From Classical to Quantum Devices By

Nonlinear periodic structures: from classical to quantum devices

Peyman Sarraﬁ

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto

Nonlinear periodic structures: from classical to quantum devices

Peyman Sarraﬁ Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2014

In this thesis, nonlinear periodic structures and their applications in both classical and quantum regime are investigated. New theoretical models are developed, and novel applications of nonlinear periodic structures are proposed and demonstrated. The theoretical studies, both design and simulation, are based on but not limited to InGaAsP material. A new method, namely the time-domain transfer-matrix (TDTM), is presented to simulate optical pulse propagation in layered media with resonant nonlinearity. As there were no satisfactory methods in the literature to model this problem, in order to validate and compare with the TDTM method, the standard FDTD method is generalized to include the rate equation in the analysis of semi-conductors. Also in this work, optical manipulation of absorption in periodic structures is studied for the ﬁrst time. Thanks to the large accessible nonlinearity that results from the absorption saturation and frequency selectivity of periodic structures, a sensitive and compact optical limiter is designed. The novel design and modeling work developed in this thesis has provided new insights and tools to the utilization of resonant nonlinearities in compact all-optical devices. The experimental studies are based on quasi-phase matched AlGaAs superlattice waveguides. These devices have been previously designed and used for classical optical wavelength conversion such as second harmonic generation and diﬀerence frequency generation. In this work, these devices are exploited for spontaneous down conversion,

ii which is a quantum eﬀect, for the ﬁrst time, through this process, entangled photon pairs are generated. Unprecedented performance, in terms of brightness and purity, of III-V semiconductor-based entangled photon sources has been demonstrated here. Moreover, the quantum properties of these entangled photons are characterized. The experimental studies presented in this thesis open up new application areas for III-V nonlinear optical devices as quantum sources and convincingly demonstrate the promising role of such devices will play in future quantum technologies.

iii Contents

Abstract ii

Contents iv

List of Tables vii

List of Figures x

1 Introduction 1 1.1 Nonlinear periodic devices ...... 2 1.1.1 Periodicity and non-phase matched ...... 3 1.1.2 Quasi phase matching ...... 4 1.2 Scope of this thesis ...... 5

2 Modeling periodic nonlinear structures 7 2.1 Finite-diﬀerence time-domain method ...... 9 2.2 Generalized time domain transfer matrix method ...... 12 2.2.1 Formulation ...... 12 2.2.1.1 Pulse Decomposition ...... 13 2.2.1.2 Time-Domain Transfer-Matrix ...... 15 2.2.1.3 Updating process ...... 19 2.2.2 Numerical result and comparison ...... 21 2.2.2.1 Linear and Nonlinear Responses of a 200fs, 800fs, and 3.2ps Gaussian Pulses ...... 22

iv 2.2.2.2 Nonlinear Responses of 50fs Incident Pulses ...... 25 2.2.2.3 Comparison between the two methods of TDTM: FT- TDTM vs. TDOTM...... 27 2.3 Conclusion ...... 28

3 Application of layered structure with resonant nonlinearity in optical limiting 29 3.1 Traditional optical limiting methods ...... 30 3.1.1 Reverse saturable absorber ...... 31 3.1.2 Two-photon absorption (TPA) ...... 31 3.1.3 Free-carrier absorption ...... 32 3.1.4 Nonlinear refraction ...... 32 3.1.5 Self-phase modulation (SPM) ...... 32 3.2 Resonant nonlinearity in InGaAsP ...... 33 3.3 Non-trivial phase shift in the interface of lossy material ...... 34 3.4 Compact optical limiter based on resonant optical nonlinearity in a layered semiconductor structure ...... 38 3.5 Conclusion ...... 44

4 Low noise on-chip source of correlated photons 45 4.1 Nonlinear optical methods for the generation of entangled photon pairs . 47 4.2 The importance of the low noise PPS ...... 50 4.3 Low noise entangled photon generation in AlGaAs superlattice Waveguides 51 4.4 Characterization of QPM AlGaAs superlattice waveguides ...... 53 4.5 Coincidence measurement ...... 56 4.6 Discussion and conclusion ...... 59

5 On-chip Time-Energy Entangled photon pair source 62 5.1 Franson Interferometry ...... 63 5.2 Franson interferometer: meeting constraints and overcoming challenges . 66 5.2.1 Matching the MZIs’ imbalance using reference ﬁber-based MZI . . 67

v 5.2.2 MZI characterization: phase-voltage relation ...... 68 5.2.3 High brightness and detector’s dead time limitation ...... 70 5.3 Time-Energy Entanglement and a measurement of visibility ...... 73 5.4 Discussion and conclusion ...... 78

6 Conclusion, contributions and future works 79 6.1 Conclusion and original contributions ...... 79 6.2 Future work ...... 81

References 83

vi List of Tables

4.1 Comparison with other AlGaAs based entangled photons sources . . . . . 61

vii List of Figures

2.1 A summary of the TDTM and updating process ...... 14 2.2 decomposition of the incident pulse to small subpulses ...... 15 2.3 Schematic plot of a spatial section in transfer matrix ...... 17 2.4 In a layered structure with resonant nonlinear material, each layer corresponds to the section illustrated in Fig. 2.3. d represents thickness, N the carrier density, n the refractive index, and α the absorption coeﬃcient. . 22

2.5 (a) Intensity linear response and (b) Intensity nonlinear response (Ipeak = 0.21GW/cm2) to a 200fs pulse. Intensities are normalized to the peak incident intensity...... 23 2.6 Normalized reﬂection and transmission intensity of a 800 fs pulse. (a)

2 linear response (b) Nonlinear response, (Ipeak = 0.21GW/cm ) ...... 24

2.7 Normalized reﬂection and transmission intensity of a 3.2ps pulse. (Ipeak = 0.21GW/cm2)...... 25 2.8 Normalized reﬂection and transmission intensity of a 50fspulse. (a) medium

2 intensity incident pulses (Ipeak = 0.21GW/cm ) (b) High intensity incident 2 pulse (Ipeak = 0.85GW/cm )...... 26 2.9 Nonlinear response to a 6.4ps pulse. (a) FT-TDTM is compared with TDOTM with 12 order polynomial (b) FT-TDTM is compared with TDOTM with 8 and 6 order polynomials ...... 27

3.1 A schematic of optical limiter eﬀect ...... 30

3.2 Absorption as a function of carrier density for In1−xGaxAs1−yPy semiconductor ...... 35

viii 3.3 Refractive index as a function of carrier density for In1−xGaxAs1−yPy semiconductor ...... 36 3.4 Schematic of incident light from medium 1 reflected from an interface with medium 2 ...... 36 3.5 The phase of reflection from a lossy medium for different refractive indexes in Fig. 3.4 ...... 37 3.6 Schematic of the proposed optical limiter consisting of a multilayered In- GaAsP structure (Type I and II alternating) sandwiched between two AR coatings. The periodicity of the InGaAsP is broken at the center, where two layers of Type II are next to each other. The bandgap energies of the Type I and II are designed to be lower and higher than the energy of the incident photons, respectively...... 40 3.7 In case of complete absorption saturation a dip appears in the reflection spectrum. The 20dB-bandwidth of this dip is approximately 4nm. . . . . 41 3.8 Exponential saturation of the reflected pulse versus incident peak intensity. 43 3.9 Reflected pulse normalized to incident pulse intensity for different incident pulse intensities...... 43

4.1 Energy level for SPDC and SFWM process ...... 49

4.2 The calculated band gaps of AlxGa1−xAs versus x ...... 52 4.3 A schematic of Quasi-Phase-Matched AlGaAs superlattice waveguides. Blue and pink beams have a wavelength around 775 nm and 1550 nm, respectively...... 55 4.4 (a) Simulated TE mode profile at 1550 nm (b) Simulated TM mode profile at 775 nm ...... 55 4.5 SHG power as a function of fundamental wavelength for a QPM waveguide with width of 2 µm and a QPM periodicity of 3.5 µm...... 56 4.6 Schematic of the setup for coincidence measurement in QPM AlGaAs superlattice waveguides. Blue lines are light propagating in free space, pink represents fibers ...... 57

ix 4.7 (a) The typical time-bin histogram for coincidence measurements (b) CAR as a function of coupled pump power...... 59 4.8 CAR versus pump detuning. Figure inset: shows the spectral location of the collected pair photons (hatch pattern) ...... 60

5.1 Model of a Franson interferometric measurement, D1 and D2 are single photon detectors...... 64 5.2 Schematic setup for comparing the path diﬀerence ...... 68 5.3 Interference pattern for (a) ﬁrst MZI and (b) second MZI as a function of delay in reference MZI...... 68 5.4 Schematic setup for MZI characterization ...... 69 5.5 Phase change in interference pattern as a function of voltage V=(0, 1.8, 3, 5, 6, 6.4, 6.5, 7, 7.5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18.6). Pink solid line shows the pattern for V=0 and as voltage increases, it shifts toward the right...... 70 5.6 Phase change as a function of voltage applied to the heater ...... 71 5.7 A time-bin histogram for a 2µm wavguide with 8 mW pum power. . . . . 72 5.8 The time-bin histogram for a 2µm wavguide with 16 mW pum power. . . 73 5.9 Schematic of the setup for Franson interferometry used to characterize the AlGaAs-waveguide-based photon pair source...... 75

o o 5.10 The result of coincidence measurements at (a) φ1 = 270 , φ2 = 180 , and o o (b) φ1 = 90 ,φ2 = 180 . The red bins are used to calculate the coincidences for Fig. ?? ...... 76 5.11 Two photon interference measurements for three phase settings on the second MZI.(a) Middle peak coincidences shown in Fig. 3 plotted as a

function of φ1 for three ﬁxed φ2 values. (b) Left and right peak coincidences o shown in Fig. 5.10 plotted as a function of φ1 for φ2 = 180 ...... 77

x Chapter 1

Introduction

While access to online information is much faster than what could be imagined not long ago, the need for faster communication and faster data processing is increasing. Still one of the ultimate goals of photonics is its incorporation into future supercomputers [1] to replace electronic processing at least in part; however, the scope of future all optical computing is still a controversial topic [2]. An optical telecommunication system where all the server-level-processes are done optically can in principle achieve ultrafast and low power operation [3]. However, this requires a new generation of novel optical components that can, among other things, memorize the data, perform Boolean operation, and mul- tiplex and demultiplex signals. Such a revolution in telecommunication is not possible without nonlinear optics [1]. Nonlinear optics includes the propagation of light in a nonlinear material. A nonlinear material is a material in which the induced polarization is not proportional to the propagating electromagnetic field that induces it. This material property is commonly used in two ways. First, it is the essential requirement for direct manipulation of a light beam by itself or by another light beam. In all-optical on-off switches or all-optical modulators, one lightwave controls the transmission (or reflection) of the other through nonlinear effects. Second, nonlinear effects have broad applications in optical frequency conversion used either to convert the data to a new frequency channel or build lasers in the spectral regions that no known material lase efficiently. Optical nonlinearity, as explained, is an interesting material property used in the de-

1 Chapter 1. Introduction 2 sign of novel signal processing components. Another degree of freedom in engineering optical devices is the design of their geometry. Periodicity is a well-studied geometrical property with applications in frequency selectivity, slow light propagation, and coupling between two lightwaves with diﬀerent propagation constants. A periodic structure consists of a repetition of identical structures, called unit cells, which are joined together. In the context of microwave and photonic devices, these unit cells have identical electromagnetic responses. They constitute a feedback system which results in wavelength selectivity. In other words, only those wavelengths which match the round trip optical length of unit cells can constructively interfere and pass through the structures. The intersection of the two areas referred to above, i.e. periodicity and nonlinearity, constitutes an exciting and fruitful domain called nonlinear periodic structures. Non- linear periodic structures have numerous applications in optical switching, and optical bi-stability, wavelength conversion, among many others. Nonlinear periodic structures are periodic structures in which the responses of the medium depends on the electromagnetic wave propagating in the structure and this dependency is also periodic. In the next sections, I present traditional nonlinear periodic applications and the new scope covered in this thesis.

1.1 Nonlinear periodic devices

Nonlinear periodic structures are at the interesting intersection between nonlinearity, which is a material property, and periodicity, which is a geometrical property. Theoretical studies of nonlinear periodic structures were initiated in the 1970s by Wilful, Garmire, et al. in the context of optical bistability [4], and by Yariv et al., in the context of nonlinear frequency generation through periodic-modulation-assisted phase matching [5]. The theoretical studies were followed by experimental demonstrations in the 1980’s [6, 7, 8], laying the foundation for all-optical devices based on periodic nonlinear structures. In this section, I discuss how periodicity can enhance the properties of nonlinear devices. In general, we can divide the applications of nonlinear devices into two categories: Chapter 1. Introduction 3 non-phase matched and phased-matched. In the first category, a lightwave has an effect on another through its manipulation of the material properties. In this case, the energy of one lightwave does not couple to the other one, and therefore it does not need to propagate with the same phase velocity. Applications of non-phase matched nonlinear optics include all-optical switching [9], passive mode locking [10], optical limiting [11], optical memory [12], modulation [13] and Boolean operations like AND and XOR gates [14]. In the second category, one or more lightwaves are mixed to create one or more lightwaves with different frequencies. Through this process, energy and momentum conservation have to be satisfied, and hence, phase-matching is essential to ensure efficient energy conversion. Frequency conversion has broad applications. Those like second harmonic generation (SHG) [15], sum frequency generation (SFG) [16] and difference frequency generation (DFG) [17] allow access to the new spectrum where direct lasing is not feasible. Another type of frequency conversion is four-wave mixing (FWM) where interactions between two wavelengths produce two extra wavelengths. Four-wave mixing [18] is a potential solution to inter-channel conversion in all-optical routing where no intermediate electronic circuit is required. Wavelength conversion has also its quantum counterpart such as parametric down conversion [19] and spontaneous four-wave mixing [20]. These have applications in the generation of entangled photons used in quantum cryptography [21] and quantum computing[22]. In all the above applications of nonlinear optics, periodicity can play an important role. In the next subsections, the advantage of using periodicity in the both categories of nonlinear devices is discussed.

1.1.1 Periodicity and non-phase matched

Here, the role of periodicity in non-phase matched nonlinear optics is brieﬂy described. Conventional nonlinear periodic structures usually consist of at least one material with Kerr nonlinearity (a third order nonlinearity). The eﬀective refractive index of these materials is the linear function of the wave intensity in the material. A periodic structure with Kerr nonlinear material can be interpreted as a tunable frequency selective device, because the resonance frequency is a function of the light intensity. Chapter 1. Introduction 4

This tunable frequency-selective structure can be used as an optical switch [23] or as an optical memory [12, 24], where another light determines the transmittance of the device at that time or for a period of time thereafter, respectively. In addition, in a nonlinear periodic structure, a beam can change its own transmittance through tuning the resonance frequency of the device like what happens in a bistable device [25] or an optical limiter [26]. A combination of both may happen, for example in creating an all-optical logic gate [27].

In all the above applications, thanks to the selectivity nature of periodic structures, low power functionality (higher sensitivity) is an advantage.

1.1.2 Quasi phase matching

As described in the previous section, periodic structures can increase sensitivity in the non-phase matched category of nonlinear optical devices. They have also been used for phase matching purposes in frequency conversion. An alternative for exact phase matching methods like birefringent phase matching and modal phase matching is quasi phase matching (QPM). This technique was first proposed by Armstrong et al. and it is based on a nonlinear periodic structure [28]. In this method, the phase mismatch resulting from the different propagation constants of two propagating beams is compensated periodically by forming a modulation on the optical nonlinear coefficient of the material and consequently increasing the interaction between the two beams. This is analogous to coupling between two counter-propagating beams in a grating structure. The former is a nonlinear grating the latter a linear one.

In various wavelength conversion schemes like SHG, SFG and DFG, phase matching plays a critical role for eﬃcient conversion and QPM has been widely used. QPM has been mainly used when the material does not have birefringence, and the transversal geometry of the waveguides is not a degree of freedom. Chapter 1. Introduction 5

1.2 Scope of this thesis

In this thesis, I study some novel applications of nonlinear periodic structures both in classical and quantum regime. In classical regime, a compact and low power optical limiter using resonant nonlinearity is investigated. In quantum regime, a entangled photon pair source is proposed.

