Towards Focusing Using Photonic Crystal Flat Lens

OPTO-ELECTRONICS REVIEW 14(3), 225–232

DOI: 10.2478/s11772-006-0030-0 Towards focusing using photonic crystal flat lens

N. FABRE*, S. FASQUEL, C. LEGRAND, X. MÉLIQUE, M. MULLER, M. FRANÇOIS, O. VANBÉSIEN, and D. LIPPENS

Institut d’Electronique de Microélectronique et de Nanotechnologie, UMR CNRS 8520 Université de Lille 1, Avenue Poincaré, BP 60069, 59652 Villeneuve d’Ascq Cedex, France

We report on the numerical simulation and fabrication of a two-dimensional flat lens based on negative refraction in photonic crystals. The slab acting as a lens is made of an hole array (operating at the wavelength of 1.5 µm) etched in a InP/InGaAsP/InP semiconductor layer. We first study the key issues for the achievement of a negative refractive index taking advantage of folding of dispersion branches with main emphasis in dispersion properties rather than the opening of forbidden gaps. The diffraction and refraction regimes are analysed according to the comparison of the wave-vector with respect to the relevant dimensions of the hole array. In the second stage, we illustrate technological challenges in terms of e-beam lithography on a sub-micron scale and deep reactive ion etching for an indium phosphide based technology.

Keywords: negative refraction, optical wavelengths, photonic crystals fabrication.

1. Introduction tures, special care has to be paid to the recognition of negative refraction effect. In fact, diffraction effects can modify Photonic crystals have received a tremendous interest [1] the direction of transmitted wave at an interface between with the prospect to use them to confine light or more gen- the PC and embedding medium with the appearance of a erally electromagnetic waves notably in 1.3–1.5 µm win- negative refraction. In other words, the angle of transmitted dow of telecommunication systems. It is well known that wave can be on the same half plane than the angle of inci- the periodical arrangement of dielectric inclusions, notably dence one due to diffraction and not to a real negative re- of dielectric rods or hole array opens forbidden gaps in fraction. One criterion which can be used in order to guar- Brillouin zone (BZ). Notably, high quality factor cavities antee that a true refraction is observed is the insensitivity of and planar waveguides have been thus fabricated using de- the corresponding refractive index to the angle of incident fected arrays paving the way of compact nanophotonics wave according to Snell-Descartes law. In BZ, this means [2]. Recently, a number of studies have also addressed the that equal-energy plots are isotropic. It will be shown that transmission properties of photonic crystals whose electro- this condition is met close to the centre of BZ namely at the magnetic properties can be tailored according to a proper vicinity of G point. choice of geometrical parameters and dielectric constants In addition, for the ground dispersion branch, the curva- [3]. This includes ultra refraction, channelling effect, and ture is such that group and phase velocities have the same negative refraction [4–5]. sign. The propagation is forward and the refractive index is In this paper, we address the latter issue with the goal to positive. In contrast, in the second band, the curvature is re- fabricate a flat focusing lens. Such a lens was primarily verse. It is thus possible to envisage the achievement of a proposed by Veselago in 1968 [6] on the basis of the as- negative refractive index with the concomitant effect of sumption of a negative refraction index material (NRIM) backward propagation in such media. Because of the rever- which composes the lens. Also recently, Pendry showed sal of phase velocity forming an indirect triplet with E that NRIM allows the amplification of evanescent waves (electric field) and H (magnetic field), NRIM’s are also within the lens with the prospect to image a point source termed left handed materials (LHM). below the diffraction limit [7–8]. Basically, two routes can Several studies have already addressed theoretically be followed in order to fabricate artificial NRIM’s, metallic and experimentally negative refraction in a flat lens. Nota- and dielectric, respectively. For the former, the term bly by taking advantage of the invariance of electromag- metamaterial or double negative media is often used. For netic concepts as a function of frequencies, scaled models the latter, this corresponds mainly to the use of photonic operating at microwave frequencies were assessed [10–11] k crystal even if high- dielectrics have been proposed for which demonstrate the possibility of focusing electro-ma- negative permeability materials [9]. For dielectric struc- gnetic waves. In optics, and more particularly at wavelengths around 1.5 µm, only a few experimental works can *e-mail: [email protected] be found in the literature owing mainly to the difficulty of

