will couple to a nearby cavity if the frequency matches, or is close to the resonant frequency of the cavity. Once in the cavity, the light can be dropped into a nearby waveguide, or even radiated out of the plane for monitoring purposes or coupling into an optical fibre. Schematics of these approaches are shown in Fig. 1.9.

(a) backward forward drop port drop port l 1

resonant cavity l l l l l 1, 2, 3 2, 3 input port transmission (b) l drop port 1 resonant cavity l l l l l 1, 2, 3 2, 3 input port transmission (c) l 1 radiative resonant drop port cavity (out-of-plane)

l l l l l 1, 2, 3 2, 3 input port transmission

Figure 1.9: Schematics showing three diﬀerent add/drop coupler geometries. (a) and (b) Two alternative methods for in-plane cavity coupled waveguides. (c) Out-of-plane channel drop ﬁlter.

A geometry for in-plane channel dropping from one waveguide to a parallel waveguide as shown in Fig. 1.9(a) was first analysed by Fan et al., [113] who showed that two resonant states with different symmetries are required for complete power transfer, and furthermore, that these states must be degenerate. These conditions are necessary to ensure light is not coupled into the backward drop port or reflected back into the input waveguide. Various PhC based designs displaying these properties have been proposed and demonstrated experimentally either using two distinct cavities or a single cavity supporting two modes [83, 114, 115]. The use of multiple coupled cavities can also be used to improve the spectral shape of the coupling response. A single cavity resonance typically exhibits a Lorentzian response, whereas multiple coupled cavities can be de- 30 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS signed to have a flat-topped response with steeper sides, thereby reducing overlap with nearby channels while improving the transmission within the desired channel [74,116]. The filter design shown in Fig. 1.9(b) works in a similar fashion to that in (a), although it is not clear from the numerical results whether mode symmetry plays the same role in determining the efficiency [83,117]. One clear advantage, however is the ability to include additional drop ports along the same input waveguide, thereby allowing multiple channels to be dropped. An alternative to coupling light into another waveguide is to radiate it out of the plane, as illustrated in Fig. 1.9(c). The main drawback of this technique is that the maximum theoretical coupling efficiency is only 50%, with the remainder coupling back into the input waveguide in both the forward and backward directions [118, 119]. Un- like the cavities required for designs (a) and (b), where radiation losses have to be minimised, the cavities in this case must be intentionally designed to radiate, while still exhibiting a sharp resonance. Experimental demonstration of a multiple wavelength filter incorporating seven different cavities was recently reported with drop efficiencies of up to 40% [120]. Thermo-optic tuning of a single cavity with a blue laser has also been used to achieve dynamic control of the coupling frequency, with shifts of nine times the resonance width demonstrated [121].

1.8 In-band applications

In this section we consider the possible applications of uniform PhCs operating at frequencies where propagating states exist. As the band diagrams in Fig. 1.5 illustrate, the Bloch modes can have complicated dispersion proﬁles, and these lead to a range of unusual propagation eﬀects, some of which are discussed in this section. Here the behaviour is illustrated by considering the transmission of light from a homogeneous dielectric into a PhC, or vice-versa, but interfaces between two PhCs or between a waveguide and a PhC have qualitatively similar properties.

1.8.1 Coupling to Bloch modes

The coupling of plane waves from a dielectric to the Bloch modes of a PhC can be understood in terms of momentum conservation or as a generalised version of Snell’s law. Figure 1.10(b) illustrates a graphical construction of Snell’s law for an interface between two dielectrics labelled 1 and 2, as shown in part (a) of the ﬁgure. A plane wave of frequency ω propagating in medium 1 is characterised in k-space by a wavevector of magnitude k1 = n1ω/c pointing in the direction of propagation. The set of all possible wavevectors at this frequency can thus be represented by a circle of radius k1, indicated by the solid green circle in the diagram. Similarly, the set of plane waves in medium 2 lie on a circle of radius k2 = n2ω/c, indicated by the dashed circle. These circles are equifrequency contours (EFCs) of the conical band surfaces of media 1 and 2. If one 1.8. IN-BAND APPLICATIONS 31 or both of the media is anisotropic, the EFCs become elliptical, rather than circular, since n depends on the propagation direction [122].

A refraction problem can be constructed for a given incident angle θi by first drawing the incident wavevector into the previous diagram, indicated by the solid green arrow. Momentum conservation requires the wavevector component parallel to the interface, kk = k1 sin(θi) to be the same in both media, as represented by the dashed-dotted line perpendicular to the interface. The wavevector of the refracted wave is given by the intersection of this line with the EFC of medium 2, indicated by the dashed blue arrow. Since the light is being transmitted through the interface, it is necessary to choose the intersection corresponding to energy flowing away from the interface. The energy flow is given by the group velocity vg = ∇k0 ω(k0), which is a vector normal to the EFC in the direction of increasing frequency. For an isotropic dielectric, the group and phase velocities are parallel, and hence vg k kpc. Snell’s law can be found using simple geometry to show n1 sin(θi) = n2 sin(θ2). A similar method is followed for an interface between a dielectric and PhC, but there are three significant differences. First, it is clear from Fig. 1.5(b) that the EFCs of PhC Bloch modes are not circular, so the refraction properties are in general anisotropic. Second, the phase and group velocities are not necessarily in the same direction. Third, the momentum conservation rule must be generalised to take account of the periodicity in the direction along the interface, dint. Hence, the construction lines perpendicular to the interface representing conservation of kk must be repeated for kk + m(2π/dint), where 2π/dint is a reciprocal lattice vector parallel to the interface and m = 0, ±1, ±2,... [123,124]. In the example shown in Fig. 1.10(d), only the first Brillouin zone is drawn, but in general the surrounding zones must also be considered.

For a dielectric, the circular EFCs have a radius proportional to the frequency, so kpc and vg are both directed radially outwards in the same direction. In a PhC, however, the situation can be very different, as illustrated by the example in Fig. 1.10(d) which shows an EFC for a band with a maximum at the Γ-point. In this case, the EFCs are almost hexagonal and their size decreases with increasing frequency so that vg is directed inwards from the contour. To ensure that energy flows away from the interface the upper intersection point is chosen, resulting in the kpc and vg indicated. θpc is given by the direction of vg. Coupling to PhCs is not always as simple as the example given here. Depending on the shape of the band surfaces, it is sometimes possible for light to couple into more than one part of a band, or into more than one band simultaneously, although these situations are not desirable for most practical applications. Similarly, coupling into higher-order Brillouin zones can sometimes occur for non-normal incidence, or when the interface is not aligned to a symmetry axis of the crystal [125] Although this construction method provides a simple and intuitive way of understanding where the light can go, like Snell’s law, it does not provide information about transmission and reflection amplitudes at the interface. For dielectric interfaces, these are characterised by the Fresnel coefficients [122]. In Chapter 2 we introduce a generalised form of Fresnel coefficient that characterises transmission between dielectric 32 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

ky y

k// q i

n1 x kx n 2 q i k1 q 2 q 2 k (a) (b) 2

kpc

q i n1 vg k q x pc q i k1

q pc (c) (d)

Figure 1.10: Diagrams of refraction at an interface between (a) two homogeneous isotropic dielectrics and (c) a dielectric and a PhC. (a), (c) Ray diagram of incident, reﬂected and refracted light for incidence from above. (b), (d): Construction diagrams for refraction in k-space showing the incident wavevectors k1, refracted wavevectors k2 and kpc, and the group velocity vectors of the refracted light vg. Solid green circles in (b) and (d) are the equifrequency contours of the medium 1 at the incident frequency. Dashed blue curves are the EFCs of the dielectric medium 2 and the PhC. and PhC or between two diﬀerent PhCs. The issue of improving transmission from a dielectric to a uniform PhC is considered in detail in Chapter 5.

1.8.2 p and q parameters

From Section 1.8.1 it is clear that θpc has a strong dependence on the local shape of the EFC where it intersects the kk conservation line in the construction diagram. Thus, changing the incident angle or the frequency can result in large variations of θpc. These two eﬀects are characterised by the parameters

∂θpc ∂θpc p ≡ , q ≡ , (1.9) ∂θ ∂ω i ω θi 1.8. IN-BAND APPLICATIONS 33 which were ﬁrst introduced by Baba et al. in the context of PhC superprisms [126]. In Sections 1.8.4 and 1.8.5 we consider two examples of unusual propagation behaviour in uniform PhCs that require careful control of p and q. First, however, we consider a more general property exhibited when light is coupled into a PhC, namely negative refraction.

1.8.3 Negative refraction Theoretical studies predict that materials with both a negative permittivity ε and permeability µ should exhibit a range of unusual properties including a negative refractive index and anti-parallel group and phase velocities [127]. Furthermore, a material with n = −1 has been shown to act as a perfect lens, capable of sub-wavelength resolution [128]. In contrast, the resolution limit of conventional optics in the far-ﬁeld is equal to the wavelength, regardless of the size and precision of the lens. Although no naturally-occurring materials are known to exhibit negative refractive indices, composite “meta-materials” with these properties have been proposed and demonstrated experimentally for microwave frequencies [129]. Photonic crystals have been known for some time to exhibit unusual refraction properties, including ultra-refraction, where a beam incident from air is refracted away from the normal, implying a refractive index of n < 1 [130, 131]. Negative refraction in PhCs, where the refracted and incident beams lie on the same side of the normal was ﬁrst observed experimentally in a 3D PhC [132] and since then has been studied in detail in both theory and experiment [123,133–136]. Negative refraction is exhibited in

PhCs wherever the local curvature of the EFC is such that vg is directed away from the normal on the same side as the incident beam. Moreover, if the EFCs are approximately circular, and vg is directed radially inwards, then the PhC displays negative refraction for all incident angles. For this situation, the concept of an effective refractive index based on the radius of the EFC has been proposed [123,133]. Although PhCs are not true negative refractive index materials, they can still exhibit some similar novel properties due to their negative refractive behaviour. A number of theoretical studies have predicted that a finite slab of PhC that exhibits negative refraction for all incident angles can focus an image to sub-wavelength resolution [137, 138]. This effect has been demonstrated experimentally with microwaves [139,140] but has not yet been reported at telecommunications wavelengths.

1.8.4 Auto-collimation Auto-collimation, or self-guiding, in photonic crystals allows a narrow beam to propagate without any significant broadening or change in the beam profile. This effect was first observed experimentally in a 3D PhC [141], where it was shown to occur at inflection points on EFCs. A subsequent theoretical analysis provided a framework for the design of auto-collimating 2D PhCs, and also demonstrated that PhCs designed for 34 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS collimation properties do not need to exhibit bandgaps [142]. This is an advantage of many in-band applications: since they rely on the shape of the bands, rather than a bandgap, many of the properties occur for lower refractive index contrasts, and hence a larger range of materials become available for fabrication. Auto-collimation of an incident beam was first demonstrated experimentally in a PhC slab for light incident from a waveguide. Relatively low-loss collimated beams were observed propagating through the crystal for a range of angles [143]. Collimation occurs when an incident beam is coupled to a Bloch mode in a region where the EFC is locally straight. Quantitatively, this can be expressed in terms of the p-parameter (1.9), which indicates the sensitivity of θpc to variations in θi at a fixed frequency. An incident Gaussian beam is not represented by a single wavevector, but can be considered as a superposition of plane waves with a small angular spread centred on the beam direction. Thus, an incident Gaussian beam in the construction diagram of Fig 1.10 becomes a narrow cone of wavevectors that couple into a region of the EFC, rather than a single point. Figure 1.11(a) illustrates the result of coupling to a region of the EFC with strong curvature (large p); the plane wave components are refracted over a range of angles ∆θpc and hence the beam broadens. In extreme cases the beam can even be split into multiple beams [125]. Therefore, collimation requires a locally flat region of the EFC where p ≈ 0, as shown in Fig. 1.11(b) [126]. A recent analytic study of the p and q parameters has shown that every PhC band has regions where p = 0, but in some cases this occurs over only a small range of incident angles [144]. If an EFC has long straight sides, rather than just local points of inflection, then collimation is possible for a range of incident angles. Square lattice PhCs exhibiting collimation for ◦ ◦ incident angles in the range −54 ≤ θi ≤ 54 have been calculated [142]. Since the collimated beams do not require a waveguide or even a bandgap for lateral confinement auto-collimation has been proposed as a novel method to achieve compact, reconfigurable photonic circuits. Multiple beams can, in principle, pass through each other without any resulting cross-talk [145, 146] and various circuit components such as beam splitters [147, 148] and mirrors have been demonstrated experimentally [146]. Circuits based on auto-collimation would also be more tolerant to lateral coupling mis- alignment than PhC waveguides which are typically only one or two lattice periods wide. In Chapter 5 we consider the issue of coupling through dielectric–PhC interfaces, and present examples of high-efficiency coupling into a PhC designed to exhibit auto-collimation for a range of incident angles.

1.8.5 Superprism effects Superprism effects, where the angle of a transmitted beam is depends strongly on the incident angle or frequency, have been proposed as a method of achieving ultra-compact WDM multiplexing and demultiplexing in photonic circuits. Superprism behaviour was first reported in 3D PhCs in a series of papers by Kosaka et al., where large beam deflections were demonstrated for changes in both the incident angle and frequency [132, 1.8. IN-BAND APPLICATIONS 35 k ky y

vg vg Dq pc Dq ~0 k pc k x k x k1 1 q q i i Dq i Dq i (a) (b)

Figure 1.11: Refraction construction diagrams similar to Fig. 1.10(d), but for an incident beam of ﬁnite width, and hence a spread of incident angles. The incident wavevector k1 and group velocity vg of the refracted beam are shown. (a) Example when |p| 1 showing the spreading of the incident beam by the sharply curved EFC. (b) Example when the EFC is straight, and hence |p| ≈ 0, resulting in collimation of the refracted beam.

149]. Wavelength separations of 1 nm were shown to produce a 3◦ deflection - around 500 times that obtained in conventional prisms. Further, the authors suggested that a PhC only 190 µm long would be sufficient to separate a WDM signal into individual channels [149]. Although studies of PhC prisms were reported earlier than the work described above, the parameters were chosen so that the EFCs were round. The prisms therefore displayed refraction properties similar to an isotropic medium, albeit with larger dispersion and a corresponding moderate improvement in the wavelength resolution [150]. The difference between this early study and those that followed was the use of non-circular EFCs to enhance the sensitivity significantly for a small range of incident angles and frequencies. The large deflections observed in Ref [149] were attributed to a sharply curved section of the EFC close to the region being accessed by the incident beam. Small changes in the input conditions moved the outgoing wavevector along the curve, resulting in rapid variation in the group velocity direction, and hence the angle of the transmitted beam. Similar results have been demonstrated in 2D PhC slabs, with angular resolutions of up to 1◦/nm at telecommunications frequencies [151,152]. In order for a superprism to resolve two closely spaced wavelengths, the angular separation must be converted into a spatial separation sufficient to resolve the two beams. The required separation therefore depends on the beam width; wider beams 36 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS require greater separation. Unfortunately the sharp curves in the EFC that produce strong angular resolution also cause spreading of narrow beams, as was discussed in Section 1.8.4, so a balance is required. To quantify these two competing effects, Baba et al. [126] introduced the collimation and angular resolution parameters, p and q respectively (1.9) and proposed that a realistic measure of resolution should be the ratio r = q/p, not simply the value of q as suggested in earlier superprism discussions [149]. This alternative measure led to the conclusion that PhCs of more than 1 cm2 would be required to resolve the channels of a DWDM network, a similar size to conventional silica-based arrayed waveguide grating filters. One reason for such long PhCs is the requirement that beams be separated spatially before leaving the PhC. Such a condition is only necessary if the input and output interfaces are parallel since in this case the separated wavelengths all leave the PhC in the same direction. This can be seen by considering again the diagram in Fig 1.10(d). The construction lines are unchanged for coupling out of the crystal, and thus the transmitted beams are composed of the same wavevectors as the incident beam. A more compact superprism is possible if the transmitted beams maintain their angular separation after leaving the PhC. One way to achieve this is to angle the second interface relative to the first, thereby changing the kk conservation condition for the output coupling. This approach has been studied in two different geometries in Refs [153] and [154].

1.9 Computational methods

In Section 1.5 the basic theory of band structures was discussed, and the eigenvalue equations (1.1) and (1.2) that determine the Bloch modes were introduced. Although the solutions and their significance were considered, we have not yet described how the equation should be solved. Furthermore, aside from identifying the bandgaps, the band structure of an infinite periodic lattice provides little information about transmission and reflection or many other aspects of PhC operation. No single formulation provides the computational tools necessary for an efficient and rigorous study of all photonic crystal devices. Hence, many different theoretical methods have been developed, some of which are discussed in this section. The most appropriate numerical method for a given problem depends largely on the structure being studied and the data that is required. Some methods provide versatility at the expense of efficiency and/or accuracy, while others are highly efficient but can only deal with a limited range of geometries or specific problems. Another consideration is whether a method solves the problem in the time or frequency domain. In time domain methods, the solution is calculated as a function of time, and can thus be used to track the propagation of a pulse or observe other transient behaviour. This is especially important when studying many nonlinear effects for which time dependent material properties must be included explicitly. A frequency spectrum can be calcu- 1.9. COMPUTATIONAL METHODS 37 lated by launching a short pulse and taking a Fourier transform of the time response, however long simulations are required to obtain the steady-state response of high-Q structures. Frequency domain methods on the other hand, calculate the solution as a function of frequency, which makes them more appropriate for calculating steady-state frequency responses and band structures, and for studying systems with strongly dispersive materials. There is also considerable overlap between the application of these two methods; while the frequency spectra can be obtained by taking a Fourier transform of a time response, it is also possible to calculate the time response of a device if the full frequency response is known. Finite-Difference-Time-Domain (FDTD) methods are the most general and widely- used tool for PhC calculations as they are flexible and can handle almost any geometry [155]. The method involves the solution of Maxwell’s equations via numerical integration on a discretised grid in both time and space. Since the grid must be fine enough to resolve the smallest feature in the problem, dealing with extended PhC structures can be very demanding of computational time and memory. FDTD can be used to simulate finite or periodic structures through the appropriate choice of absorbing or periodic boundary conditions. Another numerical propagation technique, this time in the frequency domain, is the beam propagation method (BPM) which is more commonly used for studying optical fibres and dielectric waveguides [156,157], but has also been applied to PhCs [158]. This approach involves the launch of an initial field distribution u(x, y, z = 0) which is then propagated numerically in the z direction. The method was initially implemented for forward propagation only and is most efficient for structures that vary slowly along the propagation direction. Adaptations of the original method have allowed bi-directional propagation and improved the versatility, but BPM can not deal with the same scope of problems as FDTD methods. Also in the frequency domain, plane wave methods are the most appropriate for band structure calculations and other problems that can be modelled as periodic systems [159]. In order to solve the eigenvalue form of Maxwell’s equation (1.2), the refractive index profile and the electric and magnetic fields are first decomposed into their respective Fourier components. The expansions are truncated to a finite number of plane wave orders, and substituted into Eq. (1.2), resulting in a matrix eigenvalue equation that is solved numerically. Plane wave methods are not limited to band structure calculations of simple lattices. For instance, waveguides and cavities can be simulated by creating a periodic lattice of supercells, each one containing an array of cylinders with the desired defect introduced. If the supercell is sufficiently large that defects in nearby cells do not interact significantly, then the solutions to the eigenvalue problem will approximate those for a single isolated defect. Waveguide dispersion curves are commonly calculated in this way. Supercells are also used when calculating the 3D band structure of a 2D PhC slab; the slab and surrounding cladding are replicated periodically in the vertical direction with enough cladding between each slab to ensure negligible coupling. Although this method is useful for calculating modes bound within 38 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS the slab, cladding modes can couple to nearby supercells, and are therefore not an accurate approximation of the modes of an isolated slab. As with all supercell treatments, the plane wave method is not able to calculate radiation losses, since every cell is surrounded by others, and any light leaking out of one supercell passes into the ones nearby and vice-versa. A range of alternative frequency domain methods have also been developed, with each one using a different set of orthogonal basis functions to represent the fields and the index profile of the PhC. Although the plane wave method is versatile and can be used to represent many different structures, it is not efficient for calculating complex or tightly bound defect modes as this requires many thousands of plane waves. One way to improve this is to expand the fields using functions that more closely represent the modal profiles, and therefore require fewer terms in the expansion. One approach that is most suitable for studying localised modes is based on Wannier functions which are derived from the Bloch modes of the uniform PhC and therefore exhibit many of the symmetry properties of the underlying lattice. The calculation of Wannier functions from the Bloch modes is non-trivial, but when optimised, they provide a highly-efficient basis set for PhC waveguide and defect calculations [160]. For PhCs consisting of circular cylinders, one of the most efficient modal techniques is the multipole method [161–163]. In this case, the basis functions are cylindrical harmonics centred on each of the cylinders, thereby exploiting the circular geometry and allowing the boundary conditions to be expressed analytically. For this reason, the multipole method is accurate and efficient when studying structures with very high index contrasts or absorbing materials. Multipole methods can be applied to either finite clusters of cylinders or periodic systems, with the former allowing radiation losses and edge effects to be investigated. Transfer matrix methods refer to a large class of techniques in which the structure to be modelled is considered as a stack of individual layers, where transmission and reflection from each layer can be characterised by scattering matrices. Transmission through the whole stack is calculated by multiplying the scattering matrices together in a recursive manner. Thus, if there are multiple identical layers, the scattering matrices need only be calculated once, and can then be reused every time the layer appears in the stack. Transfer matrix methods are a common technique for modelling thin- film stacks, where the layer-by-layer approach is an obvious choice [164], but they can also be applied to PhCs, which can often be treated as a periodic stack of diffraction gratings [165, 166]. Calculating the band structure of a uniform PhC thus requires only the knowledge of the scattering properties of a single layer. Various numerical methods can be used to calculate the scattering matrices, including differential Fourier methods [167,168] and finite-element methods [165]. The computational method chosen for most of the numerical calculations in this thesis is a recently developed 2D Bloch mode scattering matrix method [169, 170], which is highly efficient for studying 2D PhC structures consisting of distinct segments. The formulation is based on transfer matrix techniques where the fields are expanded in 1.10. DISCUSSION AND THESIS OUTLINE 39 terms of the natural Bloch mode basis of the PhCs. As we discuss in Chapter 2, the significant advantage and novelty of this approach is the derivation of Bloch mode scattering matrices which play the role of generalised Fresnel coefficients to describe the reflection and scattering of light at a PhC interface. This allows many complicated PhC structures to be treated using methods developed for thin-film optics, and also provides a systematic approach for deriving semianalytic models of many structures. The numerical implementation of the Bloch mode method used here also takes advantage of the multipole method for grating calculations, and as a result gains some of the additional benefits of this highly efficient and accurate technique for dealing with circular cylinders.

1.10 Discussion and thesis outline

We have reviewed in this chapter some of the unique properties of photonic crystals and the opportunities they present for the design of efficient, compact devices for a range of photonics applications. As the most recent experimental results demonstrate, photonic crystal research is entering a new phase. Fabrication techniques can now provide the quality and control of structural detail necessary for optimisation and fine-tuning of device designs, and as a consequence, modelling and simulation of new designs will be more relevant than ever. Given the inherently complex nature of photonic crystal structures, there is little doubt that they will remain more difficult to fabricate than alternative integrated photonics technologies such as high-index SOI based components. Therefore, to compete seriously with more conventional technologies, PhC devices must either provide functions that are not easy to achieve using other methods, or perform the same functions in a cheaper, more efficient and compact fashion that outweighs the additional fabrication effort. Therefore, the ultimate goal is to demonstrate compact, integrated photonic devices that satisfy both of these criteria. In the following chapters we present results and techniques that may contribute towards the future realisation of this goal. Chapter 2 provides an overview of the Bloch mode scattering matrix method which is the main computational tool that we use in later chapters. This method is not only an efficient and accurate 2D numerical tool, but can also be used to derive semianalytic treatments for many structures. We describe first the general formulation of the method, then discuss several issues regarding the application of the method to specific structures considered in this thesis. In Chapter 3 we introduce a new class of device in which directional coupling and resonant reflection effects are combined. The characteristics of these devices vary considerably with the geometry, which allows us to demonstrate a high-Q resonant filter, a wide bandwidth, high-transmission Y-junction and a coupled cavity delay line. The devices are analysed using accurate 2D numerical results obtained from the Bloch mode method, and semianalytic models that provide insight into the underlying physics. 40 CHAPTER 1. PHOTONIC CRYSTALS: PROPERTIES AND APPLICATIONS

The concepts of mode coupling and resonances are extended in Chapter 4, where we study the effects of mode recirculation in Mach-Zehnder interferometers (MZIs) and show that a perfectly reflecting cladding such as a photonic crystal can result in manifestly different transmission characteristics compared with conventional, dielectric waveguide-based MZIs. We derive a semianalytic model for this class of device and demonstrate that rather than degrading the performance, recirculation can provide an enhanced response. Numerical results are presented for both rod- and hole-type PhC structures to verify these effects. Chapter 5 concerns an entirely different type of resonance and mode coupling behaviour that we have identified in uniform rod-type PhCs. We show that scattering resonances in the high-index rods provide a mechanism for light to be coupled efficiently into the Bloch modes of the PhC for a wide range of angles. These results may lead to more efficient designs for in-band applications such as superprisms and auto-collimation devices. We demonstrate one such example by optimising a structure to exhibit both auto collimation and very low reflectance for a wide range of incident angles, and confirm the 2D results with rigorous 3D simulations. The thesis concludes with a brief outlook on the future of photonic crystal devices and a discussion of out-of-plane losses and other issues that must be considered in order for the devices we have proposed to be demonstrated experimentally. Chapter 2

Bloch mode scattering matrix method

This chapter provides an overview of the Bloch mode scattering matrix method – the main numerical tool used in this thesis. It is not the intention to provide a full derivation of the formulation, but rather to present the details of the method that relate directly to the results that follow in Chapters 3–5. The majority of this chapter is based on the work of Botten et al. in Refs. [169] and [171], with the exception of Sections 2.4 and 2.5 in which we discuss a number of issues relating to the application of the Bloch mode method to the speciﬁc structures of relevance to this thesis.

2.1 Introduction

While the calculation of 2D PhC band structures has become routine and can be performed with a minimum of computational expense, studying propagation in more complicated PhC structures remains a challenge. The main computational method used throughout this thesis is the recently developed Bloch mode scattering matrix method (BMM), which is not only an efficient 2D numerical tool but also provides a systematic and intuitive approach for deriving semianalytic models of many structures. We use both of these features here to gain insight into the underlying physics of resonances and mode coupling behaviour in photonic crystal devices. The computational efficiency of the method is a result of two key techniques that are combined in the BMM. The first of these is the transfer matrix method, which has been used for many years in the field of thin-film optics [2], and more recently has been applied to PhC problems [165,166,172]. The second technique is the use of the natural, Bloch mode basis to expand the fields. To apply transfer matrix methods to a complex structure, it is necessary to break the structure down into basic units – layers that can be characterised relatively easily in terms of their transmission and reflection properties. In a conventional thin-film stack the basic layer is typically a homogeneous dielectric material. The interfaces

41 42 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD between layers are characterised by Fresnel transmission and reflection coefficients while propagation within a layer is described by plane waves. When applied to a periodic geometry such as a slab of PhC consisting of periodic gratings as shown in Fig. 2.1, the fields are expressed in terms of the Bloch modes of the slab, and we can derive generalised Fresnel coefficients (in this case matrices) that describe transmission and reflection of these modes at the interfaces of the slab. The derivation of these Fresnel matrices is the main conceptual advance of the method we describe here, as it allows many relatively complicated PhC structures to be treated using the framework of thin- film optics. In order to represent accurately the fields in a photonic crystal device using a Bloch mode expansion, it is necessary to include all of the modes, both propagating and evanescent. Recall from Section 1.5.1 that the standard technique for solving Eq. (1.1) 2 is to calculate the eigenvalues (ω/c) ε as a function of the real Bloch vector k0, which leads to the band structure. The approach described here involves an alternative eigenvalue equation which we solve at a fixed frequency to yield the Bloch vectors. If the frequency lies in a bandgap, then all of the Bloch wavevector solutions are complex, corresponding to evanescent Bloch states that can be excited at an interface but which decay exponentially within the PhC. For frequencies where propagating modes exist, there are a finite number of solutions for which k0 is real, corresponding to these modes, and an infinite number of solutions for which k0 is complex, again corresponding to evanescent states. Bloch modes can be expanded as a superposition of plane wave orders, where both propagating and evanescent plane waves are required to represent the modes accurately. Hence, propagating Bloch modes may have evanescent plane wave components and evanescent Bloch modes may have propagating plane wave components. While the evanescent Bloch states carry no energy through an infinite structure, they can couple energy through thin PhC barriers via evanescent coupling, much like frustrated total internal reflection in conventional optics. The evanescent states are important when matching fields at an interface between two different PhCs or between a homogeneous medium and a PhC. In practice, the field expansions are truncated to include only a finite number of modes where the truncation order depends on the specific geometry and the required accuracy. For many structures, the essential physics is described accurately by the propagating modes alone, in which case the evanescent Bloch modes can be dropped from the treatment, resulting in simple expressions involving only a few modes. We provide numerical examples of this asymptotic approximation in Section 3.3.2. A typical structure that we consider is illustrated in Fig. 2.1, consisting of multiple segments, M1,M2,...,MN , each of which is formed by a stack of identical layers with a transverse period of Dx. Media M1 and MN are semi-infinite in the vertical direction and form the input and output domains of the problem. While the example shows all segments containing photonic crystal material, they may also contain a homogeneous medium such as a dielectric or free-space. Each individual layer is treated as a diffraction grating with a period Dx corresponding to the size of the supercell. A uniform 2D PhC 2.1. INTRODUCTION 43

y = y1 layer M1 y = y2 (n = 0) stack of M2 L2 layers

(n = L2)

M3 x M4 Dx

Figure 2.1: Schematic of a photonic crystal structure formed by joining segments M1 − MN , each consisting of identical grating layers (here N = 4). All layers must have the same supercell period Dx, but the positions and properties of the cylinders within a layer can all be diﬀerent. Segments M2 − MN−1 contain Lm layers, segments M1 and MN are semi-inﬁnite. Layer interfaces within each segment are numbered n = 0 to n = Lm.

structure is formed if the grating consists of a periodic array of cylinders with period d, and Dx is an integer multiple of d. Functional elements such as waveguides are created in the supercell by removing or modifying cylinders from the grating as shown. When operated in the bandgap of the bulk photonic crystal, the supercell must be sufficiently large to effectively isolate the elements in one cell from those in neighbouring cells. For many structures we find that a separation of at least 10d is sufficient for coupling effects to be neglected, but more may be required when modelling strongly resonant structures such as those studied in Chapter 3. We discuss convergence with supercell size in more detail in Section 2.4.

In Section 2.2 we describe the steps required to obtain the Bloch modes for a typical segment in Fig. 2.1 from the scattering properties of a single grating layer. We then proceed to couple segments together in Section 2.3, considering initially a single interface between two semi-inﬁnite segments which allows the derivation of generalised Bloch mode Fresnel matrices. Calculation of the complete multi-segment structure follows in direct analogy to the modelling of multi-layer thin-ﬁlm stacks via recursion. In Sections 2.4 and 2.5 we discuss several practical issues relating to the application of the Bloch mode method to modelling PhC devices. 44 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD 2.2 From a grating to a PhC

In this section we outline the derivation of the Bloch modes of a typical PhC segment in Fig. 2.1 formed from a stack of identical periodic grating layers. In the most general case, each layer is oﬀset from the previous one by δx in the horizontal direction and δy in vertical direction, as shown in Fig. 2.2. The choice of these oﬀsets determines the symmetry properties of the lattice as we discuss in Section 2.2.2.

2.2.1 Grating characterisation

Since each layer in a segment is a periodic grating with period Dx, both the layers and the segments that they form are coupled together by plane wave diﬀraction orders s, the directions θs of which are governed by the grating equation

2πs k sin θs = αs = α0 + , s = 0, ±1, ±2,... (2.1) Dx where α0 = k0x is the component of the Bloch vector along the direction of periodicity of the gratings, and k = 2π/λ is the wavenumber in the background medium. The integers s in Eq. (2.1) run over a finite set of propagating orders, and an infinite set of evanescent orders. Between each grating layer the background medium is taken to be homogeneous, and so the fields can be represented in the plane wave basis by an expansion over all plane wave orders. The response of the grating to an incident field is characterised by the plane wave reflection and transmission scattering matrices 0 0 Re , Te and Re , Te , with the two pairs corresponding to incidence from above and below, respectively. For example, the element Repq specifies the reflected amplitude in plane- wave order p due to unit amplitude incidence from above in plane-wave order q. The tilde notation is used here to differentiate between the plane-wave scattering matrices for a single grating and those of a PhC segment, which we introduce in Section 2.3. The grating scattering matrices can be calculated using a variety of techniques, including integral equation methods, differential Fourier methods and finite-element methods [165, 167, 168, 173–175]. The BMM implementation described here uses the multipole method [176], which is one of the most efficient numerical techniques for dealing with circular cylinders. The main disadvantage of the multipole method is that individual grating layers cannot interpenetrate. Although this limits somewhat the geometries that can be studied, in situations where it is valid, the multipole method provides many significant advantages over other techniques. For example, the energy conservation and reciprocity relations are embedded in the multipole formulation and these features are inherited by the scattering matrices and the Bloch modes. This issue has some important consequences regarding the verification of the method and convergence of solutions, as we discuss further in Section 2.4. 2.2. FROM A GRATING TO A PHC 45 y d x - + d e f1 f1 y 2 x

- + f2 f2

Figure 2.2: Geometry of a single grating layer in a periodic stack defined by the lattice − vector e2 = (δx, δy). The plane wave field coefficients above the grating are given by f1 + and f1 , corresponding to upward and downward propagating components respectively. − + Similarly, the fields below the grating are given by f2 and f2 .