In the first part, layered structures with resonant nonlinear material have been exploited. The novelty of this work lies in using interface phase shift instead of the conventional refractive index change. Resonant nonlinearity, contrary to the Kerr effect, is an accumulative type of nonlinearity. Resonant materials are well-known for their larger nonlinear index change and absorption saturation. Unfortunately, these materials are typically slow. However, advanced techniques in the growth of these materials can significantly reduce decaying time to a few picoseconds [29] or even subpicosecond [30].

However, simulating the resonant material is difficult due to their memory characteristics and complicated material dispersion. The simulation becomes more complicated when this resonant material is used in a periodic geometry. In Chapter Two, I propose two methods for modeling these types of structures. One method is a modification of the standard finite-difference time-domain (FDTD) and the second is a much faster method called a generalized time-domain transfer-matrix method. In the third chapter, I propose a compact and sensitive reflection-based optical limiter using a resonant material in a semi-periodic structure based on the realistic material parameter of InGaAsP. The novelty of the design is the use of absorption saturation to manipulate the interface phase shift and reduce the total reflection. This is a much larger effect and an alternative for using the Kerr effect for tunable filtering.

In the second part, I propose and demonstrate using nonlinear periodicity in III-V semiconductors to generate entangled photons. For the ﬁrst time, the QPM AlGaAs waveguides are used for this purpose. QPM AlGaAs waveguides have previously had applications in classical wavelength conversions such as SHG, and DFG [31, 32]. I chose these devices for entangled photon pair generation because of their unique features in light conﬁnement, which results in the reduction of photoluminescence noise. In Chapter Chapter 1. Introduction 6

Four of this thesis, the eﬀect of source noise in quantum communication is studied. The reason why the QPM AlGaAs waveguide is a good candidate for a pure photon pair source is explained. This claim is veriﬁed with a quantum coincidence measurement. In Chapter Five of this thesis, the time-bin entanglement property of the generated photon pairs is demonstrated. For this purpose, a Franson interferometer setup that meets the constraints for obtaining high visibility fringes is implemented. Experimental results show the violation of Bell’s inequality which implies its quantum entanglement property. Chapter 2

Modeling periodic nonlinear structures

Both resonant and non-resonant nonlinearities exist in semiconductors. Usually, non- resonant nonlinearities have small magnitudes but fast responses and resonant nonlinearities have large magnitudes and slow responses limited by carrier recombination time. Fortunately, material growth techniques have been developed to reduce the carrier recombination time down to the order of a few picoseconds or even sub-picoseconds [33, 29]. In comparison, a much larger number of devices have been conventionally designed using non-resonant nonlinearity [34, 35] than using resonant nonlinearity. One main reason is the difficulty in design and the lack of well-developed methods for the analysis of devices with resonant nonlinearity. Despite the difficulty, large nonlinearity in resonant nonlinear materials specially in III-V semiconducotrs like InGaAsP [24] and AlGaAs [36] has attracted broad attention. There are several difficulties in the analysis of devices with resonant nonlinearity. First, as the mechanisms that induce such nonlinearities can be complex, an analytical relationship between the induced nonlinear polarization and the electric field may not exist. Second, resonant nonlinearity is not instantaneous, as it depends on the history of the input excitation field more than on its current value. As a result, the frequency dependence of the nonlinear susceptibility does not have an analytical form. For the above reasons, it is conventional [37, 38] to model nonlinearity in the time domain, i.e.,

7 Chapter 2. Modeling periodic nonlinear structures 8 use rate equations to calculate the carrier densities and then relate carrier density to changes in the real and imaginary parts of the susceptibility (i.e. loss and refractive index, respectively) phenomenologically. However, time-domain methods are in general not the best candidates to model dispersion, which is important for materials with resonant nonlinearities, as they are very dispersive, especially near the resonant frequency. A third difficulty arises when such nonlinear materials are employed in layered structures, for example in order to increase sensitivity and reduce the switching energy of the device. Conventional modeling approaches, such as the Finite Difference Time Domain (FDTD) method [39] or the Split-Step Fourier (SSF) method [40], do not accurately model layered structures with high index contrasts and abrupt variations, due to either the high numerical discretization errors in the FDTD method [41] or the violation of the weak dielectric perturbation condition required by the slow-varying envelope approximation (SVEA) [42] in the SSF method. As neither standard FDTD nor SSF is adequate to model high-index-contrast layered structures with dispersion and resonant nonlinearities, a new method is required to fill in this gap in optical device modeling.

Another motivation for developing a new model derives from the consideration of the (unnecessary) computation burden of a standard FDTD method in modeling optical devices where the pulse envelope is orders of magnitude longer than the optical cycle. FDTD is a widely used method for modeling microwave and optical devices, to the extent that it has become a standard approach upon which many commercial simulation software packages are based. Its time-domain approach also appeals to the problem at hand, as it is straightforward to include rate-equations in its formalism [43]. However, FDTD introduces numerical inaccuracies (including but not limited to numerical dispersion), particularly for layered structures with abrupt and large index changes. To reduce numerical dispersion in FDTD, we need high resolution discretization in space, which in turn requires high resolution discretization in time (small time steps) in order to satisfy the stability condition [41]. Hence, the computational burden is considerable when we model device interactions with long pulses (thousands of optical cycles), which is almost always the case for optical devices. One way of reducing computational time by overcoming the stability limitations is by applying the ADi-FDTD [44, 45]. For propagation Chapter 2. Modeling periodic nonlinear structures 9 of optical pulses with long and slow-varying envelope pulses, the Envelope ADi-FDTD is much more beneficial [46] since it allows us to use much bigger FDTD time steps by modifying the stability condition of Envelop FDTD. However, it is very difficult or currently impossible to find its stability region for dispersive lossy materials [47], and materials with resonant nonlinearity are usually of this category of materials.

In short, due to the aforementioned diﬃculties, no satisfactory method has been presented to simulate optical pulse propagation in layered media having resonant nonlinearity.

In the next two sections, first the conventional finite-difference time-domain method for the analysis of structures with resonant nonlinearity is explained and then my method, which is the generalized time-domain transfer-matrix method, is introduced.

The material of this chapter was partially published in IEEE journal of Quantum Electronics [48].

2.1 Finite-diﬀerence time-domain method

The aim of this section is to analyze a one-dimensional (1D) structure with resonant nonlinear material using the FDTD method. Despite the mentioned difficulties in the introduction of this chapter, we can still add one extra equation to the conventional FDTD (without modifying the stability conditions) and simulate the dynamic interaction of optical pulses with nonlinear resonant structures. The stability equation will remain untouched to avoid the above-mentioned complexity. Consequently, analysis of long pulses (thousands of optical cycles) would have a high computational burden as explained above. I develop this modified FDTD method to later verify the accuracy of the results generated by my proposed method, which I present in the next section. Wave propagation along the z axis in a 1D structure with resonant nonlinear material has to obey both Maxwell’s equation and the rate equation. Maxwell’s equation in a 1D structure is simplified to: dE 1 dH x = − y − (J + σE ) (2.1) dt ε dz sourcex x Chapter 2. Modeling periodic nonlinear structures 10

dH 1 dE y = − x − (M + σ∗H ) (2.2) dt µ dz sourcey y

where Ex, Hy, ε and µ are the electric ﬁeld, magnetic ﬁeld, permittivity of medium, and permeability of the medium, respectively. σ is electric conductivity, and σ∗ is equivalent magnetic loss. A classical rate equation model for a resonant nonlinear material is as following: dN(t,z) = − N(t,z) + α(N(t,z),ω0) × I(t, z) dt τ ¯hω0 (2.3) 2 I(z, t) = ncε0|E(z, t)|

where N, τ, α,h ¯, ωo,n, c and ε0 are carrier density, carrier lifetime, absorption coefficient, reduced plank constant, pulse career angular frequency of pulse, refractive index, and speed of light in vacuum, respectively. In this equation, both α and n can be complicated functions of carrier density and frequency. In this thesis, the Bányai-Koch model [49] is used to formulate this relation for semiconductors. The rate equation model may not be adequate if the pulses are ultra short (10s of femto seconds or less). In such R α(N(t,z),ω) cases, the last term of the above equation may be replaced with ¯hω ×I(t, z, ω)dω to treat the absorption of the different frequency components of the field separately. We avoid such a frequency-time representation of the equation because, first, such a representation of the rate equation may not be valid for femtosecond pulses, and second, it highly increases the complexity of the mathematical problem.

∗ I assume σ , Msourcey and Jsourcex are zero, which is true in passive semiconductors. Using Yee’s centered ﬁnite diﬀerence expressions [50], (2.1), (2.2), and (2.3) are approximated as following:

E |n+1/2 − E |n−1/2 n n x i+1/2 x i+1/2 1 Hy|i+1 − Hy|i n n = − n . − σ|i+1/2E|i+1/2 (2.4) ∆t ε|i+1/2 ∆z

n+1 n n+1/2 n+1/2 ! H | − H | 1 Ex| − Ex| y i+1 y i+1 = − . i+3/2 i+1/2 (2.5) ∆t µi+1 ∆z

N|m+1 − N|m−1 N|m α(N|m ), ω ) i+1/2 i+1/2 i+1/2 i+1/2 0 m 2 0 = − + × (ncε0) (I|i+1/2) (2.6) ∆t τ hω0

n where ui,j,k = u(i∆x, j∆y, k∆z, n∆t) and u represent Ex, Hy, ε, σ or N. ∆t and ∆z Chapter 2. Modeling periodic nonlinear structures 11

are known as the time increment and lattice space increment for Maxwell’s equation. I have chosen a different time increment, ∆t0, for the rate equation, as the rate equation needs to be updated much less frequently than the field. To match the actual time in both equations, m∆t0 has to be equal to n∆t so the finite difference rate equation is only

∆t updated for those ns that m = ∆t0 n are integers. In these time steps, ﬁrst N is updated then ε and µ are updated using the B´anyai-Koch model and the Kramers-Kronig relation,

n respectively. E|i+1/2 is estimated using a semi-implicit approximation as follows:

n+1/2 n−1/2 Ex| + Ex| E|n = i+1/2 i+1/2 (2.7) i+1/2 2

n+1/2 n+1 m+1 Isolating Ex|i+1/2 , Hy|i+1 , and N|i+1/2, (2.4), (2.5) and (2.6) are reformulated so these equations can be implemented in a recursive algorithm:

n+1/2 n n−1/2 n n n Ex|i+1/2 = Ca|i+1/2Ex|i+1/2 + Cb|i+1/2. Hy|i − Hy|i+1 (2.8)

n+1 n n−1/2 n n n Hy|i+1 = Hy|i+1Ex|i+1/2 + Cb|i+1/2. Hy|i − Hy|i+1 (2.9)

0 0 0 m×∆t /∆t+∆t /∆t m+1 m−1 ∆t m ∆t m X n 2 N|i+1/2 = N|i+1/2 − ×N|i+1/2 + ×(α((N|i+1/2), ω0)ncε0) E|i+1/2 τ hω0 n=m×∆t0/∆t+1 (2.10)

where Ca, Cb, and Db are:

n n n n n Ca|i = (1 − σi ∆t/2/εi )(1 + σi ∆t/2/εi ) n n n n Cb|i = (∆t/∆z/εi )(1 + σi ∆t/2/εi ) (2.11) n Db|i = (∆t/∆z/µ)

Electric and magnetic ﬁelds evolve after each small time step ∆t but the coeﬃcients

0 Ca, Cb, and Db are updated only after each bigger time step ∆t , which saves some time without any loss of accuracy. Chapter 2. Modeling periodic nonlinear structures 12

2.2 Generalized time domain transfer matrix method

The FDTD method presented in the previous section is a time-consuming method and it can not easily deal with a non-standard model of dispersion. In this section, I adopt and generalize a Time-Domain Transfer-Matrix (TDTM) method for modeling pulse propagation in a resonant nonlinear layered structure. I demonstrate that this method is computationally efficient and can easily include dispersion. Although a TDTM method has been previously applied to model Semiconductor Optical Amplifiers (SOAs) [51], the formulation in [51] only considered pulse propagation in a homogeneous material without modeling reflections. The resulting matrices are much simpler, but the method in [51] has very limited applications due to its omission of reflections. In fact, including multiple reflections into the TDTM formalism is not trivial, as is evident in the following sections. By generalizing the TDTM method for layered media, I develop a tool for modeling high-index-contrast layered structures with a non-analytical form of dispersion and slow resonant nonlinearities for the first time.

2.2.1 Formulation

I consider a one-dimensional layered structure consisting of materials that could pos- sess carrier-induced nonlinearity, such as GaAs, InGaAsP, etc. High-contrast refractive indices, which can be obtained by varying material composition during growth, are modeled. The incident pulse has a carrier frequency near the band-edge of some (or, in some cases, all) of the layers, resulting in large, resonant, carrier-induced nonlinearities in these layers. The electric ﬁelds of the transmitted and reﬂected pulses are calculated using the generalized TDTM method, as detailed below. The starting point of our time-domain approach is to treat this nonlinear problem as a piece-wise linear problem temporally: the input pulse is decomposed into a temporal series of subpulses. We let each subpulse propagate through the entire structure one by one, modeled by the linear transfer-matrix method. After the propagation of each subpulse, the property of the material, i.e., the real and imaginary parts of the nonlinear susceptibility for each segment of the device, is updated. The whole process is Chapter 2. Modeling periodic nonlinear structures 13

summarized in the ﬂowchart Fig. 2.1.

2.2.1.1 Pulse Decomposition

The incident optical pulses can be expressed using a slow-varying, envelope function, E˜(t) , multiplied by a carrier at the optical frequency, as expressed in

E(t) = E˜(t)eiω0t + E˜∗(t)e−iω0t (2.12) where E(t) is bounded between t=0 and t=T. Our model works with the slow-varying envelope rather than the electric field of the pulse, which significantly decreases the numerical computation time compared to the FDTD method. Pulse decomposition means that we divide the input pulse, expressed by Eq. 2.12 into shorter excitations (subpulses), which are then analyzed sequentially. To limit the spectral breadth of the subpulses, we need to avoid introducing abrupt temporal changes when decomposing the input pulse. To achieve this, we multiply the pulse with bounded and shifted cosine square functions so that the original pulse envelope is the sum of M semi-cosine functions that have smooth but bounded profiles, each representing a time- shifted subpulse Fig. 2.2:

M−1 ˜ X ˜ E(t) = Em(t) (2.13) m=1 where

T πM Y T M E˜ (t) = cos2 (t − m × ) × × (t − m × ) × × E˜(t) m M 2T M 2T

 Y  1 if |x| < 0.5; (x) ≡  0 if |x| ≥ 0.5.

The required temporal resolution of the decomposition depends on several factors: (1) the rate of change of the pulse envelope; (2) the rate of change of material nonlinearity; and (3) the frequency response range used to model the material. The higher the rate of Chapter 2. Modeling periodic nonlinear structures 14

Figure 2.1: A summary of the TDTM and updating process Chapter 2. Modeling periodic nonlinear structures 15

Figure 2.2: decomposition of the incident pulse to small subpulses change of the pulse envelope or the material nonlinear response, the higher the temporal resolution of the pulse decomposition (i.e., larger M), and the larger the frequency content of the excitation pulses. However, the frequency content must be kept within the range of the frequency response used in the model, as is explained in the next section.