Opto-Electron. Rev., 14, no. 3, 2006 N. Fabre Towards focusing using photonic crystal flat lens fabrication and characterization of such prototypes. To the ogy and a targeted wavelength compatible with telecom- best of our knowledge, they are two key publications which munication systems at 1.5 µm. It will be shown that two showed experimentally focusing effect in a photonic crys- major tasks have to be conducted in connection with the tal slab either in InP-based technology [12] or making use e-beam patterning of the hole array and ridge-type wave- of silicon-based devices [13]. guide on a sub-micron scale and on the other hand with the The originality of the present work concerns on one deep reactive ion etching holes by inductively coupled hand the design of the structure which incorporates not plasma (ICP). only the photonic crystal slab itself but also a feeding wave guide. In particular, we designed it with a taper/diffracting 2. Numerical simulations hole scheme which permits us to mimic a point source while preserving a TM polarization for which focusing by 2.1. Band structure calculation negative refraction is found effective. Special care was also Figure 1(a) shows a schematic of the triangular lattice paid to the index matching issue with extensive which was chosen for the fabrication of hole array targeting equifrequency calculations from which the effective index an operation at 1.5 µm. Hole diameter is 356 nm, the pitch is deduced. In practice, this also permits us to alleviate the of the array is 496 nm. As above mentioned band structure, spurious reflecting effect at the input of the device due to calculations were carried out assuming a two-dimensional mismatch. Cavity edge effects were also avoided in the crystal with an effective refractive index of the epilayer output section taking benefit of the non invasive properties whose epitaxial sequence will be listed in Sec. 3. Typically, of SNOM probing technique. This is a definite advantage the effective refractive index value is 3.3. The main crystal with respect to a mapping via an array of output wavegu- directions are shown in Fig. 1(b), where G,K,andMare ides as it was used in both references. the high symmetry points which define the reduced BZ. On the other hand, from the fabrication viewpoint two Figure 2(a) shows band structures which were calcu- novelties were introduced with respect to the previous cited lated by means of ‘Bandsolve’ software commercialised by works. Patterning was carried out by means of a negative Rsoft for TE and TM polarization, respectively, for the HSQ resist. It appears that such a resist with respect to three main crystal directions, namely GM, GK, and MK. more conventional positive PMMA resists brings several For calculations, the filling factor is 0.47. In agreement advantages in terms of resolution and easiness in fabrica- with the issues discussed in introduction, addressing the tion notably with the prospect to avoid troublesome transfer techniques. Fabrication of holes with a high aspect ratio around 10:1, with high quality side and bottom surfaces, was also successful in terms of size accuracy and shape by optimizing the reactive ion etching resulting from the high density of plasma in ICP techniques. Section 2 deals with the numerical simulation of the effect of propagation and of negative refraction. The first stage was to calculate the band structures for TE and TM polarization for a triangular two-dimensional photonic crystal. On this basis, we discuss on left handed dispersion branch and isotropy criterion. In a second stage, we will illustrate negative refraction with two kinds of numerical experiments. The first one is an analysis of refracted waves in a prism-like structure composed of a photonic crystal lattice. The second one will be the simulation of focusing of electromagnetic waves in a flat lens in the time domain. These studies are currently carried out in order to design a prototype which was fabricated at IEMN and will be char- acterized by near field optical microscopy at the University of Bourgogne (France). As a consequence, beyond the wave propagation in NRIM, we also studied the feeding effect via the use of ridge-type waveguide with a diffracting hole at its end. Focusing will be shown numerically under strict two-dimensional conditions. Therefore we assumed that the etch depth of holes is infinite. We will discuss on the validity of this assumption in the frame of some calculations we performed recently on waveguiding structures. In Sec. 3, the technological issues will be more spe- Fig. 1. Schematic of the lattice of hole array (a) and of main crystal cially addressed by choosing an indium phosphide technol- directions (b).