2.2.2 Basic PhC lattices

Here we consider brieﬂy the steps required to form a segment of square or triangular

PhC from a stack of identical grating layers with a supercell period Dx, containing multiple cylinders. The cylinders in each supercell are typically arranged with a local period d, and Dx is chosen to be an integral multiple of d so that dislocations are not formed at the edges of the supercell. The diagrams in Fig. 2.3 illustrate how a PhC segment is formed from a stack of gratings where each successive grating is shifted by δx along the x-axis and δy along the y-axis. Stacking gratings of local period d with δx = 0 and δy = d for example, results in a square lattice oriented with the interface parallel to the Γ − X direction as shown in Fig. 2.3(a). A straight waveguide is created simply by removing one or more cylinders from each layer. As mentioned in Section 2.2.1, in order to use the multipole method for calculating the grating scattering matrices, the layers must not interpenetrate. Therefore we are restricted to cylinder radii r < δy/2. This is not a problem for the square array in Fig. 2.3(a) since the gratings do not begin to penetrate until the cylinders themselves touch at r = d/2, but as we see in the next examples, the interpenetration can limit the study of other lattice types and orientations. We note again that this is a limitation of the multipole grating formulation, and not a fundamental limitation of the Bloch mode method. The square lattice in (a) can also be oriented with the interface√ parallel to the Γ−M 0 direction by starting√ with a grating layer√ of period d = 2d, and stacking them with 0 0 δx = d /2 = 2d/2 and δy = d /2 = 2d/2 as shown in Fig.√ 2.3(b). In this case, interpenetration of layers restricts the cylinder radius to r < 2d/4. Observe that the waveguide formed by removing a single cylinder from each layer is identical to the waveguide in Fig. 2.3(a). Figures. 2.3(c) and (d) show triangular lattices of period d oriented respectively with interfaces parallel to the Γ − K and the Γ − M directions. The example in (c) is formed 46 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD

M G X G M d d X

d y d’ d (a) (b) x M K K G G M d d

d’ d (c) (d)

Figure 2.3: Diagram showing four PhC lattices with period d formed by stacking in- 0 dividual grating layers of period d with a horizontal oﬀset δx and vertical oﬀset δy as given in the text. The waveguides are formed by removing one cylinder from each layer. (a) Square lattice oriented with the interface parallel to the Γ−X direction. (b) Square lattice with interface parallel to Γ − M. (c) Triangular lattice with interface parallel to Γ − K. (d) Triangular lattice with interface parallel to Γ − M.

√ from gratings of period √d with δx = d/2 and δy = 3d/√2 and the example√ in (d) has 0 0 0 a grating period of d = 3d, stacked with δx = d /2 = 3d/2 and δy = 3d /6 = d/2. Hence the maximum√ cylinder radii to avoid interpenetration in these two cases are respectively r = 3d/4 and r = d/4.

The horizontal offset δx is included in the formulation by applying the translation operator Q = diag[exp(iαsδx/2)] to the original grating scattering matrices [177]. This translation operator adjusts the phase of each scattered plane wave order to account for the relative shift between the upper and lower interfaces. For example, the plane-wave transmission matrix Te becomes QTQe , and the reflection matrix Re becomes QRQe −1. A similar translation operator can also be defined for the vertical offset between each layer. In the following treatment, it is assumed that the plane-wave scattering matrices have already been transformed in this way.

2.2.3 Bloch mode calculation On the interface between two grating layers in a given segment, the plane wave ﬁeld expansions are represented by vectors of plane wave amplitudes f − and f +, corresponding respectively to the downward and upward propagating plane wave orders of Eq. (2.1). Both propagating and evanescent orders must be included to represent the ﬁelds accu- 2.2. FROM A GRATING TO A PHC 47

rately. The ﬁelds f1 and f2 on either side of the grating in Fig. 2.2 are related via a transfer matrix T which can be derived from the plane wave scattering matrices such that − fj f2 = T f1, where fj = + , (2.2) fj and " 0 0−1 0 0−1 # Te − Re Te Re Re Te T = 0−1 0−1 . (2.3) −Te Re Te

The periodic stack of gratings must also satisfy a 1D Bloch condition, f2 = µf1, where µ = exp(−ik0.e2), k0 is the Bloch vector and e2 is the basis vector of the lattice deﬁned in Fig. 2.2. Combining the Bloch condition with Eq. (2.2) leads to the eigenvalue equation T f = µf. (2.4) In practice, the form of the transfer matrix given in Eq. (2.3) results in a numerically unstable eigenvalue equation, so Eq. (2.4) must be expressed in an alternative, more robust form before the eigensolutions can be calculated [178, 179]. We do not discuss the details here.

2.2.4 Bloch mode partitioning The solutions to Eq. (2.4) yield the Bloch modes, expressed as a superposition of plane waves, and their associated eigenvalues, µ. In practice, the plane wave field expansions must be truncated to a finite number of orders, running from s = −Npw to s = Npw, resulting in transfer matrices of even order with dimension 2(2Npw + 1) × 2(2Npw + 1). This allows the Bloch modes to be partitioned into equal numbers of downward and upward modes. The details of the partitioning are given in Ref. [169], but for the purposes of this discussion it is sufficient to note that the non-propagating states with |µ| 6= 1 are classified according to the direction in which they decay, while the directionality of the propagating modes (|µ| = 1) is inferred from the energy flux normal to the grating. Hence the non-propagating modes with |µ| < 1 are classified as forward (downward) states, while those with |µ| > 1 decay in the opposite direction, and are thus classified as backward (upward) states. With the modes partitioned in this way, the transfer matrix can be written in its diagonalised form

T = FLF −1, (2.5) which includes the whole family of eigenvalue equations TF = FL from Eq. (2.4). Here, F is a matrix whose columns are the eigenvectors of T , and L is a diagonal matrix containing the corresponding eigenvalues µ. When partitioned using the approach described above, F and L have the forms

0 F− F− Λ 0 F = 0 , L = 0 . (2.6) F+ F+ 0 Λ 48 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD

0 The columns of the matrices F∓, F∓ are formed from the downward and upward plane- wave components of the eigenvectors for the downward and upward Bloch modes respec- 0 tively, and Λ and Λ are diagonal matrices containing the corresponding eigenvalues µj. The columns of F are scaled so that physical properties such as reciprocity and energy conservation, and modal orthogonality are represented in a physically normalised form. 0 The matrices F∓ and F∓ provide a direct mapping between the Bloch mode basis and the plane wave basis. We can thus treat propagation within a segment in terms of the Bloch modes of that segment, but transform the fields into the common plane-wave basis to match the fields at interfaces between different segments. This is the topic of the next section.

2.3 From a PhC segment to a PhC device

Propagation of light through a complex PhC structure such as that illustrated in Fig. 2.1 can be characterised by two basic processes. The first is propagation within a segment of identical gratings and the second is reflection and transmission at a single interface between two different segments. In the BMM, the first of these is described in terms of the Bloch modes of that segment, while the second is characterised in terms of Bloch mode transmission and reflection matrices that play the role of generalised Fresnel coefficients. We note that the interface could instead be characterised by plane-wave scattering matrices, but as we demonstrate in the remainder of this chapter, the Bloch mode scattering matrices allow photonic crystal structures to be treated using the formalism developed for thin-film optics.

2.3.1 Propagation in a single PhC medium Consider first the propagation of light in a single finite segment of PhC consisting of L grating layers as illustrated in Fig. 2.4. Light incident on the segment from above, whether from free-space or another PhC, excites downward propagating Bloch modes at the interface with amplitudes given by the vector c−. Similarly, light incident from below excites upward propagating Bloch modes, c+. Hence, at interface n inside the n segment, the downward and upward Bloch mode amplitudes are given by c−(n) = Λ c− 0 −(L−n) and c+(n) = Λ c+, where the Λ matrices introduced in Section 2.2.4 describe propagation through a single layer, and hence Λn describes propagation through n layers. Similarly, Λ0 is describes propagation in the backward direction. The plane-wave expansion coefficients f(n) for fields at interface n are obtained via a simple change of 0 basis using F∓ and F∓ which gives 0 f−(n) F− F− f(n) = = c−(n) + 0 c+(n). (2.7) f+(n) F+ F+ We use this plane wave expansion in Sections 2.3.2 and 2.3.3 in order to match the fields between two different segments with different Bloch mode bases. 2.3. FROM A PHC SEGMENT TO A PHC DEVICE 49

n = 0

c- n n c-(n) = Λ c-

L-n c+(n) = Λ c+ c+ L - n n = L

Figure 2.4: Downward and upward Bloch mode amplitudes c−(n) and c+(n) deﬁned at interface n in a segment of length L in terms of Bloch modes sourced respectively from the interfaces above (c−) and below (c+) the segment. Each grating layer in the segment is identical.

Although the full complement of propagating and evanescent Bloch modes may be required to match the fields at input and output interfaces, the evanescent modes excited at the top interface decay inside the segment as µn, since |µ| < 1 (see Section 2.2.4), and similarly for evanescent fields excited from below. Thus, in a long segment (typically L > 5) that supports both propagating and evanescent modes, the propagating modes dominate and in many cases the evanescent states can be dropped from the calculation. Examples of this approximation are given in Section 3.3.2 and [180], and a rigorous asymptotic analysis is described in Ref. [169]. Note that while this simplification reduces the number of Bloch modes in the analysis, a sufficiently large set of both propagating and evanescent plane wave orders is still required to represent each of the propagating Bloch modes accurately.

2.3.2 Interfacing two semi-infinite segments In this section we consider the reflection and transmission of Bloch modes at an interface between two semi-infinite media, M1 and M2, as shown in Fig. 2.5, and derive the PhC analogues of Fresnel reflection and transmission coefficients. These generalised Fresnel coefficients are Bloch mode scattering matrices with the domain and range defined by the Bloch modes of the input and output media. The interface shown in Fig. 2.5 is a fictitious line between the bottom grating layer of M1 and the top grating layer of M2. In M1, the downward and upward mode Bloch − + mode amplitudes at the interface are given by the vectors c1 and c1 respectively, and similarly in M2. Since the interface lies in the homogeneous medium between layers 50 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD

- + c1 c1

- + c2 c2

Figure 2.5: Diagram of a single interface between two semi-infinite media, M1 and M2. ∓ ∓ The Bloch mode amplitudes at the interface are given by the vectors c1 in M1 and c2 in M2. there is no change in material properties and thus the fields and their upward and downward components must be continuous across it. As the Bloch mode expansions in two different media cannot be directly compared, the fields on either side of the interface are first transformed into the common plane-wave basis. The field continuity relationship therefore becomes − −0 − −0 F1 − F1 + F2 − F2 + + c1 + +0 c1 = + c2 + +0 c2 , (2.8) F1 F1 F2 F2 ∓ where the Fj are the partitioned eigenvector matrices introduced in Section 2.2.4, defined in medium j. The generalised Fresnel (Bloch mode) transmission and reflection matrices are defined by the relations

+ def − + c1 = R12c1 + T21c2 (2.9) − def − + c2 = T12c1 + R21c2 , (2.10) which express the Bloch modes propagating away from the interface in each region in terms of the incident Bloch modes. The Bloch mode scattering matrices are set in sans serif typeface to distinguish them from the plane-wave scattering matrices, which are presented in standard Roman typeface throughout. Solving Eq. (2.8) and expressing the outgoing fields in terms of the incoming fields then leads to the following expressions for the Bloch mode reflection and transmission matrices: −1 +0 −1 ∞ ∞0 ∞ ∞ − R12 = (F1 ) I − R2 R1 (R2 − R1 ) F1 , (2.11a) −1 −0 −1 ∞0 ∞ ∞0 ∞ − T12 = (F2 ) I − R1 R2 I − R1 R1 F1 , (2.11b) where ∞ + − −1 ∞0 −0 +0 −1 Rj = Fj (Fj ) , Rj = Fj (Fj ) . (2.11c) 2.3. FROM A PHC SEGMENT TO A PHC DEVICE 51

Similar expressions may be obtained for R21 and T21. In the most general treatment, additional Bloch mode matrices are required for each propagation direction. For instance 0 T12 describes forward (downward) propagation from M1 to M2 in while T12 describes backward (upward) propagation from M1 to M2. To simplify the notation, we assume throughout this thesis that the segments are numbered in order from top to bottom as in Fig. 2.1, and thus the propagation direction is implicit in the Bloch mode matrix subscripts. ∞ In Eq. (2.11c), R1 denotes the plane-wave reflection matrix that characterises reflection of a plane-wave field incident from above on a semi-infinite slab of M1. Cor- ∞0 respondingly, R1 is the reflection matrix for plane-wave incidence from below. An arbitrary lateral shift can be introduced at the interface by applying a translation operator Q of the same form as in Section 2.2.1 to the plane-wave scattering matrices of one or both segments. This operation is used in Section 3.3.1 to study transmission through a waveguide dislocation. In Chapter 5 we study the coupling of light from free-space into the Bloch modes of a uniform PhC. For this application, R∞ proves particularly useful as it allows the coupling properties to be separated completely from the effects of finite-thickness PhCs. The Bloch modes are normalised so that the reflectance of a propagating plane wave order p into reflected order q is simply given by the square ∞ ∞ 2 modulus of the corresponding element of R , |Rqp| .

2.3.3 Propagation through three segments We now combine the results of the previous two sections to consider propagation from

M1 to M3 in the three segment structure illustrated in Fig. 2.6, where segments M1 and M3 are semi-infinite and segment M2 consists of L grating layers. This analysis reveals clearly the similarity of the Bloch mode method to the standard approach in thin-film optics. In the most general case, each interface in Fig. 2.6 is described by four Fresnel matrices, Rij, Tij, Rji, and Tji, which can be calculated from Eqs. (2.11). Here, the subscript ij corresponds to propagation from segment Mi to Mj through a single interface. Transmission from M1 to M3 is characterised by an incident field composed − of downward propagating Bloch modes with amplitudes c1 , a reflected field of up- + − ward propagating modes c1 = R13c1 , and a transmitted field of downward propagating + − modes, c3 = T13c1 in M3. In segment M2, the fields have both downward and upward Bloch mode components, − + c2 and c2 , originating from the top and bottom interfaces respectively, as shown in Fig. 2.6. The field matching equations at the two interfaces are then

+ − L + − − L + c1 = R12c1 + T12Λ c2 , c2 = T12c1 + R21Λ c2 , (2.12a) + L − − L − c2 = R23Λ c2 , c3 = T23Λ c2 , (2.12b) where Eqs. (2.12a) and (2.12b) correspond to ﬁeld matching at the top and bottom interfaces respectively. Note that Λ−1 has been substituted here for Λ0, a result due 52 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD

M1 - + c1 c1

c - LLc + M 2 2 2 L LL - + c2 c2

- + c3 c3 M3

Figure 2.6: Diagram of a three-segment structure consisting of semi-inﬁnite segments

M1 and M3 above and below segment M2, which is formed from L grating layers. to the forward/backward pairing of the Bloch mode eigenvalues that holds for all PhC lattices, as shown in Ref. [169]. Expressions for T13 and R13 are obtained by solving these four equations, which gives

L L L L−1 R13 = T21Λ R23Λ I − R21Λ R23Λ T12 + R12, (2.13a) L L L−1 T13 = T23Λ I − R21Λ R23Λ T12. (2.13b)

Given that the BMM is cast in a form that resembles thin-film optics, it is not surprising that Eqs. (2.13) are generalisations of the Airy formulae for a Fabry-Pérot L L −1 interferometer (see for example Ref. [122], Chapter 7). The term (I − R21Λ R23Λ ) in both expressions encapsulates the multiple reflections within segment M2, while the remaining terms correspond to reflection and transmission at the interfaces when the light enters and leaves M2.

2.3.4 Propagation through N segments With the results of Sections 2.3.2 and 2.3.3, we are now in a position to analyse a generic N-segment structure of the type shown in Fig. 2.7, consisting of semi-inﬁnite media M1 and MN , and ﬁnite segments M2–MN−1 of lengths L2–LN−1. This is achieved via a straightforward process of recursion, starting from the interface between segments

MN and MN−1 and working upwards to the complete structure which is characterised by the Bloch mode scattering matrices R1,N and T1,N , deﬁned according to

+ − − − c1 = R1,N c1 , cN = T1,N c1 . (2.14) 2.3. FROM A PHC SEGMENT TO A PHC DEVICE 53

M1 (semi-infinite)

¤ ¦ ¨ ¤

¢ ¢

¤ ¦ ¨ ¤

¡ £¢

M ¡

¡ ¢

n-1 £¢

¤ ¦ ¤

¡ M ¡ n ¡

Mn+1

MN-1

¤ ¦ §©¨ ¤

¥ ¢

MN (semi-infinite)

− Figure 2.7: Recursive coupling of grating stacks. An incident field from above, c1 , in + semi-infinite stack M1, is reflected back into M1 (c1 ) and transmitted through the N − stacks to semi-infinite stack MN (cN ).

We begin by considering the fields at an interface between segments Mn and Mn+1 as shown in Fig. 2.7. Following the notation used in Eq. (2.15a), we define Bloch mode scattering matrices that describe the reflection and transmission properties of the stack comprising segments Mn+1,Mn+2 ...MN , such that + − − − cn = Rn,N cn , cN = Tn,N cn . (2.15a)

Turning our attention now to the interface at the top of Mn, we consider propagation from Mn−1 through Mn and eventually into MN . Using Eq. (2.15a), we can treat the stack of segments below Mn as a single interface separating Mn and MN described by Fresnel matrices Rn,N and Tn,N which reduces the problem to the three segment structure considered in Section 2.3.3, where the three segments are Mn+1, Mn and MN . Substituting the Bloch mode scattering matrices for the two interfaces into Eqs. (2.13), we derive the recurrence relations

Ln Ln Rn−1,N = Rn−1,n + Tn,n−1Λn Rn,N Λn −1 (2.15b) Ln Ln · I − Rn,n−1Λn Rn,N Λn Tn−1,N

−1 Ln Ln Ln Tn−1,N = Tn,N Λn I − Rn,n−1Λn Rn,N Λn Tn−1,N . (2.15c)

Thus, R1,N and T1,N can be derived for any number of segments by repeated application of Eqs. (2.15b) and (2.15c), starting from the last interface and working upwards in Fig. 2.7. 54 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD 2.4 Numerical aspects

For the numerical implementation of the BMM we use a combination of Mathematica and FORTRAN code. Mathematica is used as a front-end and for all the matrix manipulation and solution of eigenvalue problems, while the more numerically intensive grating scattering calculations are implemented in FORTRAN, with the two linked by MathLink. While the code has been undergoing development during the course of this thesis, and is by no means optimised, the majority of the results presented here were calculated on a laptop computer with a 1.8 GHz Pentium 4 processor and 512 MB of memory. The numerical accuracy of the Bloch mode method depends on four main parameters: the truncation orders of the multipole, plane wave and Bloch mode expansions and the size of the supercell. Since energy conservation and reciprocity are both pre- served analytically in the formulation as a result of using the multipole method (see Section 2.2.1), these physical properties cannot be used to verify the accuracy of a calculation. Hence, the main test for accuracy is numerical convergence with respect to the truncation orders and the supercell dimension. In the current numerical implementation, an equal number of plane wave and Bloch mode orders are retained in the full calculation, so it is necessary to check convergence of only three parameters, rather than four. In practice it is often convenient to set “safe” values that guarantee convergence for a wide range of structures. Although this means sacrificing some numerical efficiency, it reduces the need for convergence tests every time a new structure is calculated. It is still important, however, to identify how the geometry of a structure influences the accuracy of the calculation, and the optimal truncation orders and supercell size.

As a general rule, the multipole truncation order Nm, required for a given accuracy depends most strongly on the relative cylinder separation within the grating; more multipole orders are needed as the relative cylinder radius, r/d increases. The BMM calculations presented throughout thesis have all been performed with a multipole truncation order of Nm = 7 ie. 2Nm + 1 = 15 multipole terms are used in the field expansions centred on each cylinder within the supercell. This is sufficient to ensure convergence for the largest cylinders considered, which have a radius of r = 0.35d. The number of plane-waves required for convergence depends on the size of the supercell since the relative spacing of orders is given by the grating equation (2.1). We have already noted that Dx must be sufficiently large to ensure that there is negligible coupling between functional elements such as waveguides or cavities in neighbouring supercells. Table 2.1 gives an example of the convergence behaviour as a function of Dx and the plane-wave truncation order, Npw. The value listed in the table is the modal propagation constant β = arg(µ) (units of d) calculated for a single waveguide in a square lattice, rod-type PhC with cylinders of radius r = 0.3d and refractive index n = 3 in air (n = 1). The data is calculated for a normalised wavelength of λ = 3.15d. Each column corresponds to a different sized supercell, and the β values are listed as a 2.5. MODELLING COMPLEX STRUCTURES 55

Dx/d Npw 11 13 15 17 19 21 23 26 1.115906 1.115406 1.115293 1.115246 1.115148 1.115033 1.115011 28 1.115907 1.115406 1.115297 1.115261 1.115240 1.115104 1.115024 30 1.115907 1.115407 1.115297 1.115271 1.115245 1.115213 1.115074 32 1.115907 1.115407 1.115297 1.115273 1.115260 1.115240 1.115169 34 1.115907 1.115407 1.115297 1.115273 1.115266 1.115250 1.115238 36 1.115907 1.115407 1.115298 1.115273 1.115268 1.115261 1.115242 38 1.115907 1.115407 1.115298 1.115273 1.115268 1.115265 1.115254 40 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115262 42 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115266 44 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115266 46 1.115907 1.115407 1.115298 1.115274 1.115268 1.115267 1.115267

Table 2.1: Convergence of the modal propagation constant β = arg(µ) (units of d), as a function of the plane-wave truncation order Npw and supercell period Dx for the waveguide described in the text.

function of Npw. Note that the plane-wave expansion is over all orders s in the range −Npw ≤ s ≤ Npw, so the total number of plane waves is 2Npw + 1. Observe that within each column of Table 2.1 β converges to 7 figures accuracy, and these converged values approach a stable solution as Dx increases. All of the BMM results given in Chapters 3 and 4 are calculated with Dx = 21d and Npw = 44. For these parameters, β is stable to 6 figures in Table 2.1. While most calculations do not require such a high level of accuracy, it can be necessary when studying strongly resonant structures with extremely narrow spectral features. We note, finally, that we have confirmed the accuracy of the BMM by comparison with a recently developed Wannier function method [160] and found agreement of results to better than 1 part in 1000.

2.5 Modelling complex structures

In this section we consider two examples that highlight several important aspects relating to the application of the Bloch mode matrix method to PhC structures consisting of multiple segments. While the Bloch mode scattering matrices allow simple expressions such as Eqs. (2.13) to be written down, the calculation of the grating scattering matrices is still a significant numerical task. Efficient modelling approaches can reduce the computational demand significantly. Any structure that is modelled using the BMM must be treated as a stack of individual segments formed from gratings. As we demonstrate in the first example there may be more than one way to do this so it is important to consider the geometry of the structure carefully before starting a calculation. For 56 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD

M1 M1

(a) (b)

Figure 2.8: Diagram showing two alternative ways to model the waveguide dislocation shown in (a) using the Bloch mode method. (a) A single interface formed from two identical waveguide segments displaced laterally. (b) The same dislocation as in (a) rotated by 45◦ to form a three-segment structure. complicated devices it is often beneﬁcial to break the structure down into basic functional components which can be characterised individually before being combined into the complete structure. We consider such an example in Section 2.5.2.

2.5.1 Lattice orientation The first example we consider is taken from Section 3.3.1, where we study transmission through a dislocation in an otherwise straight waveguide extending infinitely above and below the dislocation. The structure illustrated in Fig. 2.8(a) is an example of such a dislocation formed√ in a square lattice of period d by shifting the lower section of the PhC by 2d0 = 2 2d along the Γ − M direction, where d0 is the local period of the grating. If the structure is rotated anti-clockwise by 45◦, we obtain the structure shown in Fig. 2.8(b), which is identical to the folded directional coupler geometry considered in Section 3.4. The shading of the cylinders indicates the two different ways to model the dislocation using the BMM. The structure in (a) is treated as a single interface between two identical semi-infinite sections of PhC with a lateral displacement, whereas the structure in (b) is treated as a three-segment problem consisting of identical top and bottom segments M1 and M3 and a short segment of double waveguide M2. Consider first the structure in Fig. 2.8(a). Since there is only a single interface, the transmission and reflection characteristics are given by the Fresnel matrices defined in ∞ ∞ Eqs. (2.11) which are functions of the plane-wave scattering matrices R1 and R2 , of ∞ ∞ the semi-infinite segments. Since M1 and M2 are identical, R1 and R2 are related by a simple translation operation as described in Section 2.3.2 (see also Section 3.3.1). Thus, to calculate the transmission properties of the dislocation at one wavelength requires the scattering matrices for a single grating, after which the remaining computation involves only matrix manipulation. When the dislocation is oriented as in Fig. 2.8(b), it becomes a three-segment structure of the type treated in Section 2.3.3, where the Bloch mode transmission and re- 2.5. MODELLING COMPLEX STRUCTURES 57

M 2 L2

M3 L3

M4 L4

M5 L5

M6 L6

Figure 2.9: Diagram of a PhC Mach-Zehnder interferometer of the type studied in Chapter 4. The dashed lines indicated the six interfaces that must be considered in the BMM analysis of the device. The cylinders that form each segment are indicated by the shading.

flection properties are given by Eqs. (2.13). Modelling this structure therefore requires the characterisation of two interfaces, plus an additional grating calculation since the gratings that form segment M2 are different to those in M1 and M3. As the calculation of the grating scattering matrices is the most numerically intensive part of method, it is clearly preferable to treat the dislocation as a single interface as in Fig. 2.8(a). We have performed both calculations and confirmed that they are numerically consistent. There are, of course other advantages to the single-interface treatment, not least being the option of non-integer displacements. Quantitatively, the three-segment calculation takes approximately 70% longer than the single interface. While this is a matter of only a few seconds for the simple structure considered here, it illustrates the impor- tance of lattice orientation in the BMM, and may be important in larger, more complex structures.

2.5.2 Composite devices One of the many advantages of the Bloch mode method is that it provides a systematic approach for combining basic functional components into more complex PhC devices. We have already seen how segments consisting of identical gratings can be coupled together using Bloch mode scattering matrices and simple recursion equations (2.15), but this approach can be extended to the coupling of basic components such as junctions, PhC mirrors and waveguides. Here we consider one such example: the Mach-Zehnder interferometer that we study in detail in Chapter 4. The diagram in Fig. 2.9 shows a symmetric Mach-Zehnder interferometer formed in a triangular lattice PhC. At a basic level, a symmetric Mach-Zehnder consists of two beam splitters or junctions joined by two identical arms as shown. The details of the 58 CHAPTER 2. BLOCH MODE SCATTERING MATRIX METHOD design shown here are discussed in Chapter 4 but are unnecessary for the purposes of this section. As indicated by the dashed lines, the device can be modelled as a seven- segment structure, requiring a minimum of four distinct types of grating; gratings in

M1 and M7 are missing one cylinder; gratings in M2 and M6 are missing two cylinders; gratings in M3 and M5 are missing three cylinders; and the gratings forming M4 have two missing cylinders separated by three cylinders. Observe that segments M6 and M2 are included to ensure the structure remains symmetric with respect to the input and output waveguides. This is necessary as result of the lateral shift introduced between each grating to create the triangular lattice. An equivalent device formed in a square lattice can be modelled using only ﬁve segments and three diﬀerent types of grating, as can be seen in Fig. 4.6(a).

Propagation between M1 and M7 can be calculated using the recurrence relations (2.15), which start by characterising propagation from M5 to M7 then work backwards to the M1–M2 intersection. When designing a Mach-Zehnder interferometer, however, it may be necessary to characterise the properties of the junctions independently of the complete structure. Consider ﬁrst the input junction consisting of segments M1–M4. The transmission and reﬂection properties of the junction are given by the Bloch mode scattering matrices R14 and T14, where T14 describes the transmission amplitudes of the Bloch modes from the single input waveguide into the two arms of the interferometer in M4. These matrices, and their counterparts corresponding to propagation from M4 into M1, are calculated using Eqs. (2.15) with N = 4. Similarly, the output junction is characterised by the matrices R47 and T47. Once the complete set of scattering matrices has been calculated for the junctions, it is straightforward to calculate the properties of the complete structure using Eq. (2.3.3), where the scattering matrices of the junctions replace the scattering matrices of the interfaces, such that

−1 L4 L4 L4 L4 R17 = T41Λ4 R47Λ4 I − R41Λ4 R47Λ4 T14 + R14, (2.16a) −1 L4 L4 L4 T17 = T47Λ4 I − R41Λ4 R47Λ4 T14. (2.16b)

This approach is simply a re-ordering of the recursion process described in Section 2.3.4, and therefore does not provide any computational advantage for a single calculation. However, the advantages of this component-based approach are obtained when the Mach-Zehnder interferometer is modified but the junctions are left unchanged. This occurs in Chapter 4, for instance, when varying the length of the arms or introducing a phase delay in one arm. In this case the junction scattering matrices need only be calculated once for each wavelength, and can then be recalled whenever the junction is used, saving considerable computational expense. Even further efficiencies are gained if the device is described accurately by only the propagating Bloch modes, as then the complete junction is characterised by just nine numbers – the elements of the scattering matrices corresponding to the single propagating mode in M1 and the two modes in M4. Another example where the component-based approach is particularly useful is 2.6. DISCUSSION 59 in the analysis of the Fabry-Pérotinterferometer considered in Section 3.3.2 where we show that a PhC partial mirror is described accurately by a single pair of (complex) scalar Fresnel coefficients. As we have demonstrated here, the modelling of even relatively simple devices can be simplified further by considering first, the functional components that make up the device. Ultimately, this approach could lead to the development of a component “library” containing the Bloch mode scattering matrices for each structure. This would provide a rapid and efficient tool for developing more complex PhC circuits with many coupled components.

2.6 Discussion

In this chapter we have described the theoretical framework of the Bloch mode scattering matrix method, and discussed a number of issues relating to the practical application of the method to studying photonic crystal devices. We have shown that the structure of the formulation closely resembles that of thin-film optics, with familiar scalar quan- tities such as Fresnel coefficients being generalised to appropriate matrix forms. While the method is a highly efficient numerical tool for studying many PhC devices, there are geometrical limitations imposed by the transfer matrix approach, since all structures must be described in terms of discrete PhC segments. As we demonstrate in the remainder of this thesis, however, a wide range of PhC-based devices can be analysed in this way. A geometry that we have not discussed in detail is coupling between a homogeneous dielectric medium such as air, and a PhC. Here, the same supercell must be used in the dielectric and in the PhC so the Bloch modes outside the PhC are the diffracted plane wave orders given by the grating equation (2.2). In Chapter 5 we consider the transmission of an incident plane wave into a uniform PhC, but the method is not limited to a single plane wave order. If the supercell period Dx is large, and hence the plane wave orders are closely spaced, it is possible to simulate an incident beam of finite width by taking an appropriate superposition of plane wave orders. Perhaps the most important advantage of the Bloch mode method compared to many other numerical techniques is the physical insight that it provides as a result of using the natural Bloch mode basis. This allows many seemingly complicated scattering problems to be expressed in terms of familiar, well understood structures that can be analysed from first-principles. While we have not demonstrated this aspect here, the following chapters provide numerous examples.

Chapter 3

Coupled waveguide devices

In this chapter we introduce a new class of photonic crystal device in which directional coupling and resonant reflection effects are combined. We demonstrate, using both 2D numerical calculations and simple semianalytic models, that these devices exhibit not only high-Q reflection resonances but also high transmission over wide-bandwidths. We present three novel PhC devices that exploit these features – a notch-rejection filter, a Y-junction and a coupled waveguide delay line. Much of the work presented in this chapter has been published or is due to appear. In particular, Sections 3.3.1, 3.3.2, and 3.4–3.6 are based on Refs. [170, 171, 181, 182], with additional discussions and analysis provided where necessary.