2.2.1.2 Time-Domain Transfer-Matrix

After the input pulse is decomposed into subpulses, I model the propagation of each subpulse through the material, and its eﬀect on the material’s nonlinear property, using the Time-Domain Transfer-Matrix (TDTM) method. There are two variants of the TDTM method, and I present both in the following two sections. Fourier-Transform-based Time-domain Transfer Matrix Method (FT-TDTM) The Transfer Matrix(TM) method used in the frequency domain is a common method for analyzing layered structures. Material dispersion can naturally be included in the frequency-domain. In order to keep the advantage of easy inclusion of dispersion in the frequency-domain while carrying out the rest of the calculations in time-domain, I use Chapter 2. Modeling periodic nonlinear structures 16

Fourier Transform (FT) and inverse Fourier Transform (FT −1) to convert the formulation from the time domain to the frequency domain, and then vice versa, for each step of the matrix multiplication. This approach of using FT and FT −1 to switch between time and frequency domains is also found in the Split-Step Fourier method, but my treatment of nonlinearity is completely diﬀerent from that of SSF. The procedure of FT-TDTM is as follows: like all transfer matrix methods, I divide the structure into several sections, each with a constant refractive index. I can then express the relationships between the forward and backward propagating waves analytically, using the boundary conditions. As shown in Fig. 2.3, each spatial section (including one of the boundaries) is represented by a matrix, T j, and the relationship between the ﬁeld envelopes of the propagating waves in neighboring sections (indicated by the solid-line arrows) is given as

        ˜j,+ −ik(∆ω+ω0)L ˜j,+ Em (ω) e 0 A(∆ω + ω0) B(∆ω + ω0) Em (ω)   =       , ˜j,− +ik(∆ω+ω0)L ˜j,− Em (ω) 0 e C(∆ω + ω0) D(∆ω + ω0) Em (ω) | {z } | {z } Propagation Boundary | {z } T j (2.14) and consequently the relation of any section with the ﬁrst one in the time domain is:

     ˜j,+ j ! ˜0,+ Em (t) −1 Y k Em (t)   = FT  T FT   , (2.15) ˜j,− ˜0,− Em (t) k=1 Em (t)

where ω0 is the carrier frequency, ∆ω is the frequency detuning, which corresponds to ω in the pulse envelope, k is the spatial frequency and a function of ω, L is the length of the segment, and A, B, C, D are coeﬃcients determined by the boundary conditions of the ˜j,± th segment. Em (ω)s are the forward (+) and backward (-) waves in the j spatial section th ˜0,± th ˜ and the m time step, so obviously Em (ω) equals the m incident subpulse Em(ω). It is worth mentioning that all the above calculations are valid for a small time interval as I assumed the structure behaves linearly during this interval. This assumption is true when the subpulse is short enough not to make a major diﬀerence in carrier density. The salient feature of this method, like in SSF, is the usage of FT and FT −1 to Chapter 2. Modeling periodic nonlinear structures 17

Figure 2.3: Schematic plot of a spatial section in transfer matrix

switch between the time domain (where modeling of the nonlinear eﬀect is easier) and the frequency domain (where modeling of the dispersion eﬀect is easier). The drawback is the computational burden of FT and FT −1, which is performed for the number of spatial sections multiplied by the number of time steps. Nevertheless, because of the analytical nature of the TDTM method, the computation is still faster than a brute-force FDTD.

Time Domain Operator based Transfer Matrix Method (TDOTM)

Realizing that switching between the time and frequency domains is the most time consuming part, I introduce in this section an alternative TDTM method that uses time domain operators such that the computation remains in the time domain while still making it possible to include dispersion. Following [51, 52], the FT and FT −1 are substituted with derivative time operators. The procedure is described as follows.

˜0,− ˜0,+ First, I find the field reflected by the structure, Em (ω), as a function of Em (ω) ˜ which is just the FT of the input subpulse, or FT (Em (t)). The procedure is quite straightforward: the total transfer matrix is obtained using T = Q Tj, and then reflec- j −1 j,± ˜ j,± tion can be written in the form of FT Gm (ω) FT E (t) , where Gm (ω) s can be regarded as the impulse response in the jth spatial section and can be calculated knowing j ˜j,± the T s’ elements. It is possible to Taylor expand G (ω) and rewrite each of Em (t) as:

Np ! ! −1 j,± ˜ −1 X n ˜ FT Gm (ω) FT Em (t) = FT anω FT Em (t) (2.16) n=0

d ˜j,± Second, by replacing ω with time operator −i dt , each of Em (t) can then be expressed Chapter 2. Modeling periodic nonlinear structures 18

˜ as the sum of diﬀerent orders of time derivatives of Em (t):

Np ! ! Np n −1 X X n d E˜j,± (t) =FT a ωn FT E˜ (t) = a (−i) E˜ (t) . (2.17) m n m n dtn m n=0 n=0

Therefore, for each forward and backward wave in each spatial section, there is a corresponding Taylor expansion, which can be approximated by the polynomial numerical ﬁtting of G (ω), to save computation time. In this way, it is possible to keep the functions in time domain, although the coeﬃcients are calculated in the frequency domain.

Calculating G (ω) and finding the corresponding polynomial coefficients for each spatial section is still time consuming but the computation time can be further reduced. It ˜0,− 0,− is possible to perform the above process to find Em (t) only, so only Gm (ω) is Taylor ˜j,± expanded. The other unknown wave profile Em (t)(for j > 1) can be determined by writing Eq. 2.14 in the time domain:

    ˜j,+ Np n ˜j−1,+ Em (t) X j−1,± n d Em (t)   = Am,n (−i)  , (2.18) ˜j,− dtn ˜j−1,− Em (t) n=0 Em (t)

Np j,± P j,± n where Tm (ω) = Am,n ω , n=0

j,± j,± Am,n can be determined by ﬁtting Tm (ω) to much lower order polynomials (than 0,− j,± that for Gm (ω) ), and in many cases Am,n can be found analytically because the fre- j,± quency dependence of Tm (ω) is just a function of material dispersion. In this way, each ˜j,± Em (t)(for j > 1) can be calculated sequentially from one section to the next, in a matrix form and in the time domain.

For example, if the material dispersion is negligible, the A, B, C, D coeﬃcients in Chapter 2. Modeling periodic nonlinear structures 19

Eq. 2.14) can be treated as constants, so Eq. 2.14 can be simpliﬁed as below:

              ˜j,+  −ikL ˜j−1,+  Em (t) −1  e 0 AB Em (t)    = FT     FT   (2.19) ˜j,−  +ikL ˜j−1,−  Em (t)  0 e CD Em (t)    | {z } | {z }   Propagation Boundary  | {z } Tj

(ω0+∆ω)n α where k = c − i 2 . Here, n and α are respectively the real refractive index and absorption. Converting (2.19) into the time domain, we have:

 −iω0nL α −iω0nL α    − L n − L n ˜j,+ c 2 ˜j−1,+ c 2 ˜j−1,− Em (t)  e AEm (t) t − L + e BEm (t) t − L    =  c c  ˜j,−  +iω0nL α +iω0nL α  Em (t)  + L n + L n  e c 2 CE˜j−1,+(t) t + L + e c 2 DE˜j−1,−(t) t + L m c m c (2.20) In practice, even with a very wide input pulse bandwidth (say 100 nm) and considerable material dispersion, a third- or fourth-order polynomial would be suﬃcient, if bulk materials are used for each layer.

2.2.1.3 Updating process

˜j,± After determining the pulse envelopes Em (t) in all spatial sections at each time step, the optical property of the entire structure must be updated. In resonant nonlinear materials, the energy of the incident photons is close to the energy of the material band-gap, promoting valence electrons to the conduction band, thus changing the carrier density. The evolution of carrier density at each spatial section N(t, z) can be described by the following rate equation [38]:

dN(t, z) N(t, z) α(N (t, z) , ω ) ncε 2 = − + 0 × ( 0 ) E˜ (t, z) (2.21) dt τ hω0 2 Chapter 2. Modeling periodic nonlinear structures 20

where α (N (t, z) , ω0) is absorption and can be determined experimentally, or according to certain phenomenological models, e.g., the B´anyai-Koch model [49]. The corresponding carrier-induced refractive index change is found through the Kramers-Kronig relation. An analytical solution for the above equation can be introduced as:

(m+1) T j z Z M α N (t) ,ω Z j 2 j j −t/τ t/τ m−1 0 ncε0 1 ˜j Nm = Nm−1+e e × ×( ) Em (t, z) dzdt T hω0 2 zj − zj−1 (m−1) M zj−1 (2.22)

˜j One simple way of approximating Em (t, z) over the thickness (zj − zj−1) of the layer is: zn zn E˜j (t, z) = E˜j,− t + j + E˜j,+ t − j (2.23) m m c m c

˜j Here, possible temporal overlaps between neighboring subpulses (i.e. Em(t) and ˜j Em+1(t)) are not taken into account. This overlap is initially introduced in the pulse decomposition process (Eq. 2.13) to avoid sharp temporal rising edges of the subpulses. The overlap is further increased due to the multiple reflections of the structure, causing an effective delay in its response. If the length of each subpulse is shorter than the length of the structure, the m+1th subpulse would start to enter the structure while some part of the previous subpulse is still remaining in the structure and changing the carrier density. Without correcting for this effect, the time-domain method cannot be very accurate for these cases.

A better approximation of the carrier density is achieved by modifying the time integration limits in Eq. 2.23, so as to include the carrier density evolution from the arrival time of a pulse to the arrival time of the next pulse at every special section, as expressed here:

z (m+1) T + j−1 n˜ j z Z M c α N (t) ,ω Z j j j −t/τ t/τ m−1 0 1 ˜j Nm=Nm−1+e e × × Im (t, z) dzdt T hω0 zj − zj−1 (m−1) M zj−1 (2.24)

˜j th wheren ˜ is average refractive index and Im is the intensity of the m term plus its Chapter 2. Modeling periodic nonlinear structures 21

interference with previous terms.

ncε zn zn 2 I˜j = ( 0 ) E˜j,− t + j + E˜j,+ t − j m 2 m c m c zn zn ∗ + E˜j,− t + j + E˜j,+ t − j m c m c m−1 X zn zn . E˜j,− t + + E˜j,+ t − (2.25) p c p c p=1 zn zn + E˜j,− t + j + E˜j,+ t − j m c m c m−1 ! X zn zn∗ . E˜j,− t + + E˜j,+ t − p c p c p=1

The average refractive index is used to approximate the group index, which is suf- ﬁciently accurate in most cases. If a more accurate solution is desired, for example in structures with a very high contrast, a recursive method can be used, at a cost of computation complexity. In the recursive method, it is possible to let the m+1th subpulse

j j th propagate before updating Nm and then to update Nm by the arrival time of m+1 and again to let the m+1thpart to propagate considering the new optical property of the guiding material. This recursive procedure may be repeated as much as needed.

2.2.2 Numerical result and comparison

In this section the advantage and limitation of the proposed TDTM will be investigated through numerical examples. For short input pulses, for which FDTD is still feasible in terms of computation time, my results are compared with those of FDTD, and the pros and cons of each method are presented. For long pulses, FDTD is no longer eﬃcient, so I compare results from TDOTM to those from FT-TDTM in order to demonstrate cases where TDOTM can be both fast and accurate.

The structure under test is made up of In1−xGaxAs1−yPy with two diﬀerent compositions alternating in a periodic manner (Fig. 2.4). One composition has y =0.1 and the other has y =0.96. The corresponding bandgap energies are 1.27 eV (975nm) and 0.74 eV (1679nm), respectively. The thickness of both layers is 0.1 µm and the entire Chapter 2. Modeling periodic nonlinear structures 22

Figure 2.4: In a layered structure with resonant nonlinear material, each layer corresponds to the section illustrated in Fig. 2.3. d represents thickness, N the carrier density, n the refractive index, and α the absorption coeﬃcient. structure has 20 periods. The carrier wavelength of the excitation pulses is 1550nm, corresponding to photon energy above the bandgap of the layers with y =0.96, but much less than the energy bandgap of the layers with y =0.1. So the structure has high loss and large nonlinearity in half of the layers and has almost no loss and a linear refractive index in the rest. The electron life time is considered 1 ps, which is possible for certain engineered semiconductors [29]. Longer life times would result in more accurate simulation results because the carrier density will change more slowly. In all simulations, we assume the input excitation as Gaussian:

t−ts 2 ˜ −( t ) E = E0 × e w

where E0 is amplitude tw is pulse length and ts is the shift on the center of pulse (the analysis interval is from 0 to 2ts).

2.2.2.1 Linear and Nonlinear Responses of a 200fs, 800fs, and 3.2ps Gaussian Pulses

As mentioned, when comparing TDTM with FDTD, I chose relatively short input excitations, a regime where FDTD can be performed efficiently. In addition, as it is not possible to include non-analytical dispersion relations in FDTD, I did not include material dispersion in these simulations. I will show that the main benefit of the TDTM method stands out in the cases of long incident pico-second pulses (the equivalent spatial length is much longer than the device), whereas for some cases with femto-second Chapter 2. Modeling periodic nonlinear structures 23 incident pulses, the standard FDTD can be reliable and equally fast. Envelope of the incident pulse has a Gaussian profile shape defined as mentioned with below parameters:

2 7 with tw = 200fs , ts = 1ps , and the peak intensity is 0.21GW/cm (E0 = 4 × 10 V/m). The chosen wave amplitude in the mentioned pulse length is corresponding an energy comparable with half of the needed energy for complete loss saturation so more nonlinear properties are observed when varying the pulse length. The transmitted and reﬂected pulses for both low intensity (linear regime) and high intensity (nonlinear regime) incidence are plotted in Fig. 2.5. For the linear results, a perfect agreement between the two methods is observed.

(a) (b)

Figure 2.5: (a) Intensity linear response and (b) Intensity nonlinear response (Ipeak = 0.21GW/cm2) to a 200fs pulse. Intensities are normalized to the peak incident intensity.

For nonlinear analysis, the pulse is divided to 20fs small sub pulses and each has a temporal width comparable to its dwell time inside the device. The dwell time is the transit time of the pulse plus additional time to account for the tail of the transmitted (or reflected) pulse due to strong multiple reflections. This is the minimum recommended sub-pulse duration, because if shorter sub-pulses were used, their duration, after propagating through the structure, would not shorten due to multiple reflections. As a result, there would be a significant temporal overlap between the current subpulse and the tail of the previous subpulse after being modified by the structure (see discussion in Sec. 2.2.2.3). While our method corrects to a certain degree the effect due to this temporal overlap Chapter 2. Modeling periodic nonlinear structures 24

Sec. 2.2.2.3)), large temporal overlap would still result in noticeable inaccuracies. There- fore we recommend that the subpulse width to be larger than its dwell time inside the structure. This inaccuracy is negligible for 200fs pulse and longer pulses are discussed in this section so we come back to this point in next section. As a second example, the result for a much longer incident pulse which has the same intensity as in the ﬁrst example and

2 naturally carries more energy (with tw = 800fs , ts = 2ps, Ipeak = 0.21GW/cm ) is given in Fig. 2.6. There is a excellent agreement between FT-TDTM and FDTD results. The total energy of pulse in nonlinear case is chosen so that material loss is almost saturated after 2ps (at the center of incident pulse) so we can see the transmitted and reﬂected pulses are asymmetric (Fig. 2.6b). We can see, for long incident pulses, TDTM can be very accurate even when pulse intensity is high. Here, it is noticeable that TDTMs computational requirement is almost the same as that for the cases of shorter pulses. This is not true for FDTD of course. The computation time increases at least linearly with pulse duration for non dispersive materials. Also, the same spatial resolution gives less

(a) (b)

Figure 2.6: Normalized reﬂection and transmission intensity of a 800 fs pulse. (a) linear 2 response (b) Nonlinear response, (Ipeak = 0.21GW/cm ) accurate result for FDTD simulation of longer pulses in a periodic structure because the sum of multiple reﬂections with even a slight numerical dispersion in each one can result in high amplitude and phase inaccuracy. In third example, we investigate response to a

2 longer pulse With tw = 3.2ps, ts = 8ps, Ipeak = 0.21GW/cm . In this example, the pulse Chapter 2. Modeling periodic nonlinear structures 25

causes fast saturation of loss due to its high energy so again with increasing the pulse energy again we see symmetric reﬂection and transmission pulses. Its result is brought in Fig. 2.7. Considering the fact that evolution of carrier density has eﬀect on both loss and refractive index, Fig. 2.7. is not equal to simulating a linear lossless case.