dex can be deduced from this figure. It can be shown that n = kza/(4ðù) where kza can be the radius of circles which fits the variations of iso-frequency contours and kz is the coordinate of k vector which corresponds to the angular frequency w. At the targeted wavelength of operation, the reduced angular frequency is 0.32, kza = 3.8 and n = –0.94. Figure 2(b) shows the band structure for a square lattice. The same conclusions can be drawn. Some information which can be deduced from band structure calculation is the transition between the frequency Fig. 2. Band structure for a triangular lattice of holes for TE (blue dots) and TM polarisation (red dots) in the main directions of the band which corresponds to a guided mode and the spectral first Brillouin zone (GM, MK, and KG) calculated using Bandsolve region where the waves are leaky. In the latter situation, the from Rsoft. Rsoft convention: TE mode corresponds to E parallel waves are coupled to free space and thus, this transition is to air cylinder, resp. TM mode is H parallel to air cylinder (a), Same defined as a function of the position of dispersion branches representation for a hole array with a square lattice (directions GX, with respect to the so-called light line (w = ck) also termed XM, and MG for the first Brillouin zone). Light line is light cone in space. It is clear that such a leakage of wave superimposed in grey (b). cannot occur for the ground states since the dispersion branch is located below the light cone. In contrast, we can transmission properties, it can be noticed that almost a zero observe a cross-over between the second dispersion branch gap for TM polarization (electric field parallel to hole sym- and the light line which also corresponds to the occurrence metry axis) is achieved when considering the whole BZ. of a superluminal regime (phase velocity exceeds in this The ground band (labelled zero in the numerical code) has case light speed). At the chosen wavelength of 1.5 µm, we a conventional curvature in the GX and GK direction. Dis- are just below the light line and thus, it is expected that persion is low with a positive index which reflects the fill- such leaky waves will be avoided. It is worth mentioning, ing factor. Some reversal of the slope (negative refraction) however, that larger the wavelength with respect to the di- exists in the third crystal direction but with strong aniso- mension, the better resolution in an imaging system. In- tropy effect as shown in Ref. 9, published by our group. In deed, the most favourable situation, with respect to the res- contrast, the upper band exhibits a reverse curvature with olution issue with the prospect of sub-wavelength imaging the expectation of a mono-mode backward propagation re- would correspond to the, so-called, long wavelength regime [14]. gime. Such a denomination is often used in order to charac- Figure 3 shows the iso-frequency plot calculated for the terize metamaterials. It can be seen that photonic crystal second band (labelled 1). The frequencies are here normal- could work in the refraction regime (lg >> a) or in the dif- ized with respect to a/l where a is the pitch of hole array fraction regime (lg ~a). Negative refraction could be ob- (450 nm) and l is the wavevector in free space (l =2pc/w served, in principle, in both cases provided that isotropy with the speed of light in vacuum c). Close to centre of BZ, criterion is satisfied as discussed previously. the iso-frequency contours show round-shaped configura- In the following, we will illustrate negative refraction tions whereas they can be compared to hexagonal forms at with two typical generic situations. They can be considered the boundaries of BZ. Quantitative value of refractive in- as numerical experiments. Indeed, this kind of simulation is

Opto-Electron. Rev., 14, no. 3, 2006 N. Fabre 227 Towards focusing using photonic crystal flat lens not restricted to the calculation of eigenstates of photonic dex matching is the fact that the waves are moderately system but also takes into account other important issues. stored in the PC structure. Under continuous injection, a This is notably the case of the matching issue which is of standing wave pattern is apparent but with a limited stand- major concern for many potential applications. Ideally, 3D ing wave ratio. simulations should be carried out as it was the case in the recent paper we published on loss in photonic crystal 2.3. Flat lens waveguides [15]. Such 3D numerical simulations are very time consuming while the underlying physical effect can be One of the key issue for the demonstration of lensing via a PC pointed out by considering solely a 2D system. slab is the problem related to the light excitation of nano- structures. We learnt from numerical simulations that focus- 2.2. Prism like numerical experiment ing properties are sensitive to light polarization and to the po- sitioning of the equivalent source in front of the lensing de- Figure 4 shows the result of a simulation for a prism-like vices which require accurate control. Therefore instead of a fi- structure. We used here a square lattice with an equivalent ber excitation, as it was implemented generally, it was thus pitch having in mind that PC’s considered so far are de- decided to integrate the feeding source by means of a wave- fined with a triangular lattice. In practice, the propagation guide fabricated at the same time than the PC slab. Two of electromagnetic waves is studied by means of the FDTD waveguide configurations were envisaged, using a taper sec- code Fullwave by Rsoft. The incident wave is injected on tion (Fig. 5) or a straight guide (Figs. 6 and 7). the left hand side of prism at a wavelength of 1.5 µm. Figure 6 shows the results for a PC slab with different Within the prism-like nanostructure, it can be seen that the snapshots in order to illustrate dynamics of phenomena un- guided wavelength is larger than the relevant dimension of der transient conditions. Time spacing between each snap- nanostructure. By marking the distance between two nodes, shot is 10 fs. For the feeding, we used a cut-ridge guide an estimate of the guided wavelength is 5 µm, which has to with a width of 700 nm apparent at the bottom of figures. be compared to the relevant dimension of lattice (hole di- The first increment of time illustrates mainly the reflection ameter 200 nm, period 300 nm). The incident wave, in- of wave on the first interface of the PC slab. This gives rise jected perpendicularly to the first interface does not suffer to a complex pattern in the vicinity of the injection region diffraction effect and keeps the same direction that the one with a modulation of magnitude of electric field values of of incident wave. When the wave is impinging onto the sec- the order of the free space wavelength (l = 1.5 µm). The ond oblique interface, the wave is strongly refracted. Both second snapshot [Fig. 6(b)] illustrates mainly the formation travelling wave within the photonic crystal and refracted of a first focus inside the lens as predicted in the seminal beam have k directions in the same half plane defined by the work of Veselago. As expected, the guided wavelength is normal at this interface (dashed line in this figure). considerably reduced in the lens on a scale now compara- A rough estimate of the refraction angle by applying ble to the structuring dimension. Snell-Descartes law indicates that the effective refractive Figures 6(c) and 6(d) show how the second focus is index of this structure should be close to the ideal value formed with relatively concentric iso-phases. Figure 6(e) n = –1 (the same magnitude of incident and refractive angles with opposite sign). Another proof of a reasonable in-