3.1 Introduction

Waveguide coupling and resonant cavities were discussed separately in Sections 1.7.1 and 1.7.5, and in Section 1.7.5 we reviewed several results where a single waveguide was used to couple light to and from a resonant cavity. In contrast, here we combine waveguide coupling and resonance effects in a single structure to obtain a range of unusual transmission characteristics. We present rigorous 2D numerical calculations using the Bloch mode method, but also show that these structures can be modelled accurately using simple modal methods and semianalytic techniques. These approaches provide physical insight into the behaviour of the devices allow rapid and efficient parameter optimization. The physical interpretation involves two different forms of waveguide coupling: coupling between parallel waveguides and end-coupling between two waveguide segments. These two effects are discussed separately in Sections 3.2 and 3.3. In Section 3.2 we justify numerically the application of conventional tight-binding approximations [164] to PhC waveguides and present a detailed discussion of coupled PhC waveguide dispersion properties. Two examples of end-coupling are considered in Section 3.3: waveguide dislocations and a waveguide-based Fabry-Pérot interferometer. These examples not only provide insight into the coupling of Bloch modes through

61 62 CHAPTER 3. COUPLED WAVEGUIDE DEVICES waveguide discontinuities, but also demonstrate the efficiency of the Bloch mode method when studying such structures. The second half of the chapter consists of Sections 3.4–3.6 where we present three closely related structures that derive their properties from a combination of waveguide coupling and resonant reflection at waveguide interfaces. The first of these devices is the “folded directional coupler” (FDC) - a simple, high-Q notch-rejection filter, the second is a wide-bandwidth symmetrical Y-junction and the third is a delay line formed by coupling together multiple FDCs. The properties of all three structures are described accurately by the semianalytic model derived in Section 3.4.1.

3.1.1 Photonic crystal and numerical parameters Unless otherwise stated, the numerical examples provided throughout this chapter are based on two generic photonic crystal lattices. PC1 is a rod-type PhC consisting of a square lattice of rods of radius r = 0.3d and index ncyl = 3.0 in a background of air (nb = 1). This structure has a TM bandgap in the wavelength range 2.986 < λ/d < 3.774, corresponding to 0.265 < d/λ < 0.337. The results presented for PC1 are therefore for TM polarised light. PC2 is a hole-type PhC consisting of a triangular lattice of air holes (ncyl = 1) in a dielectric background of index nb = 3.37, corresponding to GaAs. The hole radius is r = 0.3d, which gives a TE bandgap in the range 3.335 < λ/d < 4.629 (0.216 < d/λ < 0.299). Hence the results for PC2 are for TE polarised light. We consider in this chapter a range of PhC structures and devices, and provide numerical examples of each to demonstrate the typical characteristics. Although examples are provided for either PC1 or PC2 in each case, the results are general to both rod- and hole-type PhCs. The semianalytic analysis of these structures devel- ops physical insight into device behaviour, thereby allowing simple design rules to be identified. Within the assumptions of these models, the design rules are not specific to the underlying PhC lattice and can thus be applied to a range of PhC geometries. However, this does not imply that all PhCs are equally suitable for a particular device; the dispersion and modal properties can vary widely, and it is these parameters that have a significant effect on device characteristics. The full Bloch mode calculations presented in this chapter were performed with a supercell period of Dx = 21d to ensure that coupling between supercells is negligible. Eighty-nine plane wave orders ([−44, +44]) were used to represent the fields between each layer of cylinders, and the scattering matrices were calculated with 17 multipole orders ([−8, +8]). For many of the calculations presented here, a supercell of 11d and 45 plane wave orders provides satisfactory accuracy. However, the larger supercell is required for studying the strong resonant behaviour in Section 3.4, since a very high spectral resolution is required to resolve the resonant features. 3.2. COUPLED WAVEGUIDE MODES 63 3.2 Coupled waveguide modes

The derivations of the semianalytic methods in Sections 3.4 and Chapter 4 rely on conventional weak-coupling approximations to describe the modes of parallel coupled waveguides. Although this is a standard technique for studying conventional dielectric waveguides, its application to PhC waveguides, which can exhibit strong coupling, has not been studied systematically. In this section we consider PhC waveguide coupling in detail and demonstrate numerically that weak-coupling theories predict accurately the modal fields of coupled PhC waveguides, even when the mode interaction is significant. The results also highlight important differences between conventional coupled waveguides and PhC waveguides, such as mode ordering and symmetry properties. Recall from Section 1.7 that the supermodes of a two-waveguide system are classified as either even (+) or odd (−) according to their symmetry with respect to the mirror plane between the guides. When infinitely far apart, the waveguides are isolated from each other and so the supermodes are degenerate, being characterised by the propagation constant β of the single waveguides. As the guides are brought together, the modes in each waveguide begin to interact and the supermodes split into two non-degenerate modes |ψ±i with propagation constants β±. This splitting is analogous to the splitting of atomic energy levels when two atoms approach each other. Coupling between more than two identical waveguides is essentially the same as the two-waveguide case, however the modes of the individual guides are split into N supermodes, where N is the number of waveguides. Approximate modal solutions can be derived for weakly coupled conventional waveguides using perturbation methods and a tight-binding approximation that considers only coupling between adjacent waveguides. We briefly outline here the results derived in Chapter 11 of Ref. [164], using bra-ket notation to represent the modes. The supermodes in this case are written as linear combinations of the individual waveguide modes,

1 PN |Ψsi = Cn|ψni, AN n=1 1/2 nsπ PN 2 where Cn = sin N+1 , and AN = n=1 |Cn| , (3.1) for s = 1, 2, 3,...,N and |ψni is the unperturbed mode of the nth waveguide. Here AN is a normalization factor to ensure hΨ|Ψi = 1, and hence each supermode carries unit power. In this approximation the propagation constants of the supermodes can be determined by overlap integrals of the modes in neighbouring waveguides, although we do not use this result in our analysis. The application of this simple coupled-mode-theory (CMT) to photonic crystal waveguides has not been studied rigorously, although a similar approach reported by Kuchinsky et al. showed good agreement with 2D numerical results in the case of two coupled waveguides [183]. In this chapter and in Chapter 4 we apply CMT to several PhC waveguide-based devices and derive semianalytic models that display excellent agreement with full 2D numerical calculations. The devices consist of both two and 64 CHAPTER 3. COUPLED WAVEGUIDE DEVICES three coupled waveguide systems, the properties of which we consider in more detail in Sections 3.2.1 and 3.2.2.

3.2.1 Coupling of two waveguides We consider first the simplest coupled PhC waveguide example: two identical waveguides in PC1, each formed by removing a single line of cylinders in the Γ−X direction, as shown in Figs. 3.1(e) and (f). The electric field of a single waveguide mode, calculated using the Bloch mode method, is plotted in Figs. 3.1(a) and (d) for a normalised frequency of d/λ = 0.32. The dark regions in (d) correspond to negative phase and the white regions to positive phase. The profile in (a) is calculated along the white line shown between the cylinders in (d). Note that the field extends several periods into the PhC walls of the waveguide. In the tight-binding approximation the odd and even supermodes are given by Eq. (3.1) which yields √ |ψ±i = (|ψLi ± |ψRi)/ 2 . (3.2)

Here |ψLi and |ψRi are the unperturbed modes of the left and right waveguides, respectively, which are simply obtained by applying a lateral translation to the waveguide mode in Fig. 3.1(a). Figures. 3.1(b) and (c) show the electric ﬁelds of |ψ±i calculated using Eq. (3.2) (dashed curves) and the full Bloch mode method (solid curves) for two PhC waveguides separated by a single line of cylinders as shown in Figs. 3.1(e) and (f). While the tight-binding approximation is most accurate between the layers of cylinders (red curves), reasonable agreement is also obtained for the ﬁelds in the cylinders (blue curves). Such good agreement is surprising given that the waveguides are only separated by a single line of cylinders and thus interact strongly with each other. The tight-binding treatment also provides a reasonable approximation for coupled hole-type PhC waveguides, although we do not present numerical results here. We note however, that the approximation was applied successfully to design the hole-type PhC Y-junction used in the Mach-Zehnder interferometer in Section 4.4. Figure 3.2(a) shows the projected band structure and dispersion curve for the single waveguide shown in the inset. The PhC bands are shaded dark grey and the bandgap is shaded light grey. In contrast to the waveguide modes of the hole-type PhC shown in Fig. 1.8, there is at most one guided mode for all frequencies in the bandgap. This is a property common to most single line defect waveguides in rod-type PhCs, and one of their advantages over hole-type PhCs. Observe also that the dispersion slope of the rod- type PhC waveguide is positive, while the even mode of the hole-type PhC waveguide has a negative dispersion slope. Figures 3.2(b)–(f) show the supermode dispersion curves for coupled waveguides with separations of Nc = 5 to Nc = 1 respectively, where Nc is the number of cylinder layers between the waveguides, as shown in the inset diagrams. The even and odd modes are indicated by dotted and dashed curves respectively. For comparison, the single waveguide dispersion curve from (a) is also plotted in each of (b)–(f), clearly showing that the mode splitting is almost symmetric 3.2. COUPLED WAVEGUIDE MODES 65

(a)(b) (c)

(d) (e) (f)

Figure 3.1: Electric ﬁelds of the modes in a single waveguide (a,d) and two waveguides separated by a row of cylinders (b,c,e,f). The solid curves in (a)–(c) are calculated along the corresponding red and blue lines in (d)–(f) using the full Bloch mode method. The dashed curves in (b) and (c) are the approximate result given by Eq. (3.2). The light and dark regions in (d)–(f) correspond respectively to positive and negative phases.

about the single waveguide mode, and hence the average of the two modes β = (β+ + β−)/2 ≈ β. This feature is important for the semianalytic analysis in Section 3.4. The strong mode coupling for Nc = 1 and Nc = 2 results in large mode splitting ∆β = |β+ − β−|/2, a feature that has attracted much attention recently. Recall from Section 1.7.5 that large mode splitting results in short coupling lengths (1.8) and therefore provides the possibility of compact coupler-based devices [105–107,110–112]. The dispersion curves in Fig. 3.2 also illustrate an unusual mode ordering property of coupled PhC waveguides which occurs when the dominant Bloch factor of the bulk crystal, µ, is negative [184]. When Nc = 1, 3, 5, the lowest frequency (fundamental) mode has odd symmetry, whereas for Nc = 2, 4, the fundamental mode is even. In contrast, the fundamental mode of a symmetric coupler formed in conventional waveguides is always even and the second mode is odd. An interesting consequence of this is the ability to design waveguide structures that support only an odd mode. Consider, for instance, the dispersion curves for the Nc = 1 case in Fig. 3.2(f). The even mode, indicated by the dotted curve, is cut oﬀ at a frequency of d/λ = 0.305, below which 66 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

0.36 (a) 0.36 (b) 0.34 0.34 Nc=5 0.32 0.32 d/l 0.3 d/l 0.3 0.28 0.28 0.26 0.26 0.24 0.24 0.22 0.22 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 bd/2p bd/2p 0.36 (c) 0.36 (d) 0.34 Nc=4 0.34 Nc=3 0.32 0.32 d/l 0.3 d/l 0.3 0.28 0.28 0.26 0.26 0.24 0.24 0.22 0.22 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 bd/2p bd/2p 0.36 (e) 0.36 (f) 0.34 Nc=2 0.34 Nc=1 0.32 0.32 d/l 0.3 d/l 0.3 0.28 0.28 0.26 0.26 0.24 0.24 0.22 0.22 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 bd/2p bd/2p

Figure 3.2: Projected band structures and dispersion curves for the various waveguide structures shown in the insets. Dark grey regions represent PhC bands, while the light grey region shows the bandgap. (a) Single waveguide (b)-(f): Two coupled waveguides separated by Nc = 5 − 1 rows of cylinders respectively. The dotted curve corresponds to the even supermode, and the dashed curve corresponds to the odd supermode. The solid curve is the single waveguide dispersion plotted in (a). only the odd supermode can propagate. In an ideal symmetric structure, an even mode incident from another waveguide would be perfectly reﬂected by the waveguide pair. Similar concepts of mode symmetry ﬁltering play an important role in the recirculating Mach-Zehnder interferometer studied in Chapter 4.

3.2.2 Coupling of three waveguides We now consider the eﬀect of adding a third waveguide to the system in Section 3.2.1 to form a waveguide structure that supports three supermodes, |ψ1,2,3i respectively. In this case the correct linear superpositions of waveguide modes is not as obvious as in the two waveguide example. These are calculated by substituting N = 3 into Eq. (3.1) 3.2. COUPLED WAVEGUIDE MODES 67

(a) (b) (c)

(d) (e) (f)

Figure 3.3: Contour and line plots of the supermode electric fields in a three waveguide system with Nc = 1(a–c) and Nc = 2 (d–f). The line plots above each figure show the electric field profile along the white lines in the contour plots. The solid curves show the rigorous numerical field calculation and the dashed curves are the coupled mode approximation (3.1). which gives √ |ψ i = (|ψ i + 2 |ψ i + |ψ i)/2, 1 L C√ R |ψ i = (|ψ i − |ψ i)/ 2, (3.3) 2 L √R |ψ3i = −(|ψLi − 2 |ψC i + |ψRi)/2, where |ψLi, |ψC i and |ψRi are the unperturbed modes of the left, centre and right waveguides respectively. Observe that |ψ1i and |ψ3i are even modes, whereas |ψ2i is odd. For conventional waveguides, |ψ1i is the fundamental mode, followed by |ψ2i and |ψ3i but this is not necessarily the case for PhC waveguides. The contour plots in Fig. 3.3 show the mode fields for Nc = 1 (a–c) and Nc = 2 (d–f) at d/λ = 0.32. The modes are listed in order of decreasing β, where β = 1.473, 1.171, 0.377 for modes (a)– (c) and β = 1.320, 1.164, 0.907 for (d)–(f). As for double waveguides, the mode order 68 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

0.36 (a) 0.36 (b) 0.34 0.34 0.32 0.32 d/l 0.3 d/l 0.3 0.28 0.28 0.26 0.26 0.24 0.24 0.22 0.22 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 bd/2p bd/2p (a) (b)

Figure 3.4: Dispersion curves for three coupled PhC waveguides with (a) Nc = 1 and (b) Nc = 2. The dotted and dashed-dotted represent the two even modes |ψ1i and |ψ3i respectively, and the dashed curve corresponds to the odd mode |ψ2i. The solid line in both plots is the dispersion of a single waveguide mode.

changes depending on whether Nc is even or odd, but here it is the two even modes, |ψ1i and |ψ3i, that swap positions. This is clearly seen in the line plots above each of the ﬁgures which show the electric ﬁeld along the white lines in each of the contour plots (solid curves) and the approximate result given by Eqs. (3.3) (dashed lines). Once again the agreement between the rigorous numerical calculation and the coupled mode approximation is excellent, with the two curves being indistinguishable for Nc = 2. The supermode dispersion curves are shown in Fig. 3.4 for Nc = 1 and Nc = 2, where the dotted and dashed-dotted curves correspond to the two even modes |ψ1i and |ψ3i respectively, and the dashed curves corresponds to the odd mode |ψ2i. The single waveguide dispersion is also indicated by the solid curve. Since the three supermodes are split almost symmetrically relative to the single waveguide mode, the dispersion curve of |ψ2i is very close to that of the unperturbed waveguide. Note that in contrast to double waveguides, it is not possible to design a triple waveguide structure that only supports an odd mode since |ψ2i always lies between the two even modes.

3.3 Waveguide coupling at interfaces

The second type of waveguide coupling that plays an important role in the context of this thesis is mode coupling through an interface between the ends of two PhC waveguides. In this case, the coupling properties are often a combination of direct transmission and resonant coupling eﬀects. The Bloch mode scattering matrix method is an ideal numerical tool for studying this form of coupling as it allows interface eﬀects to be treated separately from propagation along sections of straight waveguide. In particular, the generalised Fresnel matrices introduced in Section 2.3.2 describe the coupling from one Bloch mode basis to another. Here we present two examples that provide physical insight into Bloch mode coupling behaviour, and at the same time demonstrate the advantages of the Bloch mode method for studying such problems. In 3.3. WAVEGUIDE COUPLING AT INTERFACES 69

Section 3.3.1 we consider a lateral dislocation in a waveguide that can be treated as a single interface between two identical PhC stacks and therefore requires the calculation of only one set of scattering matrices. Coupling through multiple interfaces in the form of a Fabry-P´erot cavity is considered in Section 3.3.2 where we show that the cavity resonance is described accurately by considering only the propagating Bloch modes.

3.3.1 Waveguide dislocations Recall from Section 1.7.2 that an ideal, straight PhC waveguide can guide light without loss or reflection. In practical situations, however, non-uniformities in the crystal or waveguide structure introduce losses due to reflection and scattering. In this section we consider the effect of a linear dislocation through the crystal and waveguide. Two structures are considered: a dislocation perpendicular to the guide direction (along the Γ − X axis of the crystal), as in Fig. 3.5(a), and a dislocation at a 45◦ angle to the waveguide (along the Γ−M crystal axis) (Fig. 3.5(b)). The bulk crystal in Fig. 3.5(b) is the same square lattice of period d as in (a), but rotated by 45◦. Thus, when calculating the scattering√ matrices for the rotated crystal, the lattice period within each grating layer is d0 = 2d. The dislocation is modelled as an interface between two identical semi-infinite crystals as discussed in Section 2.5.1. To introduce a lateral shift of sx parallel to the iαpsx/2 grating layers as shown in Fig. 3.5, we apply a translation operator Qs = diag e ∞ ∞ −1 ∞ ∞0 ∞0 −1 to the appropriate R matrices and define Rsx = Qs R Qs and Rsx = QsR Qs as −1 described in Section 2.3.2. Here, the switching of Qs and Qs simply changes the sign of sx depending on whether the dislocation is viewed from above or below. The Fresnel ∞ ∞ matrices are then calculated using Eqs. (2.11) with Rsx replacing R . Figure 3.6 shows the transmission through the dislocated waveguides in Fig. 3.5 as a function of displacement sx. The wavelength is fixed at λ = 3.25d. Since the period of the supercell used in these calculations is Dx = 21d, the transmission curves have the same periodicity. Most of the features in the spectra of Fig. 3.6 are easily understood. Both cases exhibit 100% transmission for sx = 0, since this corresponds to an infinite guide with no dislocation. However, the two curves show very different behaviour when a dislocation is introduced. In the case of the perpendicular dislocation (Fig. 3.5(a)), the transmission curve is symmetric about sx = 0, as expected since the guide is symmetric about the axis transverse to the interface. The transmission peaks correspond to local maxima in the overlap of the fields in the two waveguide sections. These peaks occur close to integer values of sx/d since at these points the cylinders are aligned at the interface between the two sections so the only dislocation occurs at the waveguide ends. ◦ For the 45 dislocation, the transmission curve is strongly asymmetric in sx, which can be understood from the diagram in Fig. 3.5(b). When sx > 0 (as in the diagram), the structure resembles two overlapping waveguides at 45◦ separated by an arrangement of cylinders. In this case, the overlap region allows strong coupling between the two waveguides, and hence strong resonant transmission peaks. The first two of these 70 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

Dx=21d (a) d

sx ’ Dx=21d (b)

’ sx d’= 2 d

Figure 3.5: (a) Straight waveguide with dislocation along the Γ − X crystal axis. The circles represent the cross-section of the dielectric cylinders forming the 2D photonic crystal. The cylinder shading indicates the two distinct waveguide sections considered in the calculations. (b) Identical waveguide, but with dislocation along the Γ − M crystal axis. Both diagrams show the supercell used in the calculations, and the dash- dot line indicates the line of dislocation. The waveguides extend inﬁnitely in the vertical direction.

0 0 peaks occur at sx ≈ 1d and sx ≈ 2d , for which the transmission exceeds 99.5%. For 0 integer values of sx/d , the structure resembles the folded directional coupler discussed in Section 3.4. If the guide is displaced in the sx < 0 direction, however, the two guides are eﬀectively being pulled apart in both directions, and so the transmission is seen to decrease rapidly below 1% and stay low as |sx| is increased. Such an asymmetry in transmission with respect to displacement could potentially be used in a sensitive directional motion detector [185].

3.3.2 Fabry-Pérotcavity Resonant devices are an essential component in modern optics, being used in filters, switches, couplers and many other other applications. We reviewed in Section 1.7 some of the properties and applications of resonant PhC cavities, most of which were formed by removing or modifying one or several cylinders in an otherwise uniform PhC. Here we consider a more conventional resonant cavity design – a Fabry-Pérot cavity – and show that it can be modelled accurately using semianalytic methods. The Fabry-Pérot (FP) resonator is one of the most basic resonant devices, consist- 3.3. WAVEGUIDE COUPLING AT INTERFACES 71

0.8

0.6 T 0.4

0.2

10 5 0 5 10 Displacement, sx (units of d)

Figure 3.6: Transmission of dislocated straight guides, for dislocations parallel to the Γ − X (solid curve) and Γ − M (dashed curve) crystal axes at λ/d = 3.25. ing of two high-reflectivity mirrors on either end of a cavity. Away from resonance, the transmission of the Fabry-Pérotdevice is typically very small, depending on the reflectivity of the individual mirrors. On resonance, however, the small transmitted amplitude through the first mirror interferes constructively inside the cavity, and high transmission is possible. If the mirrors are identical (balanced), the resonant transmission can reach 100% but this is reduced if the reflectivities are unequal. The finesse, F, of a balanced FP cavity is defined as the ratio of the fringe separation (free spectral range) to the full width at half maximum (FWHM) of the transmission peak. It is a function of the mirror reflectivity only, given by √ F = π R/(1 − R), (3.4) where R = |r|2 is the reflectance of each mirror (Ref. [122], Chapter 7). In a physical sense, the finesse is a measure of how many times light circulates through the cavity before it has all leaked away. Thus, for high finesse cavities, the mirrors must be strongly reflecting, requiring R > 0.9985 for F > 2000. The resonance width is characterised by the “quality factor” Q, defined in Section 1.7.1 as the ratio of the resonant frequency

(ω0) to the FWHM (∆ω) of the transmission intensity peak (1.6). In practical PhC devices, however, radiation losses can reduce Q signiﬁcantly. As we discussed in Sec- tion 1.7.1, cavity losses can be minimised with careful cavity design, for instance by modifying the cavity walls to reduce coupling to radiation modes [54]. Since the issue of losses is relevant to all of the structures considered in this thesis, we provide a general discussion of the matter in Chapter 6. 72 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

d air holes

mirror depth

Figure 3.7: Diagram of the Fabry-P´erotresonator formed in a single-mode waveguide in PC2. The mirrors are formed by “plugs” of cylinders. For the example here, all cylinders are identical.

Mirror properties Fabry-Pérotdevices in photonic crystals have been studied previously in various con- texts [112,186–188]. Here we apply the Bloch mode method to the simple Fabry-Pérot cavity shown in Fig. 3.7, formed by placing two “plugs” of Nplug cylinders in a single- mode waveguide formed in the hole-type PhC, PC2. Although their general behaviour is similar to that of conventional Fabry-Pérot resonators, PhC-based Fabry-Pérot devices have one significant difference that arises because the mirrors in a PhC FP are distributed reflectors, in which the light penetrates several layers into the crystal structure. This introduces additional spectral features such as resonances in the mirror reflectivity due to reflections within the mirror itself. The reflectivity of the mirrors can be increased by adding more cylinders, or by changing the properties of the cylinders forming the mirror, such as the radius [186]. Figure 3.8(a) shows the reflectivity of a single mirror as Nplug is increased from 1 to 6 cylinders. As expected, the reflectivity increases with the number of cylinders, however the effect is not monotonic at all wavelengths due to resonant reflections from either end of the plug, as seen in the curves corresponding to Nplug = 3 − 6. The properties of an individual mirror can be calculated with the Bloch mode method as a 3-section waveguide structure, where the two end sections are identical 3.3. WAVEGUIDE COUPLING AT INTERFACES 73

10-1

10-3 T 10-5

10-7 Mirror thickness (cylinders) 1 4 2 5 (a) 3 6

10-2

10-5

T 10-8

10-11 (b) 10-14 3.4 3.5 3.6 3.7 3.8 d/l

Figure 3.8: (a) Transmittance (log scale) as a function of normalised wavelength, through a single mirror (plug) formed in a single-mode waveguide in PC2 for mirrors with Nplug = 1 − 6. (b) Transmittance of a Fabry-Perot cavity of length L = 5d for mirrors with Nplug = 1 − 6. single waveguides, and the central section is composed of complete layers of cylinders (possibly with changes made to the central one). The transmission and reflection of the mirror are therefore described by Eqs. (2.13). The scattering matrices are calculated for each of the interfaces, with any symmetries being taken into account to save computation time. In this case the mirror is symmetric with identical waveguides on both sides, so T21 = T23 and R21 = R23, which reduces significantly the required scattering matrix calculations. Many calculations can benefit from an understanding of the symmetry properties of the structure as demonstrated by the examples in this chapter and Chapter 4. Unless the central blocking cylinder is made very small, the plug section does not support any propagating modes, and hence the reflection of the propagating modes at the 1−2 interface is 100%. However, the mirror is of finite thickness so energy is coupled into the cavity by evanescent modes within the plug, giving a non-zero transmission of 74 CHAPTER 3. COUPLED WAVEGUIDE DEVICES the propagating Bloch mode from section 1 to section 3.

Fabry-Pérot properties As shown in Fig. 3.7, the FP cavity is formed by placing two such mirrors in a single- mode waveguide, to form a cavity of length L. The transmission through the complete structure can be calculated in two ways. The first approach is to apply Eq. (2.15) with N = 5, thereby treating it as a 5 layer structure consisting of the input and output waveguides, the two mirrors and the cavity. The second, and more efficient, method is to treat each mirror as a single interface characterised by the scattering matrices

Rplug = R and Tplug = T which are calculated from Eqs. (2.13) as described above. This allows the whole FP structure to again be treated as a 3-layer problem using Eqs. (2.13), where the matrices T12 = T21 = T23 = Tplug describe the transmission of Bloch modes through the mirror, and R12 = R21 = R23 = Rplug describe reflection from the mirror. Thus, once the mirror properties have been calculated, the length of the cavity can be varied without the need to repeat any scattering matrix calculations. Figure 3.8(b) shows the transmission spectra for a cavity of length L = 5d with increasing mirror thickness. As expected from Eq. (3.4), the peak width decreases (Q increases) as the reflectivity of the mirrors is increased. At frequencies far from the reflection resonances shown in Fig. 3.8(a), the reflectivity of the mirrors increases exponentially with Nplug so Q also increases rapidly with the size of the mirror. The transmission spectra become more complicated in the vicinity of a mirror reflection resonance, since the reflectance changes significantly across the width of the cavity resonance. As a consequence of this, Eq. (3.4), which is calculated for a fixed R loses some accuracy. Table 3.1 shows the quality factor of the Fabry-Pérot resonance near d/λ = 0.369 as a function of the number of cylinders in the mirror for Nplug = 1 − 4. Since these calculations are purely 2D, out-of-plane losses would have to be controlled to achieve such large Q values in realistic slab geometries. We discuss this issue in Chapter 6. Experimental measurements of Q = 7300 have been reported for a similar FP cavity [83] and Q = 1.2 × 104 was measured for a structure with multi-mode rather than single- mode waveguides [188]. Although these values are several orders of magnitude lower than the highest-Q PhC cavities discussed in Section 1.7.1 [75], the in-line geometry of the FP structure provides a simple design for an integrated notch-pass filter.

Propagating mode analysis The results presented so far include both propagating and evanescent Bloch mode orders. It is not always necessary, however, to include the evanescent orders and in these cases, many of the complicated matrix expressions can be replaced with semianalytic scalar forms [180]. For the FP cavity described above, the mirror sections have no propagating modes and therefore rely solely on evanescent modes to carry energy through the reﬂecting layers into the cavity. Between the mirrors, however, most of the energy 3.3. WAVEGUIDE COUPLING AT INTERFACES 75 √ Nplug λ0(d) (full) λ0(d) (prop) Q (full) Q (prop) F[= π R/(1 − R)] 1 3.68692 3.68692 78 78 6.8 2 3.69090 3.69091 1200 1200 97 3 3.690642 3.690656 6.6 × 104 6.4 × 104 5.4 × 103 4 3.6905974 3.6906108 1.3 × 106 1.1 × 106 1.1 × 105

Table 3.1: Resonant wavelength (units of d), Q and F for a Fabry-Pérot cavity of length L = 5d in PC2 for mirrors of thickness Nplug. Resonant wavelength and Q values are given for both the full numerical calculation (full) and the propagating mode approximation (prop). is carried by the single propagating Bloch mode, suggesting that a propagating mode analysis might be appropriate. We describe here the application of this approach to the FP structure, although we note that a rigorous asymptotic analysis for a general class of three layer structures was reported by White et al. [170]. The most significant advantage in using Bloch mode methods derives from their capacity to model efficiently, long propagation spans in a fixed medium. Since this is characterised by the matrix ΛL in Eqs. (2.13), it follows that for sufficiently long spans (typically L > 5), the evanescent modes can be disregarded since they decay exponentially with L and therefore the calculations are dominated by the propagating states. To apply this to the Fabry-Pérotresonator in Fig. 3.7, we first calculate the transmission and reflection matrices through a single mirror using the full Bloch mode calculation. Since these describe coupling between single-mode waveguides on either side of the mirror, they reduce to complex scalar transmission and reflection coefficients when only the single propagating mode is considered in each segment. Similarly, the matrix Λ in Eqs. (2.13) reduces to the eigenvalue µ corresponding to the single propagating mode of the waveguide. Substituting these scalar values into Eq. (2.13b) results in a simple scalar equation that gives the transmission through the complete FP structure as a function of the cavity length, L. Since the properties of the mirror are wavelength-dependent, the scattering matrices must be calculated for each wavelength of interest. In practice, once the matrices have been calculated in full, only the elements corresponding to the propagating mode(s) need to be saved. The effect of neglecting the evanescent modes within the cavity can be seen in Table 3.1, where the resonant wavelength and Q for the FP cavity of length 5d are compared for the full evanescent field calculation and the reduced, propagating mode calculation. For all but the highest Q cavities, the agreement is excellent. If the cavity length is reduced, it might be expected that the evanescent fields would have a more significant effect on the results. However, even for cavity lengths of 2 lattice periods, the difference between the propagating mode result and the full result is relatively small. For mirrors that are 3 layers thick, the difference in resonant frequency calculated with the two methods is less than 0.05%, and the Q values agree to 2 figures. Table 3.1 also shows the finesse calculated using the standard FP equation (3.4) with the reflectance 76 CHAPTER 3. COUPLED WAVEGUIDE DEVICES values R = 1 − T obtained from the transmission data plotted in Fig. 3.8(a). The two examples in this section demonstrate clearly the advantages of the Bloch mode matrix method for studying PhC waveguide structures consisting of a few different waveguide sections. We have shown that the calculational demands can be reduced by taking into account the symmetry properties of the structures being considered, and even greater reductions can be made by considering only the propagating modes. When these two simplifications are combined with the coupled mode theory results of Section 3.2, they provide a powerful set of tools for developing simple semianalytic models to describe relatively complex photonic crystal structures. In the remainder of this chapter and in Chapter 4, we study several different PhC devices that exhibit the two types of coupling discussed in Sections 3.2 and 3.3 in addition to Fabry-Pérot-like resonances. Elegant semianalytic models are derived for each device, and the results are compared to the rigorous 2D numerical results obtained with the full Bloch mode method.