Figure 2.7: Normalized reﬂection and transmission intensity of a 3.2ps pulse. (Ipeak = 0.21GW/cm2)

2.2.2.2 Nonlinear Responses of 50fs Incident Pulses

To illustrate the effect of incident pulse duration on the accuracy of the TDTM method, this example involves cases with much shorter incident pulses (tw = 50fs). Fig. 2.8a and Fig. 2.8b show two cases of different input intensities. In Fig. 2.8a pulse has the same maximum intensity of previous examples and obviously steeper intensity profile but because the total energy of the pulse is not enough to saturate the loss and multiple reflection does not happen still we have accurate result. In other words the total energy of the pulse is not enough to induce high nonlinearity and this result almost corresponds the linear result. The only situation that inaccuracy may arise in the proposed method is when the incident pulse has steep profile and high pulse energy. Fig. 2.8b is such a case where the maximum intensity is 0.85GW/cm2. In this figure some disagreements between FDTD and TDTM methods is observed. For such short pulses FDTD is fast enough and TDTM is not beneficial. To look into the limitation of method carefully we should say more important than Chapter 2. Modeling periodic nonlinear structures 26 steep intensity profile is where there is steep changes in optical properties of the material. It is difficult to give analytical formula to clarify this limit because changes in optical properties are function of optical properties (mostly loss) themselves in addition of intensity and its variation. As loss changes in time we cannot introduce an analytical formula. This inaccuracy can be overcome by considering initial condition in the begin- ning of each sub-pulses analysis. However it is theoretically possible but practically it is time consuming.

(a) (b)

Figure 2.8: Normalized reﬂection and transmission intensity of a 50fspulse. (a) medium 2 intensity incident pulses (Ipeak = 0.21GW/cm ) (b) High intensity incident pulse (Ipeak = 0.85GW/cm2)

Due to mentioned constraint in Sec. 2.2.2.1 on the minimum duration of the subpulse, for incident pulses with steep temporal profiles, shorter subpulses cannot be used to reduce the intensity variation between adjacent subpulses, which may lead to large variation in carrier density between adjacent time steps. Consequently, inaccuracies may result for the cases of large nonlinearity coupled with incident pulses with steep temporal profiles, such as ultrashort, high-intensity pulses. The discrepancies in Fig. 2.8b between FDTD and TDTM is limitation manifestation of this effect. Chapter 2. Modeling periodic nonlinear structures 27

2.2.2.3 Comparison between the two methods of TDTM: FT-TDTM vs. TDOTM.

So far, we compared the TDTM with FDTD. Now we present a comparison between FT-TDTM and TDOTM. Theoretically, they should be equivalent, but as we can keep only a finite number of orders in Eq. 2.17, there is a discrepancy. As we already justified FTTM for different pulse lengths (Sec. 2.2.2.1 and Sec. 2.2.2.2), we need only compare FTTM and TDOTM for longer pulses, where TDTM offers a distinct advantage over

FDTD. In Fig. 2.9 the result of such a comparison is shown for a long pulse with tw = 2 6.4ps, ts = 16ps, Ipeak = 0.21GW/cm . As mentioned a high order polynomial is required in the calculation of the ﬁrst spatial section and much lower order polynomial in all other sections is suﬃcient. Shorter time steps increase the needed spectral bandwidth so higher order polynomial is needed to approximate the dispersion property.

(a) (b)

Figure 2.9: Nonlinear response to a 6.4ps pulse. (a) FT-TDTM is compared with TDOTM with 12 order polynomial (b) FT-TDTM is compared with TDOTM with 8 and 6 order polynomials

An twelve order polynomial is used and length of each time step is 160 fs to generate results are shown in Fig. 2.9a. Of course, for diﬀerent cases, the order of polynomial completely depends on the complexity of dispersion relation (mostly structural dispersion) around carrier frequency. Fig. 2.9a shows a prefect agreement between FT-TDTM and TDOTM for reﬂection Chapter 2. Modeling periodic nonlinear structures 28 and transmission of a 6.4 ps pulse. If we reduce the order of polynomial to six or eight, the inaccuracy in Fig. 2.9b) will be observed. . The computation load of TDTM is 2 order of magnitude less than FDTD.

2.3 Conclusion

The proposed method in this section is a novel method for analyzing pulse propagation in a nonlinear layered structure where resonant nonlinearities and dispersion can both be in non-analytical forms. In summary, the nonlinearity is treated as a piece-wise linear problem in the time domain, and the dispersion is included in a frequency-domain transfer-matrix formulation. Then, using either F T/F T −1 or a time-domain operator approach, one can convert frequency-domain analysis to time-domain and vice versa. I verified the accuracy of my method using the FDTD method formulated in the first section. Comparison of the two methods is undertaken for various pulse intensities and durations. As with all numerical methods, the accuracy of my method is limited by temporal and spatial discretizations. The accuracy is additionally affected when the excitation pulse is short in duration (compared to the device length) and the reflections at the layer interfaces are strong. In such cases, FDTD does show an advantage. Conversely, the advantages of my method are shown most distinctively in the cases where relatively long excitation pulses were used, where FDTD and similar time domain methods suffer more computational burdens. The proposed method is the only effective method that is able to model simultaneously slow nonlinearity and material dispersion, both in non-analytical forms, as is generally the case for semiconductor-based resonant nonlinear devices. In addition, the transfer- matrix based method is computationally efficient. I presented only a one-dimensional model here, but as with other transfer matrix methods, my method can be generalized to solve higher dimension problems by using Galerkin’s expansions in other directions [53, 54], which has been left for future work. Chapter 3

Application of layered structure with resonant nonlinearity in optical limiting

With the rapid growth in optical communication networks, all-optical signal processing (AOSP), i.e., controlling light by light, has been suggested as an alternative to overcome the bandwidth limitations imposed by current electronic components [55]. AOSP circuits employ optical components that are intrinsically lossy and nonlinear. As a result, intensity variations are imparted to the pulses, which can lead to detrimental effects for subsequent signal processing. Therefore, there is a need to equalize the intensities of pulses at various stages to guarantee the performance of components in the next level of processing. This functionality can be achieved by the use of optical limiters (See Fig. 3.1). Optical limiter devices employ materials that exhibit optical nonlinearity. Despite the fact that all materials have a certain degree of nonlinearity, a choice of a practical material which is sensitive enough to the change in optical excitation remains a major challenge. Moreover, the material response speed and the recovery time of the nonlinear effects are important criteria for engineering practical devices. In addition, to complete the feedback loop, the change in material has to have an affect on the light so the light can itself manipulate its transmission or reflection. In this chapter, I first review the history of optical limiters. Second, I briefly review the resonant nonlinearity in InGaAsP

29 Chapter 3. Optical limiters 30

Figure 3.1: A schematic of optical limiter eﬀect materials. Third, I discuss the non-trivial phase shift in the interface between lossy materials. Fourth, I introduce my contribution, which is a proposal for optical limiting using resonant nonlinearity and interface phase shift.

3.1 Traditional optical limiting methods

All optical limiter devices with various materials and techniques have one thing in common: they all have nonlinear properties. The origin of these nonlinearities may be varied. Two-photon absorption, free-carrier absorption, and the Kerr effect are examples of exploited nonlinear properties for optical limiting. All optical limiter devices can be divided to two main categories based on their type of nonlinearity: instantaneous or accumulative nonlinearity. In the former, the polarization density is an instantaneous result of an applied electric field. It may have a second, third or higher order relationship presented by χ(2), χ(3), etc. χ(2) has usually been used for frequency mixing while χ(3) has been widely used for optical limiting. χ(3) can be imaginary or real, representing, for example, two photon absorption or the Kerr effect, respectively. On the other hand, accumulative nonlinearities have memory. This means the polarization density is a weighted sum of all applied electric fields in the past (a causal weighted integral). In other words, the effect of the applied field on polarization density decays over time. This nonlinearity is usually based on energy deposited in the medium. Chapter 3. Optical limiters 31

Examples of this nonlinearity are excited state absorption and free carrier absorption.

Accumulative nonlinearities are used for optical limiters working in longer pulse regimes (> ns). Usually, they limit the pulse energy rather than the intensities if the pulse length is shorter than the decay time of the carriers. Optical limiters based on instantaneous nonlinearities usually need higher intensity and work for short pulses.

Some of the techniques for all optical limiting are discussed in the next subsections.

3.1.1 Reverse saturable absorber

Materials which become transparent due to intense illumination are widely known as saturable absorbers. Guiliano and Hess investigated vat dyes which become more opaque at high intensities [56]. These materials are called reverse saturable absorbers. In these materials, the excited-state absorption cross section is larger than that of the ground state. As light is absorbed by the material, the ﬁrst excited state becomes populated and this causes an increase in the total absorption cross section. Various organic and inorganic materials, such as phthalocyanines [57, 58], porphyrins [59, 60], fullerenes [61], carbon nanotubes (CNTs) [62, 63], and inorganic nanoparticles [64] exhibit reverse saturable absorption and can be used for practical optical limiters.

3.1.2 Two-photon absorption (TPA)

Unlike with reverse saturable absorbers, two-photon absorption is an instantaneous (femto second) nonlinearity where two photons almost simultaneously interact with an electron at the initial state and send it to the ﬁnal state. TPA is an intensity-dependent absorption and it changes the intensity of the traversing beam as follows:

dI = −(α + βI)I (3.1) dz

where α is the linear absorption coeﬃcient and β is the TPA coeﬃcient. β is β =

3ω (3) 2 2 Im[χ ] where ω is angular frequency, c is speed of light in vacuum, n0 is refractive 20c n0 Chapter 3. Optical limiters 32

index of medium, and 0 is the vacuum permeability. The solution of (3.1) for α = 0 is:

I I(L) = 0 (3.2) 1 + I0βL where L is the length of the sample. According to (3.2) the transmission coeﬃcient decreases as the input intensity increases. This is the desired behavior for an optical limiter. For TPA, the material response is instantaneous and the optical limiter based on TPA is pulse intensity sensitive rather than pulse energy sensitive. As a result, these types of devices are usually suitable for short pulses with high energy density. Examples of these optical limiters are [65, 66, 67].

3.1.3 Free-carrier absorption

The carriers generated in semiconductors can go to higher states in the conduction band by absorbing additional photons [68]. This is the equivalent to excited-state absorption in a molecule. Accumulative absorption in this material provides the condition for optical pulse-energy limiting.

3.1.4 Nonlinear refraction

This type of optical limiting is based on self-focusing and defocusing. In this type of device, light is refracted away from the detection point when the intensity is high. This can be a result of the Kerr effect which modifies the refractive index of the device and redirects the beam. Jensen pointed out that the power dependence of Kerr nonlinearity can be used for all-optical devices for the first time [69]. These devices potentially have a larger dynamic range and higher damage threshold compared to absorbing devices [70, 71].

3.1.5 Self-phase modulation (SPM)

In optical systems like fiber where space is not a degree of freedom, optical limiting based SPM are conventionally considered. This type of optical limiter is being used for stabi- Chapter 3. Optical limiters 33 lized generation of wavelength conversion. Also it has been utilized for compensation of pulse intensity fluctuation. Optical limiting relies on a SPM-induced spectral broadening, and spectral filtering. In this technique, the input pulses are amplified with an EDFA. Then the pulses are sent through a dispersion compensation fiber, a highly nonlinear fiber and an optical band-pass filter. The nonlinearity causes spectral broadening but due to the pre-chirping procedure, the intensity remains constant at the center of the wavelength component, which results in an active but low power optical limiter [55, 72].

3.2 Resonant nonlinearity in InGaAsP

Among all other methods used for optical limiting, I focus on the application of resonant nonlinearity in optical limiting. Resonant nonlinearity in InGaAsP is introduced here but will be used later for the design of the optical limiter. Here, the criteria for a good optical limiter material is discussed and the advantage of using resonant nonlinear materials is described. Usually, the main determining factors for the choice of the material for an all optical device are: first, the intensity of light required for a noticeable change in the optical property (sensitivity) of the material in the spectrum of interest, second, the time required for the above change, and third, for operating in a high speed circuit, the recovery time (τ, the required time for the material to return to the state before triggering). The energy and recovery time requirements vary on device configuration. For example, if one can confine the light or has access to high power light, the sensitivity of the material may matter less. Also, if the pulses are short but the repetition rate is low, as far as the material’s response is instantaneous, the recovery time is much less a limiting factor. Therefore, different materials are used for different applications. As explained in the previous chapter, resonant nonlinearities are typically much larger than non-resonant nonlinearities so they are attractive for compact devices. Non-resonant nonlinearities usually have fast responses but resonant nonlinearities are usually limited by a slow recombination process, although this process can be accelerated by engineering the material. This is done by adding defects and impurities and creating new states in the electronic bandgap of the semiconductors. These intermediate states fasten the Chapter 3. Optical limiters 34 recombination process. These defects can be created naturally or they can be produced using different methods including doping, low-temperature beam epitaxial growth, ion implantation, or Helium-plasma-assisted molecular beam epitaxial (HELP) growth. The latter is a growth process in which the material is exposed to a high-energy helium plasma flow. HELP-grown InGaAsP produces deep-level carrier traps in its bandgap. Experiments have previously verified the picosecond carrier lifetime in InGaAsP [29, 30]. Also, all optical switches have been implemented based on HELP-grown InGaAsP [73]. In HELP InGaAsP, both the absorption and refractive index are functions of carrier density. This material can be modeled using a rate equation as previously explained in chapter 2, which shows the evolution of carrier density. A model is needed to find the material absorption and refractive index for different carrier densities. The Banyai-Koch model is a partly phenomenological theory developed to state the material absorption coefficient as a function of the electron-hole-pair concentration [49]. Knowing the absorption for different wavelengths, the refractive index of the material can be calculated based on the Kramers-Kroning relation. We have developed a code to calculate the absorption and refractive index of InGaAsP based on the Banyai-Koch and Kramers-Kroning relation, respectively. These two are plotted as a function of carrier density in 1550 nm wavelength Fig. 3.2, Fig. 3.3.

3.3 Non-trivial phase shift in the interface of lossy material

Non-trivial interface phase shift in an electromagnetic wave is another concept used to design an optical limiter in the next section. In this section, I discuss the eﬀect of a phase shift in the propagating wave due to an interface between two materials where at least one of them is lossy. This concept has recently found some attention where plasmonic material or highly absorptive semiconductors play a role [74, 75]. Reﬂection from the interface can be calculated using Maxwell’s equation and the Chapter 3. Optical limiters 35

Figure 3.2: Absorption as a function of carrier density for In1−xGaxAs1−yPy semiconductor

boundary equation at the interface and is equal to n¯1−n¯2 for normal incidence, where n¯1+¯n2

n¯1 andn ¯2 are the complex refractive indices of the first ans second layers. Assuming illumination from a lossless medium 1, the phase of the reflected wave has been plotted for different ratio of real and imaginary parts of the refractive indices in Fig. 3.5. As shown in this figure, theoretically, any desired interface phase shift can be extracted with the right choice of refractive indices.

Many optical devices, such as the Mach Zehnder interferometer, are based on the optical phase shift. This phase shift has been conventionally created by a longer optical path. The above phase shift resulting from reﬂection from an interface can be equivalent to the phase shift resulting from an optical path. One application of using the non-trivial phase shift is nanometer anti-reﬂection coating [75]. Here, the behavior of the incident

light from the air onto a dielectric ﬁlm with a complex refractive index ofn ¯2 = n2 + ik2

deposited on a metallic layer with a complex indexn ¯3 has been discussed. I follow the same formulation to elaborate the point. The reﬂection coeﬃcient from the device at a Chapter 3. Optical limiters 36

Figure 3.3: Refractive index as a function of carrier density for In1−xGaxAs1−yPy semiconductor

Figure 3.4: Schematic of incident light from medium 1 reﬂected from an interface with medium 2

θ1 angle of incidence for transverse-electric light is

r¯ +r ¯ e2iβ¯ r¯ = 12 23 (3.3) 2iβ¯ 1 +r ¯12r¯23e

¯ ¯ ¯ wherer ¯mn = (¯pm − p¯n)/(¯pm +p ¯n),p ¯m =n ¯mcos(θm), β = (2π/λ)¯n2hcos(θ2), and ¯ −1 θm = sin (sin(θ1)/n¯m), h is the thickness of the lossy layer.p ¯m is replaced byq ¯m = Chapter 3. Optical limiters 37

Figure 3.5: The phase of reﬂection from a lossy medium for diﬀerent refractive indexes in Fig. 3.4

¯ cos(θm/n¯m) for transverse magnetic light.r ¯mns include a phase component which are created by complex refractive contrasts at the interfaces. This structure has been known as an asymmetric Fabry-Perot cavity. Conventionally, the loss in the dielectric film has been treated as a perturbation and the thickness of the layer has been chosen to be the odd multiple of λ/4/n2. In those wavelengths where loss is not large and the metal substrate can be considered a perfect electrical conductor (PEC), the phase shifts at both interfaces are 0 or π. In this case, the dielectric film provides the required optical path for phase change. As a result, the partial reflection waves may have destructive interferences and the device may have zero reflection. In a more general case, the metal is not a PEC or the dielectric film is highly absorptive. In such a case, other factors contribute to the reflection. The transmission and reflection phase shifts at the boundary between air and dielectric film and between metal substrate and dielectric film are no longer limited to 0 and π. This interface phase shift can replace the well known phase shift conventionally created by the optical path in the thin film. So, for example, nanometer thick anti-reflection coating has been proposed [75]. In next session, Chapter 3. Optical limiters 38

I have exploited the nonlinear ramiﬁcation of this phase shift. The manipulation of the interface shift in comparison with the optical path can have very broad applications in optical signal processing.