Fig. 4. Simulation of negative refraction in a prism experiment using a square lattice. 2D-plot of the real part of Hx. An incident Fig. 5. Simulation of a tapering waveguide to mimic a point source. wave is defined on the left-hand side. Propagation in the crystal, A circular wave front is clearly generated by the diffracting hole along x axis. etched at the end of the guide.

228 Opto-Electron. Rev., 14, no. 3, 2006 © 2006 COSiW SEP, Warsaw Fig. 6. Snapshots illustrating focusing effect in a PC flat lens. Time spacing between each time step is 10 fs. corresponds to quasi-steady state conditions. At this stage, 3. Technological issues it seems interesting to see the information which could be drawn from ray-tracing as it was made in the original The general scheme of fabrication of the lens whose prop- work of Veselago. Figure 7 illustrates such an approach erties have been discussed in the previous section consists on the basis of the general trend of iso-phases. Despite the of three major stages: fact that colour contrast was attenuated in order to stress • growth of the epilayer by molecular beam epitaxy, ray tracing, it can be shown that the positions of various • nanoscale patterning by e-beam lithography, focus are in agreement with Snell-Descartes law for a re- • fabrication of hole array along with ridge waveguide by fractive index which is negative with a magnitude slightly inductively coupled plasma (ICP) etching. below the relative value of embedding region (n = 1). An order of magnitude would be –0.9 but it seems, however, 3.1. Epitaxial growth difficult to really assess quantitatively the negative value of the refractive index of the PC lens considered here by Figure 8 is a cross section of 2D confinement layer which this method. is imperative to grow prior to the fabrication of the lens. The quaternary layer is used to confine light at 1.5 µm in a direction parallel to the growth direction and thus, transverse to propagation direction. We choose the concentration of various elements [Ga (27%), In (73%), As (60%), P (40%)] so that, the material is transparent at the wavelength of 1.5 µm and for lattice matching conditions. The thickness of this layer is 500 nm so that, monomode propagation can be expected in the frequency range of interest. The capping InP layer has a thickness of 200 nm. The confinement layer is thus relatively shallow with the goal to reduce, as far as possible, the depth of holes which will be subsequently etched. Indeed, a free surface with a high index contrast (nInP/nAir = 3.3) permits one to limit dramatically the spreading optical mode above

Fig. 7. Illustration of ray tracing. Using Gauss approximation of small angles, a negative refractive index close to –0.9 can be Fig. 8. Epitaxial sequence for transverse confinement on InP deduced from the source/lens and lens/focal point distances. substrate. The lateral confinement is ensured by the InGaAsP layer.

Opto-Electron. Rev., 14, no. 3, 2006 N. Fabre 229 Towards focusing using photonic crystal flat lens the quaternary layer. In contrast, the slight index difference between the confinement layer and the buried InP layer yields a large evanescence of waves toward the substrate and special attention has to be paid to the tail of optical mode in the substrate [15]. From the material point of view, because the quaternary confinement layer, as well as the capping layer content phosphorus element, we grew the epilayers by means of a gas source MBE system (Riber 32P).