3.4 Folded directional coupler

In this section we discuss the folded directional coupler (FDC): a novel notch-rejection filter that exploits the mode coupling properties of a directional coupler and the sharp resonances of a Fabry-Pérot resonator. This same concept is extended in Sections 3.5 and 3.6 to design an efficient, wide-bandwidth junction and a novel coupled resonator waveguide that could be used as a delay line. Figure 3.9(a) illustrates the basic geometry of the FDC which consists of a single- mode input waveguide (region M1), a single-mode output waveguide (M3), and an overlap region (M2), of length L, where the modes of the two guides couple. The similarity to a directional coupler is obvious, but in addition to coupling behaviour, reflections from the closed waveguide ends result in resonant transmission characteristics. We demonstrate that an ideal FDC structure that is less than 3 µm long can be tuned to have a notch-rejection width of less than 10 GHz, corresponding to Q > 104, and with minor structural modifications, this can be increased to Q > 6 × 105. The concept of combining Fabry-Perot effects in a directional coupler geometry was studied previously by Safaai-Jazi et al. [189] as a way to improve mode discrimination and longitudinal mode spacing in resonant cavities. Although the FDC shares similarities with the earlier device, there are also distinct differences. In Ref. [189] the resonant cavity is formed by placing two partially reflecting mirrors across a pair of coupled waveguides to form a four-port device. In contrast, the FDC is a two-port device since one of the waveguides at each end of the cavity is blocked by a fully reflective photonic crystal mirror, while the other waveguide is left open. This difference in the reflection mechanism allows the FDC to exhibit complementary transmission behaviour to FP devices. FP structures are normally highly reflective, only transmitting when a resonance condition is satisfied (notch-pass filter), whereas the FDC can be designed to

3.4. FOLDED DIRECTIONAL COUPLER 77

¢¡

M1 ¥

¢¡¨§ ¥

M2 L

¢¡¤£¦¥ ¢¡ © M3 ¥

(a) (b)

Figure 3.9: (a) Geometry of the FDC consisting of three sections M1, M2 and M3, of identical cylinders. The shading indicates the three waveguide different sections. (b) Schematics showing all of the propagating modes in the structure. The input and output guides each support a single mode, while region M2 supports an even and an odd supermode, |ψ±i, indicated by the dashed and dotted curves respectively. transmit at most frequencies and only reflect on resonance (notch-rejection filter). In general, FDC structures exhibit a complicated and varied range of transmission spectra that depend strongly on the structural parameters. Several examples are shown in Fig. 3.10, where the transmission is plotted as a function of frequency for different cavity lengths (a) and guide separations (b). These curves were calculated using the full numerical formulation of the Bloch Mode method. Observe that there are several common features in the spectra in Fig. 3.10 such as resonances where the transmission vanishes, and regions where the transmission is flat and close to 100%. The solid curves in both parts of Fig. 3.10 correspond to the same FDC with a cavity length of

L = 5d and guide separation of Nc = 1. This structure exhibits a sharp resonance at d/λ = 0.331 where the transmittance drops from almost 100% to < 0.1% for a very narrow range of frequencies. In Section 3.4.3 we use the semianalytic model derived in Section 3.4.1 to identify the origin of this resonance, and find an expression for the resonance width in terms of the waveguide dispersion parameters. In Sections 3.4.1 and 3.4.2 we describe two alternative but equivalent derivations of a simple semianalytic model for the FDC structure illustrated in Fig. 3.9(a), both of which are based on mode coupling and symmetry arguments. In the first method, we begin from the coupled mode approximation for two identical coupled waveguides (3.2) and follow the propagation of an input mode through the FDC. The second derivation is based on the Bloch mode scattering matrix formulation, which is first reduced to include only the propagating modes, as in Section 3.3.2, then simplified further by replacing the numerically calculated scattering matrices by idealised expressions. The 78 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

1 1

0.8 0.8

0.6 0.6 T T 0.4 0.4

0.2 0.2

0 0 0.29 0.3 0.31 0.32 0.33 0.29 0.3 0.31 0.32 0.33 d/l d/l (a) (b)

Figure 3.10: FDC transmission spectra for (a) FDCs with Nc = 1 and cavity lengths L = 4d (dashed), L = 5d (solid) and L = 6d (dashed-dotted) and (b) FDCs of length

L = 5d and guide separations of Nc = 1 (solid), Nc = 2 (dashed) and Nc = 3 (dashed- dotted). resulting model requires numerical input of only the waveguide dispersion properties and the structure length to describe accurately the transmission properties of the FDC. It also provides physical insight into many of the spectral features observed in Fig. 3.10. We compare the results to full numerical calculations using the Bloch mode method.

3.4.1 Semianalytic model: coupling analysis Consider ﬁrst the modes in each section of the FDC shown in Fig. 3.9(b). The input and output waveguides each support a single mode, |ψini and |ψouti, respectively, while the coupling region supports even and odd supermodes |ψ±i with propagation constants β±. As in Sections 1.7.5 and 3.2.1, we deﬁne ∆β = |β+ − β−|/2 and β = (β+ + β−)/2, so that the coupling length Lc = π/(2∆β). Recall from Section 3.2.1 that the supermodes of two identical coupled waveguides can be approximated by linear superpositions of the individual waveguide modes using Eq. (3.2). The comparison with rigorous numerical results in Fig. 3.1 clearly demon- strates that this approximation remains accurate even for PhC waveguides separated by a single line of cylinders and can therefore be applied to the modes in region M2 of the FDC. Since the input and output waveguides continue into region M2, the coupled mode result provides a simple relationship between the modes in each of the three regions. From Eq. (3.2) we can write √ √ |ψin/outi = (|ψ+i ± |ψ−i)/ 2, |ψ±i = (|ψini ± |ψouti)/ 2 . (3.5)

We now follow the propagation of a mode as it passes through the FDC. Light entering from the input waveguide in mode |ψini is coupled into the supermodes of region M2 in the ratio given by Eq. (3.5), where we ignore back reflection at this first 3.4. FOLDED DIRECTIONAL COUPLER 79 interface since the input waveguide is continuous. The even and odd supermodes then propagate a distance L through region M2, advancing in phase by factors exp(iβ+L) and exp(iβ−L) respectively so that the field at the output interface is given by √ (1) iβ+L iβ−L Ψ(L) = e |ψ+i + e |ψ−i / 2 , iβ+L iβ−L iβ+L iβ−L = e + e |ψini/2 + e − e |ψouti/2 , iβL = e (cos(∆βL)|ψini + i sin(∆βL)|ψouti) , (3.6) where we have used the definitions of β and ∆β to write β± = β ± ∆β. The superscript (1) on the left hand side of equation (3.6) indicates that this is the first pass of the light through M2. Notice that the modes of the two waveguides have a relative phase difference of π/2 due to the mode coupling, while they share a common phase term exp(iβL) which depends on the average propagation constant. At the M2–M3 interface, the right-hand waveguide continues into M3 without in- terruption, whereas the left-hand waveguide ends. We therefore assume that the field component in mode |ψouti is transmitted with amplitude τ1 = i exp(iβL) sin(∆βL), while the remaining field in mode |ψini is reflected by the photonic crystal back into the left-hand waveguide. For the purposes of this model, we neglect any phase change that may occur when |ψini is reflected, and so the reflected mode has an amplitude of exp(iβL) cos(∆βL). Note that if the FDC was formed from conventional dielectric waveguides, most of the light in the closed waveguide would be radiated into the cladding and lost. The reflected field propagates back through region M2 to the first interface, where it is again expressed in terms of the individual waveguide modes as

(1) 2iβL 2 Ψ(0) = e cos (∆βL)|ψini + i cos(∆βL) sin(∆βL)|ψouti . (3.7)

Here, as at the output interface, the field in the open guide is transmitted into the input 2 waveguide with amplitude ρ1 = exp(2iβL) cos (∆βL) while the field in the right-hand waveguide is reflected back into region M2. We again see that both terms in Eq. (3.7) have gained an additional factor of exp(iβL) and the individual waveguide modes are again π/2 out of phase. The total transmission and reflection amplitudes of the FDC, denoted τ and ρ respectively, are found by repeating this calculation for each pass through M2 and adding coherently the transmitted and reflected components, τn and ρn respectively, 80 CHAPTER 3. COUPLED WAVEGUIDE DEVICES which yields τ = ieiβL sin(∆βL) + ie3iβL cos2(∆βL) sin(∆βL) − ie5iβL cos2(∆βL) sin3(∆βL) + ie7iβL cos2(∆βL) sin5(∆βL) − ... (3.8a) 1 + exp(2iβL) = i exp(iβL) sin(∆βL) 1 + sin2(∆βL) exp(2iβL)

ρ = e2iβL cos2(∆βL) − e4iβL cos2(∆βL) sin2(∆βL) + ie6iβL cos2(∆βL) sin4(∆βL) − e8iβL cos2(∆βL) sin6(∆βL) − ... (3.8b) cos2(∆βL) exp(2iβL) = . 1 + sin2(∆βL) exp(2iβ)L Hence the reflectance is cos4(∆βL) R = |ρ|2 = , (3.9) cos4(∆βL) + 4 sin2(∆βL) cos2(βL) and the transmittance is simply T = 1−R. Note that the summation for τ in Eq. (3.8a) is a geometric series from the second term onwards, whereas the summation for ρ in Eq. (3.8b) is geometric series from the first term, as for a standard FP interferometer [122]. The relative phase of each term is determined by the value of exp(iβL), which is associated with a single pass through M2. We return to this property when considering reflection resonances in Section 3.4.3. Here we have derived a semianalytic model to describe the transmission properties of the FDC that requires numerical input of only the modal dispersion properties in region M2. These are calculated easily using the Bloch mode method or another PhC calculation tool, and substituted into Eq. (3.9) to provide a very efficient formulation to study FDC behaviour. The properties predicted by the model are discussed in Section 3.4.3, but first we outline an alternative derivation of Eqs. (3.8) based on Bloch mode scattering matrices.

3.4.2 Semianalytic model: Bloch mode matrix method The FDC shown in Fig. 3.9(a) can be analysed using the Bloch mode method as a three layer structure, and is therefore described by Eqs. (2.13) in terms of the Fresnel matrices at each interface. Propagation from M1 to M3 is characterised by six Fresnel matrices, R12, R21, R23, T12, T21 and T23. We showed in Section 3.3.2 that it is not always necessary to retain the evanescent Bloch modes in the calculation if the interfaces in the structure are suﬃciently far apart. In this limit, accurate results can often be obtained by including only the propagating modes in each section. This approach can be applied to the FDC, however in contrast to the Fabry-P´erot example, there are two propagating modes in the central section, rather than one. 3.4. FOLDED DIRECTIONAL COUPLER 81

The truncated matrix dimensions are determined by the number of propagating modes on either side of the relevant interface. For instance, R12 describes modes incident and reflected in section M1, which only supports a single mode, hence R12 is a 1 × 1 matrix, and for the same reason, R21 and R23 are 2 × 2 matrices. T12 describes transmission of the single mode in M1 into the two modes of M2, and hence it is a column vector with two rows (2 × 1). Similarly, T21 and T23 are 1 × 2 row vectors. The matrix Λ, which represents propagation through region M2, reduces to a 2×2 diagonal matrix of eigenvalues µi = exp(iβjd), where j = 1, 2. When these reduced matrices are substituted into Eq. (2.13b), the resulting transmission coefficient is a complex scalar quantity. While the transmission properties of the FDC are described accurately by such a propagating mode treatment, the computational requirements are only modestly reduced since the full plane wave basis is still required to represent the Bloch modes. We can however, determine approximate analytic expressions for the Fresnel matrices using the same assumptions as in Section 3.4.1. At the first interface, |ψini is perfectly transmitted into |ψ±i with amplitudes given by Eq. (3.5) which gives √ 1/ 2 R = 0 , T = √ . (3.10) 12 12 1/ 2

At the M2–M3 interface, the ﬁeld in the right-hand guide is transmitted freely into the output guide, so we can again use Eq. (3.5) to write √ √ T23 = [ 1/ 2 −1/ 2 ]. (3.11)

The reflection matrix at the second interface is a 2 × 2 matrix, where element (i, j) is the reflection coefficient for the reflection of mode j into mode i (travelling in the opposite direction), and modes 1 and 2 are the even and odd supermodes, respectively. Once again, assuming that a mode in the right guide is completely transmitted while a mode in the left guide is completely reflected back into the left guide, we can write √ √ √ 1/ 2 1/ 2 1/ 2 0 1/2 1/2 R . √ √ = √ ⇒ R = . (3.12) 23 1/ 2 −1/ 2 1/ 2 0 23 1/2 1/2

Similar arguments are used to ﬁnd the ﬁnal two scattering matrices,

1/2 −1/2 √ √ R = and T = [ 1/ 2 1/ 2 ] . (3.13) 21 −1/2 1/2 21

Equations (3.8) are derived by substituting these ideal Fresnel matrices into Eqs. (2.13) and making the appropriate substitutions for β and ∆β. Although the result is identical to that derived using the coupled mode analysis in Section 3.4.1, the technique described in this section provides a systematic approach that is more easily applied to complicated structures. In addition, it is possible to improve the accuracy of the model by replacing the approximate matrix entries derived above by more accurate estimates. 82 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

For instance, it is straightforward to include a small reflection at the first interface by setting R12 6= 0 and making the appropriate modifications to T12 to maintain energy conservation.

3.4.3 Transmission properties In this section we investigate the transmission characteristics of the FDC and compare the semianalytic model derived in Section 3.4.1 to full numerical calculations using the Bloch mode method. The model is used to identify parameters for which the FDC operates as a high-Q notch-rejection filter with a resonance width determined largely by the mode splitting parameter ∆β. We also discuss methods of tuning the FDC geometry to increase Q and reduced the device length. The transmission spectra in Fig. 3.10 exhibit a number of high reflection and high transmission features that we now identify using the model derived in Section 3.4.1. Each of these features is driven by one of two physical effects; the features associated with particular values of ∆βL result from coupling between the waveguides in M2, while those features associated with particular βL values are caused by mode recirculation in M2. We consider first the high transmission features. From Eq. (3.9) it can be seen that all incident light is transmitted when cos(∆βL) = 0 which corresponds to L = Lc. The physical explanation for this is clear from Fig. 3.9: light incident in mode |ψini is completely coupled into |ψouti at the end of M2 where it is transmitted into the output waveguide with minimal reflection. This single-pass behaviour can be seen in the field intensity plot of Fig. 3.11(a), calculated for a FDC with L = 4d and Nc = 1 at d/λ = 0.319. For these parameters, T > 99.9%. Provided that cos(βLc) 6= 0, the reflection zero of Eq. (3.9) has a quartic dependence on cos(∆βL), resulting in the flat- topped, high-transmission parts of the spectra in Fig. 3.10(a). The Nc = 2 and Nc = 3 curves in Fig. 3.10(b) do not exhibit this behaviour since the coupling is much weaker and hence L < Lc. If the cavity length is doubled to L = 2Lc, then all of the light is coupled back into mode |ψini by the end of M2, where it is reflected back through M2 and out of the input guide. From Eq. (3.9) it is seen that L = 2Lc gives sin(∆βL) = 0, and hence R = 1. Figure 3.11(b) shows the field intensity in a FDC with Nc = 1 and L = 8d, calculated at d/λ = 0.317, for which T < 1%. The small frequency difference is attributed to an additional phase shift introduced when the mode is reflected – recall that we ignored any phase change on reflection by setting the reflection coefficient equal to 1 in Section 3.4.1. For the purpose of designing a notch-rejection filter, we are interested in the second condition for which R = 1 in Eq. (3.9), namely when cos(βL) = 0, or exp(iβL) = ± i. Unlike the transmission and reflection features that we have just discussed, this reflection resonance is a result of mode recirculation through the double guides in M2. To see this, consider again the series for ρ and τ in Eqs. (3.8a) and (3.8b) which correspond to the transmitted and reflected field amplitudes on each pass through section 3.4. FOLDED DIRECTIONAL COUPLER 83

(a) (b) 0 0.2 0.4 0.6 0.8 1 Field intensity (a.u.)

Figure 3.11: (a) Field intensity in a FDC of length L = 4d at d/λ = 0.319, for which the reﬂected power is less than 0.1%. (b) Field intensity for a FDC of length L = 8d at d/λ = 0.317d, for which R > 99.9%. The geometry of the two devices is identical aside from the length.

M2. When the resonance condition is satisfied the field transmitted on the first pass, ◦ τ1 is 180 out-of-phase relative to transmitted fields on each subsequent pass. On the other hand, the reflection terms ρn are all in-phase and hence they combine constructively to produce the resonant reflection. This resonance is the cause of all of the local transmission minima seen in Fig. 3.10 since L < 2Lc in each example and hence the first R = 1 condition identified above is not satisfied. Observe that the resonance width varies considerably for the different structures, but appears to be narrowest when the transmission on either side of the resonance is close to 1. This is not a coincidence, as we now show.

Suppose that the resonance occurs at a frequency ω = ω0, and denote the values of

∆β and β at resonance by ∆β0 and β0. For frequencies ω = ω0 + δω close to ω0, ∆β and β are expanded to ﬁrst order as

∆β = ∆β0 + δω∆β1 , β = β0 + δωβ1 , (3.14) where d∆β dβ ∆β = and β = . 1 dω 1 dω ω=ω0 ω=ω0

Since we are interested in the resonance related to the zeros of cos(βL), we set β0 = nπ/(2L) for n = 1, 3, 5,... and substitute Eqs. (3.14) into Eq. (3.8b) to obtain −1 1 ρ = ,R = , (3.15) 1 − 2iδω/∆ω 1 + (2δω/∆ω)2 84 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

where R = 1/2 at ω = ω0 ± ∆ω/2 and cos2(∆β L) ∆ω = 0 . (3.16) β1L Note that Eqs. (3.15) describe a classic Lorentzian reflection resonance, as expected for a simple resonant cavity [113,190]. Our earlier observation that the resonance is narrowest when the surrounding transmission is high is verified by Eq. (3.16), since both R → 0 and ∆ω → 0 as cos(∆β0L) → 0. This is a desirable feature for notch-rejection filters which ideally transmit all frequencies ω 6= ω0. We note also that the β1 term in Eq. (3.16) is a function of the group velocities of |ψ+i and |ψ−i. Since β is well approximated by the dispersion of a single waveguide for frequencies well above cutoff (see Section 3.2.1), we have β1 ≈ 1/vg. Thus Q is approximately inversely proportional to the group velocity of the waveguides at the resonant frequency. A similar dependence is displayed by other coupled-cavity designs, and this has been used to improve their resonant characteristics [83, 191]. Equations (3.9),(3.15) and (3.16) also show that the resonant frequency for a FDC of length L is determined largely by β, whereas the resonance width depends strongly on ∆β. Since ∆β is more sensitive than β to changes in the coupling conditions, it is therefore possible to tune the resonance width by modifying the structure of region M2, for instance by changing the waveguide separation or modifying the cylinders between the two guides. An example of the latter method is given in Section 3.4.4, where we demonstrate tuning of Q by almost four orders of magnitude. The results of Eqs. (3.9) and (3.15) (dashed curves) are compared to the full numerical calculations (solid curves) in Figs. 3.12(a) and (b) for a FDC of length L = 5 and Nc = 1. The dashed curve in (a) is given by Eq. (3.9) with L = 6.067d, which was chosen to give the best agreement with the numerical result. Although L was defined in Fig. 3.9 as the number of PhC layers forming region M2, this is not necessarily the most accurate value to use in Eq. (3.9) since the waveguide ends act as distributed reflectors and hence the phase change on reflection can be relatively large. This effect was neglected in Section 3.4.1, but can be partly corrected for by adjusting L. In many cases a fixed L value gives reasonable agreement for a small range of frequencies, as can be seen in Figs. 3.12(a) and (b), but this does not take into account the wavelength dependence of the phase change. Attempts to include a wavelength-dependent length correction by considering the phase of the reflected mode have proved no more accurate than the simple result, suggesting that there are additional physical effects that are not accounted for in our model. In most cases, the length adjustment necessary for the best agreement with numerical results is approximately one lattice period or less. The transmission spectrum in Fig. 3.12(b) is a high-resolution plot of the sharp reflection resonance observed at d/λ = 0.33107 in (a) where the resonant reflection exceeds 99.95%. For operation at λ = 1.55 µm, corresponding to d = 513 nm, the length of M2 is only L = 5d = 2.6 µm or 1.7 wavelengths, and the resonance width is approximately 0.11 nm. From the resonance we calculate Q = 1.3×104 for the numerical 3.4. FOLDED DIRECTIONAL COUPLER 85

1 (a) 0.8

0.6 T 0.4

0.2

0 0.31 0.315 0.32 0.325 0.33 d/l 1 (b) 0.8

0.6

T 0.4 0.2 100 101 102 103 104 105 0 Field intensity (a.u) 0.33102 0.33106 0.33110 0.33114 d/l (c)

Figure 3.12: (a) Transmission spectrum as a function of normalised frequency for an un-optimised FDC of length L = 5d and Nc = 1. The solid curve in (a) and (b) is the full numerical result and the dashed curve is given by Eq. (3.9) with L = 6.066. (b) High resolution plot of the resonance in (a) centred at d/λ = 0.33107. (c) Field intensity in the FDC on resonance. Note that the colour scaling is logarithmic and spans five orders of magnitude. result (solid curve) and Q = 2.4 × 104 for the approximate result (dashed curve). As expected, there is a strong field enhancement in region M2 when the resonance condition is satisfied, as shown in Fig. 3.12(c), where the field intensity on resonance is plotted on a log scale. The peak intensity is more than 4 orders of magnitude greater than the input intensity, which is consistent with the Q of the resonance. Despite the factor of two difference in the peak width, the simple model predicts the broader characteristics of the FDC with reasonable accuracy, and as we show in Section 3.4.4, it can be used as an efficient aid to device optimisation.

3.4.4 High-Q tuning In this section we demonstrate ﬁne-tuning of the FDC response by changing the radius of the cylinders that separate the two guides in region M2. The resonance width is varied by almost four orders of magnitude to obtain an optimised FDC design of length L = 4d that exhibits Q > 6×105. The change in Q is predicted accurately by Eq. (3.16). We begin with the FDC illustrated in Fig. 3.11 (a), which has a length of L = 4d. 86 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

rc=0.25d

0.8 rc=0.26d

rc=0.27d

rc=0.28d 0.6 r =0.29d T c rc=0.30d r =0.31d 0.4 c rc=0.32d

rc=0.33d

0.2 rc=0.34d (a) rc=0.35d 0 0.31 0.312 0.314 0.316 0.318 0.32 d/l

6 Bloch mode method (L=4d) 10 Model (L = 4.83d) Model (L = 4.90d) 105 Q 104 103

102 (b) 10 0.26 0.28 0.3 0.32 0.34

Figure 3.13: (a) Transmission spectra for a FDC of length L = 4d as the cylinder radius between the two waveguides (rc) is changed between 0.25d and 0.35d. The black curve corresponds to the original structure with rc = 0.34d. (b) Variation of Q with rc. The black points correspond to the curves in (a), while the red and blue points are given by Eq. (3.16) for two diﬀerent value of L.

The transmission spectrum for the unmodiﬁed structure is indicated by the dashed line in Fig. 3.10(a) and exhibits a resonant reﬂection close to d/λ = 0.315 with Q = 1300. The same curve is shown in Fig. 3.13(a) for a narrower range of frequencies, where the black curve corresponds to the original FDC with a cylinder radius of r = 0.3d. The remaining curves in Fig. 3.13(a) show the transmission spectrum as the radius of the four cylinders between the guides is varied between rc = 0.25d and rc = 0.35d. Observe that the resonant frequency decreases with increasing radius, while the peak width initially narrows to a minimum at rc = 0.27d, then broadens again. Figure 3.13(b) is a plot of Q as a function of rc, where the black circles are calculated directly from the transmission spectra in (a). Observe that Q varies by almost four 5 orders of magnitude, and reaches a maximum of Q = 6 × 10 when rc = 0.27d. In this 3.4. FOLDED DIRECTIONAL COUPLER 87

100 101 102 103 104 105 106 Field intensity (a.u)

Figure 3.14: Resonant ﬁeld intensity in the optimised FDC with L = 4d and Nc = 1. The four central cylinders have a radius of rc = 0.27d: 0.03d smaller than the other cylinders. Note that the colour scaling is logarithmic and spans six orders of magnitude.

purely 2D geometry, Q can be made arbitrarily large in principle by adjusting rc to match the ∆ω = 0 condition given by Eq. (3.16). The red and blue points plotted in

Fig. 3.13 are given by Eq. (3.16) for L = 4.83d and L = 4.90d respectively, where ∆β0 and β1 have been calculated for each value of rc. The choice of L = 4.83d gives the best agreement between Eq. (3.9) and the rc = 0.3d curve in Fig. 3.13, whereas L = 4.9d provides the best agreement with the numerically calculated Q values. The resonant field intensity inside the optimised FDC is shown in Fig. 3.14. Observe that the peak intensity inside region M2 is more than five orders of magnitude larger than the incident field, clearly illustrating the high-Q properties of the FDC. The results in this section provide a convincing demonstration of how the semianalytic model can be used to great advantage when optimizing design parameters. In this example, the results are summarised by the plot in Fig. 3.13(b) which indicates the variation of Q as a function of rc. Each black point in (b) corresponds to a single curve in (a), each of which must be calculated with enough points to resolve the resonance. The curves in (a) each consist of approximately 250 points, and many more would be required if the frequency grid was uniform. One data point takes approximately 17 s to 88 CHAPTER 3. COUPLED WAVEGUIDE DEVICES calculate using the Bloch mode method on a Pentium 4 (2.8 GHz) desktop computer, which gives a total computation time of 13 hours to generate the 11 curves in Fig. 3.13. In contrast, the blue and red data points in Fig. 3.13(b) are given by Eq. (3.16), for which we only need to calculate the coupled waveguide dispersion curves for each value of rc. Since β and ∆β are slowly varying functions of frequency only 11 data points were used to calculate the dispersion curves and then a polynomial function was fitted to allow interpolation and differentiation. Each point takes approximately 7 s to calculate, so the total calculation time for the 11 different rc values was approximately 810 s – fifty-five times faster than the full calculation. Once the dispersion data was calculated, the red and blue data points in Fig. 3.13(b) were generated in a few seconds from Eq. (3.16). Although the accuracy of the model depends on the choice of L, this can be esti- mated using a number of methods. For instance, the red points in Fig. 3.13(b) were calculated using the L value that gave the best match between Eq. (3.9) and the original spectrum. Even if this meant one full transmission spectrum was required, the reduction in computation time is still significant. Approximate methods do not replace full numerical calculations, which would still be required for the final optimization process, however they can be used to restrict the computational demand to a minimum.

3.4.5 Comparison with side-coupled cavity Although it is not immediately obvious from the geometry, the FDC is closely related to other waveguide-coupled cavities such as those in Refs. [115, 120, 121, 192], in which a short line-defect is coupled by evanescent fields to a straight PhC waveguide as illustrated in Fig. 3.15(a). A number of semianalytic theories have been developed for describing the coupling in this type of structure based on tight-binding approximations and scattering-theory [113,190,193,194]. Here we show that the simple mode-coupling analysis developed in Section 3.4.1 also provides an accurate approximation for simple side-coupled cavity geometries. A feature of these designs is that the in-plane Q, increases exponentially with the distance between the cavity and the waveguide [192]. In contrast, it is clear from Fig. 3.10 that the FDC resonances start to disappear as the waveguide separation increases. To understand this difference we consider the simple side-coupled cavity (SCC) design illustrated in Fig. 3.15(a), which resembles the FDC in that it consists of single input and output waveguides coupled into a section of double waveguide. This structure can be modelled using the same semianalytic method applied to the FDC in Section 3.4.1. The only difference in the analysis is that the light in the right-hand waveguide is reflected at the second interface of the SCC, whereas in the FDC the light in the left-hand waveguide was reflected. The equation for the reflected power of the SCC is thus found to be

sin4(∆βL) R = , (3.17) sin4(∆βL) + 4 cos2(∆βL) sin2(βL) 3.4. FOLDED DIRECTIONAL COUPLER 89

106

105

Q 4 L 10

103

102 2 3 4 5 6 7 N c Nc (a) (b)

Figure 3.15: (a) Diagram of the side-coupled cavity geometry considered in Sec- tion 3.4.5. The example shown here is for Nc = 6. (b) Plot of Q as a function of Nc for a cavity with L = 2, as in (a). Black points are calculated using the Bloch mode method and the red points are given by Eq. (3.18) for L = 2.882. which is identical in form to Eq. (3.9) but with the cos(∆βL) and sin(∆βL) terms swapped and sin(βL) in place of cos(βL). In analogy to the FDC, the resonance of interest occurs in the SCC for sin(βL) = 0 and has a width approximated by

sin2(∆β L) ∆ω = 0 , (3.18) β1L which is the equivalent of Eq. (3.16). Let us now compare Eqs. (3.16) and (3.18) to identify the diﬀerences between the FDC and the SCC. For both geometries, the position of the resonance is determined by

βL whereas the resonance width depends on ∆βL and β1L. Recall from Section 3.4.3, that the maximum Q of the FDC occurs when cos(∆βL) ≈ 0, and thus we were able to tune the Q in Section 3.4.4 by modifying the coupling conditions, and hence the value of ∆β in the cavity region. Similarly, to maximise the Q of the SCC, we want sin(∆βL) = 0 in Eq. (3.18). This can be achieved in one of two ways. The ﬁrst is to design the cavity such that L = 2mLc, where m is an integer, and hence ∆βL = mπ, but this requires a cavity twice the length of the equivalent FDC. The second option is to let ∆β → 0, which can be done by increasing the waveguide separation. This latter approach for tuning Q is the most common technique used in the literature (see for example Ref. [192]). While we have only considered a basic SCC geometry, it is interesting to compare the relative sizes of a FDC and a SCC with the same Q. In Section 3.4.4 we designed a FDC with L = 4d and Q = 6 × 105. This resonance corresponded to βL = 3π/2 – the second zero of cos(βL). A shorter FDC would not have been possible in this geometry, since we require L = Lc in order to maximise Q, and hence the shortest length is determined by the maximum ∆β that can be achieved. The SCC does not have the same length limitation since Q can always be increased by decreasing ∆β. Hence, the 90 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

ﬁrst practical resonance occurs when βL = π. This condition is satisﬁed for d/λ ≈ 0.316 for a cavity of L = 2d as illustrated in Fig. 3.15(a). Figure 3.15(b) is a plot of Q as a function of Nc for Nc = 1 − 7 where the black points were calculated using the full Bloch mode method and the red points are given by Eq. (3.18) with L = 2.882d. This value of L was chosen so that the resonant frequency predicted by Eq. (3.17) matched the numerical results for Nc = 4. As in the FDC example, the semianalytic model provides an accurate estimate of the resonant properties of the cavity.

Despite the exponential increase of Q with Nc, the SCC still requires Nc > 6 for Q > 105, which corresponds to a distance of more than 7d between the cavity and the waveguide. This is approximately twice the size of the FDC design considered in Section 3.4.4. Although a rigorous comparison of relative sizes would include the area of PhC required around each design to conﬁne the light in the plane, the simple comparison here suggests that the FDC compares favourably with alternative high-Q ﬁlter designs.

3.4.6 Discussion In this section we have demonstrated a novel filter design that exhibits high Q resonances in a compact geometry. Since it is essentially an in-line device it can be easily coupled to, and the resonance properties can be optimised by modifying the geometry of the coupling region. As with any resonant cavity, however, the control of out-of- plane losses is critical if such high-Q values are to be achieved in practice. Although we have not considered losses in our analysis, it is expected that low-loss designs could be optimised in much the same way as the cavities discussed in Section 1.7.1. We return to this issue in Chapter6. The comparison between the FDC and a simple side-coupled cavity filter in Sec- tion 3.4.5 shows that the two devices have similar resonance characteristics when operated as narrow-band filters, but we have not compared other functions such as add-drop behaviour or nonlinear switching characteristics. A more detailed study of these properties would be required in order to establish the relative merits of the two geometries. Although we do not consider the structure here, we note that the modification proposed by Fan [195] to achieve asymmetric line shapes in side-coupled cavities can also be applied in the case of the FDC. In addition to the high-Q resonances studied in this section, the transmission spectra in Fig. 3.10(a) also exhibit regions of flat, almost perfect transmission. This characteristic behaviour is exploited in Section 3.5 to design an efficient wide-bandwidth symmetric Y-junction that is incorporated into a PhC-based Mach-Zehnder interferometer in Chapter 4. We also study the transmission properties and band structure of a chain of identical FDCs in Section 3.6. 3.5. COUPLED Y-JUNCTION 91 3.5 Coupled Y-junction

Efficient, wide-bandwidth Y-junctions or beam splitters are required for compact integration of multiple optical devices on photonic chips. As a basic component of many integrated optical devices, these junctions must be designed with a minimum back reflection and maximum transmission over the operating bandwidth. A number of studies have been made into the optimal method of splitting a single guided mode into two modes, and the approaches taken can be broadly classified into two groups. The first design approach has been to optimise simple Y-junction designs, while the second approach has been to design novel alternative structures exploiting the properties of photonic crystals. Good transmission has been demonstrated in Y-junctions both theoretically [77,196–198], and experimentally [198,199] using optimisation techniques such as placing one, or several, “tuning” cylinders near the junction of three waveguides. High transmission junctions, with calculated transmissions up to 99% have been designed, however in most cases, the transmission bandwidth decreases rapidly as the maximum transmission approaches 100% [77]. Typical bandwidths reported in the literature for hole-type PhC junctions are on the order of 10 − 40 nm for 95% transmission.