3.4 Compact optical limiter based on resonant optical nonlinearity in a layered semiconductor structure

There are various types of optical limiters such as those based on self-focusing and self- defocusing [76, 77, 78], two-photon absorption [79], nonlinear scattering [80], self-phase modulation [55, 81], or reverse saturable absorption [82, 83]. These diverse types of optical limiters can be mainly divided into two categories. In the first category, the real part of the refractive index is intensity dependent and any alteration in the refractive index causes spatial (e.g. those based on self-defocusing) or temporal (e.g. those based on self-phase modulation) phase variation in the incident wave. Subsequently, the use of a spatial or spectral filter offers an intensity dependent transmission. In the second category, material absorption is proportional to the incident light intensity (e.g. those based on reverse saturable absorption). Unfortunately, both categories have noticeable draw- backs for integrated AOSP applications. Those in the first category are not sufficiently sensitive or compact for integrated computing applications, which typically require sen- sitivities in the order of 10W/µm2 and device dimensions in the order of µm to mm. In the second category, though devices are more sensitive, there still has not been any reported device achieving 10W/µm2 sensitivity. More importantly, irreversible damage of heat accumulation is a main drawback for devices in the second category, especially for high-density integrated systems. Here, I propose a reflective optical limiter that uti- lizes intensity-dependent phase shifts at multiple interfaces of a periodic structure. This phase shift is caused by a change in the imaginary part of the refractive index due to absorption saturation in a resonant nonlinear semiconductor. In contrast to the devices in the first category, mine is more sensitive and compact because I use the large resonant Chapter 3. Optical limiters 39 nonlinearity. In contrast to the devices in the second category, loss in my proposed device is inversely proportional to the incident pulse intensity, and therefore my device does not suffer from heat accumulation while fulfilling its optical limiting function. Instead of being absorbed, most of the pulses energy in high intensity pulses is transmitted through the structure. This compact device has a high sensitivity owing to the accessible high resonant nonlinearities in many well-studied material platforms (e.g. InGaAsP used here as an example).

In resonant nonlinear devices, the imaginary part of the refractive index not only reduces the wave amplitude along the propagation direction, but also adds non-trivial phase shifts at the interfaces between two media [75]. Therefore, a considerable amount of loss may alter the conditions for the constructive or destructive interferences, which would be significantly different from those observed in a lossless medium. This effect of absorption on phase has been initially utilized for an anti-reflection coating application in a linear optics scheme [75]. However, the same phenomenon can be exploited for designing a broad range of nonlinear optical components. Optically manipulating the absorption, in a periodic structure whose reflection and transmission are sensitive to the above-mentioned phase shift at its interfaces, makes many all-optical functional- ities achievable. To obtain an optical limiting effect using this characteristic, I design a structure consisting of distributed Bragg reflectors forming a FabryProt cavity. The structure works in the following way: under high intensity illumination, the absorption of the structure is saturated, and I design the structure to have zero reflection due to complete destructive interference. Under low intensity illumination, the complete destructive interference conditions break down mainly because there are interface phase shifts arising from the imaginary permittivity of the layers. This phase shift disappears in the case of high intensity illumination when the layers become lossless due to complete absorption saturation. In short, the combination of absorption saturation and structural resonance (discrete phase shifts) creates a dip in the bandgap reflection spectrum of the structure and suppresses the reflection of high intensity incident beams. I use a recursive Finite-Difference Time-Domain (FDTD) method to design such a device. In my proposed design, the reflection dip in the bandgap is controlled with a change in Chapter 3. Optical limiters 40

Figure 3.6: Schematic of the proposed optical limiter consisting of a multilayered In- GaAsP structure (Type I and II alternating) sandwiched between two AR coatings. The periodicity of the InGaAsP is broken at the center, where two layers of Type II are next to each other. The bandgap energies of the Type I and II are designed to be lower and higher than the energy of the incident photons, respectively. absorption in carrier-induced nonlinear semiconductors, resulting in much more compact and sensitive devices.

My design, shown in Fig. 3.6, is made up of In1−xGaxAs1−yPy with two different compositions alternating periodically and with a breaking of periodicity (repeated layer) in the center. I call the first and second layers Type I and Type II layers, respectively. Type I and II layers have different compositions: y =1 (x=0.47) and y =0.1 (x=0.05), with corresponding bandgap energies of 0.75eV (1726nm) and 1.27eV (975nm), respectively. (The x-values of the two layers are determined to ensure both layers are lattice matched to InP.) The thicknesses of the Type I and Type II layers are 120 nm and 109 nm, respectively, which are equal to a quarter wavelength in the respective layers (designed for 1550nm). The entire structure includes 40 layers. All layers alternate in composition except for the 20th and 21st layers, which are both Type II layers. Together, they create a nonlinear resonator with a thin cavity in the middle of the structure sandwiched by mirrors made of quarter-wave stacks. Then the whole structure is wrapped between two anti-reflection layers. A single-layer anti-reflector (AR) is used on each side of the device but multi-layers AR can be used to broaden the matching bandwidth further [84]. To illustrate its optical limiting effect numerically, I use 3ps (FWHM) incident Gaus- sian pulses at 1550nm, which correspond to a photon energy above the electronic bandgap of Type I layers, but much lower than the bandgap of Type II layers (see Fig. 3.6). Con- Chapter 3. Optical limiters 41

Figure 3.7: In case of complete absorption saturation a dip appears in the reflection spectrum. The 20dB-bandwidth of this dip is approximately 4nm. sequently, the structure has a high loss and a relatively large nonlinearity in half of the layers but almost no loss and a linear refractive index in the rest. If both layers were lossless, a dip would appear in the middle of the reflection band. However, the absorption of the Type I layers breaks the condition for constructive interference in reflection. Consequently, the dip in the reflection spectrum disappears for low-intensities. The dip appears and grows in depth as the incident intensity increases due to absorption saturation in the Type I layers (see Fig. 3.7). In case of complete absorption saturation, a dip appears in the reflection spectrum. The 20dB-bandwidth of this dip is around 4nm . A FDTD simulation has been carried out to investigate the time-domain evolution of the pulses reflected by the device. To account for the materials resonant nonlinear property, a rate equation is added to the set of standard FDTD equations to update the carrier densities for all layers as the pulse propagates through the structure. Carrier- density-dependent absorption is then calculated via the Banyai-Koch model [49], with a saturation carrier density of 1018cm−3 which agrees well with experimentally observed saturation density in bulk InGaAsP. The material permittivity is then updated based on Chapter 3. Optical limiters 42 absorption via the Kramers-Kronig relation, and the standard FDTD procedure is carried out with the updated absorption and permittivity. The peak intensities of reflected pulses are plotted versus peak intensities of incident pulses in Fig. 3.8. This figure shows an exponential limiting effect and at the incident peak intensity of 0.05GW/cm2, the rate of change in normalized reflection is halved. In Fig. 3.9, the reflected pulses normalized to their incident peak intensities for different incident peak intensities 0.013GW/cm2, 0.052GW/cm2, 0.208GW/cm2, are plotted. This figure shows a drop in reflection starting at an incident peak intensity as low as 0.013GW/cm2. When a more intense pulse is incident on the structure, the leading part of the incident pulse experiences non-saturated loss (this is the same as in the low intensity case). As explained previously, the structure is designed for high reflection when the Type I layers are highly absorbing, with the presence of a phase shift at the interfaces due to its complex permittivity. However, the interaction of this leading part of an intense pulse with the device results in partial absorption saturation, which modifies the interface phase shift and leads to a reduced reflectivity. Consequently, more transmission occurs and leads to more absorption saturation, further reducing reflectivity. Therefore, the tail of the pulse transmits through the structure and observes the lowest loss and almost no reflection. As a result, the reflected pulse intensity above an incident intensity threshold is almost independent of the incident intensity and is proportional to the energy that is needed to saturate the absorption. This limiting value depends on the linear loss at the operating wavelength, the size of the structure and the saturation carrier density.

It is worth noting that the absorption saturation also has an effect on the real part of the refractive index via the Kramers-Kronig relation. This resonant nonlinearity is a much larger effect than the non-resonant nonlinearity (the Kerr effect). Nevertheless, our simulation shows that its effect on the reflection spectrum is still negligible compared to the effect of the changing phase shift at the interfaces due to absorption saturation. This explains why some optical power limiters previously developed based on nonlinear 1D photonic crystal (e.g. [85]) are less sensitive. In those designs, a nonlinear modulation of the refractive index, manipulating the photonic bandgap, results in a limiting effect. These devices use Kerr nonlinearity, which is relatively weak. Therefore, achieving a high Chapter 3. Optical limiters 43

Figure 3.8: Exponential saturation of the reﬂected pulse versus incident peak intensity.

Figure 3.9: Reflected pulse normalized to incident pulse intensity for different incident pulse intensities. Chapter 3. Optical limiters 44 sensitivity needs fairly large (millimeters or longer) devices. The idea of manipulating reflection utilizing the non-trivial interface phase shift at interfaces between lossy media by thermal tuning of the loss was previously suggested in [74].

3.5 Conclusion

A new optical limiter design based on resonant nonlinearity and a photonic bandgap structure is proposed. An optical limiting eﬀect is observed by exploiting band-to-band absorption in a periodic structure with a discrete phase shift. The simulation is based on realistic semiconductor properties, and the proposed device is much more compact (a few micrometers) and the limiting eﬀect starts at a much lower intensity (0.013 GW/cm2) compared to other optical limiting devices proposed in the literature. Chapter 4

Low noise on-chip source of correlated photons

As discussed in the introduction of this thesis, frequency conversion is a major application of nonlinear optics. For an efficient frequency conversion, a phase matching method is required. Periodic structures have played a critical role in phase matching required by frequency conversion. In this chapter, a periodic variation in the nonlinear property of AlGaAs has been exploited for a specific type of frequency conversion called spontaneous parametric down conversion (SPDC). This process is used for the low-noise generation of entangled photons. First, I describe what entangled photons are and present a brief review for the generation of entangled photons. When two particles are generated in such a way that the quantum state of each particle cannot be described independently, they are called quantum entangled. Position, momentum, spin and polarization are among the physical properties that show correlation when a measurement is performed on entangled particles. Albert Einstein, Boris Podolsky and Nathan Rosen (EPR) [86] described that the existence of these particles violates the local realist view of causality. This is known as the EPR paradox. However, a series of polarization or spin measurements on entangled particles proved that the local realist view of causality cannot be true. In particular, John Bell proposes his inequality in 1964 [87]. He provided the groundbreaking mathematics which allows researchers to experimentally exclude the effect of any local hidden variables on the outcomes obtained

45 Chapter 4. Low noise on-chip source of correlated photons 46 through measuring entangled particles. If the inequality holds, then entanglement can be a result of local effects. If violated, consideration of nonlocality is a must which is in agreement with standard quantum mechanics interpretations. Later in 1972 [88], Stuart Freedman and John Clauser experimentally tested Bell’s theorem for the first time by measuring the polarizations of a pair of photons, although the experiment had a few loop holes. This was followed by Aspect’s measurement which presented a stronger verification of Bell’s entanglement [89]. Despite the fact that efforts for demonstrating a loophole-free Bell’s inequality test has continued to the present day [90], each experiment has at least one loophole that questions the validity of the results. Although the majority of experts in the field agree on the concept of non-locality, there is no scientific consensus.

Aside from the philosophical debate, the existence of quantum entanglement has found broad applications, namely quantum communication and quantum computing. Some of the new products of ID Quantique, MagiQ, and D-wave companies are evidences of com- mercialization of quantum technologies. The first theoretical application was proposed for quantum cryptography in 1984 which uses photons in a superposition of states to create a secure key [91]. Slightly over 5 years later, the first experimental quantum key distribution prototype worked and was published in Sigact news [92]. In 1997, the first quantum teleportation experiment was performed and played an important role in quantum computation [93]. In 2007, entanglement-based quantum communication extended to over 144 km [94].

In many applications of quantum entanglement, a low noise and bright entangled photon pair source (PPS) plays a critical role. In addition, advances in technology have opened the door to on-chip entanglement. Initially, the planar waveguide based PPS has mostly been based on the periodically poled lithium niobate (PPLN) waveguides [95, 96, 97] but by now, other materials such as silicon [98], chalcogenide [99], and (Al)GaAs [100] have been adopted for this purpose and opened the way for on-chip PPS. In particular, (Al)GaAs based devices have recently attracted more attention [101, 102]. The III-V semiconductor AlGaAs has been shown to be the leading material candidate for nonlinear optical interactions at the telecom wavelength (1.55 microns). AlGaAs-based waveguides offer the opportunity for combining well-developed fabrication technologies to incorporate Chapter 4. Low noise on-chip source of correlated photons 47 laser pump source integration with comparatively large second-order nonlinearity, low linear loss and low two-photon absorption making it a promising candidate for highly- integrated photon pair sources. However, while the AlGaAs source of entangled photons have shown advantages of potential high brightness and potential monolithic integration with the pump, the signal-to-noise ratios for previous sources of entangled photons are limited. In this chapter, I investigate quasi-phase-matched (QPM) AlGaAs superlattice waveguides for quantum photonic sources. This device has two major advantages over the current AlGaAs based PPS: first, a high spatial overlap between the pump and signal/idler beams increase the efficiency and reduces the noise level as will be discussed later in this chapter. Second, the post-growth lithography process provides the flexibility to choose different phase matching wavelengths. This facilitates achieving an appropriate overlap between phase-matching bandwidth and the lasing spectrum of the pump potentially built on the chip which is essential for efficient photon pair generation. In this chapter, we focus on low noise generation of photon pairs, and in the next chapter, the entanglement properties of these photon pairs will be investigated. In the first section, device requirements for efficient photon pair generation will be explained. In the second section, the importance of noise reduction is described. In the third section, I discuss why QPM superlattice waveguides are the best candidate for low noise generation of entangled photons. In the forth section, the result of classical characterization of these waveguides is presented which supports my claim in the third section. In the fifth section, my experimental results for low noise generation of photon pairs are given. The material of this chapter was partially published in applied physics letters [103].