3.2. Electron beam lithography

Electron-beam lithography is a necessary stage in order to pattern the, so-called, transfer layer. In fact, as seen above, a deep highly anisotropic etching (aspect ratio of 10:1, typically) is performed in the next stage. It requires a mask Fig. 9. Illustration of mask transfer using e-beam lithography. which can resist to the chemical etching by ions. This is the key role of this layer. In practice, we chose Si3N4 film and high resolution lithography is used to transfer, into this di- 3.3. Inductively coupled plasma etching electric film, the footprint of the array of holes with a low tolerance about the possible change in shape and dimension The last stage in fabrication of flat lens is the deep etching such as oversize of holes. of holes. Over the past for this task, we used a conventional In previous works [2], we used PMMA (poly-methyl- reactive ion etching (RIE) with two main limitations. The methacrylate) resists. For the present work, hydrogen silse- first one was the rate of etching which was relatively low squioxane (HSQ) was used. It is a negative resist (removal with respect to the depth. Indeed, as mentioned in Sec. 3.1, of the non exposed regions) with the prospect to a high res- the tail of the wave-function is important towards the sub- olution, owing to high contrast [16–17]. In addition, we strate. This can give rise to substrate propagation mode if could expect to use it directly as a mask. However, here we the depth of hole is insufficient for the confinement of the kept the Si3N4 layer transfer whose flow chart was outlined optical mode in the transverse direction. Numerical simula- in the previous section. tions indicate that a hole depth of about 2 µm is sufficient One of the key issue in patterning of hole array, or more for good confinement. With respect to the hole diameter of generally of photonic crystal for nanophotonics, is the correc- the order of 356 nm, this means that etching has to be con- tion of proximity effects. In short forward and back, scatte- ducted not only on a sub-micron scale but also with an as- rings spread the initial bunch of electrons. As a consequence, pect ratio of the order of 10:1. The second disadvantage of there is an increase in the zone which is written in practice RIE is the formation of carbon polymer during etching. We with respect to the initial diameter of the electron spot. One solved this problem by several oxygen plasma etchings solution in order to correct such enlarging of dimensions during the process. could be to undersize the relevant dimension but we preferred In order to increase the etching and also to solve the to use a multi-dose process on the basis of a simulation of problem of polymers, which act as micro-masks, we thus electrons diffusion in the layered substrate constituted notably decided to develop etching by inductively coupled plasma by the stacking of (400 nm-thick HSQ/350 nm-thick Si3N4 (ICP) using an Oxford equipment (plasma lab system 100) film and 200 nm-thick InP semiconductor layer). These simu- [18–19]. ICP power was 500 W, for an RF power of 75 W, lations first start from the modelling of diffusion effects by a pressure of 4 mTorr and a table temperature of 20°C. We means of a Monte Carlo method. To this aim, we used the used the following gas: CH4 (7 sccm), H2 (8 sccm) and Cl2 commercial software Skeleton by Synopsys. We then use the (8 sccm). After completing the etching (5 min duration), data of spreading electron beam in order to calculate a proximity the Si3N4 thickness was measured to be 190 nm for an ini- function at various depths in the resist layer. This function per- tial thickness of 350 nm. This means that the etching rate of mits us to define the different doses which are used to respect, as Si3N4 is around 30 nm per minute. In contrast, a depth of 2 far as possible, the targeted dimensions of hole array. This is per- µm was achieved for InP layer which yields an etching rate formed using the software Proxecco by Synopsis too. of 400 nm/minute. We thus obtained an etching selectivity Electron writing was carried out using high-resolution of 13:1. pattern generator equipment (EBPG LEICA 5000) with a Figure 10 shows a scanning electron micrograph of the voltage of 50 kV. Sixteen or even thirty two doses are used completed device. There is no micro-masking effect with for the correction of proximity effects. Figure 9 is a scanning grass-like pattern. The sidewalls of the ridge guide are very electron micrograph of the edge of a flat lens in front of a sharp and well-defined along with the hole array. Figure 11 feeding ridge waveguide. It can be noticed, at the end of this is a close view of an edge of flat lens. The roughness of guide, a diffracting hole whose topology was engineered by side wall is good along the round-shape and dimension of numerical modelling to create a quasi point source. holes is 350 nm.

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