An alternative to the Y-junction is essentially a directional coupler with L = Lc/2 so that light incident in one of the guides is split evenly between the two guides. Such a structure was proposed by Martinez et al. [200] as a means of splitting an input beam for a Mach-Zehnder interferometer application. The optimal coupling length is wavelength dependent, but may be robust enough to give a satisfactory bandwidth. One of the big advantages of a coupler-based beam splitter is that the back reflection can be negligible. If the coupler is not perfectly matched to the wavelength, the splitting between the output guides changes, but the total transmission of the splitter remains high. This is an important issue in compact devices where back reflections can cause unwanted interference effects. A modified version of the directional coupler junction was demonstrated by Sugimoto et al. [109] in which a cavity was placed between the two waveguides to create a similar geometry to the coupled cavity add-drop filter illustrated in Fig. 1.9(a). Couplers of this type have been successfully integrated into PhC Mach- Zehnder interferometers [201]. Here, we present a novel coupled Y-junction design which operates similarly to a directional coupler junction, but with the symmetry properties of a Y-junction that ensure the transmission into each output arm is identical even when the coupling length is not perfectly matched to the wavelength. In addition, recirculation inside the junction results in high transmission over a wide bandwidth. We show that the junction is closely related to the folded directional coupler, and a modal analysis similar to that in Section 3.4 yields identical expressions for the transmission and reflection. While similar structures have been studied in dielectric waveguides [202, 203], the PhC-based device gains a significant advantage from the perfectly reflecting cladding which provides mode recirculation that increases the transmission bandwidth.

92 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

¡ ¢¡¨§©¥

¢¡¤£¦¥ ¢¡ ¥

(a) (b)

Figure 3.16: (a) Geometry of the coupled Y-junction. (b) Schematics of the symmetric propagating modes in the three waveguide sections.

The general geometry of the coupled Y-junction is illustrated in Fig. 3.16(a). A single input guide enters a coupling region of length L consisting of three identical waveguides. At the output interface, the central waveguide ends and the two remaining waveguides continue as outputs. Light entering the triple guide region in the central waveguide is coupled into the two guides on either side. If the length of the triple guide region is such that all of the light has been transferred into the outer guides at the end of the triple guide region, then almost all light propagates into the two output guides. If the output guides are sufficiently far apart, the coupling between them is negligible. In the junction shown in Fig. 3.16 the output guides are separated by five layers of cylinders so the coupling length is several hundred lattice periods. Hence, for compact devices the guides can be treated as isolated from each other. A semianalytic model for the coupled Y-junction is derived in the same way as the FDC model in Sec. 3.4.1, and we show here the two structures are described by an identical set of equations. We must first consider the propagating modes within each of the three regions. The input guide supports a single even mode, the output guides support an odd and an even mode, and the central triple guide section supports three modes - two even and one odd - as discussed in Section 3.2.2. However, since the structure is symmetric and the input waveguide only supports a single even mode, light can only couple to the even modes in each section. Hence, the single input mode couples into two modes in the central region, which in turn couple into a single output mode, in much the same way as in the FDC. Schematics of the even modes in each region of the coupled Y-junction are shown in Fig. 3.16(b). Recall from Section 3.2.2 that the supermodes of the triple guide region can be approximated by linear superpositions of the individual waveguide modes. Hence we 3.5. COUPLED Y-JUNCTION 93 can use Eq. (3.3) to write √ √ |ψ1i = (|ψLi + 2 |ψini + |ψRi)/2 = (|ψini + |ψouti)/ 2, (3.19) √ √ |ψ3i = −(|ψLi − 2 |ψini + |ψRi)/2 = (|ψini − |ψouti)/ 2, where |ψLi, |ψini and |ψRi are the modes√ of the left, centre (input) and right waveguides respectively, and |ψouti = (|ψLi + |ψRi)/ 2 is the even mode of the output waveguides. As in the case of the FDC, the propagation constants of |ψ1i and |ψ3i are expressed as β1 = β + ∆β and β3 = β − ∆β, where β = (β1 + β3)/2 and ∆β = |β1 − β3|/2. Equations (3.19) reveal the direct mapping between the coupled Y-junction and the FDC; |ψ1i and |ψ3i are equivalent to |ψ+i and |ψ−i in Eq. (3.5) while the input and output modes of the two structures are also equivalent. Hence, the transmission properties of the coupled Y-junction are described by Eqs. (3.8) and (3.9) with β and ∆β calculated for the even modes of the triple guide section. Although the coupled Y-junction also exhibits the sharp resonances of the FDC, it is the flat, high-transmission regions of the spectrum that we wish to exploit for the purposes of this device. In Section 3.4.3 we identified this feature with the condition cos(∆βL) = 0, which corresponds to L = Lc. When this condition is satisfied, all of the incident light is coupled from the central input waveguide into the output waveguides on the first pass through the triple guide section, and so the transmission is high. Since the reflectance given by Eq. (3.9) is quartic in cos(∆βL), the high transmission is typically flat-topped and broad. In order to maximise the transmission bandwidth, it is desirable to have cos(βLc) ≈ 1 so that the reflection resonances are far from the maximum transmission frequency, although reasonable bandwidths are possible without this condition, as demonstrated in Section 4.4.1. Figures 3.17(a) and (b) show the transmission through two different coupled Y- junctions, with lengths L = 5d and L = 7d respectively. The junction is formed in the PC1 lattice, with the output guides separated by 5 rows of cylinders, and each of the guides in the central region separated by 2 rows of cylinders. The lattice period is d = 491 nm in both examples, corresponding to a cylinder radius of 173 nm. For lengths of L = 5d and L = 7d, the Y-junction has 95% transmission bandwidths of approximately 92 nm and 83 nm, respectively at λ = 1.55 µm, and 99% transmission bandwidths of about 80 nm, and 76 nm. The dashed curves in Figs. 3.17(a) and (b) are the semianalytic result (3.9) calculated with L = 5.75d and L = 7.80d respectively for the two devices. The solid curves show the transmission calculated using the full Bloch mode method. As in the examples of Section 3.4, the difference between the length defined in Fig. 3.16 and the length parameter in the model results from phase changes upon reflection, and the ambiguity in defining a reflection plane at either end of the triple guide region. The semianalytic model was used to design the two junctions, and it can be seen that the transmission bandwidth obtained from the full Bloch mode calculation exceeds the bandwidth predicted by the model. In Section 4.4.1 we present two other coupled Y-junction designs - one in a rod-type PhC and the other in a triangular 94 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

(a) 1 92nm 0.8

0.6 T 0.4

0.2 L = 5d (b) 1 83nm 0.8

0.6 T 0.4

0.2 L = 7d 0 1.48 1.5 1.52 1.54 1.56 1.58 1.6 λ (µm)

Figure 3.17: Transmission curves for the coupled Y-junction with (a) L = 5d and (b) L = 7d. Wavelengths have been scaled so that the high transmission bands occur near λ = 1.55 µm. The 95% transmission bandwidths are 92 nm and 83 nm respectively. The solid curves are calculated using the full Bloch mode method, and the dashed curves are the semianalytic result given by Eq. (3.9) for L = 5.75d and L = 7.80d. lattice hole-type PhC. The hole-type PhC junction has a 95% transmission bandwidth of 39 nm, which compares favourably with alternative junction designs reported in the literature [77,197,198]. It is expected that the bandwidths could be further increased by tuning the structure of the triple guide region as demonstrated in Section 3.4.4 for the FDC. We note that since this work was reported in Ref. [204], similar junction designs have been proposed by other authors [205, 206], however the structures we present here exhibit superior bandwidth and transmission properties. There have also been several experimental demonstrations of efficient junction designs obtained via numerical optimisation of the structure [199,207]. When designing the coupled Y-junction for maximum bandwidth, the splitting of the modes in the triple guide region must be considered. Since the successful operation of the device requires two propagating even modes to exist in this region, we must operate in the frequency range above the cut-off of the second even mode, and below 3.6. SERPENTINE WAVEGUIDE 95 the high-frequency edge of the bandgap. If the three guides are too close together, the mode splitting becomes very large, and the frequency range for operation becomes limited. This is clearly seen in the triple waveguide dispersion curves for Nc = 1 in Fig. 3.4. On the other hand, if the guides are too far apart, the coupling length becomes prohibitively large. Thus, the choice of guide separation and its effect on mode splitting must be considered. In the examples chosen above, the second even mode cutoff occurs at λ ≈ 3.32d, giving a maximum bandwidth of ∆λ = 0.33d corresponding to approximately 160 nm at λ = 1.55 µm. The transmission bandwidth can also be increased by operating at frequencies where the dispersion curves of all three modes are approximately parallel, such that ∆β and hence Lc are very weak functions of wavelength [205]. The coupled Y-junction design provides a novel solution to the problem of achieving high-transmission, wide-bandwidth beamsplitting, by combining the efficient coupling properties of a directional coupler with the symmetrical splitting of more conventional Y-junction designs. Such properties make this junction ideal for use in PhC-based Mach-Zehnder interferometers where high performance junctions are required. We demonstrate this application in Chapter 4. While the 2D results presented here are promising, further work is required to give an accurate comparison with alternative Y-junction designs, not only in terms of transmission and bandwidth, but also regarding fabrication tolerance. One issue of particular interest is the sensitivity to structural variations that break the symmetry properties of the junction. We have only considered the even modes in our analysis, however any breaking of the symmetry allows light to also couple into the odd modes. Although this is likely to result in unequal splitting between the two output waveguides, it is not clear whether the back reflection would increase significantly.

3.6 Serpentine waveguide

In this section we consider the serpentine waveguide – a period coupled-cavity waveguide formed by cascading multiple FDC structures as shown in Fig. 3.18. Periodic coupled resonators such as coupled-resonator optical waveguides (CROWs) and coupled ring resonators have been studied previously for a range of applications, including add-drop filtering, slow-light propagation, and nonlinear switching [101, 102, 208–213]. The serpentine waveguide is of particular interest as a slow-light structure due to the existence of very low group velocity bands as we demonstrate here. In Section 3.6.1 we derive a dispersion relation for the serpentine waveguide in terms of the transmission and reflection properties of a single FDC. The dispersion properties of the periodic structure can thus be evaluated using either the full Bloch mode method or the semianalytic expressions in Eq. (3.8). We calculate the band structure and transmission properties of two different serpentine waveguides in Section 3.6.2 and discuss their general properties. 96 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

L/22

y2 L Dy 2

L/22 y3

(a) (b)

Figure 3.18: Two equivalent serpentine waveguide geometries (assuming no tunnelling through the guide ends). Both are characterised by the period Dy and the double and single guide lengths L1 and L2.

3.6.1 Derivation of the dispersion relation The unit cell of the serpentine waveguide is a pair of coupled FDC devices joined input- to-output by a single waveguide of length L2 to create a periodic structure with a period Dy = 2(L2 + L1) as shown in Fig. 3.18(a). Each of the individual components of this structure is essentially the same as those of the isolated FDC device. In the sections of length L1 there are two parallel waveguides that support a pair of propagating modes which allow light to couple from one guide to the other. The single guide sections of length L2 act as the input/output guides to the next/previous coupling region. If L2 is suﬃciently long that evanescent tunnelling between the blocked waveguide ends is negligible, the only connection between the coupling regions is via the one propagating mode of the single guide. In this limit, the structures shown in Figs. 3.18(a) and (b) are equivalent and hence a single FDC, rather than a complete unit cell, is the minimum segment of the serpentine required to determine the band structure. As we show in Section 3.6.3, this property holds for the structure in Fig. 3.18(a) even when the single mode approximation is invalid. We now proceed to analyse the structure using a single mode approximation which is 3.6. SERPENTINE WAVEGUIDE 97

valid when the L2 is sufficiently long to ensure coupling between the double waveguide segments in Fig. 3.18 occurs only via the propagating mode in the single waveguide section. Consider first the transmission and reflection characteristics of a single period of the serpentine waveguide, bounded by the interfaces y1 and y3 shown in Fig. 3.18. Using the notation of Chapter 2 we can express the fields at interface yj in terms of + − their upward and downward components bj and bj respectively. In the propagating ± mode analysis here, the bj are complex amplitudes of the single propagating waveguide mode, which are related via a transfer matrix according to

− − b3 b1 + = Ts + . (3.20) b3 b1 Denoting the complex transmission and reﬂection amplitudes of a single period by

τs and ρs respectively, we can write down the transfer matrix

2 τs − ρs/τs ρs/τs Ts = . (3.21) −ρs/τs 1/τs

Since the basic unit of the serpentine is the FDC bounded by y1 and y2 in Fig. 3.18, we can rewrite Eq. (3.20) as

2 2 τf − ρf /τf ρf /τf Ts = Tf , where Tf = (3.22) −ρf /τf 1/τf is the transfer matrix for a FDC with input and output guides of length L2/2. Here, τf and ρf are the transmission and reﬂection coeﬃcients of a FDC with an additional phase term exp(iβL2) to include propagation between the FDC and interfaces y1 and y2. β is the propagation constant of the single input and output mode. Note that a more general relationship between Ts and Tf , valid for all L2, is derived in Section 3.6.3. The dispersion relation for the periodic serpentine structure is obtained by solving the eigenvalue problem Tsb = µb, where the eigenvalues are µ = exp(iqDy) and q is 2 the Bloch factor in the longitudinal direction of the serpentine. Since Ts = Tf , the 1/2 eigenvalue equation can be written as Tf b = ±µ b, which has solutions that satisfy

2 2 µ + 1 τf − ρf + 1 1/2 = . (3.23) ∓µ τf This expression can be further simpliﬁed using the energy conservation properties 2 2 |ρf | + |τf | = 1 and arg(τf ) − arg(ρf ) = ±π/2, to yield the dispersion relation

∓ cos(qDy/2) = <(1/τf ). (3.24)

The right hand side of Eq. (3.24) can be calculated using either the full Bloch mode method or the semianalytic expression in Eq. (3.8), with τf = τ exp(iβL2) to include the additional phase due to propagation through the single waveguide sections. The latter 98 CHAPTER 3. COUPLED WAVEGUIDE DEVICES method results in a semianalytic form of the right hand side, which can provide a better understanding of the serpentine waveguide properties. Substituting the semianalytic expression for τf into Eq. (3.24) yields

cos(∆βL1) sin(2βL1 + βL2) cos(qDy/2) = sin(∆βL) sin(βL1 + βL2) + , (3.25) 2 sin(∆βL1) cos(βL1) which can be evaluated after numerical input of the waveguide dispersion properties and the lengths L1 and L2. In the next section we consider the properties predicted by this dispersion relation and compare the results to full numerical calculations.

3.6.2 Serpentine waveguide properties We consider first the different solutions to Eq. (3.25), and identify band features that are directly related to the properties of a single FDC, and others that result from the periodicity. The serpentine waveguide exhibits transmission bands for frequencies where the right hand side of Eq. (3.25), <(1/τf ), has magnitude less than unity. Correspond- ingly, band gaps, which lie between the transmission bands, occur when |<(1/τf )| > 1. There are three distinct types of band gap, each originating through a distinct physical effect. Following from Eq. (3.25), or from the analysis of a single FDC in Section 3.4.3, there are two conditions for which τf = 0. The first occurs when sin(∆βL1) = 0, which corresponds to L1 being an integer multiple of the coupling length Lc. The second condition occurs when cos(βL) = 0, corresponding to the reflection resonance identified in Section 3.4.3 as being suitable for narrow-band filtering. When either of these conditions is satisfied, the right hand side of Eq. (3.25) diverges, and a band gap appears in the serpentine band diagram. Similar resonator band gaps are displayed by coupled ring-resonator structures [211] and other periodic coupled resonator systems [193]. We label the two types of resonator band gaps identified here as RG1 and RG2 respectively. The third class of band gap that appears in the serpentine band diagram is a Bragg gap [211] that occurs as a result of the overall periodicity of the serpentine structure, rather than a single feature of the FDC.

Figure 3.19(a) shows the band diagram of a serpentine waveguide with L1 = L2 = 5d, formed in the square rod-type PhC, PC1. The FDCs making up this device are identical to the L = 5d structure shown in Fig. 3.12(c), which has the transmission properties plotted in Fig. 3.12(a) and (b). The very narrow RG2-type band gap near d/λ = 0.33 corresponds to the sharp reﬂection resonance of the FDC. In Fig. 3.19(b), the intensity transmission is shown for the single FDC (half a period), a single whole period and two whole periods of the serpentine waveguide. The band diagram was calculated using

τf obtained from the full Bloch mode method and the transmission spectra were also calculated using the full numerical approach. Counting from the top of Fig. 3.19(a), we see that top two resonator gaps are both of type RG2, and hence they correspond to strong features in all three of the transmission curves. In contrast, the Bragg gaps do not correspond to any obvious features in the FDC transmission spectrum, and 3.6. SERPENTINE WAVEGUIDE 99

0.33 Resonator gap type RG2

0.32 Bragg gap λ / Resonator gap type RG2 d 0.31

0.3 Bragg gap

0.29 Bragg gap

Figure 3.19: (a) Band diagram for a serpentine waveguide with L1 = L2 = 5d, where q is the Bloch coeﬃcient along the waveguide and Dy = L1 + L2. (b) Transmission through a FDC with the same parameters (dashed), one period of the serpentine waveguide (dotted) and two periods (solid). Note that for frequencies below d/λ, the double guide cavity only supports a single mode (see Fig. 3.2(f)), and thus the analytic result of Eq. (3.25) does not apply. only begin to appear in the multiple FDC serpentine structure. Given the resonance conditions for RG1 and RG2, and since β > ∆β, it is apparent that resonator gaps of type RG1 occur less frequently than RG2-type gaps. If the structure is made slightly longer, a RG1-type gap appears, as shown in

Fig. 3.20(a), which shows the band structure for a serpentine waveguide with L1 = L2 = 7d. The solid curves in (a) are the full numerical result given by Eq. (3.24) calculated using the Bloch mode method, and the dashed curves are calculated using the semianalytic expression (3.25), with L1 = 7.5d and L2 = 6.7d, which were chosen to give the best agreement to the numerical result. Figure 3.20(b) shows the transmission spectrum through a single FDC with L = 7d (dashed curve), and two and three periods of the serpentine waveguide, indicated by the dotted and solid curves respectively. Observe that for these parameters, the resonator gaps are relatively strong even for such a small number of periods, whereas the Bragg gaps only appear as very weak perturbations in the transmission spectrum for two periods. Another feature associated with both types of resonator band gap is the positioning of the bands above and below the gaps. In a one-dimensional Kronig-Penney model, the right hand side (RHS) of the dispersion relation is continuous, and hence only direct gaps exist since the RHS cannot change sign without passing through zero (see, for example Ref. [164], chapter 6). In the case of the resonator gaps however, the RHS of the dispersion relation (3.25) diverges when either of the resonance conditions is satisﬁed. When this happens, the RHS can change sign without passing back through zero, resulting in an indirect band gap. Indirect band gaps can be seen in Figs. 3.19(a) and Fig. 3.20(a) at each of the resonance gaps. Although indirect band gaps occur 100 CHAPTER 3. COUPLED WAVEGUIDE DEVICES

0.33

0.325 Bragg gap λ

/ 0.32 Resonator gap type RG2 d 0.315 Resonator gap type RG1 0.31 Bragg gap 0 π/4 π/2 3π/4 π0 0.2 0.4 0.6 0.8 1 ¢¡¤£¦¥¨§ Transmission (a) (b)

Figure 3.20: (a) Band diagram for a serpentine waveguide with L1 = L2 = 7d. The solid curve is calculated with the full numerical simulation while the dashed curve is calculated using Eq. (3.25). (b) Transmission through a FDC with L = 7d (dashed), 2 periods of the serpentine guide (dotted) and 3 periods of the serpentine guide. commonly in two-dimensional photonic crystals, such features have only recently been observed in other coupled resonator waveguides consisting of a linear periodic array of resonant elements [211]. The serpentine waveguide also exhibits a number of other interesting band structures that can be created with appropriate choices of the lengths L1 and L2 and waveguide separation, Nc. For example, the sharp resonance at d/λ = 0.331 in Fig. 3.19(a) creates very flat bands which correspond to low group velocities, and hence large group delays. We showed in Section 3.4.4 that the width of a RG2-type resonance is very sensitive to structural changes in the double waveguide section, so a similar tuning method could be applied to adjust the group velocity response. An alternative mechanism for achieving low group velocity bands is discussed below. Another unusual band feature occurs when two or more consecutive band gaps are resonator gaps, as in Fig. 3.20 where the band gaps at centre frequencies d/λ = 0.3135 and d/λ = 0.3183 are resonator gaps of type RG1 and RG2 respectively. Since each of these gaps is indirect, the three bands above, below and between the gaps are almost parallel to one other and all have a positive slope. With optimisation of the geometry it may be possible to design a band structure with several equally-spaced bands, each exhibiting a similar group-velocity profile. The width of the central band is also very sensitive to the relative positions of the two resonances. Since the RG1 gap depends on ∆β and the RG2 gap depends on β, it is possible to change their relative positions by changing L1 or tuning the geometry of the double guide region. If the two gaps are brought close together, the band between them becomes flat, and as the two resonances pass through each other, the slope of the band, and hence the group velocity changes sign. The group velocity could thus be controlled by tuning the relative resonance positions. Similar properties have been identified in other coupled-resonator 3.6. SERPENTINE WAVEGUIDE 101

Forward Backward ~ d R1d ~ T1’ d y1 y1

y1 L/22

L1 FDC1

y2 y2 ~ d ~R T1d 1’ d L2 y2

~ L1 d R d ~ FDC2 2 T2’ d

y2 y2

L/22 y3

y y ~ 3 ~ 3 T2d d R2’ d

(a) (b)

Figure 3.21: (a) Schematic of the serpentine waveguide structure shown in Fig. 3.18(a). (b) The FDC structures that are combined to form the structure in (a). Each FDC is characterised by plane wave scattering matrices for a ﬁeld δ incident from above (Te, Re ) 0 0 and below (Te , Re ). waveguides, and have been proposed as a basis for all-optical trapping of light [102,214]. Further work is required to determine whether a serpentine structure can be designed to exhibit the same properties.

3.6.3 Symmetry properties of the serpentine waveguide In Section 3.6.1 we derived the dispersion properties of the serpentine waveguide in the limit where the single waveguide regions in Figs. 3.18(a) and (b) were suﬃciently long to neglect evanescent coupling between the ends of the blocked waveguides. In this limit, the Bloch mode transfer matrix of the full serpentine period is equal to the square of the transfer matrix of a single FDC (3.22). Here we use symmetry relations to derive a more general form of Eq. (3.22) for the structure in Fig. 3.18(a) which is valid for all L2. Consider ﬁrst the serpentine structure shown in Fig. 3.21 which consists of two

FDCs (FDC1 and FDC2) joined by a single waveguide. Since FDC2 is identical to FDC1 after reﬂection about the dashed-dotted line in Fig. 3.21(a), the plane wave 102 CHAPTER 3. COUPLED WAVEGUIDE DEVICES scattering matrices of the two structures are related according to

Re 1 = SRe 2S Te 1 = STe 2S 0 0 0 0 Re 1 = SRe 2S Te 1 = STe 2S, (3.26a) where S is the reversing permutation matrix

 0 ··· 1   ··· 1 ·    S =  ·····  .    · 1 ···  1 ··· 0

Here, the plane wave scattering matrices Te 1, Re 1, Te 2, Re 2 correspond to transmission 0 0 0 0 and reﬂection of a ﬁeld δ incident in the forward (downward) direction and Te 1, Re 1, Te 2, Re 2 correspond to incidence in the backward (upward) direction as shown in Fig. 3.21(b).

Furthermore, it can be seen from Fig. 3.21(a) that forward propagation from y1 to y2 is equivalent to backward propagation from y3 to y2 such that

0 0 Te 1 = Te 2 Te 2 = Te 1 0 0 Re 1 = Re 2 Re 2 = Re 1. (3.26b)

Following the notation of Chapter 2, the plane wave ﬁelds at either end of FDC1 and FDC2 are related by the transfer matrices

 0 0 −1 0 0 −1  Te 1 − Re 1 Te 1 Re 1 Re 1 Te 1 T pw = , (3.27a) f1  0 −1 0 −1  −Te 1 Re 1 Te 1

 0 0 −1 0 0 −1  Te 2 − Re 2 Te 2 Re 2 Re 2 Te 2 T pw = . (3.27b) f2  0 −1 0 −1  −Te 2 Re 2 Te 2 Hence, given the relations (3.26a) and (3.26b), it follows that

S T pw = ST pw S, where S = . (3.28) f2 f1 S

We now convert the above transfer matrix formulation from the plane wave basis to the waveguide mode basis (Bloch mode basis). At interface yi in Fig. 3.22, the ﬁeld ± can be represented as an expansion of plane waves with amplitudes fi or Bloch modes ± with amplitudes ci . At y1 and y2 these expansions are related according to

− − − − f1 c1 f2 c2 + = F 1 + + = F 2 + , (3.29a) f1 c1 f2 c2 3.6. SERPENTINE WAVEGUIDE 103

± ± f1 c1 y1

pw BM T 1 T f f1

f± c± 2 2 T BM y2 s

T pw T BM f2 f2

± ± f3 c3 y3

Figure 3.22: Schematic of the unit cell of the serpentine waveguide formed from two

FDC structures. The forward and backward propagating ﬁelds at interfaces y1 − y3 ± ± are represented in the basis of plane waves fi or Bloch modes ci . The ﬁelds at each interface are related by plane wave and Bloch mode transfer matrices, T pw and T BM respectively.

where the F i provide the change of basis from plane waves to Bloch modes as deﬁned Section 2.2.4. pw By deﬁnition, the plane wave transfer matrix Tf1 relates − − f2 pw f1 + = Tf1 + , (3.29b) f2 f1 which can be converted to Bloch mode form using Eqs. (3.29a) such that − − − c2 −1 pw c1 BM c1 + = F 2 Tf1 F 1 + = Tf1 + , (3.29c) c2 c1 c1 where BM −1 pw Tf1 = F 2 Tf1 F 1. (3.29d)

Similarly, the equivalent Bloch mode transfer matrix for FDC2 is

BM −1 pw Tf2 = F 1 Tf2 F 2, (3.29e)

BM which can be expressed in terms of Tf1 using Eqs. (3.28) and (3.29d) as BM −1 pw −1 BM −1 Tf2 = F 1 STf1 SF 2 = F 1 SF 2Tf1 F 1 SF 2. (3.29f) Therefore the Bloch mode transfer matrix for the full period of the serpentine waveguide in Fig. 3.22 is

BM BM BM BM BM BM 2 Ts = Tf2 Tf1 = QTf1 QTf1 = QTf1 , (3.30) 104 CHAPTER 3. COUPLED WAVEGUIDE DEVICES where −1 Q = F 1 SF 2. (3.31) Recall from Section 3.6.1 that the dispersion relation for the serpentine waveguide BM is obtained by solving for the eigenvalues of Ts . In the monomodal approximation BM BM 2 we observed that Ts = Tf1 and hence the eigenvalues µ were found by solving the BM 1/2 eigenvalue equation Tf1 b = ±µ b. Equation (3.30) shows that the general solution can also be found in terms of a single FDC, where the relevant eigenvalue equation is

BM 1/2 QTf1 B = µ B. (3.32)

Here B is a matrix of eigenvectors expanded in the full Bloch mode basis containing both propagating and evanescent modes. Although we do not derive the result here, Q 2 is a diagonal matrix of elements ±1, and so (Q) = I. In the limit where L2 is large, BM Tf1 is dominated by the single propagating Bloch mode in the waveguide, and hence Q has little eﬀect on the solutions of Eq. (3.32), which then reduces to the single mode result derived in Section 3.6.1.

3.7 Conclusion

We have covered in this chapter a range of topics associated with directional coupling and resonant reflection in photonic crystals, and have presented several devices that exhibit unusual transmission characteristics as a result of combining the two effects. These devices are described accurately by semianalytic models which allow rapid and efficient optimization and provide physical insight into the properties observed in full numerical simulations. The semianalytic formulations and the physical interpretation of the numerical results are based on the two coupling effects considered in Sections 3.2 and 3.2. Coupling between parallel waveguides was considered in Section 3.2 where we demonstrated that weak-coupling approximations typically used for dielectric waveguides can also be applied to photonic crystal waveguides. The second type of coupling occurs when two different waveguide sections are interfaced end-to-end, as in the two examples presented in Section 3.3. In this case, the coupling properties are influenced not only by direct propagation from one waveguide to the other, but also by resonant coupling through the photonic crystal barrier, should one exist. The devices presented in Sections 3.4 — 3.6 derive their properties from a combination of these two types of coupling. Chapter 4

Recirculating Mach-Zehnder interferometer

In this chapter we use many of the techniques and results from Chapter 3 to investigate the characteristics of photonic crystal Mach-Zehnder interferometers (MZIs). We show that the MZIs formed from waveguides in a perfectly reflecting cladding can display manifestly different transmission characteristics from conventional MZIs due to mode recirculation and resonant reflection. In devices of this type, recirculation effects can result in a significantly sharper switching response than in conventional interferometers. A simple and accurate analytic model is derived to describe this behaviour and we propose specific PhC structures in both rod-type and hole-type geometries that display these properties. This chapter is based largely on the work published in Ref. [204], with some additional details included relating to model in Section 4.2 and the design and simulation of the photonic crystal based Mach-Zehnder interferometer in Section 4.4. The discussion in Section 4.6 is based on work to be published in Ref. [182].

4.1 Introduction

Mach-Zehnder interferometers (MZIs) are used extensively in optical systems as filtering and switching devices and in phase measurement and detection applications. Their many uses and functions have led to the fabrication of MZI devices in bulk-optics [103], fibres [215], planar dielectric waveguides [216], rib waveguides [217], and more recently 2D PhCs [200,201,218–220]. Although the morphology of all MZIs is the same regardless of the type of waveguide in which they are formed, the operational characteristics can be strongly influenced by the waveguide and cladding properties. Three generic MZI designs are illustrated in Figure 4.1. The example in (a) is a bulk-optics MZI, while those in (b) and (c) are waveguide-based designs that can be formed in either dielectric or PhC waveguides. In each case, an incident beam is split into two beams at the first junction, with each one taking a different path through

105 106 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

in in 1 out 1

in out

out 1 in 2 out 2 out 2 (a) (b) (c)

Figure 4.1: Examples of three generic symmetric MZI designs with different types of junction. (a) Bulk-optics MZI using 50/50 beam-splitters and mirrors. (b) Directional coupler-based MZI with two possible input and output waveguides. (c) Y-junction MZI with single input and output waveguides. the interferometer before being recombined at the second junction. In a symmetric, or balanced, MZI, the two paths are identical as in the diagram, whereas the paths of an unbalanced MZI are different so as to introduce a relative phase delay. The devices illustrated in Fig. 4.1 (a) and (b) each have two output ports where the power coupled into each depends on the relative phase delay between the two arms of the interferometer. For an ideal device, the combined output power is equal to the input power, and the light only passes through the interferometer once. In contrast, the MZI in Fig. 4.1(c) only has a single output waveguide. In this case, any light that is not coupled into the output waveguide must either be radiated out of the device or reflected back into the interferometer. If the junction is formed in a dielectric waveguide relying on TIR for guidance, most of the non-transmitted light is radiated into the cladding, as happened with the terminated waveguides discussed in Chapter 3. For a PhC waveguide junction however, the cladding is perfectly reflecting so radiation cannot occur and thus any light that is not transmitted must be reflected. The resulting recirculating MZI exhibits strong resonance effects that are not usually observed in conventional dielectric MZIs or coupler-based MZIs of the type illustrated in Figs. 4.1(a) and (b). The clear observation of these effects relies critically on the use of efficient, symmetric beam splitters that exhibit very low reflectance (< 2%). In Section 4.2 we present a simple modal analysis to explain the transmission characteristics of recirculating MZIs and derive an elegant semianalytic expression that accurately describes their behaviour. The properties of these devices are strikingly different from conventional MZIs, and our analysis shows that they are typical of a general class of MZI with perfectly reflecting cladding and single-mode input and output waveguides. One feature of these devices is an almost square transmission response with phase and thus a significantly sharper switching response than conventional MZIs. Such behaviour is a highly desirable characteristic for both linear and nonlinear switching applications. As in Chapter 3, the analysis and numerical results are purely 2D, but the results hold generally if radiation losses are low. This issue is discussed in more 4.2. MODAL ANALYSIS 107 detail in Chapter 6. Two specific examples of PhC waveguide-based recirculating MZIs are proposed in Section 4.4. Both examples use the coupled Y-junction design presented in Section 3.5 which provides the necessary high-transmission and mode rejection properties to observe the recirculation effects. In this case, the perfectly reflecting cladding is provided by the photonic bandgap of the PhC waveguide walls. While it has been proposed that the strong reflections and associated interference observed in many PhC based devices may limit their practical use in optical systems [221], we show here that these effects can also be used to great advantage. Similar recirculation characteristics can be obtained in a bulk-optics geometry by including two additional mirrors in the system illustrated in Fig. 4.1(a). We analyse such a system in Section 4.6 and show that the transmission properties are analogous to those of the PhC MZIs.