4.1 Nonlinear optical methods for the generation of entangled photon pairs

Entangled photon pair sources are of primary importance in quantum communication [21] and quantum computing [104]. Many previous experimental demonstrations of these Chapter 4. Low noise on-chip source of correlated photons 48

sources have been reported in ferroelectric crystals and waveguides [95, 96], fibers [105, 106], and silicon waveguides [107, 108, 109, 110]. There are two main schemes to generate correlated photon pairs in nonlinear waveguides: spontaneous four-wave-mixing (SFWM), a third-order nonlinear process relying on χ(3) nonlinearity; and spontaneous parametric down conversion (SPDC), also known as parametric fluorescence, based on χ(2) nonlinearity. In the case of SFWM, the process is mediated by a χ(3) nonlinear susceptibility, wherein two pump photons that are not necessarily degenerate are annihilated, and then produce signal and idler photons. In the SPDC process, one photon (as opposed to two photons in SFWM) is annihilated and two potentially indistinguishable photons are created that meet the criteria of energy conservation and phase-matching. The schematic illustration of these processes are shown in Fig. 4.1. There are two main challenges associated with the SFWM scheme. The first one is the suppression of Raman scattering noise. Due to the broadband nature of Raman scattering in many materials, filtering this noise is impossible when the spectrum of signal and Raman noise has noticeable overlap [111]. Second, SPM of pump photons causes overlap of the pump spectrum with those of the signal and the idler. Both of the above factors may significantly decrease the purity of pair generation as the pump frequency and the signal/idler frequencies are located within a limited spectral range. In contrast, SPDC-based correlated photon sources emit correlated photons with frequencies around half that of the pump frequency. Therefore the SPDC approach allows for the possibility of the same wafer providing a pump source at the fundamental bandgap, whilst exploiting the low linear and nonlinear losses associated with half-bandgap nonlinearities. Since the cubic symmetry of most common semiconductors, including AlGaAs, results in the lack of a natural birefringence, an artificial means for phase-matching must be devised in order to obtain efficient SPDC in the material. This has been previously achieved in AlGaAs structures using Bragg reflection waveguides [101, 112] or vertical quasi-phase matching (QPM) by alternating layers of different aluminum fraction [100]. In Bragg reflection waveguides [113] a modal phase matching method has been used where concentric waveguides guide different propagating modes using a combination of Chapter 4. Low noise on-chip source of correlated photons 49

Figure 4.1: Energy level for SPDC and SFWM process total internal reflection (TIR) and the bandgap effects due to Bragg structure. The drastically different dispersion characteristics of Bragg the reflection mode in comparison with the TIR mode result in the possibility of such phase matching. In these structures a wave with frequency of ω in the TIR mode can phase-match a wave with frequency of 2ω in the Bragg mode. Unfortunately, the Bragg mode is not well-localized in the core, therefore the coupling to this mode is very difficult and not efficient. In vertical quasi-phase matching semiconductor sources [100] counter-propagating twin photons are generated by a pump beam impinging on top of an AlGaAs waveguide. Such a structure was first proposed and demonstrated by Cada et al. for SHG in InGaAsP [114]. Here, the phase matching between the vertically illuminating pump and counter-propagating twin photons is maintained using a periodic layered structure. This type of source is more difficult to integrate as it needs a vertical pump and it can not be integrated with a planar pump laser.

In this thesis, I have used domain-disordered quasi phase matching (DS-QPM) in AlGaAs superlattice waveguides. The periodicity along the direction of the propagation in waveguides guarantees the phase matching, between the pump, signal and idler Chapter 4. Low noise on-chip source of correlated photons 50 wavelength. This type of phase matching has previously been used for second harmonic generation (SHG) [32] and diﬀerence frequency generation (DFG) [31]. As explained, one of the major advantage of the investigated device is its low noise property. In the next section, I discuss the importance of PPS with high purity.

4.2 The importance of the low noise PPS

Applications of entanglement sources include quantum computing [115], quantum cryptography [116], quantum imaging [117]. In all of the above applications, low noise (high purity) entanglement is highly desired. Take, for example, quantum cryptography which has attracted increasing attention as a secure way of key distribution on optical networks [118]. The security of the method is guaranteed by the laws of quantum mechanics. In short, any observation of the transmitting keys by an eavesdropper has unavoidable effects on keys and this eﬀect can be monitored by the sender and transmitter. To minimize the information known to an eavesdropper, the quantum bit error rate (QBER) has to be more than a certain threshold [119]. One important factor in determining the QBER is the visibility (V = Imax−Imin ) of the quantum interference fringes. For example, for a Imax+Imin BB84 protocol the QBER is:

1 − V QBER ≈ . (4.1) 2 [120] and if QBER is less than 11%, the sender and receiver will be able to extract ﬁnite amount of secure key without leaking any information to the eavesdropper. Also, to relate QBER to secure key rate (R), one uses:

R = 1 − H2(δ) − H2(δp), (4.2)

where δ is the bit error rate of the key tranmission basis, δp is the bit error rate of the test basis and H2 is the binary entropy function [121]. Theoretically, a pure source of entangled photons has a visibility of 1, but in practice this number is reduced by a few factors including the noise level and loss of transmission Chapter 4. Low noise on-chip source of correlated photons 51

link (the latter one is eﬀective in existance of detectors’ dark counts). As a result, reducing the inherent noise level in PPSs is very critical for their applications in quantum key distribution systems. In the next section, the ﬂuorescence noise generation in the AlGaAs material and a method for reducing this noise is discussed.

4.3 Low noise entangled photon generation in Al- GaAs superlattice Waveguides

In the last section, the effect of noise photons in reducing the visibility and as a conse- quence increasing the QBER in the receiver side of an optical network was described. A major source of these noise photons in a direct bandgap semiconductor is the fluorescence emission. Fluorescence (PL) emission normally happens when the semi-conductors are excited with a light source with an energy larger than the material’s band-gap energy [122]. In this phenomenon, the excited photons are absorbed and form electrons and holes in the conduction and valence bands, respectively. Through a Coulomb scattering or the interaction with phonons, the excited electrons or holes decay, emitting photons. Finally, the electrons and holes recombine, which results in photon emission. In general, many other factors such as electric fields or neighbor dielectric materials affect PL properties. A more accurate description of PL properties is explained with semiconductor luminescence equations [123]. Here, the details of this equation are not discussed, but it is important to note that to suppress the effect, the band gap of the material has to be

o larger than the pump photon energy. The AlxGa1−xAs material band-gap in 300K is a function of the Al concentration. This band-gap is direct when x ≤ 0.45 and indirect when x > 0.45[124]. This band-gap as a function of Al concentration is shown in Fig. 4.2 [125]. If one wants to generate entangled photons in the standard optical communication band, the pump wavelength can be as low as .75µm. Therefore, to avoid the PL the band-gap of the material has to be adequately high. Also, due to the defects in the fabrication procedure of a multi-layer structure, more intermediate states may appear. Chapter 4. Low noise on-chip source of correlated photons 52

Figure 4.2: The calculated band gaps of AlxGa1−xAs versus x [125]

To be in a safe distance of any PL eﬀect, it is recommended to use AlGaAs in its indirect band-gap region (x > 0.45). Restricting the Al concentration to a certain region limits the accessible range of the AlGaAs refractive index. For instance, in a 1550nm wavelength, the available range of refractive indices reduces from [2.94,3.43] to [2.94,3.16]. Therefore, those methods which used the ﬂexible range of AlGaAs face some limitations.

For example, using Al0.2Ga0.8As results in noticeable amount of noise in the generation of entangled photons [101]. In QPM AlGaAs superlattice waveguides, the light is conﬁned in the AlGaAs superlattice material in the core of the waveguide (as will be shown in section Sec. 4.4). As a result, due to the large energy band-gap of this material, the PL intensity will be suppressed. This has been previously shown in chapter 2 of [126]. The less the PL intensity, the lower the noise level in the coincidence measurement, as will be discussed in Sec. 4.5. Chapter 4. Low noise on-chip source of correlated photons 53

4.4 Characterization of QPM AlGaAs superlattice waveguides

As discussed in the last section, quasi phase matching is an alternative phase matching used specifically when one wants to avoid PL noise. In this method, different wavelengths of lightwaves are propagating with different speeds in each section, but the accumulated phase difference between lightwaves is periodically compensated by a modulation of χ(2). In the case of three wave mixing, QPM adds an additional term to the momentum conservation equation. The modulation depth of χ(2) can vary. If the sign of χ(2) changes from positive to negative, it is called domain-reversal QPM. This type is usually more efficient, but in practice it is difficult to reverse the sign of nonlinearity for most materials. Instead, one can periodically suppress the χ(2). This method of phase matching is used in this thesis. The momentum conservation formula for the devices with QPM is as follows:

2πm ∆β = β − β − β − (4.3) pump signal idler Λ where Λ is the period of χ(2) modulation and m can be any integer. When ∆β = 0, the wavelength conversion eﬃciency is at its highest.

One approach to DS-QPM is using quantum well structure for the core of the waveguides and modifying the nonlinear property of the quantum well using quantum well intermixing (QWI) [127]. This approach gives the flexibility of post-growth lithography and selecting the phase-matching wavelength which extends the possibility of integration. QWI increases the energy band gap, and as a result modifies the optical properties of the material, for instance that of second order nonlinearity. To maximize the extinction ratio, the target fundamental phase matching wavelength has to be close to the half- bandgap energy of the material before QWI, called as-grown material. Therefore, χ(2) can be significantly suppressed. Using this method, a periodic modulation in χ(2) of the material can be achieved.

The wafer has been grown in [001] direction of the lattice but the direction of propagation is [110].¯ Due to the symmetry of the 43¯ m cubic lattice structure, there are only Chapter 4. Low noise on-chip source of correlated photons 54

six non-zero elements in the susceptibility tensors, which are as follow:

(2) (2) (2) (2) (2) (2) χxyz = χxzy = χyzx = χyxz = χzxy = χzyx (4.4)

The device consists of a waveguide with a 0.6-m-thick core layer of 14:14 monolayer

GaAs/Al0.85Ga0.15As superlattice, with buﬀer layers of 300 nm Al0.56Ga0.44As on both

sides, and cladding layers of 800 nm Al0.6Ga0.4As. There is an additional 1-µm-thick layer of Al0.85Ga0.15As beneath the lower cladding in order to avoid field leakage into the GaAs substrate. This heterostructure ensures a low two photon absorption and good mode confinement for the 1550 nm band. All sample fabrication processes were developed and carried out by colleagues at the University of Glasgow. The fabrication procedure is as follows. A layer of poly-methyl-metha-crylate (PMMA) was spin-coated on top of the AlGaAs substrate, initially coated with a Ti-Au seed layer. Then the gratings, with different periods and duty cycles, were defined in the PMMA mask using a Vistec VB6 electron beam lithography system. After that, a 2.3 µm-thick Au implantation mask was grown in the PMMA gaps using electroplating, followed by the removal of the PMMA mask. The ion implantation was carried out with 4 MeV As2+ ions at a dosage of 2×1013cm2 to create point defects in selected areas, followed by removal of the Au mask. Rapid thermal annealing (RTA) at 775oC for 60 s provides QWI through the point defect diffusion. Ridge waveguides of different widths, with a height of 1 µm, were fabricated by reactive ion etching and the resulting sample was cleaved to a length of 3.5 mm. A schematic representation of the device is depicted in Fig. 4.3.

This waveguide supports the fundamental TE mode in 1550 nm and the second harmonic TM mode in 775 nm. I have simulated one of our waveguides with 2 µm width and a QPM periodicity of 3.5 µm, The mode proﬁles are shown in Fig. 4.4. It is worth noting that the superlattice breaks the symmetry in the vertical (z) direction, and as a result, the material birefringence should have been taken into account [128].

I have performed a Type I SHG measurement to ﬁnd the accurate phase matching wavelength for the above mentioned waveguide. In this experiment, a CW pump laser beam at 1550 nm is used. The laser beam passes through a quarter wave plate, half Chapter 4. Low noise on-chip source of correlated photons 55

Figure 4.3: A schematic of Quasi-Phase-Matched AlGaAs superlattice waveguides. Blue and pink beams have a wavelength around 775 nm and 1550 nm, respectively.

(a) (b)

Figure 4.4: (a) Simulated TE mode profile at 1550 nm (b) Simulated TM mode profile at 775 nm wave plate, and a polarizer to obtain a linear TE polarization. Then, the laser beam is coupled to the waveguide using a 40x objective lens. The second harmonic is collected and collimated with another 20x objective lens. Using a free space long pass filter, the Chapter 4. Low noise on-chip source of correlated photons 56

Figure 4.5: SHG power as a function of fundamental wavelength for a QPM waveguide with width of 2 µm and a QPM periodicity of 3.5 µm. second harmonic is redirected and detected by a Silicon detector. The SHG power for a 10 mW CW input pump (in free space) has been shown in Fig. 4.5 as a function of wavelength. The observable narrow peaks are due to the Fabry-Perot property of the waveguides. The fringe period of 0.1 nm agrees with the waveguide length of 3.5 mm. As this ﬁgure shows, the phase-matching wavelength is around 1546 nm and the SHG has a bandwidth of less than 1 nm.

4.5 Coincidence measurement

In this section, I present the correlated photon pair generation experiment and coincidence count using the experimental setup depicted in Fig. 4.6. For the purpose of this section a 3 µm-wide waveguide with a QPM periodicity of 3.5 µm and duty cycle of a 60 percent has been used. A TM-polarized 772.8 nm CW source was used as a pump source for producing parametrically down converted correlated photon pairs, centered around twice the pump wavelength 1545.6 nm. This light was ﬁltered by a polarizer (passing TE Chapter 4. Low noise on-chip source of correlated photons 57

Figure 4.6: Schematic of the setup for coincidence measurement in QPM AlGaAs superlattice waveguides. Blue lines are light propagating in free space, pink represents ﬁbers

polarization only), a long-pass filter, and three cascaded fiber-based pump suppression filters. The down-converted photons were then deterministically separated into two spectral bands (1562.0-1578.0 nm and 1522.0-1538.0 nm) using two fiber-pigtailed bandpass filters (BPFs) with 16.0 nm bandwidths, centered around 1570.0 and 1530.0 nm. The available BPFs were non-optimal in the sense that they were not frequency-conjugated around the degenerate phase matching wavelength, and consequently correlated photons could only be detected across half the bandwidth of the BPFs (1562.0-1569.2 nm and 1522.0-1529.2 nm), reducing the achievable coincidence-to-accidental ratio (CAR). A schematic of the filters is presented in Fig. 4.8.

Two free-running id Quantique id220 single photon detectors (SPDs), with measured quantum efficiencies of 20% at 1550.0 nm were used for coincidence measurements. The total estimated loss from the input objective to the coupling-out fiber was 17 dB at 1550.0 nm. There was a 5-dB loss associated with the coupling of the pump into the waveguide. The pump suppression filters had a total loss of 4.9 dB. The insertion losses of the 1570.0 and 1530.0 nm BPFs were 0.67 and 0.97 dB, respectively. Finally, an Field-programmable gate array (FPGA)-based time interval analyzer (TIA) allowed us to measure the coincidence as a function of the difference in the arrival times of the SPD signals. Chapter 4. Low noise on-chip source of correlated photons 58

A histogram of the raw coincidence measurement with a time-bin of 500 ps is depicted in Fig. 4.7a. Here, the input CW pump power inside the sample is estimated to be ∼ 8mW , and the integration time is 200 seconds. Coincidence versus the diﬀerence in arrival time is plotted, and a peak at 26 ns is observed, which corresponds to the relative optical path delay of the two arms of the BPFs, demonstrating true coincidence counts due to SPDC. The ﬁnite width of the peak is due to the electronic jitter of the detection system. The total electronic jitter is estimated to be 380 ps. A detailed study of the electronic jitter has shown that each detector had a 200-250 ps jitter, and that there was another 200 ps jitter due to the FPGA. As shown in Fig. 4.7a, the true coincidence is much higher than the accidental coincidence (including the contributions from the dark counts and all other noise photons), and a high CAR (113) was measured.

In Fig. 4.7b, the CAR is plotted versus the pump power inside the sample (blue cir- cles). One can observe a positive slope for the low input powers. In order to interpret this result, we calculate the ratio of recorded true coincidences fαnl2 to accidental coincidences, due to single photons remaining from lost pairs and dark counts (αnl + d)2R, assuming that there is no fluorescence or other type of optical noise passed by the BPFs, where α is the conversion efficiency, n is the number of pump photons per second, l is the ratio of detected photons to generated photons (10−3 corresponding to the overall loss coefficient), d is the dark count of the detectors (measured to be around 2000 counts per second) and R is the resolution of the TIA (500 ps). The factor of f (for our filter pass bands and pump wavelength, 0.45) accounts for the reduction in effective bandwidth for coincidence measurements due to the non-optimal BPFs. This ratio is as follows:

αnl2 CAR = f (4.5) (αnl + d)2R

The pump power at which the maximum CAR occurs is used to extract a brightness of 1.87 × 106 down converted photon pairs per second collected within the 7.2-nm eﬀec- tive bandwidth. An excellent agreement between theory and measurement is obtained, despite the fact no other normalization factor is used. I experimentally conﬁrmed that the photon pairs are generated through type-I phase Chapter 4. Low noise on-chip source of correlated photons 59

(a) (b)

Figure 4.7: (a) The typical time-bin histogram for coincidence measurements (b) CAR as a function of coupled pump power. matching. When the output is passed through a polarizer with its axis aligned to TM- polarization, the CAR dropped to one, indicating that no true coincidence events were observed. The phase-matching bandwidth was investigated by performing the CAR measurement as a function of the pump wavelength (at a pump power of 6 mW), with the results shown in Fig. 4.8 Increasing the pump wavelength beyond the degeneracy point (corresponding to SHG) results in the phase-matching condition no longer being satisfied. Decreasing the pump wavelength from the degeneracy point by only around 1 nm results in a tuning of the signal and idler wavelengths of ∼100 nm. Therefore, the signal and idler wavelengths no longer coincide with the effective passband of the BPFs. Both the phase-matching bandwidth and its asymmetry are in accord with the predicted tuning curves for the difference frequency generation reported in reference [31]. To show the importance of this work, I compare our results with those recently reported in [101, 112, 100] in table I. It is shown that this work offers higher CAR and brightness compared with all other photon sources in AlGaAs platform.

4.6 Discussion and conclusion

In this chapter, the QWI quasi-phase-matching method in a GaAs/AlGaAs superlattice waveguide has been used to implement a source of low-noise correlated photon pairs with Chapter 4. Low noise on-chip source of correlated photons 60

Figure 4.8: CAR versus pump detuning. Figure inset: shows the spectral location of the collected pair photons (hatch pattern)

CW pumping. There are three main advantages for using QPM AlGaAs superlattive waveguides. First, the large band-gap of AlGaAs superlattice reduces the PL intensity. As a result, the noise photons level is also reduced. Second, with confining both pump photons and down converted photons in the core of the waveguide, a high overlap is achieved. It increases the conversion efficiency. Third, the periodicity in the waveguides will be determined post growth so PPS with different emmission wavelengths can be built on the same wafer. Thanks to these advantages, a CAR as high as 115 and a brightness over one million pairs per second is achievable. There are some modifications for improving the rate of production of the correlated photon pairs which can be considered for future works. Selecting BPFs that are frequency-conjugated at double the pump wavelength would significantly improve the measured CAR. Using a 3 dB coupler instead of 1570.0 nm/1530.0 nm BPFs resulted in about an 8-fold improvement in the brightness of the SPDC source at the expense of lower CAR due to an increase in noise photon detection events. The number of total true coincidences per unit time can be significantly increased by reducing the overall Chapter 4. Low noise on-chip source of correlated photons 61

Table 4.1: Comparison with other AlGaAs based entangled photons sources

This work [101] [112] [100] Planar Bragg- Bragg- Multilayer Structural GaAs/AlGaAs reflection reflection AlGaAs ridge Type superlattice waveguide waveguide waveguide waveguide 2 @ 113 @ Not reported 6.43 × 105 19.25 @ 700 1.87 × 106 (very low by CAR @ given pairs/s And pairs/s Or @ pairs/s Or @ inferring from brightness 20 @ 583 2.6 × 105 supplied ∼ 6.4 × 104 pairs/s/nm pairs/s/nm data) (a) pairs/s CW laser Pulsed laser CW laser Pulsed laser Pump source source source source Potential for High Moderate Moderate Low integration optical losses in the system and, in particular, improving the optical coupling between the waveguide and the collection fiber. In addition to these quick modifications for en- hancing the performance of the system, one can integrate the pump laser on the chip. This is possible as compatible laser sources have already been built on the same type of wafers [129]. This makes the reported technique a practical solution for a future on-chip, electrically-pumped, planar integrated, high-quality, turn-key CW entangled photon pair source. Chapter 5

On-chip Time-Energy Entangled photon pair source

Quantum information science has been developed rapidly during the last couple of decades due to the unique potential of quantum systems. Today, quantum states can be stored, processed or transmitted and thanks to their special quantum properties, many classically impossible tasks are now possible. For example, a quantum system allows for polynomial time algorithms for solving some problems which are computationally infeasi- ble to solve by using an ordinary computer. Another application is secure communication. Today, commercially available quantum key distribution systems are offering enhanced security in communication guaranteed by the laws of quantum mechanics. To match the momentum in the technological evolution, quantum systems need to be integrated on chip. This offers compactness, robustness and reliability. In many quantum systems, entanglement plays an important role. Super-dense coding and quantum teleportation are two of the most well-known applications of entanglement [93]. This validates the increasing effort behind the generation of entangled photons witnessed in the last couple of decades. More recently, on-chip generation of entangled photons attracted broad attention. This entanglement can be encoded in polarizations, paths and times of arrival. The time-bin entangled photon pairs has the advantage of being tolerant to polarization drifts in the optical fibers [130] while in polarization entangled photon pairs, the quantum state can be manipulated more easily. Attempts to

62 Chapter 5. On-chip Time-Energy Entangled photon pair source 63 build both of theses sources on-chip on diﬀerent platforms are growing. For example, time-bin entanglement [98, 131] and polarization entanglement [107] has recently been demonstrated on-chip in a silicon platform. The other interesting material is AlGaAs, which was introduced in the previous chapter, and I discussed how a low noise correlated PPS can be built using this material. This material has also been used for the purpose of polarization entanglement elsewhere. The high nonlinearity in AlGaAs allows for a highly bright [132] source of polarization-entangled photons. Also, the direct bandgap in this material permitted the monolithic integration of a pump laser, which results in an electrically injected photon-pair source at room temperature [102]. These last two works arguably demonstrate that AlGaAs is the best platform for on-chip entangled photon generation. Despite all the recent developments, prior to this work, no time-bin entanglement has been demonstrated based on an AlGaAs platform. In this chapter, time-bin entanglement is demonstrated using QPM AlGaAs superlattice waveguides. A two photon interference is implemented using Franson interferometry. In the next sections, ﬁrst the theory for Frasnon interferometer is studied. Second, the constraints and challenges for building a Franson interferometer are discussed. Third, time-bin entanglement in the proposed PPS using the Franson interferometer is validated. The material of this chapter was partially published in Optics Letters [133].

5.1 Franson Interferometry

After Bell’s inequality was proposed in 1964 [87], many attempts were made to measure the time correlation of down-converted photons. In 1987, the Hong-Ou-Mandel eﬀect, which is a two-photon interference eﬀect, was demonstrated. However, in this method both photons need to be present at the same location and, therefore, it is called a local measurement. Later, a truly nonlocal experiment using fourth order interference in a Franson interferometer [134] was proposed where two photons under experiment were spatially far away. In short, the two photons generated in an EPR source are sent to two imbalance Mach-Zehnder interferometers (MZIs) and the coincidence measurement at the output of the two MZIs will be carried out. In this work, a non-local Franson Chapter 5. On-chip Time-Energy Entangled photon pair source 64

Figure 5.1: Model of a Franson interferometric measurement, D1 and D2 are single photon detectors. interferometry measurement for characterizing the quantum eﬀect of the down-converted photons has been performed. A schematic model of this setup is shown in Fig. 5.1.

There are some constraints in Franson interferometry experiment that have to be satisﬁed. First, there need to be a large enough imbalance between MZI’s arms to prevent any single photon interference. Second, this imbalance should be large enough that the delay between two photons, one taking the shorter arm and the other taking the longer arm, is observable with single photon detectors (SPDs) and the time interval analyzer. Third, the pump coherence time has to be larger than the MZIs’ imbalance. The way each constraint is treated will be explained in the next sections. One ﬁnal important constraint is derived following the derivation in [134], which will be explained below.

† † Down-converted photons with ﬁeld operator a1 and a3 are passed through the MZIs,

† iφ12 † † iφ34 † and the field operators of 0.5(a1(t−τ1)−e a1(t−τ2)) and 0.5(a3(t−τ3)−e a3(t−τ4)) are obtained, where τ1 and τ3 are the imbalance in the first and second MZI, respectively, and φ1 and φ2 are the relative phases added to the longer arm of the first and second MZI, respectively. The wavefunction at the output of the MZIs is given by

1 ZZ |ψi = A(t, t0)(a†(t−τ )−eiφ1 a†(t−τ ))(a†(t−τ )−eiφ2 a†(t−τ ) |0i dt dt0 (5.1) 4 1 1 1 2 3 3 3 4 1 1 Chapter 5. On-chip Time-Energy Entangled photon pair source 65

where |0 > is the vacuum function and A(t, t0) is the correlation function between † † a1 and a3. If only two indistinguishable cases, where either both photons propagate through the longer arms or both propagate through the shorter arms, are considered and all remaining cases are ﬁltered, the wavefunction can be simpliﬁed to:

1 ZZ |ψi = A(t, t0)(a†(t − τ )a†(t − τ ) + eiφ1 eiφ2 a†(t − τ )a†(t − τ )) |0i dt dt0 (5.2) 4 1 1 3 3 2 2 4 4 1 1

Here, the importance of the ﬁrst constraint emerges. In case the MZI’s imbalance is smaller than the photon coherence length, extra terms corresponding to the interference of each photon with itself appears in the above equation and perturbs the calculation.

If the coherence time of the pump photon is larger than the MZI’s imbalance, two terms can cancel each other out depending on the value of φ1 + φ2. The fourth order interference between two detectors is:

† † † P1,3 ∝ hψ| a1(t1)a2(t1 − τ)a2(t1 − τ)a1(t1) |ψi (5.3)

Using [a(t), a†] = δ(t−t0), and dropping the integral due to long detection response of the detector (much longer than the correlation time), the above equation can be further simpliﬁed to:

2 2 P1,3 ∝ |A(τ1 − τ3 − τ)| + A(τ2 − τ4 − τ)|

i(φ1+φ2) ∗ −i(φ1+φ2) ∗ + e A (τ1 − τ3 − τ)A(τ2 − τ4 − τ) + e A (τ2 − τ4 − τ)A(τ1 − τ3 − τ) (5.4)

(τ1−τ3)+(τ2−τ4) (τ1−τ3)−(τ2−τ4) Deﬁning ∆τ = 2 , δτ = 2 and φ = φ1 + φ2, the two-photon detection rate is:

R dτA(∆τ − τ − δτ)A(∆τ − τ + δτ) R ∝ 1 + cosφ (5.5) 1,3 R dτ|A(∆τ − τ)|2

Assuming the spectral passbands of the interference filters is Gaussian, the temporal Chapter 5. On-chip Time-Energy Entangled photon pair source 66 correlation function is A(t, t0) = e−(t−t0)2∆2/2, where ∆ is the spectral bandwidth of the filters. So the two-photon detection rate is further simplified to:

−δτ 2∆2 R1,3 = 1 + cosφe (5.6)

If the mismatch between two MZIs is equal to zero (δτ = 0), a 100% visibility fringes is achievable. However, if δτ >> 1/∆, the fringe disappears. To violate any classical interpretation of this experiment, high visibility is needed, so it is very important to adjust the MZI’s mismatch to be near zero. This actually adds a forth constraint if one uses Franson interferometry to verify the entanglement property. As will be explained in the next sections, this has been done experimentally with accurate global heat control of both MZIs.

5.2 Franson interferometer: meeting constraints and overcoming challenges

As theoretically discussed in the last section, performing a Franson interferometer’s measurement needs careful consideration of its constraints. Also, there are some experimental challenges which have to be overcome. The first theoretical constraint mentioned in the previous section can be satisfied relatively easy. Assuming the single photon bandwidth is more than 0.1 nm in the telecom frequency, a MZI’s imbalance of 100 ps is enough. The single photon bandwidth is usually much wider (1-100 nm), in my experiment, the filter bandwidth is about 8 nm. In practice, being able to satisfy the second constraint is dependent on the equipment’s quality. In my experiment, TIA has a good resolution of 64 ps but there is around 200 ps jitter applied by the detector. Therefore, choosing a minimum MZI’s imbalance of 500 ps in needed to satisfy this requirement. The third requirement can be met by using a low bandwidth pump laser. As the lower band of the MZI’s imbalance is about 500 ps in our case, the pump laser bandwidth has to be narrower than 2 GHz. The last constraint mentioned at the end of the previous section is much more difficult to meet. Chapter 5. On-chip Time-Energy Entangled photon pair source 67

This constraint says the mismatch between the two MZI’s imbalance has to be much smaller than the coherent time of the single photon. Assuming a bandwidth of 10 nm (in a 1550 nm band) for the single photon, the coherent time is a fraction of a ps. As shown above, the MZI’s imbalance has to be 500 ps or larger. In the next subsection, I discuss how to meet this criteria. In the subsequent sections, the experimental complexities, and solutions to these complexities are described.

5.2.1 Matching the MZIs’ imbalance using reference ﬁber-based MZI

As theoretically veriﬁed, one of the Franson interferometer requirements to achieve ac-

(τ1−τ3)−(τ2−τ4) ceptable visibility is |δτ| << 1/∆ [134], where δτ = 2 and ∆ is the spectral

bandwidth of the ﬁlters. τ1 and τ2 are the delay of the short and long arm of one of the

MZIs, respectively, and τ3 and τ4 are the delay of the short and long arm of the other MZI, respectively. Assuming the spectral bandwidth of each filter equals 7.2 nm (this will later be used in an actual experiment), the MZI’s imbalance mismatch, τ, has to be much less than 1.1ps. To balance the MZIs, they are compared with a reference fiber- based MZI and their imbalance is adjusted to the same value. The fiber-based MZI has a measurable tunable delay on one arm. To do this, a configuration shown in Fig. 5.2 was built. The white light source is a broad-band source (> 100nm) with a certain coherence length, so in case τ varies by a sufficient amount, the interference pattern observed by the detector will be suppressed. Using temperature controllers on the MZIs under the test, the peak of the interference pattern can be tuned. In Fig. 5.3, the interference pattern for each MZI, recorded by an IR detector, is plotted versus the delay in the reference MZI. A visibility of ∼ 99% is observed. In this figure, both peaks occur when the delay on the reference MZI is 86.75ps . Therefore, MZIs are balanced by an accuracy of ∼ 50fs, which is much smaller than the maximum mismatch time, limited by the coherence time of down-converted photons (1.1ps). Therefore, it satisfies the fourth constraint. Chapter 5. On-chip Time-Energy Entangled photon pair source 68

Figure 5.2: Schematic setup for comparing the path diﬀerence

(a) (b)

Figure 5.3: Interference pattern for (a) ﬁrst MZI and (b) second MZI as a function of delay in reference MZI.

5.2.2 MZI characterization: phase-voltage relation

The MZI’s phase delays are controllable using built-in heaters, yet the phase-voltage relationship has to be characterized. To ﬁnd the phase-voltage relationship, a setup as shown in Fig. 5.4 has been conﬁgured. In this setup, a tunable laser swept the wavelength of the input light and the output was recorded using an IR detector. The MZI was stabilized using the temperature controller system, which controls the global temperature of the MZI. The interference pattern at the output of the MZIs is recorded as a function of wavelength for a set of voltages applied to the heater. In Fig. 5.5 the interference pattern Chapter 5. On-chip Time-Energy Entangled photon pair source 69

Figure 5.4: Schematic setup for MZI characterization Chapter 5. On-chip Time-Energy Entangled photon pair source 70

Figure 5.5: Phase change in interference pattern as a function of voltage V=(0, 1.8, 3, 5, 6, 6.4, 6.5, 7, 7.5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18.6). Pink solid line shows the pattern for V=0 and as voltage increases, it shifts toward the right. for many diﬀerent voltages is shown. The phases of these sinusoides are extracted and plotted in Fig. 5.6. The best quadratic ﬁt to the data is φ = v2 + 0.13v + 0.68.