4.2 Modal analysis

Consider first the MZI illustrated in Fig. 4.2(a), consisting of single input and output waveguides, and two junctions Y1 and Y2 joined by waveguide arms A1 and A2 of length L. In this analysis we assume that the Y-junctions are ideal and split light equally between A1 and A2. If an element is placed in A2 to introduce a phase difference ϕ between the two arms, the intensity of light transmitted into the output depends on ϕ. Active devices such as switches and filters can be made by varying ϕ to modulate the output [218, 220, 222], or in passive devices, the measured output can be used to determine ϕ [215]. In a symmetric, or balanced, MZI ϕ = 0, so light entering from the input guide recombines in-phase at Y2 and is transmitted into the output guide - a balanced MZI transmits equally well at all wavelengths.

A2 ϕ

input waveguide output Y Y 1 junctions 2

(a) (b)

Figure 4.2: (a) Schematic of a simple MZI with single input and output waveguides joined by arms A1 and A2 via Y-junctions Y1 and Y2. (b) Schematic of the propagating modes in each of the waveguide sections in (a). The green and red curves show the even and odd superpositions of the individual waveguide modes.

The transmission properties of the MZI in Fig. 4.2(a) can be derived by considering the propagating modes in each of the waveguide sections, in a procedure similar to that 108 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER used to analyse the coupled waveguide devices in Chapter 3. An alternative derivation based on ideal Bloch mode scattering matrices is outlined in Section 4.5. All of the waveguides in Fig. 4.2 are taken to be identical, with each supporting a single mode of even symmetry with respect to the waveguide axis. Thus, the input and output guides support one mode each, denoted by |ψini and |ψouti respectively, and the two- guide region supports one mode in each of A1 and A2, denoted |ψ1i and |ψ2i, as shown in Fig. 4.2(b). Alternatively, the modes in A1 and A2 are well approximated as a combination of two supermodes of even (+) and odd (−) symmetry with respect to the axis of the interferometer, indicated by the green and red curves in Fig. 4.2(b). The supermodes are approximately related to the modes in the individual guides by 1 1 |ψ±i = √ (|ψ1i ± |ψ2i) , |ψ1,2i = √ (|ψ+i ± |ψ−i) . (4.1) 2 2

In an ideal structure, A1 and A2 are separated suﬃciently that coupling between the two guides is negligible, in which case the propagation constants of |ψ±i are equal to the propagation constant of an isolated waveguide, β+ = β− = β respectively. This approximation is not always accurate however, in which case an alternative approximation can be made to the semianalytic model, as discussed in Section 4.4.3.

Since mode |ψini is symmetric with respect to A1 and A2, it can only couple to the supermode with the same symmetry, |ψ+i, so for an ideal Y1 junction, mode |ψini is fully transmitted into |ψ+i. When this mode propagates through A1 and A2, the effect of ϕ is to introduce an asymmetric field component, equivalent to transferring some of the energy into |ψ−i. As at Y1, the |ψ+i component of the field entering Y2 can couple into the output guide, but the |ψ−i component cannot and must be either radiated or reflected. To this point, the modal description holds for MZIs in general. The essential difference lies in what happens to |ψ−i at Y2. In dielectric waveguides, most of the light in |ψ−i is radiated into the cladding and lost, and only a very small amount is reflected back into the arms of the interferometer [223]. However, if light is unable to radiate into the cladding, any light in |ψ−i must be reflected at Y2, back into the interferometer. This odd mode reflection results in the unique resonant behaviour of recirculating MZIs with single-mode input and output waveguides. We now follow the propagation of a mode Ψ with unit energy hΨ|Ψi = 1 entering an unbalanced MZI from the input guide and consider the effect of odd mode reflection at

Y2. After passing through Y1, the resulting symmetric ﬁeld distribution can be expressed in terms of |ψ1i and |ψ2i using Eq. (4.1) to give

(1) 1 |Ψ(0)i = |ψ+i = √ (|ψ1i + |ψ2i) . 2 Here the superscript (1) indicates that we are considering the ﬁelds on the ﬁrst pass through the interferometer. The modes propagate through each arm essentially independently of each other, advancing in phase as they propagate. At Y2, the combined 4.2. MODAL ANALYSIS 109

field in A1 and A2 is √ (1) iβL i(βL+ϕ) iχ |Ψ(L)i = e |ψ1i + e |ψ2i / 2 = e (cos(ϕ/2)|ψ+i − i sin(ϕ/2)|ψ−i) , (4.2) where χ = βL + ϕ/2 is the average phase of the two arms and β is the propagation constant of the single propagating mode of an isolated waveguide. At Y2, the even field component is transmitted into the output guide, while the odd mode is reflected back into A1 and A2. Note that in a conventional MZI the odd mode is radiated into the cladding and the transmission is simply given by the even mode term in Eq. (4.2), 1 T = |eiχ cos(ϕ/2)|2 = cos2(ϕ/2) = . (4.3) 1 + tan2(ϕ/2)

In the PhC structure, however, the reflected odd field component, −i exp(iχ) sin(ϕ/2)|ψ−i, propagates back through the two arms, accumulating a further phase ϕ so that the field arriving back at Y1 is √ (2) iχ iβL i(βL+ϕ) |Ψ(0)i = −ie sin(ϕ/2) e |ψ1i − e |ψ2i / 2 2iχ = −ie sin(ϕ/2) (cos(ϕ/2)|ψ−i − i sin(ϕ/2)|ψ+i) . (4.4)

Once again the odd field component is reflected back into the interferometer, while the even component is transmitted through the junction, in this case back into the input waveguide. The total transmission is given by the superposition of the fields transmitted through

Y2 each time the odd mode is reﬂected. Denoting the contribution after n passes through the interferometer by tn, the transmitted ﬁeld is

∞ X t = tn (4.5) n=1 = eiχ cos(ϕ/2) − e3iχ sin2(ϕ/2) cos(ϕ/2) − e5iχ sin2(ϕ/2) cos3(ϕ/2) − ... (4.6) e3iχ sin2(ϕ/2) cos(ϕ/2) = eiχ cos(ϕ/2) − , (4.7) 1 − e2iχ cos2(ϕ/2)

P∞ where the second term in (4.7) is the sum of the geometric series n=2 tn. Hence, the transmitted intensity is given by

4 sin2(χ) cos2(ϕ/2)/ sin4(ϕ/2) T = |t|2 = . (4.8) 1 + 4 sin2(χ) cos2(ϕ/2)/ sin4(ϕ/2)

Note that the form of Eq. (4.7) closely resembles the Airy-formula [122] for reflection from a Fabry-Pérot cavity – a feature we return to in Section 4.3. The reflected intensity (R = 1 − T ) can be derived in the same way by summing the field amplitudes reflected back into the input waveguide. Observe that Eq. (4.8) is a function of both ϕ and βL = β(λ)L, in contrast to the conventional MZI transmission (4.3), which is a 110 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER function of ϕ only. Thus, the waveguide dispersion properties are especially important in recirculating MZIs. Apart from the different functional forms of Eqs. (4.3) and (4.8), the additional dependence on wavelength and structure length leads to significantly different transmission properties. The explicit dependence on structure length is a direct result of the mode recirculation inside the device.

4.3 Transmission characteristics

In this section we study the transmission characteristics of recirculating MZIs described by Eqs. (4.7) and (4.8). For generality the normalised units βL are used, although we note that when modelling a realistic device, the waveguide dispersion β = β(λ) must be taken into account. Hence for a specific MZI design, the dispersion properties provide a direct mapping between βL and wavelength. The effects of waveguide dispersion are discussed in Section 4.4, where the transmission properties of two PhC-based MZIs are presented in terms of wavelength. Figures 4.3(a) and (b) are contour plots of transmittance as a function of βL and ϕ for a conventional and a recirculating MZI respectively, where the red areas correspond to high transmittance and the blue areas correspond to low transmittance. The transmittance of the conventional MZI is given by Eq. (4.3) so there is no variation with βL, but only with ϕ. For the recirculating MZI however, there is a strong dependence on both parameters but it can be seen that T = 1 when ϕ = 0 and T = 0 when ϕ = π, as for the conventional MZI. In (b) the slightly tilted, low transmittance ‘valleys’ correspond to strong resonances due to the multiple reflections of the odd mode. At the centre of the valleys, the transmittance drops to zero and the valleys become rapidly narrower as ϕ approaches 0 or 2π. The slight tilt of the valleys from vertical is due to the dependence of χ on both βL and ϕ in Eq. (4.8). We consider first the properties of Eq. (4.8) as ϕ is varied at a fixed βL, corresponding to traversing a vertical line across Fig. 4.3(b). Figures 4.4(a)–(e) show the transmittance of a recirculating MZI as a function of ϕ for βL = 0, π/8, π/4, 3π/8 and π/2 respectively, indicated by the solid curves. The corresponding transmittance of a conventional MZI (4.3) is indicated in each plot by the dashed curves. Observe from Fig. 4.3(b) that for most values of βL, a vertical line of varying ϕ intersects one end of a low transmission valley, resulting in a response with a strong resonant reflection as seen in Fig. 4.4(b). The width of the resonance decreases to zero as ϕ → 0 or ϕ → 2π, at which point T = 1. Of particular interest is the response of the MZI when βL = π/2 , 3π/2 , 5π/2 ..., where the diagonal valleys in Fig. 4.3(b) intersect the low transmittance band at ϕ = π. The solid curve in Fig. 4.4(e) shows the response as a function of ϕ when βL = π/2. Comparing this with the dashed curve of the conventional MZI, it is seen that the recirculating MZI response exhibits a significantly sharper transition between ‘on’ and ‘off’ states. Substituting the condition identified 4.3. TRANSMISSION CHARACTERISTICS 111

2p 2p

3p 3p 2 2

j p j p

p p 2 2

0 0 0 2p 4p 6p 0 2p 4p 6p bL bL (a) 0 0.2 0.4 0.6 0.8 1 (b) Transmission

Figure 4.3: Contour plots of transmittance as a function of the dimensionless parameter βL and the phase diﬀerence ϕ for (a) a conventional MZI described by Eq. (4.3) and (b) a recirculating MZI described by Eq. (4.8) above for βL into Eq. (4.8), we ﬁnd the transmittance to be 4 cot4(ϕ/2) 1 T = = . (4.9) 1 + 4 cot4(ϕ/2) 1 + tan4(ϕ/2)/4

Comparing Eqs. (4.3) and (4.9), we see that the recirculating MZI response is quartic in tan(ϕ/2), compared with the quadratic response of a conventional MZI. The change in ϕ required to switch the transmittance from 0.9 to 0.1 in the recirculating MZI is approximately 0.32π, whereas a conventional MZI would require a change of 0.59π. Although cos4(ϕ/2) responses can be achieved with a pair of conventional MZIs, and higher order behaviour occurs for a chain of such devices [224], we find that at least six identical conventional MZIs would be required to obtain a superior switching performance to that of a single recirculating MZI. Note also, that the transmittance given by Eq. (4.9) has a fourth-order variation near both T = 0 and T = 1, whereas a chain of MZIs always exhibits a quadratic variation in the region around T = 1. The steep, almost square response of the recirculating MZI is highly desirable for switching applications, and could be used to improve performance in both linear and nonlinear devices. If βL is varied while ϕ is fixed, as in Figs. 4.4(f)–(j), the transmission properties are very different from those shown in Figs. 4.4(a)–(e). For a given ϕ, the transmittance T = T (βL) given by Eq. (4.8) resembles the expression for the reflectance of a Fabry- Pérotetalon [122] with finesse π| cos(ϕ/2)| F = . (4.10) 1 − | cos(ϕ/2)| 112 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

1 bL=0 j=0 0.8 0.6 T 0.4 0.2 (a) (f) 0 1 bL=p/8 j=p/4 0.8 0.6 T 0.4 0.2 (b) (g) 0 1 bL=p/4 j=p/2 0.8 0.6 T 0.4 0.2 (c) (h) 0 1 bL=3p/8 j=3p/4 0.8 0.6 T 0.4 0.2 (d) (i) 0 1 bL=p/2 j=p 0.8 0.6 T 0.4 0.2 (e) (j) 0 0 p p 3p 2p 0 p 2p 3p 4p 5p 6p 2 2 j bL

Figure 4.4: Solid curves: (a)–(e): Power transmitted by an ideal recirculating MZI as a function of ϕ for βL = 0, π/8, π/4, 3π/8, π/2 respectively. (f)–(j): Power transmitted by an ideal recirculating MZI as a function of βL for ϕ = 0, π/4, π/2, 3π/4, π respectively. Dashed curves: Power transmitted by a conventional MZI for the same parameters.

Thus, the transmission behaviour of the recirculating MZI is essentially the reverse of a Fabry-Pérot etalon, exhibiting resonant reflection, rather than resonant transmission. This is seen by considering horizontal lines of constant ϕ in Fig. 4.3(b), and the examples in Figs. 4.4 (f)–(j). As ϕ approaches 0 or 2π, the resonances become narrower, and the finesse increases according to Eq. (4.10). As ϕ approaches π, both the fringe visibility and the finesse decrease. These reversed Fabry-Pérot transmission characteristics could be used in the design of a tunable notch rejection filter where the finesse and resonance position could be adjusted by varying β and ϕ. In practice both β and ϕ are functions of wavelength, and so it is difficult to design a device to exhibit transmission spectra identical to those shown in Figs. 4.3(f)–(j). Any realistic measurement of transmission spectra is likely to correspond to the transmission along a curved line in Fig. 4.3(b), with the exact shape of such a curve being determined 4.4. PHOTONIC CRYSTAL RECIRCULATING MZI STRUCTURES 113 by the method used to induce ϕ. This aspect is discussed further in Section 4.4.3.

4.4 Photonic crystal recirculating MZI structures

In Section 4.2 we derived a simple model for the transmission of an ideal MZI with a perfectly reflecting cladding and single-mode input and output guides, and showed that recirculation and resonant properties occur due to the reflection of modes with odd symmetry at the waveguide junctions. In this section we present both rod- and hole-type PhC MZI designs that display these characteristics and compare full numerical calculations with the semianalytic results of Eq. (4.8). For these comparisons, the waveguide dispersion properties are required as numerical input into Eq. (4.8), however this is a relatively simple calculation that does not require additional knowledge of the MZI structure. The rod-type PhC considered throughout this section consists of a square lattice of rods with index ncyl = 3.4 and radius r = 0.18d in air. This structure has a TM bandgap in the normalised frequency range 0.302 ≤ d/λ ≤ 0.445. We present results for a structure with lattice period d = 570 nm, chosen so that the MZI operates close to λ = 1.55 µm, within the bandgap spanning 1.28 µm ≤ λ ≤ 1.88 µm. A waveguide formed by removing a single line of cylinders supports a single propagating mode of even symmetry in the wavelength range 1.28 µm ≤ λ ≤ 1.83 µm. For the hole-type PhC we use the same parameters as in Section 1.5.3, namely a triangular lattice of air holes with radius r = 0.32d in a silicon background of nb = 3.4. A lattice period of 350 nm is chosen so that the TE bandgap in the range 0.216 < d/λ < 0.298 corresponds to a wavelength range 1.17 µm ≤ λ ≤ 1.62 µm. Since a W 1 waveguide can support two modes in the bandgap (see Fig. 1.8), we operate only in the low-frequency region of the bandgap where a single mode is supported for 1.435 µm ≤ λ ≤ 1.580 µm. This mode has a large group velocity dispersion close to cutoff at the low frequency end of this range which distorts the transmission profile of the MZI considerably, as seen in Section 4.4.3.

4.4.1 Design The most important components in a PhC-based recirculating MZI are the Y-junctions or beam splitters. Recall from Section 4.2 that the Y-junctions in the model were taken to be ideal, transmitting modes of even symmetry and reflecting modes of odd symmetry. In practice, the even mode is not perfectly transmitted, and the additional reflected field components can result in extra Fabry-Pérot effects [218] that make the odd mode resonances difficult to identify. From our numerical studies, we find that the odd mode effects are most clearly observed when the even mode reflectance is less than 2%. Thus, to observe the properties predicted by Eq. (4.8), the junction design is critical. In Section 3.5 we proposed a PhC coupled Y-junction design that exhibits very high transmission over a wide bandwidth due to a combination of directional coupling 114 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER and mode recirculation behaviour. This type of junction is ideal for use in recirculating PhC MZI designs as demonstrated by the examples in this section. We note that Wang et al. proposed a similar coupled junction design for a surface plasmon MZI based on metal gap waveguides, although results were only reported for a balanced MZI so recirculation effects were not observed [225].

Ly=5d Ly=11d

(a) (b) 1 1

0.99 0.99

0.98 0.98 T 88nm 67nm T 36nm 0.97 0.97

0.96 0.96

0.95 0.95 1.5 1.55 1.6 1.65 1.54 1.55 1.56 1.57 l (mm) l (mm) (c) (d)

Figure 4.5: (a) and (b): Y-junction designs used for the rod- and hole-type PhC MZIs illustrated in Figs. 4.6(a) and (b) respectively. (c) Solid curve: transmission spectrum of the Y-junction shown in (a). Dashed curve: transmission of the PhC MZI in Fig. 4.6(a) with ϕ = 0 and LMZ = 15d. (d) Solid curve: transmission spectrum of the Y-junction shown in (b). Dashed curve: Transmission of the PhC MZI in Fig. 4.6(b) with ϕ = 0 and LMZ = 32d. Note that the transmission axis starts at 95% in both plots.

Figures 4.5 (a) and (b) show two coupled Y-junctions designed for wide bandwidth transmission in the rod- and hole- type PhCs defined at the start of Section 4.4. The transmission spectra for these junctions are plotted in (c) and (d) of the same figure (solid curves). The spectrum in (c) has two transmission bands of widths 88 nm and 67 nm respectively where T > 98%, but between these two regions is a sharp resonant reflection, the cause of which is discussed in Section 3.5. Transmission spectra are calculated over the combined wavelength range of the two bands and thus the resonant reflection feature can be seen in some of the results for the rod-type PhC MZI. Since the resonance is sharp, however, the MZI properties are only affected in a very narrow wavelength region. The hole-type PhC junction shown in Fig. 4.5(b) has a 98% transmission bandwidth of 36 nm, as shown by the solid curve in Fig. 4.5(d). When 4.4. PHOTONIC CRYSTAL RECIRCULATING MZI STRUCTURES 115 the junctions are operated in reverse as beam combiners, the even mode is transmitted into the single output waveguide with the same efficiencies shown in Figs. 4.5(c) and (d), while the symmetrical design of the junctions provides the necessary odd mode reflection properties. The PhC junctions illustrated in Figs. 4.5(a) and (b) are formed into MZIs by joining two identical junctions together with PhC waveguides as shown in Fig. 4.6. The total length of the device, including the junctions of length LY is given by LMZ as shown. In Section 4.4.2 we demonstrate that a phase difference can be introduced between the two arms by modifying the radius of several cylinders on the edge of one waveguide, thereby changing the optical path length relative to the other arm. The number of cylinders modified on each edge is given by the parameter Lϕ, defined in Fig. 4.6, where the modified cylinders are shaded a different colour for illustration purposes. Recall that we discussed in Section 2.5.2 some of the issues involved with modelling a balanced PhC MZI using the Bloch mode method. To introduce the modified cylinders in one arm we must include a new segment in the structure, and thus an additional grating calculation is required. To ensure that the modified segment does not interfere with the junction operation, at least two layers of symmetric double waveguide are retained where the arms enter the junctions. Thus, the MZI in Fig. 4.6(a) must be calculated as a seven segment structure, while the one in Fig. 4.6(b) is calculated as a nine segment structure by adding an extra segment in the middle of M4 in Fig. 2.9. The Bloch mode method results presented in this chapter we calculated with a grating supercell period of Dx = 21d and a plane-wave truncation order of Npw = 44, where Dx and Npw are defined in Chapter 2. Recall from Section 4.2 that when ϕ = 0 an MZI is symmetric (balanced) and therefore transmits equally well at all wavelengths. In a practical PhC based device, this is only expected for wavelengths within the transmission band of the junction. The dashed curves in Figs. 4.5(c) and (d) show the transmission spectra for balanced

MZIs of length LMZ = 15d (rod-type PhC) and LMZ = 32d (hole-type PhC) respectively. Note that although the transmission bandwidth is reduced slightly relative to the single junction in both cases, the transmission remains high across the band with only a few shallow ripples due to weak Fabry-P´erot resonances of the even mode.

4.4.2 Tuning ϕ Most MZI applications require ϕ to be tuned for the purposes of modulating or switching the output signal. This has been demonstrated in PhC based MZIs using thermo- optic [218] and nonlinear [220] eﬀects to induce a small refractive index change in one arm. Liquid crystal tuning of the refractive index has also been proposed [222]. For the purposes of demonstrating the response of recirculating MZIs, here we generate ϕ by changing the radii of Lϕ/d pairs of cylinders on either side of one waveguide from r tor ˜ as indicated in Fig. 4.6. This approach is chosen for numerical convenience as it is easier to modify the properties of individual cylinders rather than the background 116 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

LY Lj (a)

LMZ

(b) LY Lj

LMZ

Figure 4.6: Diagrams showing the basic geometry of the PhC MZIs studied in this chapter. (a) Rod-type PhC MZI formed from the junction shown in Fig. 4.5(a). (b) Hole-type PhC MZI formed from the junction shown in Fig. 4.5(b). The cylinders highlighted in (a) and (b) are modified to generate ϕ as described in Section 4.4.2. refractive index in the Bloch-mode scattering matrix method. The modal propagation constant in the modified waveguide section differs from the original waveguide by ∆β β, so a mode propagating through the modified waveguide arm accumulates a total phase difference of approximately ϕ = ∆βLϕ relative to the other arm. Thus, the maximum ϕ that can be introduced depends on Lϕ and the change in radius ∆r =r ˜ − r. However ∆r must be sufficiently small to avoid significant reflection caused by the change in waveguide properties and hence many cylinders typically need to be modified to achieve even a relatively small phase change.

This is shown in Fig. 4.7(a) where ϕ is plotted as a function of ∆r for Lϕ = 5d and Lϕ = 10d for the rod-type PhC waveguides used in Fig. 4.1(a) at λ = 1.553 µm. Hole-type PhC waveguides exhibit similar behaviour when tuned in the same way.

Observe that ϕ is doubled when Lϕ is increased from 5d to 10d as expected. The total reflected power from the modified waveguide section is also plotted for the two lengths in Fig. 4.7(b). The reflectance does not increase linearly with Lϕ because the reflections originate only at the two ends of the modified section. Since any additional reflection 4.4. PHOTONIC CRYSTAL RECIRCULATING MZI STRUCTURES 117 effects could interfere with the recirculation properties of the MZI, we limit ∆r such that the R . 1%. In Section 4.4.3 we use the Bloch mode method to calculate the transmission spectra of the PhC MZIs shown in Fig. 4.6 for different values of Lϕ. Since ∆β and hence ϕ are both functions of wavelength, this calculation is not directly equivalent to the results plotted in Fig. 4.4(f)–(j), where ϕ was held constant. In order to compare Eq. (4.8) with the full numerical results, the wavelength dependence variation of ϕ must be included. The dashed and dash-dot curves plotted in Fig. 4.8 show ϕ as a function of wavelength for the different Lϕ used in the numerical calculations of Section 4.4.3.

p 2 0.014 L d (a) (b) j=5 0.012 Lj=10d 0.01

j 0 R 0.008 0.006 0.004 Lj=5d 0.002 Lj=10d p 2 -0.02 0 0.02 0.04 -0.02 0 0.02 0.04 Dr/d Dr/d

Figure 4.7: (a) Phase difference induced by changing the radii of Lϕ pairs of cylinders by ∆r in one arm of the rod-type PhC MZI in Fig. 4.6(a). (b) Reflectance of the modified waveguide section for the same parameters. Note that both ϕ = 0 and R = 0 whenr ˜ = r = 0.18d (∆r = 0). The calculations are for λ = 1.553 µm, corresponding to λ/d = 2.689.

4.4.3 Numerical results Before proceeding to full numerical calculations, we consider first the transmission properties predicted by Eq. (4.8) for the two MZIs illustrated in Fig. 4.6. Figures 4.8(a) and (b) are contour plots of transmittance as a function of wavelength and ϕ calculated using Eq. (4.8) for rod- and hole-type PhC MZIs of length L = 36d and L = 40d respectively. The waveguide dispersion β = β(λ) was calculated using the Bloch mode method and used as numerical input. These plots are essentially the same as Fig. 4.3(b), where the horizontal axis has been rescaled to take into account the length and dispersion parameters of the devices in Fig. 4.6. There are some clear differences between the contour plots in Figs. 4.8(a) and (b). First, the low transmittance valleys run in the opposite diagonal direction because of the different dispersion slopes of the waveguide modes in the two PhC structures. Sec- ond, in (b) the tilt and the separation of the resonances changes significantly at longer wavelengths, whereas in (a) the valleys remain almost parallel and equally spaced. This is also a result of the different dispersion slopes. The waveguide modes of the MZI in 118 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

2p 2p

3p 3p 2 2 j j p p

p p 2 2

0 0 1.5 1.55 1.6 1.52 1.54 1.56 1.58 l (mm) l (mm) 0 0.2 0.4 0.6 0.8 1 Transmission

Figure 4.8: (a) Contour plot of transmittance as a function of λ and ϕ calculated using

Eq. (4.8) for the PhC waveguides in Fig. 4.6(a) with LMZ = 36d. The dashed and dashed-dotted curves show the phase change induced by ∆r = 0.02d for Lϕ = 8d and Lϕ = 16d respectively as described in Section 4.4.3. (b) As in (a) but for the waveguides in Fig. 4.6(b) with LMZ = 40d. The dashed and dashed-dotted curves correspond to the phase change for ∆r = −0.02d and Lϕ = 4d and Lϕ = 8d respectively.

Fig. 4.6(a) are well away from cutoff, so the dispersion slope is relatively constant, resulting in the regular features of Fig. 4.8(a). In contrast, the waveguide modes of the MZI in Fig. 4.6(b) are closer to cutoff and the resulting strong dispersion distorts the transmission profile. The spacing of the resonances decreases with increasing wavelength, while the resonances become narrower, maintaining a constant finesse. Despite this distortion, the square response predicted by Eq. (4.9) still occurs when βL satisfies the conditions identified in Section 4.3.

Rod-type PhC MZI Figures 4.9(a) and (b) show the transmission spectra of two MZIs of the type shown in Fig. 4.6(a) with Lϕ = 8d and Lϕ = 16d respectively forr ˜ = 0.2d (∆r = 0.02d) and LMZ = 36d. The solid curves are calculated using the full Bloch mode numerical calculation, and the dashed and dashed-dotted curves are the result of Eq. (4.8), calculated along the corresponding curves in Fig. 4.8(a) to account for the wavelength dependence of ϕ. Both spectra exhibit the reversed Fabry-Pérot-like behaviour predicted by Eq. (4.8) with the finesse of the reflection resonances decreasing with increasing phase according to Eq. (4.10). Note also that the sharp reflection feature exhibited by the Y-junction in Fig. 4.5(a) can just be seen in Fig. 4.9(a) at λ ≈ 1.57 µm, very close to a resonance of the MZI. The two resonances coincide exactly in Fig. 4.9(b) and cannot 4.4. PHOTONIC CRYSTAL RECIRCULATING MZI STRUCTURES 119

1 1

0.8 0.8

0.6 0.6 T T 0.4 0.4

0.2 0.2 (a) (b) 0 0 1.5 1.54 1.58 1.62 1.5 1.54 1.58 1.62 l (mm) l (mm) 1 1

0.8 0.8

0.6 0.6 T T 0.4 0.4

0.2 0.2 (c) (d) 0 0 1.54 1.55 1.56 1.57 1.54 1.55 1.56 1.57 l (mm) l (mm)

Figure 4.9: (a) and (b): Transmission spectra of the rod-type PhC MZI (Fig. 4.6(a)) calculated using the Bloch mode matrix method with LMZ = 36d and ∆r = 0.02d for (a) Lϕ = 8d and (b) Lϕ = 16d (solid curve). Dashed and dashed-dotted curves are given by Eq. (4.8) calculated along the corresponding curves in Fig. 4.8(a). (c) and (d): Transmission spectra of the hole-type PhC MZI (Fig. 4.6(b)) calculated using the

Bloch mode matrix method with LMZ = 40d and ∆r = −0.02d for (a) Lϕ = 4d and (b) Lϕ = 8d (solid curve). Dashed and dashed-dotted curves are given by Eq. (4.8) calculated along the corresponding curves in Fig. 4.8(b). be distinguished.

The parameters L and Lϕ used in Eq. (4.8) are not necessarily identical to those defining the structure in Fig. 4.3(d). Since PhCs behave as distributed reflectors, defining a structure length is somewhat ambiguous. We find that the best agreement in resonance position occurs when we take L in Eq. (4.8) to be the total length of the structure including the junctions, LMZ, as shown in Fig. 4.6, with a slight correction for the phase change introduced by reflection from the PhC. For the comparisons in Figs. 4.9(a) and (b) we found the best agreement for L = 36.75d. Under these conditions, the agreement between the rigorous numerical result and the semianalytic model is excellent. In particular, the model predicts the resonance widths to a high accuracy, and with the corrected length parameter the resonance positions are also closely matched. 120 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

In Section 4.3 we showed that recirculating MZIs exhibit a considerably sharper response to ϕ than conventional, single-pass MZIs, with that response described by Eq. (4.9) when βL = π/2 , 3π/2 , 5π/2 ... To demonstrate this property in a full numerical calculation ϕ must be varied while keeping the wavelength fixed. Recall from Section 4.4.2 that the maximum ϕ that can be induced by varying the cylinder radii is limited by Lϕ and the maximum ∆r that can be used without introducing additional reflections into the system. From Fig. 4.7(b) it can be seen that the reflectance remains below approximately 1% in the range −0.025 ≤ ∆r/d ≤ 0.035. This results in a phase variation of −0.36π . ϕ . 0.36π for an MZI with Lϕ = 10d, as shown by the dashed curve in Fig. 4.7(a).

p 1

0.8 p/2

0.6 j T 0 0.4

-p/2 0.2

0 -p -0.02 -0.01 0 0.01 0.02 0.03 Dr

Figure 4.10: Solid (blue) curve: Transmission as a function of ∆r for a rod-type PhC

MZI (Fig. 4.6(a)) with LMZ = 36d and Lϕ = 22d at λ = 1.553 µm. Dashed-dotted curve: phase diﬀerence induced by the radius modiﬁcation ∆r. Dashed (red) curve: Quartic response given by Eq. (4.9).

The range of ϕ is maximised in the rod-type PhC MZI of length LMZ = 36d by choosing Lϕ = 22d so that there are only two lattice periods of undisturbed waveguide separating the modiﬁed waveguide section and junctions at either end. The dashed- dotted curve in Fig. 4.10 shows ϕ as a function of ∆r for λ = 1.553 µm and the solid curve shows the MZI transmittance calculated using the Bloch mode method for the same wavelength. The dashed curve is calculated using Eq. (4.9) with ϕ = ϕ(∆r) given by the dashed-dotted curve. Again there is very good agreement between the rigorous numerical calculation and the semianalytic model derived in Section 4.2. Given the restrictions placed on ϕ by the choice of tuning method, a longer structure would be required to obtain a full 2π phase change, but shorter devices may be possible with alternative tuning methods. 4.4. PHOTONIC CRYSTAL RECIRCULATING MZI STRUCTURES 121

Hole-type PhC MZI Figures 4.9(c) and (d) show the transmittance as a function of wavelength for two

MZIs of the type shown in Fig. 4.6(b) with Lϕ = 4d and Lϕ = 8d respectively for r˜ = 0.30d (∆r = −0.02d) and LMZ = 40d. The solid curves are the full numerical result and the dashed and dashed-dotted curves are given by Eq. (4.8), calculated along the dashed (Lϕ = 4d) and dashed-dotted (Lϕ = 8d) curves in Fig. 4.8(b), corresponding to the wavelength variation of ϕ = ∆βLϕ. As in the rod-type PhC example, the parameter L used in Eq. (4.8) does not match LMZ exactly. For the comparisons in Figs. 4.9(c) and (d) the best agreement was found for L = 39.26d. Once again we see that the semianalytic model predicts the finesse of the resonances with good accuracy. The resonance positions line up closely at the shorter wavelength end of the transmission region, however the agreement is poorer at longer wavelengths, as a result of the strong modal dispersion close to cutoff. We have found that the agreement is improved at longer wavelengths if β is replaced by the propagation constant of the odd supermode β− in Eq. (4.8). This modification is reasonable since the resonances inside the interferometer are driven by multiple reflections of the odd mode. For the rod-type PhC examples in Figs. 4.9(a) and (b), the modes are well away from cutoff and β and β− are almost equal, so the comparison is not improved significantly by this modification. For consistency however, the semianalytic curves plotted in Figs. 4.9(a) and (b) are also calculated using β− in place of β. 1 0

0.8 -p/2 0.6 T -p j 0.4 -3p/2 0.2 0 -2p 0 5 10 15 20 25 30

Lj/d

Figure 4.11: Blue data points: transmission as a function of Lϕ for a hole-type PhC MZI (Fig. 4.6(b)) with LMZ = 72d andr ˜ = 0.266 d (∆r = −0.054) at λ = 1.5392 µm. The dotted line between the points is an aid to the eye. Dashed-dotted curve: phase diﬀerence induced by ∆r. Dashed (red) curve: quartic response given by Eq. (4.9).

In order to demonstrate the response of the hole-type MZI as ϕ is varied, we now consider a similar hole-type PhC MZI to the previous example, but with a length of 122 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER

LMZ = 72 d. This longer structure allows us to change ϕ through a complete 2π period. For this example we fix the radius of the modified cylinders tor ˜ = 0.266 d, corresponding to ∆r = −0.054 d, then vary ϕ by changing the number of modified cylinders, Lϕ. Figure 4.11 shows the transmitted power as a function of Lϕ for 0d ≤ Lϕ/2 ≤ 32d. The blue data points were calculated using the full Bloch mode method, and the dashed red curve is the quartic response given by Eq. (4.9), where the dependence of ϕ on Lϕ is given by the dashed-dotted curve and the right-hand axis. Observe that there is excellent agreement between the approximate and the numerical calculation for Lϕ < 15d, but for Lϕ > 15d the numerical results appear to be shifted relative to the model. We attribute this to the change in ∆β that results from the radius change in the MZI arm. In the derivation of the semianalytic model in Section 4.2, we assume that β changes by ϕ/2, but that ∆β remains small and approximately zero. This is valid when ϕ is small, but becomes less accurate as ϕ → π. Although this may make it difficult to obtain the predicted quartic response over a full 2π change in ϕ, the steep switching behaviour between T = 0 and T = 1 is still displayed in the example shown in Fig. 4.11. Alternative tuning methods may also make it possible to reduce this effect.

4.5 Semianalytic Bloch mode matrix derivation

In Section 4.2, the transmission equations (4.7) and (4.8) were derived from a simple modal analysis and compared in Section 4.4.3 with full numerical results calculated using the Bloch mode method. Identical expressions can be obtained via the Bloch mode matrix method by considering only the propagating modes in each region, and ﬁlling the Bloch mode scattering matrices with idealised values corresponding to the assumptions of Section 4.2. This alternative approach is outlined here. The MZI of Fig. 4.2 can be modelled as a three layer structure as shown in Fig. 4.12. Two semi-inﬁnite regions form the input and output waveguides on either end of the central section which consists of the two arms of the interferometer. Transmission through the three-layer structure is given by the matrix equation

−1 t = T23P (I − R21PR23P) T21, (4.11) where the operator P describes Bloch mode propagation between interfaces 1 and 2. We can write this as √ √ −1 √ √ 1/ 2 1/ 2 ei(χ−ϕ/2) 0 1/ 2 1/ 2 P = √ √ √ √ (4.12a) 1/ 2 −1/ 2 0 ei(χ+ϕ/2) 1/ 2 −1/ 2 eiχ cos(ϕ/2) −ieiχ sin(ϕ/2) = , (4.12b) −ieiχ sin(ϕ/2) eiχ cos(ϕ/2) where the diagonal matrix in Eq. (4.12a) describes propagation through the two arms of the interferometer, while the other two matrices provide a change of basis between the odd/even supermodes and the modes of the individual arms. The diagonal elements 4.5. SEMIANALYTIC BLOCH MODE MATRIX DERIVATION 123

1 2 3

T12 T23 R23 R12 P

T21

R21

Interface 1 Interface 2

Figure 4.12: Diagram showing the three layer structure considered in Section 4.5 to derive a semianalytic Bloch mode scattering matrix model of a recirculating MZI. In- terface 1 lies between the input waveguide and the two arms and is characterised by the scattering matrices R12, T12, R21 and T21. Interface 2 lies between the two arms and the output waveguide and is characterised by the scattering matrices R23 and T23. of the matrix in Eq. (4.12b) can also be obtained directly from Eqs. (4.2) and (4.4) by extracting the odd and even mode amplitudes before and after a single pass through the interferometer. Analytic approximations to the scattering matrices in Eq. (4.11) are obtained by considering the transmission and reﬂection of modes at the interfaces between the input and output waveguides and interferometer arms. In Section 4.2 we assumed that a mode in the input guide is perfectly transmitted into the even mode inside the interferometer.

In the Bloch mode matrix notation, this corresponds to setting R12 = 0 and

1 T = , 12 0 where the first and second elements of T12 represent transmission into the even and odd supermodes respectively. Light incident on either interface from inside the interferometer is filtered by the junction according to the odd and even mode components; the even mode is perfectly transmitted, while the odd mode is reflected. Hence, the remaining Bloch mode scattering matrices in Eq. (4.11) can be written as

0 0 R = R = T = T = [1 0] . 21 23 0 1 21 23

Equations (4.7) and (4.8) can be derived by substituting the analytic matrix expressions above into Eq. (4.11). Although this method of deriving the semianalytic result may seem identical to the modal method in Section 4.2, it provides an intermediate step between an ideal model 124 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER and the a full numerical treatment. For instance, if the full Bloch mode matrices are truncated to include only the propagating modes, the complex reflection and transmission coefficients are retained but the matrices have the dimensions of the idealised version above and can thus be manipulated very efficiently. Alternatively, the effects of non-ideal transmission and reflection can be included in an intuitive way by modifying the analytic matrix expressions appropriately.

4.6 Recirculation in coupler-based MZIs

Recall from Section 4.2 that mode recirculation occurred because the input and output waveguides of the MZI only supported a single mode of even symmetry, and thus any odd field component created inside the interferometer is reflected by both junctions. In Section 4.3 we showed that rather than degrading the performance of the device, recirculation enhances the response for some wavelengths, and so it may be a desirable feature to generate in other MZI geometries. While mode recirculation is likely to occur in PhC-based MZIs of the type Fig. 4.1(c), MZIs of the type illustrated in Figs. 4.1(a) and (b) do not exhibit the same behaviour because the beam splitter and coupler based junctions can transmit all incident fields. Mode recirculation can be achieved in a such MZIs, however, by blocking one each of the input and output ports and reflecting the light back into the interferometer. The resulting transmission characteristics are identical to those derived in Section 4.2. Here we use a conventional MZI analysis with the addition of reflections from mirrors m1 and m2 to model the bulk-optics recirculating MZI of the type illustrated in Fig. 4.1(a). A similar approach could also be applied to the coupler-based MZI in (b).

B1 B h in in 1

L1 L1

50/50 beamsplitters 50/50 beamsplitters

L2 out 1 L2 out B B2 h 2 out 2 (a) (b) m1

Figure 4.13: (a) Conventional bulk-optics MZI using beam splitters. (b) Modified bulk- optics MZI designed to exhibit recirculation effects. The two additional mirrors m1 and m2 reflect light back into the interferometer.

Figures 4.13(a) and (b) respectively show the conventional bulk optics MZI from 4.7. DISCUSSION 125

Fig. 4.1(a) and a modified version designed to exhibit mode recirculation. Consider first the design illustrated in (a). Light entering via the input port is split at the first beam splitter B1 and the two beams travel different paths before being recombined at the second beam splitter B2. Suppose that the two arms of the interferometer have lengths L1 and L2 = L1 + δL respectively, as shown in Fig. 4.13. A beam propagating along path 1 therefore advances in phase by Φ1 = k0L1, while a beam propagating along path 2 advances by Φ2 = k0(L1 + δL) = Φ1 + ϕ, where k0 = 2π/λ. The light transmitted into output 1 is a superposition of two beams; the first is transmitted through B1 then propagates a distance L1 and is reflected at B2; the second beam is reflected at B1 then propagates a distance L2 and is transmitted through B2. The superposition of these two beams and their accumulated phases is given by

iΦ1 iΦ2 iΦ1 t1 = τe ρ + ρe τ = 2ρτe cos(ϕ/2), (4.13) √ √ where we have assumed that ρ = 1/ 2 exp(iφr) and τ = 1/ 2 exp(iφt) are the reflection and transmission coefficients of the beam splitters B1 and B2. Therefore, the 2 transmitted intensity for the single pass MZI is T1 = cos (ϕ/2), identical to Eq. (4.3). Similarly, the field transmitted into output 2 is given by

iΦ1 iΦ2 2iφr Φ1 t2 = τe τ + ρe ρ = ie e sin(ϕ/2), (4.14) where we have taken the fields reflected by the beam splitters to be π/2 out of phase relative to the transmitted fields i.e. φt = φr + π/2. Comparing Eqs. (4.13) and (4.14) with Eq. (4.2) it can be seen that the field transmitted at output 1 is equivalent to the even mode component in Eq. (4.2), while the field transmitted at output 2 is equivalent to the odd mode component.

Now consider the effect of placing two additional mirrors m1 and m2 in the positions indicated in Fig. 4.13(b). Mirror m1 reflects the field t2 back into the interferometer with an additional phase of exp(2ik0h) due to propagation between B2 and m1. The reflected beam splits and propagates through the interferometer back to B1 where some is transmitted into the input port, while the rest is reflected at m2. This selective reflection at either end of the interferometer is analogous to the odd-mode recirculation between the junctions of the PhC MZIs in Section 4.4, and results in identical recirculation properties. By tracking the recirculating beam back and forth through the interferometer, the total transmitted power is found to be given by Eq. (4.8), where

χ = βL + ϕ/2 is replaced by χ = k0(L1 + 2h) + φr + ϕ/2.

4.7 Discussion

In this chapter we have shown that MZIs formed from waveguides with a perfectly reflecting cladding display significantly different transmission characteristics to conventional MZIs. In particular, reflections due to modal symmetry mismatches result in mode recirculation and resonant transmission behaviour. Photonic crystals provide one 126 CHAPTER 4. RECIRCULATING MACH-ZEHNDER INTERFEROMETER means of obtaining a reflecting cladding with the required properties and we have proposed two PhC MZI designs that exhibit the predicted mode recirculation behaviour. The semianalytic model derived in Section 4.2 gives insight into the physics of MZIs and provides a powerful design tool for predicting the properties of relatively complicated devices knowing only the dispersion properties of a single waveguide. When combined with rigorous numerical techniques such as the Bloch mode matrix method, new designs can be optimised rapidly over a broad parameter space. An example of this was given in Sections 4.3 and 4.4.3 where we first identified using Eq. (4.8), the conditions under which a recirculating MZI exhibits a quartic phase dependence (4.9), then demonstrated the predicted behaviour in a PhC MZI (Fig. 4.10). The model provided an accurate estimate of the optimum wavelength and structural parameters, thereby saving a considerable amount of computational time compared to an all-numerical parameter search. Although several PhC MZI designs have been fabricated, the recirculation properties that we predict here have not been identified experimentally for several reasons. First, the devices studied in Refs. [200, 201, 220] all use directional coupler-based junctions that transmit both odd and even symmetry modes, and are thus inherently single-pass MZIs. Second, although the designs in Refs [218] and [219] both have single input and output waveguides, and would thus be expected to exhibit recirculation properties, the junctions do not have the necessary high transmission properties identified in Sec- tion 4.4.1 to observe clearly the odd-mode resonance effects. The coupled Y-junction proposed in Chapter 3 exhibits high transmission over a wide bandwidth and is thus an ideal junction for designing efficient MZIs. A theoretical and experimental investigation of an MZI with a realistic tuning mechanism, rather than the model adopted for convenience in Section 4.4.2 would be of interest. Among other things, this would provide a better estimate of the length requirements for tuning the phase. Three-dimensional modelling in a slab design would also be valuable. Chapter 5

Eﬃcient wide-angle transmission into rod-type photonic crystals

In this chapter we demonstrate highly efficient coupling into uniform rod-type photonic crystals with unmodified interfaces. Reflected powers of less than 1% are calculated for two-dimensional structures at incident angles up to 80◦. These properties are closely related to forward scattering resonances of the high index cylinders, and as a consequence, are not displayed by rod-type photonic crystals. We use these results to optimise a realistic PhC slab structure to exhibit both high efficiency coupling and auto-collimation and present numerical results for two- and three-dimensional finite difference time domain calculations. The results could also be applied to the design of efficient superprisms and other in-band dispersion based PhC devices. This chapter contains a substantial amount of unpublished work in Sections 5.1–5.4 which is currently being prepared for publication. The results presented in Section 5.5 are based on Ref. [226] which is due to appear in Applied Physics Letters in September 2005.

5.1 Introduction

We now turn our attention away from photonic bandgap devices and consider what is perhaps a more fundamental photonic crystal problem, that of coupling light from a homogeneous dielectric material such as free-space into the propagating Bloch modes of a uniform PhC. As we saw in Section 1.8, the unusual dispersion properties of PhC bands lead to a range of interesting effects such as superprism behaviour and auto-collimation. To use these effects in practical devices it is essential that light can be coupled efficiently into and out of the PhC structure with minimal back-reflection. This is necessary not only to keep the insertion losses low, but also to prevent light from being scattered into other parts of a device where it may cause interference and cross-talk. The latter issue will especially important in compact integrated optical circuits consisting of many components on a single chip.

127 128 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION

While one of the main attractions of PhC devices is their compact size, this also presents many practical challenges when interfacing these PhC components to conventional optical systems. Several efficient methods have been developed for coupling to waveguide-based devices [8, 227–230], but the challenge remains to find a practical solution for coupling light into the extended Bloch modes of a uniform PhC lattice for in- band applications. The main approach reported in the literature for air-hole type PhCs has been to modify one or several of the interface layers to produce an anti-reflection coating on the front and back faces of the structure. One such method is analogous to the application of thin-film anti-reflecting coatings to optical elements, where resonant effects are used to enhance the transmission properties over a narrow frequency bandwidth, although the bandwidth can be improved with multi-layered gratings [231]. A second approach, proposed by Baba et al. [232] is to change the size and/or shape of the air holes to gradually modify the field profile as it approaches the PhC slab, effectively creating an apodised interface. A third possibility suggested by Witzens et al. [233] is to place a waveguide above a PhC slab and use evanescent coupling to transfer light from the waveguide into counter-propagating Bloch modes of the PhC. Here we study the transmission properties of rod-type, rather than hole-type, PhCs and demonstrate that it is possible to couple more than 99% of incident light into such structures for a wide range of incident angles without making any modifications to the interface. We start in Section 5.2 by considering the basic 2D geometry illustrated in Fig. 5.1, in which a plane wave is incident on a semi-infinite PhC from a homogeneous, semi-infinite region above. The incident medium is taken to continue into the PhC as the background material. Both square and triangular rod-type PhCs oriented as in Figs. 5.1(a) and (b) respectively, are found to exhibit almost perfect wide-angle transmission features for a wide range of structural parameters. In Sections 5.3 and 5.4 we show that this behaviour is closely related to the scattering properties of the individual cylinders which in turn influence the transmission properties of the gratings that form the PhC lattice. We identify the conditions under which high efficiency coupling can occur and show in Section 5.4 that these provide a simple explanation as to why efficient coupling is achieved more easily in rod-type PhCs than in hole-type PhCs. In Section 5.5 we optimise a purely 2D rod-type PhC to exhibit simultaneously very low reflectance and auto-collimation. The successful combination of these two properties is demonstrated for an incident Gaussian beam using Finite Difference Time Domain (FDTD) simulations, first for a purely 2D geometry, and finally in a realistic rod-type PhC slab consisting of finite length silicon rods in a background of silica.

5.2 Plane wave coupling to semi-inﬁnite PhCs

In this section we consider the two geometries shown in Figs. 5.1(a) and (b), where a plane wave is incident at an angle θi onto a semi-inﬁnite 2D PhC consisting of dielectric 5.2. PLANE WAVE COUPLING TO SEMI-INFINITE PHCS 129

M K q M q i i G X G

(a) (b) 0.7 0.7

0.6 0.6

0.5 0.5 d/l d/l 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 0 20 40 60 80 0 20 40 60 80 q q i (degrees) i (degrees) (c) 0 0.2 0.4 0.6 0.8 1 (d) Transmission

Figure 5.1: Geometry and transmittance contour plots for a plane wave incident on a semi-inﬁnite PhC of rods (ncyl = 3) in air (n = 1). (a) Square lattice oriented with Γ − X direction normal to the interface. (b) Triangular lattice with Γ − M direction normal to the interface. (c) and (d) Transmittance contour plots for varying θi and d/λ for the PhCs in (a) and (b) respectively. Below the dashed white line there is only a single propagating plane wave grating order.

rods of refractive index ncyl, and radius r embedded in a dielectric background nb. To calculate the transmission into the PhC, we use the Bloch mode method to calculate the matrix R∞, defined in Section 2.3.2 as the plane wave reflection scattering matrix for a plane wave incident from free-space onto a semi-infinite PhC. Since we are interested in the reflected power, and no energy is carried by the evanescent states, R∞ is truncated to contain only those elements corresponding to reflection into propagating plane wave orders. Thus, when only a single reflected order is supported, R∞ can be treated as a scalar and the total reflected power is R = |R∞|2. When multiple reflected orders are present, the total reflectance is calculated as a sum over the reflectance into each order. For the purely 2D structures that we consider first, the transmittance is then simply calculated as T = 1 − R. The Bloch mode method is ideal for these calculations, as it is very efficient and can 130 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION

1 80 0.998 60 0.996 T 40

0.994 (degrees) x T>99%

ma 0.992 q 20 T>95% T>90% 0.99 0 0 20 40 60 80 0.25 0.3 0.35 0.4 q i (degrees) r/d (a) (b)

Figure 5.2: (a) Transmittance as a function of incident angle for the square (solid curve) and triangular (dashed curve) lattices in Fig. 5.1. Note the vertical axis starts at T = 99%. (b) Maximum acceptance angle θmax plotted as a function of normalised radius for a square lattice of rods with ncyl = 3.0 in air. The three curves correspond to T > 90%, 95%, and 99% in the range 0 ≤ θi ≤ θmax. deal with semi-infinite structures without the need for absorbing boundary conditions or other techniques required by many other simulation methods to avoid reflection from the rear surface of finite structures. In addition, the method implicitly calculates the scattering properties of a single grating, from which the PhC lattices are formed. As we show in Section 5.4, studying the properties of a single grating provides an intermediate case between a single cylinder and the properties of a full 2D lattice. The contour plots in Figs. 5.1(c) and (d) show the transmittance as a function of incident angle and normalised frequency for a TM-polarised plane wave incident on the semi-infinite PhCs shown in (a) and (b) respectively. The PhCs both have period d, and consist of rods with ncyl = 3.0 and r = 0.34d embedded in free-space (nb = 1). Recall from Section 1.8.1 that when coupling into a PhC, the incident wavevector component parallel to the interface (kx here) must be conserved and so the incident field can only couple to specific regions of each PhC band. We can therefore understand the relatively complex structure of the transmission plots in Figs. 5.1(c) and (d) as a projection of the band diagram onto the kx component of the incident wavevector, where kx = 2π/λ sin(θi). Hence, the dark blue regions of zero transmission correspond to full and partial bandgaps in the projected band structure. Note that the transmission into the lowest frequency band does not exceed 90%, but almost perfect transmission is possible into some of the higher order bands. One of the most striking features in Fig. 5.1 is the dark red, high transmission ‘finger’ extending almost horizontally across both transmission plots at approximately d/λ = 0.34, corresponding to transmission into the second band in both cases. At this frequency, indicated by the dotted green lines in Figs. 5.1(c) and (d), the transmittance ◦ ◦ into the square lattice PhC exceeds 99% for incident angles 0 ≤ θi ≤ 80 . This can 5.2. PLANE WAVE COUPLING TO SEMI-INFINITE PHCS 131 be seen in Fig. 5.2(a), which shows the transmittance as a function of incident angle at d/λ = 0.34 for the two lattices. The triangular lattice exhibits T > 99% for all angles ◦ ◦ in the range 0 ≤ θi ≤ 37 . Observe that a second region of almost perfect transmission occurs at frequencies close to d/λ = 0.6, for which the transmission exceeds 99% for ◦ θi ≤ 25 in both cases. Such efficient coupling over a wide range of angles is far superior to the published results for air hole PhCs. In the remainder of this section and Sections 5.3–5.4 we identify the physical origin of this behaviour and use the results to explain why hole-type PhCs do not have similar properties. The high transmission features exhibited by the PhCs in Fig. 5.1 are not unique to the structural parameters chosen for the examples. In fact, highly efficient, wide- angle coupling appears to be a common feature of rod-type PhCs, occurring for incident fields of both TE and TM polarisations and for hexagonal and square lattices with a range of refractive indices and radii. As an example of the robust nature of the high transmission, Fig. 5.2(b) shows the maximum acceptance angle θmax for transmission into the second PhC band, plotted as a function of cylinder radius for a square lattice of rods with ncyl = 3.0 in air. The three curves correspond to T > 90%, 95%, and 99% for all incident angles 0 ≤ θi ≤ θmax.

0.24

0.22 1

0.2 0.8

0.18 0.6 r/l 0.16 T 0.4 0.14 0.2 0.12 0 0.1

0.08 2.5 3.0 3.5 4.0 normalised lattice period (d/r)

Figure 5.3: Plane wave transmission for normal incidence into a square lattice of rods with ﬁxed radius r and ncyl = 3.0 plotted as a function of normalised lattice period d/r and normalised frequency r/λ. The dashed black curves are the T = 98% contours. The green dotted lines at r/λ = 0.115 and r/λ = 0.204 indicate the maximum transmission frequencies for r = 0.34d (d/r = 2.94) as shown in Figs. 5.1(c) and (d).

Another unusual property of the high transmission features is weak dependence on the lattice geometry. While the frequency at which they occur is a strong function of the cylinder radius and refractive index, it is a much weaker function of the cylinder 132 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION position. Evidence of this is seen in Figs. 5.1(c) and (d) where the high transmission bands occur at d/λ = 0.34 and d/λ = 0.6 for both the square and the triangular lattice. Perhaps more surprising is the observation that the maximum transmission frequencies change very little when the cylinder separation (lattice period) is varied but the cylinders remain unchanged. This is demonstrated in Fig. 5.3, which is a contour plot of the plane wave transmittance for normal incidence into a square lattice similar to that of Fig. 5.1(a), but with a fixed cylinder radius and varying lattice period d. The horizontal axis is expressed as the normalised lattice period d/r and the vertical axis is the frequency r/λ, also normalised to the cylinder radius. In these units, the structure calculated in Fig. 5.1(c) with r = 0.34d corresponds to d/r = 2.94. Observe that the two high transmission bands occur at approximately r/λ = 0.115 and r/λ = 0.205 and do not change significantly as the lattice period is varied by more than 60%. Other features in the same transmission plot are clearly influenced more strongly by a change in lattice period. This very weak dependence on structure suggests that the efficient coupling in rod-type PhCs is driven by a property of the individual rods, rather than by a property of the periodic lattice. We investigate this further in Sections 5.3 and 5.4.

5.3 Single cylinder scattering

In this section we show that the high transmission features of rod-type PhCs are closely related to the scattering characteristics of the individual rods. We identify two scattering parameters, namely the scattering cross-section and the asymmetry parameter, that provide an intuitive physical interpretation of the results in Section 5.2. Although photonic crystals rely intrinsically on coherent scattering for many of their unique properties, the individual scattering elements contribute to the detailed characteristics of the structure. In Refs. [234,235] it was shown that there is a correspondence between the position of bandgaps in three-dimensional (3D) PhCs formed from dielectric spheres, and the Mie resonances of the spheres. Similar results have been shown for 2D PhCs consisting of dielectric cylinders [236] but to our knowledge, the relationship between scattering resonances and PhC transmittance has not been directly studied. Here we show that there is not only a correspondence between cylinder resonances and band position, but also a direct relationship between forward scattering and the transmittance of plane waves incident on a PhC. We note that although the single cylinder treatment presented in this section is strictly 2D, the results are also applicable to realistic PhC slab structures, as demonstrated in Section 5.5. The scattering characteristics of an inﬁnite dielectric cylinder are well understood, and can be calculated for non-conical incidence following the treatment of van de Hulst [237]. Results for TM polarization are presented here, but we note that similar scattering behaviour also occurs for TE polarization. Consider ﬁrst a TM polarised wave in a background medium of index (nb = 1), incident on a dielectric cylinder of 5.3. SINGLE CYLINDER SCATTERING 133

r r q

Figure 5.4: Single cylinder scattering geometry. An incident plane wave from above is scattered in directions θ, where θ = 0 corresponds to forward scattering.

radius r and index ncyl in the plane perpendicular to cylinder axis as shown in Fig. 5.4. The total field outside the cylinder is expressed as the sum of the incident and scattered components, which allows weak scattering effects to be separated from the dominant incident field. At large distances ρ from the cylinder, the scattered field can be written as [237] r 2 E ∝ exp [−ikρ + iωt − i3π/4] T (θ), (5.1) z πkρ where k = 2π/λ, ∞ X T (θ) = b0 + 2 bn cos(nθ), (5.2) n=1 and 0 0 1 ncylJn(ncylkr)Yn(kr) − Jn(ncylkr)Yn(kr) = 1 + i 0 0 (5.3) bn ncylJn(ncylkr)Jn(kr) − Jn(ncylkr)Jn(kr). Note that these equations can be applied to an arbitrary index contrast after appropriate scaling of ncyl and k. Two scattering parameters relevant to this discussion are the scattering cross-section per unit length of the cylinder,

∞ 2 Z 2π 4 X c = |T (θ)|2dθ = |b |2, (5.4) scat πk k n 0 n=−∞ and the asymmetry parameter, g = hcos(θ)i, the average cosine of the scattering angle. The scattering cross-section is a measure of the total energy scattered by the cylinder in all directions, while the asymmetry parameter gives an indication of whether the scattered energy is in the forward or backward direction. Here, |g| < 1 and g > 0 (g < 0) corresponds to dominant forward (backward) scattering. In the long wavelength 134 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION

2(n+1) limit, as k → 0, the bn coeﬃcients (5.3) scale approximately as k , and hence the ﬁeld expansion (5.1) is dominated by the b0 term, and cscat → 0.

(a) 1

0.8

Square 0.6 T Triangular 0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 d/l (b)

cscat g

(a.u)

sca

0 0 0.05 0.1 0.15 0.2 r/l

Figure 5.5: (a) Normal incidence transmission spectrum into the square (solid curve) and the triangular (dashed curve) PhCs in Fig. 5.2.(b) Single-cylinder scattering cross- section (dashed curve) and asymmetry parameter g (dashed-dotted curve) plotted as a function of r/λ for the same frequency range as (a). (c)–(g) Polar plots the scattering pattern |T (θ)|2/k for r/λ = 0.034, 0.085, 0.116, 0.163 and 0.204. The light is incident from above.

Figure 5.5(a) shows the normal incidence transmittance spectrum for the square and triangular lattices considered in Fig. 5.1. Note that the transmission peaks at d/λ = 0.34 and d/λ = 0.6 almost coincide for the two structures, even though the band gap frequencies shift significantly. The curves plotted in Fig. 5.5(b) show the scattering cross-section and asymmetry parameter as a function of r/λ for light scattered from a single cylinder of index ncyl = 3.0 in air. Figures 5.5 (c)-(g) are polar plots of the scattering profile given by |T (θ)|2/k, where T (θ) is defined in Eq. (5.2). The profiles are plotted for frequencies in the first band, the first gap, the second band, the second gap and the third band respectively. Observe from Eq. (5.4) that the area enclosed by these scattering diagrams is proportional to the scattering cross-section. Figure 5.5 shows a clear correspondence between the position of the bands (indicated by non-zero transmittance) and peaks in cscat, as has been observed by others [235,236]. 5.4. GRATING PROPERTIES 135

Observe also that the transmittance into the second and third bands is better than 99%, whereas transmittance into the first band does not exceed 90% – a feature we also identified in Section 5.2. The asymmetry parameter curve and the polar plots provide an explanation for this difference. In the second and third bands, where the transmittance is highest, the peak in cscat coincides approximately with a peak in g, indicating a forward scattering resonance. In the first band g remains positive but approaches zero, despite there being a small, broad peak in cscat. Thus, although there is still significant scattering from each cylinder at low frequencies, the scattering direction is almost isotropic, with Eqs. (5.1), (5.2), and (5.4) being dominated by the b0 term as described above. This is clearly seen in the scattering patterns plotted in Figs. 5.5(c) and (d). As the frequency is increased, higher-order scattering terms become significant, thus introducing dipole-like and higher order scattering behaviour which can result in strong directional scattering. For example, the cscat peaks near d/λ = 0.36 and d/λ = 0.60 correspond to maxima in b1 and b2 respectively, with both resonances producing strong forward scattering, indicated by the g curve of Fig. 5.5(b) and the scattering patterns in Figs. 5.5(e) and (g). The observation that forward scattering resonances enhance the transmission into a PhC is a physically intuitive explanation for the results presented in Sec. 5.2, however we have not discussed the possibility of coupling between closely spaced cylinders. In Section 5.4 we consider the properties of a grating formed from a single line of cylinders, which provides an intermediate case between the single cylinder results of Section 5.3 and the full 2D PhC results in Section 5.2.

5.4 Grating properties

Recall from Chapter 2 that we consider a PhC to be a stack of identical cylinder grating layers that are coupled together by diffracted plane wave orders given by the grating equation (2.2). In this chapter we consider only gratings with period d consisting of identical cylinders and so we would expect the scattering properties of a grating to approach those of the individual cylinders as d/λ → ∞. For finite lattice periods we would still expect the forward scattering resonances of the single cylinders identified in Section 5.3 to have strong influence on the scattering properties of a grating formed from those cylinders. We consider first the transmission properties of the grating that is used to form the PhCs shown in Fig. 5.1. The square and triangular lattices are both formed from the same grating by stacking each layer as described in Section 2.2.2. Figure 5.6(a) shows the transmittance as a function of θi and d/λ for the grating of period d formed from rods with ncyl = 3.0 and r = 0.34d. The results are calculated directly from the plane-wave scattering matrix Te, where we sum the total transmittance over all propagating grating orders. Observe that the two high-transmission features of the grating at d/λ = 0.34 and d/λ = 0.6 resemble closely those in Figs. 5.1 (c) and (d). The agreement is even 136 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION more apparent in Fig. 5.6(b) where we compare the normal incidence reflectance of the grating and the square PhC with the same parameters. The strong correlation between low grating reflectance and efficient coupling into the PhC further reinforces the single cylinder scattering explanation. It also explains why the transmission peaks occur at the same frequencies in the square and triangular lattices.

0.7 1 1 0.6 0.8 0.5 0.1 0.6 d/l 0.4 T R 0.4 0.3 0.01 0.2 0.2 0 0.001 0.1 0 0.2 0.4 0.6 0 20 40 60 80 q d/l i (a) (b)

Figure 5.6: (a) Plane wave transmittance through a grating of period d consisting of rods with ncyl = 3.0 and r = 0.34d, plotted as a function of incident angle and normalised frequency. (b) Plane wave reﬂectance spectra (log scale) for normal incidence on the grating (dashed red curve) and a semi-inﬁnite square PhC lattice formed from the grating (solid blue curve).

The challenge of establishing a rigorous proof of the relationship between the forward scattering resonances of a cylinder and high grating transmission is a topic of ongoing research, and is not solved in this thesis. The multipole method may be particularly helpful in this regard as it includes implicitly the scattering properties of the individual cylinders in the grating scattering formulation [176] thereby providing a mathematical connection between the two structures. An alternative approach may be to consider first the interaction of the scattered fields from two neighbouring cylinders, then use a tight-binding approximation similar to that of Lidorikis et al. [236] to characterise the scattering from a grating. We show here, however that the single-cylinder interpretation provides a simple physical explanation as to why rod-type PhCs exhibit superior coupling properties compared to hole-type PhCs in agreement with our earlier results. This argument relies on key observations: the first is that strong forward scattering is required from each cylinder, and the second is that there must only be a single propagating diffracted grating order. The second condition can be understood in the following way. When a plane-wave is incident on a grating with period d, the number of propagating grating orders depends on both the ratio of the incident wavelength to the period and the incident angle. The condition for only a single reflected and single transmitted grating order to exist can be derived from the grating equation (2.1). If the background 5.5. EFFICIENT COUPLING FOR AUTO-COLLIMATION 137

material has a refractive index nb, then the single order condition can be expressed as d 1 < , (5.5) λ nb(1 + sin(θi)) where λ is the free-space wavelength. Consider now the properties of the grating at a forward scattering resonance of the cylinders. If Eq. (5.5) is satisfied, the light scattered forward by each cylinder can only couple into the zeroth (specular) grating order, and thus there is very little overlap between the scattered fields resulting in high transmission through the grating. If multiple grating orders are present, they provide a means for the fields scattered by each cylinder to interfere with each other via coupling to higher diffraction orders. When this occurs, phase differences between the zeroth and higher orders become an additional complicating factor. Moreover, it is considerably easier to satisfy the conditions for zero reflectance into a single grating order than into multiple reflected orders simultaneously. The relationship between single cylinder scattering and grating transmission thus becomes much more complex and the simple arguments that we applied in Section 5.3 are no longer valid. This interpretation is consistent with our observations that the zero reflectance features in Fig. 5.6(b) disappear at frequencies where multiple grating orders exist. The results and observations of this section and Section 5.3 provide a simple explanation as to why rod-type PhCs typically exhibit highly efficient coupling over a wide range of incident angles, whereas air-hole PhCs may require interface modifications in order to achieve satisfactory results [231–233]. In Section 5.3 we showed that the resonance condition for strong forward scattering is not satisfied at low frequencies where light is scattered almost isotropically. The same applies to scattering from cylinders with a low refractive index contrast relative to the background. Efficient coupling is therefore more likely to occur in the second PhC band or higher, while additional grating orders appear if the frequency is too high. For rod-type PhCs, where the background refractive index is lower than that of the rods, these two conditions are relatively easy to satisfy, as shown in Figs. 5.1(c) and (d) where the dashed white lines indicate indicate the upper frequency limit given by Eq. (5.5). In air hole PC structures it is much more difficult to satisfy both conditions simultaneously since non-isotropic scattering from air holes only occurs at relatively high frequencies but at the same time the high index background material reduces the frequency at which multiple plane wave orders appear.