5.2.3 High brightness and detector’s dead time limitation

In this section we discuss two more experimental challenges. The ﬁrst one is the added loss due to MZIs. The MZIs used for the Franson interferometry add 6 dB of loss. During the coincidence measurements in the last chapter, about two photon pairs are generated each second for the highest CAR. It is always possible to increase the pump power but the CAR is reduced due to multi-pair generation (4.5). It is also possible to perform the experiment with the added loss, but the integration time (the time needed for each measurement) will increase. As tens of data points are needed for a Franson Chapter 5. On-chip Time-Energy Entangled photon pair source 71

Figure 5.6: Phase change as a function of voltage applied to the heater

measurement, this makes the experiment very time consuming. One alternative is using a waveguide with a higher conversion rate. A waveguide with a smaller width of 2µm is used for this purpose. However, it compromises the CAR. The reason can be that the waveguide mode spreads out and interferes with defect states created between the lower layers. Also, to increase the efficiency, an Au ion implantation mask with a duty cycle of 50 percent (instead of 60 percent) is used. This waveguide was pumped with a 8 mW laser beam, and its coincidence histogram is shown in Fig. 5.7. This figure shows a 5-fold enhancement in brightness (8 × 106 pairs/s in a 8-nm bandwidth). The CAR, however, is 44.5 which is less than half of what was previously measured. The pump power is increased further to achieve even higher brightness. A histogram for 16 mW pump power (twice of the previous one) is plotted in Fig. 5.8. On first reflection, it seems surprising that the number of coincidences did not double, since during a SPDC process, the number of down-converted photons has to be a linear function of pump power. Further investigation showed another practical limitation due to single photon detectors (SPDs). In fact, each SPD has a dead time. The dead time is the time after Chapter 5. On-chip Time-Energy Entangled photon pair source 72

Figure 5.7: A time-bin histogram for a 2µm wavguide with 8 mW pum power.

each photon detection during which the SPD is not able to record another event. The coincidence count can be corrected as follows:

Cm Cc = (5.7) (1 − N1τ)(1 − N2τ) where, Cc and Cm are the corrected and measured coincidences, respectively, N1 and

N2 are single counts per second for the first and second SPDs, respectively, and τ is the detector’s dead time. In this measurement (using the 16 mW pump power), Cm is measured to be 1939. N1 and N2 are 32.6 K and 28.8 K, respectively. Regarding the 15µs dead time for detectors, the Cm is calculated to be 6680. I should note that all reported results are based on measured coincidences. Also, the pump power has been held low enough to avoid any significant effect of the detectors’ dead time. Given the detector deadtime and the CAR and brightness dependence on pump power, the optimal pump power has to be around 8 mW, so one can maximize the brightness without a significant reduction of CAR. Chapter 5. On-chip Time-Energy Entangled photon pair source 73

Figure 5.8: The time-bin histogram for a 2µm wavguide with 16 mW pum power.

5.3 Time-Energy Entanglement and a measurement of visibility

In previous sections, all the requirements and constraints for a Franson interferometry experiment are discussed. Moreover, the coincidence measurement results for a low noise but bright PPS were reported. However, the quantum properties of generated entangled photons were not yet characterized. In this section, the time entanglement property of the down-converted photons is demonstrated using a Franson interferometer [134]. All the constraints are met according to the calculations in the previous sections. For the following experiment we have used a more tightly conﬁned waveguide with 2 µm width and a QPM periodicity of 3.5 µm and a duty cycle of 50 percent. As a result, we achieved a 5-fold enhancement in brightness (8 × 106 pairs/s) for the same pump power (8 mW).

The schematic of our experimental setup is shown in Fig. 5.9. The pump is a 773 nm CW laser (Toptica DL pro) with a linewidth of smaller than 1 MHz, and is launched into the waveguide at TM polarization. The waveguide output, containing type I down- Chapter 5. On-chip Time-Energy Entangled photon pair source 74 converted photon pairs, is sent through a polarizer (passing TE) and a pump-suppression filter. The pump suppression filter can be a fiber based filter or more simply a 1 mm-thick layer of GaAs in free space. Note that in comparison to four-wave mixing (χ(3)) processes (for example, silicon-on-insulator platforms), in three-wave mixing (χ(2)) processes, as exploited here, pump suppression filtering is far simpler as the pump wavelength is not in proximity to the signal and idler wavelengths [107, 108]. After pump filtering, fiber- based band-pass filters (BPF) [107, 108] are then used to deterministically separate the photon pairs into two 16-nm spectral bands, centered on 1570 nm and 1530 nm. (Note the choice of the BPFs is limited by availability, and the BPFs are not exactly symmetric with respect to the degenerate wavelength of 1546.0 nm.) As a result, the portion of the filter passbands that are frequency conjugate reduce the effective bandwidths to 8 nm.

Each of the BPFs sends its output to a commercial unbalanced silica Planar Light- wave Circuit (PLC) MZI. The two MZIs form a Franson interferometer which is used to characterize the entangled nature of the photon pairs. The delay difference between the short and long arms in both MZIs is 500 ps (2 GHz). Each of the MZIs is equipped with a Peltier cell to adjust and stabilize the overall device temperature. These Peltier cells have been used to match the time delay differences between the MZIs to an accuracy of better than 50 fs. This difference has been measured in a separate setup using white light interferometry (Sec. 5.2.1) and utilizing a third reference MZI with a tunable delay on one arm. Additionally, on the long arm of each MZI, a local heater is used to accurately control the phase difference over a range of 0 − 360o. Characterization of these heaters shows that the relative phase is proportional to the square of the applied voltage, as expected (Sec. 5.2.2). The output of these two MZIs are connected to two, free-running, single photon detectors (SPDs, IDQ id220), with measured quantum efficiencies of 20% at 1550 nm. The SPD output electrical signals are sent to a commercial time interval analyzer (TIA, Picoquant Hydraharp) to record the difference in photon arrival time. The TIA has a measurement resolution of 64 ps which acts as a narrow time domain filter [135].

The total estimated loss associated with the waveguide-to-free-space coupling, TE polarizer, free- space-to-ﬁber coupling, BPF and MZI is ∼ 20 dB (each branch) at 1550 Chapter 5. On-chip Time-Energy Entangled photon pair source 75

Figure 5.9: Schematic of the setup for Franson interferometry used to characterize the AlGaAs-waveguide-based photon pair source.

nm. There is also a ∼ 5 dB pump-to-waveguide coupling loss. The total length of fibers for each branch is less than 8 metres, thus the effects of dispersion can be neglected. Ad- ditionally, Franson interferometry requirements are satisfied [134], as the pump linewidth (1 MHz) is smaller than the free-spectral range of the MZIs (2 GHz), which is smaller than the spectral extent of the signal or idler photon (∼ 10 nm at 1550 nm, or 1.3 THz). In Fig. 5.10, two typical histograms show the coincidence measurements recorded by the TIA, corresponding to two different sets of voltages applied to the MZI heaters. For the data in Fig. 5.10a, the applied phase differences to long arms of MZI 1 and 2 are 270 and 180 degrees, respectively. For the data in Fig. 5.10b), the applied phase differences to long arms of MZI 1 and 2 are 90 and 180 degrees, respectively. As shown in Fig. 5.10, the recorded histogram has three coincidence peaks. The left and right peaks correspond to a state when one photon goes through the short arm in one MZI and the other through the long arm in the other MZI. The middle peak results from the interference between the state where both photons traverse the short arms and the state where both traverse the long arms of MZIs. This peak can be interpreted as the result of fourth order quantum interference of the following post-selected state [134]:

(a) (b)

o o Figure 5.10: The result of coincidence measurements at (a) φ1 = 270 , φ2 = 180 , and o o (b) φ1 = 90 ,φ2 = 180 . The red bins are used to calculate the coincidences for Fig. ?? of coincidences in the middle peak exhibits an interference fringe of unity visibility when the relative phases of the two MZIs are varied [134]. The two phases can be chosen to maximize (Fig. 5.10a) or minimize (Fig. 5.10b) the middle peak height. A 212 ps electronic jitter, dominated by the 150 ps jitter of each SPD, limits the measurement accuracy of the arrival time diﬀerences between photons pairs. In Fig. 5.11, the coincidence counts corresponding to the middle peak, as indicated by the 3 red time-bins

(each 64 ps) in Fig. 5.10, have been plotted as a function of φ1 for set values of φ2 of 0, 90 and 180 degrees. Choosing three bins is a result of compromise between maximizing the coincidence counts of the central peak and minimizing the inﬁltration of counts from the adjacent peaks. The total width of the 3 bins is also equal to total electronic jitter. Also, I have used chi-square data ﬁtting to plot the solid lines in Fig. 5.11. Sinusoidal

o o visibilities, V (φ2), of 96.0±0.7% for φ2 = 180 and 94.3±0.7% for φ2 = 90 are obtained. The tolarance in visibility was calculated using the fitting error. The observed difference from 100% visibility is mainly due to the accidental coincidence (shown as dashed line in Fig. 5.11), and would increase to ∼ 99% with the subtraction of a constant accidental background. Note here the contributions to the noise include dark counts (2 × 103 per second per detector) and background fluorescence (8 × 103 per second per detector). The coincidence counts corresponding to the left (short-long arm interference) and Chapter 5. On-chip Time-Energy Entangled photon pair source 77

Figure 5.11: Two photon interference measurements for three phase settings on the second MZI.(a) Middle peak coincidences shown in Fig. 3 plotted as a function of φ1 for three ﬁxed φ2 values. (b) Left and right peak coincidences shown in Fig. 5.10 plotted as o a function of φ1 for φ2 = 180 Chapter 5. On-chip Time-Energy Entangled photon pair source 78 right (long-short arm interference) peaks in Fig. 5.10 remained essentially unchanged during the experiment (Fig. 5.11b), verifying that the sinusoidal variation seen in the middle peak (Fig. 5.11a) is due to quantum interference. A Clauser-Horne-Shimony-Holt (CHSH)-Bell inequality [136] of S = 2.687 ± 0.013 is obtained from the raw visibilities, √ o o V (φ2),S = 2(V (90 ) + V (180 )). It demonstrates the violation of Bell’s inequality by more than 52 standard deviations, conﬁrming that down-converted photon pairs produced by the quasi-phase-matched AlGaAs superlattice waveguides are time-energy entangled.

5.4 Discussion and conclusion

In this chapter, first, the theoretical background for the Franson interferometer is presented. Second, both experimental and theoretical constraints for a Franson interferometry experiment are investigated. The main difficulty is compensating for the imbalance mismatch between two MZIs. Both MZIs have been aligned to a reference MZI using TECs. Following the optimization of experimental conditions, two photon interference measurements obtained from a Franson interferometer have been used to characterize a CW- pumped, QPM, AlGaAs superlattice entangled photon pair source. The high raw visibility of approximately 95% (without background subtraction) indicates the high-purity, low-noise feature of this source, which has not been observed to date in any other type of AlGaAs-based photon pair source. Given its high brightness (8 × 106pairs/s), achieved with relatively low pump power (8 mW), its potential to be integrated planarly with a pump source, and its well-behaved mode profiles for fiber or waveguide coupling, this source proves a practical solution for a future on-chip, electrically-pumped, planar integrated, high-quality, turn-key CW entangled photon pair source. Chapter 6

Conclusion, contributions and future works

6.1 Conclusion and original contributions

In this thesis, a broad range of applications of periodic nonlinear structures both in the classical and quantum regime have been studied. The theoretical and experimental studies are based on resonant nonlinearity and non-resonant nonlinearity in semiconductors, respectively. The purpose of this study has been to introduce the potential for broader application of nonlinear periodic structures in classical optical signal processing and quantum optics circuits. I proposed and developed the Time-Domain Transfer-Matrix method for the analysis of semiconductors with resonant nonlinearity. The time-domain property of this method helps to deal with the rate equation, which models the material property of semi-conductors. Also, the frequency-domain nature of the standard transfer matrix method facilitates inclusion of the material dispersion property. This method is ideal for relatively long pulse propagation and outperforms the FDTD method for all pulses longer than 200 fs. I have proposed and designed a new optical limiter based on resonant nonlinearity in semi-periodic structures. The concept of a non-trivial interface shift has been used for this design. This optical limiter is as compact as a few micro-meters and as sensitive as

79 Chapter 6. Conclusion, contributions and future works 80

50MW/cm2. These properties are accessible due to the large nonlinear absorption of the materials and the frequency selectivity of the geometry. In a separate work, for the first time, I proposed and demonstrated the idea of using QPM AlGaAs superlattice waveguides for a quantum photonic source has been conceived. These waveguides can be fabricated post-growth next to other components including the pump laser built on the same wafer. This is the only practical and suitable solution for planar pump integration on the same chip. Moreover, I experimentally demonstrated that the proposed entangled photon pair source offers the highest coincidence to accidental ratio (∼113) in AlGaAs systems. The highest internal brightness of the source is 10 million photon pairs per second in a 7.2 nm bandwidth. A Franson interferometer has been used to investigate the entanglement property of the photon pairs generated in the above source. A visibility of 96±0.7 and 94.3±0.7 has been achieved, which shows the quantum interference. The Bell-inequality was violated by 52 standard deviations, which is the highest among all AlGaAs devices. My work described above resulted in the following peer-reviewed publications: Journal papers: P. Sarrafi, Eric Y. Zhu, Barry M. Holmes, David C. Hutchings, J. Stewart Aitchison and Li Qian, ”High visibility two-photon interference of time entangled photons in a quasi-phase-matched AlGaAs waveguide” Optics letters, vol. 39, no. 17, pp. 5188-5191, 2014. P. Sarrafi, L. Qian, ”Optical limiting effect in layered structure based resonant nonlinearity”, pending. P. Sarrafi, Eric Y. Zhu, Ksenia Dolgaleva, Barry M. Holmes, David C. Hutchings, J. Stewart Aitchison and Li Qian ”Continuous-wave quasi-phase-matched waveguide correlated photon pair source on a III-V chip”, Applied Physics Letters, vol. 103, no. 25, p. 251115, 2013. P. Sarrafi, L. Qian, ”Modeling of Pulse Propagation in Layered Structures with Reso- nant Nonlinearities Using a Generalized Time-Domain Transfer Matrix Method ” Quan- tum Electronics, IEEE Journal of , vol.48, no.5, pp.559-567, May 2012. Conference papers: Chapter 6. Conclusion, contributions and future works 81

L Qian, E. Y. Zhu and P. Sarrafi, ”Practical Entangled Photon Sources” Invited paper LPHYS’14. P. Sarrafi, Eric Y. Zhu, Barry M. Holmes, David C. Hutchings, J. Stewart Aitchison and Li Qian ”High visibility two-photon interference of time entangled photons in a quasi-phase-matched AlGaAs waveguide” CLEO 2014. P. Sarrafi, Eric Y. Zhu, Barry M. Holmes, David C. Hutchings, J. Stewart Aitchison and Li Qian ”Generation of Entangeled photons in QPM AlGaAs Waveguides” photonic north 2014. J. S. Aitchison, P. Sarrafi, et al. ”Wavelength Conversion and Correlated Photon Generation in Quasi Phase matched AlGaAs Waveguides.” Invited, Photonics North 2013. P. Sarrafi, Eric Y. Zhu, K. Dolgaleva, Barry M. Holmes, David C. Hutchings, J. Stewart Aitchison and Li Qian ” Continuous wave correlated photon pairs generation in quasi-phase-matched superlattice AlGaAs waveguides ” CLEO 2013. K. Dolgaleva, P. Sarrafi, P. Kultavewuti, J. Aitchison, L. Qian, M. Volatier, R. Ares, and V. Aimez, ”Highly Efficient Broadly Tunable Four-Wave Mixing in AlGaAs Nanowires,” CLEO 2013. K. Dolgaleva, P. Sarrafi et al. ”Efficient Tunable Four-Wave Mixing in AlGaAs Nanowires” Photonics North 2013. P. Sarrafi, L. Qian, ”Time-domain analysis of pulse propagation in high-contrast layered structures with resonant nonlinearities,” CLEO, May 2011.

6.2 Future work

I have shown resonant nonlinear periodic structures have great potential and I have provided accurate and fast modeling tools for future device design. My model can be used in the future for sophisticated all-optical device design, such as optical logic XORs. Moreover, the design of proposed optical limiter is a proof of concept that the novel nonlinear ramification of interface phase shift can increase sensitivity and decrease the device size significantly. The same concept can be exploited for the design of other useful Chapter 6. Conclusion, contributions and future works 82 optical signal processing devices. In addition, I have experimentally demonstrated that QPM-AlGaAs is a viable platform for quantum sources, and has superior performance compared to alternative devices based on AlGaAs. My work can be expanded to create polarization entangled on-chip photon sources using type-II phase matching. Also, fabrication of both the pump laser and filters on the same chip is a feasible and interesting extension of this work. References

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