For example, if the background material is silicon (nb = 3.4) the ﬁrst non-zero grating order appears for large angle of incidence at frequencies above d/λ = 0.15, which is well below the second band for most air-hole PhCs.

5.5 Eﬃcient coupling for auto-collimation

So far in this chapter we have demonstrated very low reflection coupling to rod-type PhCs over a wide range of incident angles, however we have not considered how the light 138 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION propagates once inside the structure. In order to exploit the efficient coupling for in- band applications such as superprism effects and auto-collimation, we must optimise the PhC structure so that the light is coupled into bands which have the desired dispersion properties for these effects. In this section we present a realistic rod-type PhC slab design that exhibits auto-collimation and less than 1.5% reflection loss at each interface ◦ for angles of incidence up to θi = 25 . We first describe in Section 5.5.1 the optimisation process using 2D calculations, then in Sections 5.5.2 and 5.5.3 we present both 2D and 3D results that demonstrate the successful combination of the two effects. We also show that this simple PhC structure can be used as a collimating beam combiner.

Si rods

n ncyl = 3.4 clad = 1 .458

h =3d y q z i n slab = 1 x .460 n clad = 1 .458

Figure 5.7: Geometry of the rod-type PhC slab consisting of a triangular lattice of silicon rods (ncyl = 3.4) in a slab of silica (nslab = 1.460) of height h = 3d where d is the lattice period. The structure is clad above and below by a cladding with refractive index nclad = 1.458.

Figure 5.7 shows the PhC slab geometry considered here, consisting of a triangular lattice of silicon rods (ncyl = 3.4) embedded in a slab of silica with index nslab = 1.46. The lattice is oriented such that the interface is parallel to the Γ − M direction. The slab and the cylinders have a finite height of h = 3d, and they are capped above and below with a cladding of index nclad = 1.458. As we demonstrate in this section a cylinder radius of r = 0.35d results in both low reflectance and auto-collimation properties. This geometry resembles that proposed by Martinez et al. [50], although the slab height is increased to improve the mode overlap with the incident Gaussian beam in the 3D simulations. The design and modelling of this structure is performed using a combination of 2D and 3D Finite Difference Time Domain (FDTD) simulations, in addition to Bloch mode method calculations and the use of commercial plane-wave band structure software. 5.5. EFFICIENT COUPLING FOR AUTO-COLLIMATION 139

5.5.1 Design We describe here the process of optimising the PhC in Fig. 5.7 for auto-collimation ◦ and transmittance at incident angles of θi = 22.5 , as this allows the PhC to be used as a simple beam combiner for two incident beams with an angular separation of 45◦. Although designed for a specific incident angle, the properties of the optimised structure are relatively robust. The reflectance remains below 1.5% for all incident angles θi < 25◦, and auto-collimation is observed throughout this range. ◦ We consider first the transmission of a plane wave incident at θi = 22.5 on the PhC in Fig. 5.7, which we approximate as a 2D PhC lattice of rods with index ncyl = 3.4 in a background of index nb = 1.458. As we demonstrate in Sections 5.5.2 and 5.5.3, the 2D results provide a good approximation to the 3D slab structure, thus providing an efficient tool for optimizing parameters before launching computationally intensive 3D simulations. The contour plot in Fig. 5.8(a) shows the transmittance into the second band of the PhC as a function of the cylinder radius, where the contours correspond to transmittances of T = 90%, 95%, 98% and 99%. These results were obtained using the Bloch mode method as described in Section 5.2. Observe that for all cylinder radii in the range 0.31 ≤ r ≤ 0.4 there are frequencies for which the transmittance exceeds 99%. This region defines the ideal part of parameter space for obtaining efficient auto- collimation.

0.34 p

0.32 T > p 9 /2 9 0.3 % l k d/ y 0 0.28

-p/2 0.26

-p 0.24 0.3 0.32 0.34 0.36 0.38 0.4 -p -p/2 0 p/2 p k r/d x (a) (b)

◦ Figure 5.8: (a) Transmittance contour plot for a plane wave incident at θi = 22.5 onto a 2D triangular PhC lattice of silicon rods embedded in silica. The contour levels are at T = 90%, 95%, 98% and 99%. The green points represent the optimum frequency ◦ for collimation at θi = 22.5 . (b) Background: contour plot of 1/p calculated over the Brillouin zone for the second band of the PhC in (a) with r = 0.35d. White regions correspond to maximum 1/p (strong collimation). Blue curves: equi-incident angle ◦ contours for θi = ±22.5 . Red curve: d/λ = 0.29 equifrequency contour. 140 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION

Recall from Section 1.8.4 that auto-collimation occurs when an incident beam couples into a Bloch mode where there is a point of inflection in the equifrequency curve [141]. At this point, the small range of incident k-vectors present in the beam all propagate in approximately the same direction within the crystal, and thus the beam is collimated. If the equifrequency curve is relatively flat, then two beams can be launched into the crystal at different angles, but once inside, they are collimated and propagate through the crystal in approximately the same direction. Quantitatively, collimation is measured by the parameter p defined in Eq. (1.9) [126]. For good collimation, we require small p values so that the beam direction inside the crystal θc, varies slowly with θi. The green circles plotted on top of the transmission contours in Fig. 5.8 (a) show ◦ the optimum collimation frequency as a function of the cylinder radius, for θi = 22.5 . The diagram in Fig. 5.8(b) illustrates how the optimum collimation frequency is calculated for each cylinder radius using a similar approach to Baba et al. [238]. We first calculate the 2D band surface for the second band using the commercial software BandSOLVE from which we evaluate the p parameter over the Brillouin zone as shown in Fig. 5.8(b), where the grey contours in the background represent the value of 1/p, calculated for a cylinder radius of r = 0.35d. The narrow white regions of the 1/p plot correspond to large 1/p values, and hence indicate the regions where collimation occurs

(the horizontal white line on the ky = 0 axis indicates the interface). The blue curves in Fig. 5.8(b) are the equi-incident angle contours [126] that indicate the regions of the ◦ Brillouin zone that can be accessed by varying the frequency while θi = 22.5 . The green circle indicates the region we wish to couple into with the incident beam, deﬁned by the intersection of the blue equi-incident angle contour and the white collimation ◦ region. Thus, the optimum frequency for collimation at θi = 22.5 is given by the frequency of the second band at this intersection point. In the example shown here, the optimum collimation frequency for r = 0.35d is approximately d/λ = 0.290, indicated by the red equifrequency curve in Fig. 5.8(b). To plot the green line in Fig. 5.8(a) we repeat the process described above for a range of cylinder radii. The results show that while high transmittance and auto-collimation are independent properties, they are not incompatible. Figure 5.8(a) shows that both properties occur simultaneously for cylinder radii in the range 0.335d ≤ r ≤ 0.382. For the results presented in the remainder of this chapter we consider rods with r = 0.35d, for which the optimum collimation occurs at approximately d/λ = 0.29, as shown in Fig. 5.8(b).

5.5.2 Results: 2D vs 3D coupling properties Before presenting the collimation results, we demonstrate first that the efficient coupling conditions identified with the 2D calculations in Section 5.5.1 provide a good approximation to the slab structure in Fig. 5.7, which requires full 3D numerical simulations. The commercial FDTD software FullWave is used for the 3D calculations with perfectly matched layer boundary conditions on all sides of the computational domain to simulate a semi-infinite cladding above and below the slab. We consider an incident Gaussian 5.5. EFFICIENT COUPLING FOR AUTO-COLLIMATION 141 beam with an elliptical cross-section with full widths at half maximum (FWHM) of wxy = 5d and wz = 2.5d, launched with TM polarization. While there is some loss at the interface of the PhC due to a mismatch in the beam profile, the reflected power remains very low, which is essential to avoid interference effects in compact devices. ◦ Figure 5.9 shows the 3D reflectance spectra for Gaussian beams with θi = 0 (solid ◦ curve) and θi = 22.5 (long dashed curve) incident on 16 layers of the PhC in Fig. 5.7. Also shown are the equivalent reflectance spectra for plane waves incident on a 2D semi-infinite PhC, calculated using the Bloch mode matrix method. On all curves the minimum reflected power occurs for scaled frequencies between 0.29 < d/λ < 0.295, with bandwidths of more than 15% of the central frequency in which the reflectance is ◦ below 10%. The 3D results show a minimum reflectance of 0.2% and 1.4% for θi = 0 ◦ and θi = 22.5 respectively, compared to < 0.1% in the 2D plane-wave results, a variation attributed to the angular spread of the Gaussian beam in the 3D calculations. Note also that the Fabry-Perot fringes in the FDTD results are much stronger for the normally incident beam since the finite beam width weakens multiple reflection interference for oblique incidence.

1 q o 2D R∞, i = 0 0.9 q o 2D R∞, i = 22.5 3D FDTD, 16 layers, q = 0o 0.8 i q o 3D FDTD, 16 layers, i = 22.5 0.7 0.6 0.5 0.4 0.3

Reflected power 0.2 0.1 0 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 d/l

Figure 5.9: 2D plane wave reflectance spectra (R∞: Bloch mode method) for a semi- infinite PC and 3D FDTD reflectance spectra for a Gaussian beam of width wxy = 5d ◦ and height wz = 2.5d incident on 16 layers of cylinders. Results are shown for θi = 0 ◦ and θi = 22.5 .

The agreement between the 2D and 3D calculations in Fig. 5.9 is surprisingly good given that the 2D calculation is for plane waves incident on a semi-inﬁnite 2D PhC, 142 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION while the 3D results are calculated for a relatively narrow Gaussian beam incident on a PhC slab consisting of only 16 grating layers. This means that the 2D calculations can provide an eﬃcient tool for initial optimization calculations over a broad parameter space as demonstrated in Sec. 5.5.1. The results in Section 5.5 show that the 2D optimisation of the collimation parameter p is also a good approximation to the 3D results.

5.5.3 Results: collimation We consider now the second optimised feature of the PhC – auto-collimation. Ob- serve that the approximately hexagonal equifrequency curve in Fig. 5.8(b) has slightly concave, rather than ﬂat sides. This is a consequence of optimising the structure for collimation at non-normal incidence. We have found however, that reasonable collimation ◦ is achieved for all incident angles θi ≤ 25 .

(a) (b)

Figure 5.10: (a) Electric field from a 2D FDTD simulation for an incident Gaussian ◦ beam launched from below at a frequency of d/λ at θi = 22.5 incident on 30 layers of PhC. The dashed lines indicate the upper and lower interfaces of the lattice. The cylinders are omitted so that the field inside the PhC can be seen more clearly. (b) Electric field for the same Gaussian beam propagating in the uniform background material with no PhC present.

◦ Figure 5.10(a) shows the electric field for an incident Gaussian beam at θi = 22.5 launched from below with a FWHM of 5.7d onto a PhC consisting of 30 grating layers. The cylinders have not been drawn in the figure so that the fields inside the PhC can be more easily seen, but the dashed lines indicate the upper and lower boundaries of the structure. The results were calculated using the commercial 2D FDTD software FullWAVE for a frequency of d/λ = 0.29. For comparison, Fig. 5.10(b) shows the fields in the background medium when the PhC is removed. The collimation effects of the PhC can be clearly seen; the beam is deflected when it enters the PhC, and propagates through the structure at an angle θc ≈ 0. At the rear interface the beam leaves the 5.5. EFFICIENT COUPLING FOR AUTO-COLLIMATION 143

PhC and continues to propagate in the direction of the original beam. The beam on either side of the PhC does not broaden significantly, and remains narrower than the beam propagating in the dielectric background. As expected from the 2D calculations in Fig. 5.9, the reflectance is less than 1%, and no reflected fields can be observed Fig. 5.10(a).

45o 45o (a) (b)

Figure 5.11: (a) Electric field calculated using the 2D FDTD method for a PhC beam combiner. Two Gaussian beams of width 5.7d are incident on the front face of a 40-layer ◦ PhC with θi = ±22.5 . (b) Electric field for the same parameters as (a) but without the PhC. Since the beam direction inside the crystal is approximately normal to the interface ◦ of the PhC, two beams incident at θi = ±22.5 will propagate in approximately the same direction through the crystal. We use this property to demonstrate a simple PhC beam combiner exhibiting low reflection, wide angle coupling and collimation that could be used to interact multiple beams for studying nonlinear effects in PCs. In Fig. 5.11(a) we present 2D results for the case where two Gaussian beams with the same parameters as in Fig. 5.10 are focused onto the front face of a 40-layer PC. Interference of the beams results in the three-lobed pattern in front of the interface in Fig. 5.11(a). Inside the PC, the beams are collimated and propagate in approximately the same direction, emerging 144 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION through the rear interface where they continue to propagate in their original directions. For comparison, Fig. 5.11(b) shows the propagation of the beams in the background material without the PhC. As we demonstrate in the 3D examples in Fig. 5.12, the field profile at the input interface is effectively transferred through the crystal to the rear interface with only minor distortion. This behaviour is maintained regardless of the relative phase difference between the beams. PhC designs of this type could potentially be used for transferring more complex field modulations such as images from one side of the crystal to the other. In contrast, if the two beams were coupled into a single finite waveguide, the conversion between waveguide and free space modes would destroy the original field profile without careful waveguide design. In this final example we demonstrate auto-collimation behaviour using full 3D numerical simulations for the PhC slab illustrated in Fig. 5.7 with the parameters optimised for the 2D PhC in Section 5.5.1. Figure 5.9 shows that the minimum reflectance for the 3D structure occurs for a frequency of approximately d/λ = 0.295, slightly higher than for the purely 2D lattice. Hence the results that follow are calculated at this frequency, for which the reflectance is approximately 1.4%. Figure 5.12(a) shows the fields inside a PhC of 16 layers calculated using 3D FDTD software for two incident

Gaussian beams as in Fig. 5.11. The beams have a width of wxy = 5d and height of wz = 2.5d and are launched in phase with other. These 3D simulations are computationally intensive and are limited by the memory requirements for large structures. We are thus limited to relatively small computational domains that do not allow us to consider long propagation regions before and after the PhC. Instead, the overlapping Gaussian beams are launched just before the front face of the PhC to reproduce the input field of the two converging beams in Fig. 5.11. With this modification, we have been able to simulate propagation through 16 layers of PhC, but this is close to the limit of current computational resources. As in the 2D results in Fig. 5.11(a), the fields at the front and rear interfaces of the PhC in Fig. 5.12(a) have a very similar profile. This can be seen more clearly in Fig. 5.12(b) and (c) where the field intensities at the front and rear interfaces are plotted for different phase differences between the two input beams. The dashed curve in Fig. 5.12(b) is a Gaussian profile with a width of 5d cos(22.5◦), corresponding to the envelope of the two input beams, while the dashed curve in Fig. 5.12(c) is a Gaussian of width 5.8d cos(22.5◦), indicating that the beams are not perfectly collimated. Apart from this slight broadening, the field profiles at the back interface closely resemble those at the front interface for each of the relative phase differences. We have confirmed that this occurs regardless of the lateral alignment of the beams with the cylinders of the interface. While we did not explicitly consider the out-of-plane losses in the optimisation, the operating frequency of d/λ = 0.29 lies close to the light line, and so the light is relatively well confined once it enters the PhC region where the high index rods provide a large index contrast in the vertical direction. The beam profiles in Figs. 5.12 however indicate a decrease in the peak intensity. This is due in part to the spreading of the beam as it 5.5. EFFICIENT COUPLING FOR AUTO-COLLIMATION 145 propagates, but also to a mismatch between the incident Gaussian beam profile and the vertical width of the guided mode, which results in additional scattering losses. The good agreement between the 2D and 3D reflectance spectra in Fig. 5.9 suggests that improved mode matching in the vertical plane would increase the transmission into the PhC without a significant increase in the reflected power.

(a) 1 1 j = 0 j = p/2 0.8 0.8 j = p j = 3p/2 0.6 0.6

0.4 0.4

Intensity (a.u.) Intensity (a.u.) 0.2 0.2

0 0 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x (d) x (d) (b) (c)

Figure 5.12: (a) 3D FDTD simulation showing the |Ey| field component in the PC ◦ for two in-phase Gaussian beams launched with θi = ±22.5 . (b) Field intensity (normalised) at the input interface for beams with phase differences ϕ = 0, π/2, π, and 3π/2. ◦ The dashed curve is a Gaussian profile of width wxy = 5d cos(22.5 ). (c) Intensity at the output interface for the phase differences in (b). The dashed curve is a Gaussian profile of width 5.8d cos(22.5◦). 146 CHAPTER 5. EFFICIENT WIDE-ANGLE TRANSMISSION 5.6 Discussion

In this chapter we have studied the coupling of light from a homogeneous dielectric into the Bloch modes of uniform rod-type PhCs and have shown that very efficient coupling is possible over a wide range of angles without modifying the interface of the crystal. Numerical studies reveal that this behaviour is a generic property of many rod-type PhCs, and thus they present an attractive class of structure for studying in-band PhC applications such as auto-collimation and superprism effects. In contrast, coupling to air-hole type PhCs is typically much less efficient and requires interface modifications or novel coupling techniques such as those proposed in Refs. [231–233]. While a rigorous proof of the single cylinder scattering interpretation is still being developed, there is sufficient evidence to connect the efficient coupling properties of the PhCs with forward scattering resonances of the cylinders. We have shown that this interpretation provides a simple and intuitive physical explanation for the coupling properties of rod-type PhCs and the difference between these and hole-type PhCs. The recognition that forward scattering resonances play a large rope in determining the transmission properties of a PhC could also lead to improved designs for hole-type PhCs in which the hole shape is optimised to produce strong forward scattering. If this was to work in practice, the modifications would require careful design so as not to interfere with other desirable properties. While the additional fabrication challenges may be comparable to the interface modifications that have been proposed, a theoretical study of this approach may be worthwhile. Analogies can be made between the transmission properties observed here and loss mechanism in a new class of microstructured optical fibre consisting of high refractive index cylindrical inclusions in a low index background [239]. These fibres rely on “anti-resonant reflection” and exhibit high loss features in the transmission spectrum that correspond to forward scattering resonances of the individual cylinders [240]. We note also that there are similarities between our results and recent studies of perfect transmission through periodically structured metallic films [241–243]. In particular, Garc´ıa et al. identified Mie resonances as the cause of enhanced transmission through periodic arrays of silica spheres embedded in a metallic film [242]. In another recent paper, Mandatori et al. demonstrated theoretically, wide-angle transmission into 1D multilayer structures consisting of birefringent materials [244]. High transmissions were obtained for incident angles up to 80◦ by choosing the birefringent properties of each layer to balance the angular transmission dependence due to increased path length and variation of Fresnel coefficients. While it is not immediately clear whether the wide- angle PhC transmission investigated in this chapter can be interpreted in a similar way, a future comparison of the two structures is warranted. Since rod-type PhCs exhibit efficient coupling over a wide parameter space, this provides significant flexibility to optimise a structure for additional characteristics as demonstrated in Section 5.5 for auto-collimation. These results have been used to design an efficient PC self-collimating beam combiner that could be used for studying 5.6. DISCUSSION 147 nonlinear interactions in PhCs. Many other in-band applications could benefit from these design techniques and results. For example PhC superprisms require light to be coupled efficiently through both interfaces while maintaining a good beam shape [126] and there are also proposals for optical chips based on self collimated beams [146,245], for which this work would clearly be of great benefit. The ability to combine auto-collimation and efficient coupling without complex interface modification is surprising if considered from the viewpoint of conventional optics. At a dielectric interface, strong beam deflections require a large index contrast but this is accompanied by high reflectance. In contrast, here we have demonstrated deflections of more than 20◦ with a reflectance of less than 1%. This is only possible as a result of the unique dispersion properties of photonic crystals. In conclusion, we have found that coupling into regular rod-type PhCs can be significantly more efficient over a wide range of incident angles than for PCs with air holes, and this can be achieved without modifications to the interface. We have demonstrated that 2D simulations of this behaviour can be used to optimise the parameters for realistic PC slab geometries, and that these properties are retained by the 3D structures.

Chapter 6

Discussion and conclusions

In this thesis we have considered a number of photonic crystal structures and devices and identified the basic physical effects that determine their behaviour. While we have focussed on two aspects of these unique structure – resonances and mode coupling properties – the theoretical approach taken here is much more widely applicable. In particular, the combined use of simple physical models and accurate numerical methods provides a powerful and efficient array of tools for studying many structures. As we demonstrated, this is aided greatly by the Bloch mode scattering matrix method, which is not only an efficient numerical method but also provides framework for deriving semi-analytic models. Perhaps one of the greatest advantages to be gained from this approach is the physical insight, and as a consequence, intuition which can lead to simple design rules. These will become ever more important as photonic crystal devices enter commercial applications. In Chapters 3 and 4 we presented several PhC waveguide based devices designed for operation in a photonic bandgap. The three main structures we considered are the FDC, the coupled Y-junction and the recirculating Mach-Zehnder interferometer. While these devices have been demonstrated only in a purely 2D geometry, the underlying concepts are general and rely predominantly on modal dispersion properties and characteristics common to most PhCs rather than specific details of the waveguides. The geometries that we used for the numerical demonstration of these concepts were designed without considering the out-of-plane losses that are present in realistic PhC slab structures. As such, the specific design parameters are intended as examples only, and do not necessarily correspond to practical devices. The logical next step, therefore, is to use the insights gained from this research to develop complete designs for experimental demonstration. As we have mentioned several times throughout this thesis, one of the important challenges in designing efficient resonant devices is controlling out-of-plane losses. Al- though the Y-junction and the Mach-Zehnder interferometer presented here are not high-Q resonant structures like the FDC, the unique properties of all three structures are a result of multiple reflections between the ends of blocked waveguides, and so the

149 150 CHAPTER 6. DISCUSSION AND CONCLUSIONS design considerations are common. We discuss here three distinct issues that must be addressed in order to achieve realistic device designs. They are:

1. Control of waveguide dispersion properties for modes guided below the lightline.

2. Out-of-plane scattering losses at the reﬂective interfaces.

3. Tolerance to symmetry breaking and other fabrication variations.

These issues are common to many PhC slab-based devices, and thus there are many examples in the literature where they have been successfully addressed. Let us consider first the waveguide dispersion properties. As the semi-analytic models derived in Sections 3.4.1 and 4.2 show, the performance of these devices depends strongly on the dispersion. While dispersion tuning is relatively straightforward in the 2D analysis where radiation losses are ignored, the available parameter space is reduced considerably in slab structures by the requirement to operate below the lightline. How- ever, techniques such as changing the waveguide width [97], or tuning the radius of the cylinders close to waveguide [98] provide significant control over the dispersion. We expect that tuning mechanisms such as these can be used do optimise the waveguide design to permit our devices to operate below the lightline. The second issue – reducing scattering losses at the reflective interfaces – is somewhat more of a challenge. While waveguide modes that lie below the cladding lightline are theoretically lossless in a straight waveguide, considerable losses may still occur when additional features such as junctions, bends or mirrors are introduced. Since the devices we have proposed all require light to be reflected by the PhC at the end of a blocked waveguide, the control of losses at these mirrors is essential. While there has been considerable effort to reduce the radiation losses of high-Q PhC cavities via various structural tuning methods as discussed in Section 1.7.1, [54,61,68,70], the radiation losses that occur at a single reflective interface have received less attention. The physical mechanisms of this loss have been studied in some detail in Refs. [61,246,247], and a number of approaches have been proposed to minimise the effects. Although the results were reported for waveguides with a 1D Bragg mirror, the physical interpretation applies equally to 2D PhC mirrors in a slab geometry. In Ref. [247], Bragg reflection from a PhC mirror is described as a three-step process; first the forward propagating mode in the waveguide must couple to the fundamental forward (non-propagating) Bloch mode of the PhC; second, light is coupled from the forward travelling Bloch mode into its counterpropagating pair; and finally the backward travelling Bloch mode is coupled into the backward propagating waveguide mode. While the second process is lossless provided the Bloch modes are guided by the slab, radiation losses may occur during the first and last stages of the process due to possible mode mismatches between the modes of the PhC and the modes of the waveguide. Sev- eral adiabatic tapering techniques have been proposed to optimise the mode-matching in 1D periodic structures [246, 247] for high reflectance and low radiation losses. A 151 similar approach could be applied to the blocked waveguides in the FDC and the other devices to minimise the radiation loss at these interfaces. An alternative highly efficient waveguide mirror was demonstrated recently by Song et al. [248], formed by coupling two slightly different waveguides. Almost perfect reflection occurs at frequencies where the input waveguide supports a propagating mode but the output waveguide supports no propagating modes (ie: in the frequency interval between the waveguide mode cutoff and the bottom of the bandgap). These results are consistent with a mode-matching interpretation since the very similar geometry of the two waveguides is likely to allow efficient coupling between the propagating and nonpropagating modes. While this design is suitable for a single waveguide mirror, some modification would be required to apply to the FDC or the Y-junction, where waveguides on both sides of the interface are required to support propagating modes. Reducing the scattering losses upon reflection is likely to result in efficient Y-junction and Mach-Zehnder designs, but in order to achieve the predicted high-Q resonances in the FDC, it may be necessary to consider the complete cavity design. While reducing the out-of-plane scattering losses at each reflective interface is critical to achieve a high- Q response, a second mechanism due to “radiation-loss recycling” has also been shown to play a significant role in these structures [61]. Since this effect depends on the details of the cavity as well as the interfaces, more complicated optimisation approaches may be required. In Section 3.4.5 we show that the FDC is closely related to other common PhC cavity designs, so the same optimisation techniques developed for those structures could be applied [54,67,68,70,74]. The third and final consideration for developing practical PhC device designs is their tolerance to fabrication errors, although recent demonstrations of ultrahigh-Q cavities [74] and low-loss waveguides [83–85] indicate that these errors can be tightly controlled. While we cannot predict the sensitivity of the devices proposed here without a numerical study, we can make some general observations. One aspect common to all of our structures, is their dependence on modal symmetry, a property we used to develop the semi-analytic models in Sections 3.4.1 and 4.2. Although it is unlikely that small structural variations which break the symmetry will destroy the properties of these devices, they may be more sensitive to this type of fabrication error. The Y-junction in particular requires further study in this regard, since an asymmetric structure would allow light to couple into the odd supermode of the triple waveguide section – a possibility forbidden in the symmetric geometry considered here. To summarise the results of this discussion, the waveguide-based devices that we propose in this thesis are expected to present the same challenges as many other PhC based structures that have been successfully demonstrated in experiment. By addressing the three issues listed above, we hope to develop realistic designs for future fabrication. The PhC structures that we study in Chapter 5 face similar loss-related issues which we have already discussed briefly in Section 5.6. Provided the Bloch modes being coupled into lie below the cladding lightline of the slab, the dominant cause of loss is the vertical mode-mismatch at the interface. The width of the incident Gaussian 152 CHAPTER 6. DISCUSSION AND CONCLUSIONS beam in the 3D simulations was chosen to match approximately the observed width of the guided mode in the slab, but no further attempt was made to optimise the coupling. To minimise the radiation loss at the interface one could either tune the incident beam parameters, or alternatively couple light into the crystal via a dielectric waveguide [143,249], which may provide better mode matching conditions. In conclusion, we have studied in this thesis a number of photonic crystal structures and devices that derive their characteristic properties from resonance and mode coupling effects that are common to most PhC geometries. The combination of these two features in a single device results in a range of different behaviours that we have exploited to design several novel and promising devices. In Chapter 3 we presented three such structures: a high-Q resonant filter, a wide-bandwidth symmetric Y-junction and a periodic coupled waveguide that can be tuned to exhibit very low group velocities. These same design concepts and analysis methods were extended in Chapter 4 where we showed that mode recirculation in PhC Mach-Zehnder interferometers can be harnessed to enhance the response of these devices. In Chapter 5 we considered an entirely different aspect of resonance and mode coupling: the transmission of light from free-space into the propagating Bloch modes of a uniform PhC. The demonstration of highly efficient wide-angle coupling in rod-type geometries, and the combination of this property with auto-collimation effects could motivate further research efforts to fabricate rod-type PhCs for in-band applications. Finally, it should be noted that the structures we have studied are intended to exploit the unique properties of photonic crystals, rather than to replicate conventional devices on a smaller scale. As we have demonstrated, this design philosophy, combined with physical insight can lead to simple but efficient devices with highly attractive properties. The combination of powerful general methods and semi-analytic solutions for particular structures is one which will help the rapid progress in photonic crystal optics to continue. Bibliography

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List of author’s publications

Journal papers

[1] M. J. Steel, T. P. White, C. M. de Sterke, and R. C. McPhedran, “Symmetry and degeneracy in microstructured optical ﬁbers,” Opt. Lett. 26, 488-490 (2001).

[2] T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Conﬁnement losses in microstructured optical ﬁbers,” Opt. Lett. 26, 1660-1662 (2001).

[3] T. P. White, R. C. McPhedran, L. C. Botten, G. H. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal ﬁbers using the multipole method,” Opt. Express 9, 721-732 (2001).

[4] T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical ﬁbers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322-2330 (2002).

[5] B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical ﬁbers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331-2340 (2002).

[6] T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser and B. J. Eggleton, “Resonance and scattering in microstructured optical ﬁbers,” Opt. Lett. 27, 1977-1979 (2002).

[7] M. Sumetsky, Y. Dulashko, T. P. White, T. Olsen, P. S. Westbrook, and B. J. Eggleton, “Interferometric characterization of phase masks,” Appl. Opt. 42, 2336- 2340 (2003).

[8] L. C. Botten, A. A. Asatryan, T. N. Langtry, T. P. White, C. M. de Sterke, and R. C. McPhedran, “Semianalytic treatment for propagation in ﬁnite photonic crystal waveguides,” Opt. Lett. 28, 854–856 (2003).

173 174 APPENDIX A. LIST OF AUTHOR’S PUBLICATIONS

[9] N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhe- dran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 10, 1243-1251 (2003).

[10] T. P. White, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Ultracompact resonant ﬁlters in photonic crystals,” Opt. Lett. 28, 2452–2454 (2003).

[11] N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540-1550 (2004).

[12] L. C. Botten, T. P. White, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Photonic crystal devices modelled as grating stacks: matrix generalizations of thin ﬁlm optics,” Opt. Express 12, 1592–1604 (2004).

[13] C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384–1386 (2004).

[14] T. P. White, C. M. de Sterke, R. C. McPhedran, T. Huang, and L. C. Botten, “Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferometers,” Opt. Express 12, 3035–3045 (2004).

[15] M. Sumetsky, T. P. White, H. M. Dyson, P. S. Westbrook, and B. J. Eggleton, “Interferometric characterization of nonﬂat transmission diﬀraction gratings,” IEEE Photon. Technol. Lett. 16, 2314-2316 (2004).

[16] L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part I: Theory,” Phys. Rev. E 70, 056606 (2004).

[17] T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part II: Applications,” Phys. Rev. E 70, 056607 (2004).

[18] P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, “Long wavelength anti-resonant guidance in high index inclusion microstructured ﬁbers,” Opt. Express 12, 5424-5433 (2004).

[19] T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Highly eﬃcient wide-angle coupling into uniform rod-type photonic crystals,” Appl. Phys. Lett. 87, 111107 (2005). 175 Book chapter

[20] L. C. Botten, R. C. McPhedran, C. M. de Sterke, N. A. Nicorovici, A. A. Asatryan, G. H. Smith, T. N. Langtry, T. P. White, D. P. Fussell, and B. T. Kuhlmey, “From multipole methods to photonic crystal device modeling,” in Electromagnetic theory and applications for photonic crystals, K. Yasomoto, ed. (CRC Taylor and Francis, Boca Raton 2006).