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Classical and quantum nonlinear optics in confined photonic structures Ghafari Banaee, Mohamadreza 2007

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Classical and Quantum Nonlinear Optics in Confined Photonic Structures by Mohamadreza Ghafari Banaee B.Sc., Tehran University Iran, 1991 M.Sc., The University of British Columbia, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Physics) The University of British Columbia November 2007 Mohamadreza Ghafari Banaee, 2007 Abstract Nonlinear optical phenomena associated with high-order soliton breakup in photonic crystal fibres and squeezed state generation in three dimensional photonic crystal microcavities are investigated. In both cases, the properties of periodically patterned, high-index contrast dielectric structures are engineered to control the dispersion and local field enhancements of the electromagnetic field. Ultra-short pulse propagation in a polarization-maintaining microstructured fibre (with 1 um core diameter and 1.1 m length) is investigated experimentally and the- oretically. For an 80 MHz train of 130 fs pulses with average propagating powers in the fibre up to 13.8 mW, the output spectra consist of multiple discrete solitons that shift continuously to lower energies as they propagate in the lowest transverse mode of the fibre. The number of solitons and the amount that they shift both increase with the launched power. All of the data is quantitatively consistent with solutions of the nonlinear SchrOdinger equation, but only when the Raman nonlinearity is treated without approximation, and self-steepening is included. The feasibility of using a parametric down-conversion process to generate squeezed electromagnetic states in 3D photonic crystal microcavity structures is investigated for the first time. The spectrum of the squeezed light is theoretically calculated by using an open cavity quantum mechanical formalism. The cavity communicates with two main channels, which model vertical radiation losses and coupling into a single- mode waveguide respectively. The amount of squeezing is determined by the correla- tion functions relating the field quadratures of light coupled into the waveguide. All of the relevant model parameters are realistically estimated using 3D finite-difference 11 Abstract^ iii time-domain (FDTD) simulations. Squeezing up to ,20% below the shot noise level is predicted for reasonable optical excitation levels. To preserve the squeezed nature of the light generated in the microcavity, a uni- directional coupling geometry from the microcavity to a ridge waveguide in a slab photonic crystal structure is studied. The structure was successfully fabricated in a silicon membrane, and experimental measurements of the efficiency for the signal cou- pled out of the structure are in good agreement with the result of FDTD simulations. The coupling efficiency of the cavity mode to the output channel is ,s,60%. Contents Abstract  ^ii Contents ^  iv Abbreviat ions  vii List of Tables ^  viii List of Figures  ix Acknowledgements ^  xiv Co-Authorship Statement ^  xv 1 Introduction ^1 Bibliography ^  12 2 High-order soliton breakup in a microstructured optical fibre . . . 15 2.1 Experimental setup   17 2.2 Output spectral dependence on input power and propagation distance 20 2.3 Numerical simulation   24 Bibliography ^  34 iv 3 Squeezed state generation in photonic crystal microcavities . . . . 38 3.1 Field quantization in an open optical cavity ^  40 Contents^ v 3.2 Parametric down-conversion in an open cavity ^ 43 3.3 Numerical estimation of the squeezing spectrum  52 3.3.1 Degenerate parametric down-conversion ^ 54 Bibliography ^  63 4 Efficient coupling of photonic crystal microcavity modes to a ridge waveguide ^  67 4.1 Design and numerical simulation of the waveguide-microcavity structure 69 4.1.1 Power coupled out of the grating coupler ^ 73 4.2 Fabrication procedure^  75 4.3 Optical Characterization  78 Bibliography ^  83 5 Summary and conclusions ^  85 A Third-order nonlinear effects in optical fibres ^  88 A.1 Self-phase modulation ^  91 A.2 Raman scattering  93 A.3 Generalized nonlinear SchrOdinger equation^  93 A.4 Self-steepening ^  99 A.5 Solitonic solutions of the GNLS equation ^  101 A.6 Perturbation of solitons ^  104 A.6.1 Soliton self-frequency shift ^  106 B Frequency resolved optical gating  108 B.1 Generalized projection algorithm for pulse retrieval in the FROG method ^  111 Contents^ vi C Field quantization in open optical cavities ^  115 C.1 Feshbach projection ^  116 C.2 Hamiltonian of the system-reservoir ^  119 D Quadrature correlations in optical parametric down-conversion . 122 D.1 Squeezing of the quantum fluctuations ^  122 D.2 Strong correlation with squeezed states  126 E Spectrum of squeezing ^  129 Abbreviations CW: Continuous Wave DPDC: Degenerate Parametric Down-Conversion DWDM: Dense Wavelength Division Multiplexed EPR: Einstein-Podolsky-Rosen FDTD: Finite-Difference Time-Domain FROG: Frequency Resolved Optical Gating FT: Fourier Transformation GNLS: Generalized Nonlinear SchrOdinger GVD: Group Velocity Dispersion NLS: Nonlinear SchrOdinger NPDC: Nondegenerate Parametric Down-Conversion OPO; Optical Parametric Oscillator PCF: Photonic Crystal Fibre PDC: Parametric Down-Conversion PhC: Photonic Crystal RSSS: Raman Self-Shifting Soliton SEM: Scanning Electron Microscope SHG: Second Harmonic Generation SOI: Silicon On Insulator SPM: Self-Phase Modulation SSF: Split-Step Fourier SSFS: Soliton Self-Frequency Shift vii List of Tables viii 2.1 Optical elements used in the experiment^  19 List of Figures 1.1 A scanning electron micrograph, in cross section, of the photonic crys- tal fibre used in this thesis.  ^4 1.2 Down-conversion process in a 2D photonic crystal waveguide-microcavity structure.  ^8 1.3 SEM image of the fabricated microcavity-waveguide structure^ 10 2.1 Experimental setup.   18 2.2 Experimental output spectra of the photonic crystal fiber pumped at different average propagating powers (shown at right). The spectra marked with a) and b) have the most shifted and the second most shifted solitons centred at 850 nm respectively  20 2.3 Frequency shifts of the experimental peaks in Fig.(2.2), red squares, and corresponding simulation results from Fig.(2.8), blue circles, versus average propagating power.   21 2.4 Spectrum of the PCF bent at a) 30 cm, b) 50 cm, c) 70 cm, and d) the whole length of the fibre with 13.8 mW of average power launched into the fundamental mode.   22 2.5 Frequency shifts of peaks in Fig. (2.4), and Fig.(2.9) versus propaga- tion length. Squares are the frequency shifts of spectral components observed in the experiment, and circles are the simulation values for shifts.   22 ix List of Figures 2.6 Spatial profile of the fibre output when a) the most shifted soliton is centred at 850 nm and b) the second most shifted soliton is at 850 nm. These correspond to spectrum a) and b) of Fig.(2.2) respectively. . . 23 2.7 a) Discretizing the entire fiber length and for each segment, b) applying linear and nonlinear operators.   28 2.8 Simulated spectra after propagating 1.1 m at different average prop- agating powers. Each spectrum was calculated for the same average propagating powers as in the experimental results shown in Fig.(2) The input pulses are taken to be 130 fs in duration  29 2.9 Numerical solution of Eq. (2.6) at four different propagation lengths with 13.8 mW average propagating power and a 130 fs pulse duration launched into the fibre  30 2.10 Group velocity dispersion provided by the fibre manufacturer (blue solid line), which is the dispersion calculated by using a fully-vectorial beam propagation method in the absence of any manufacturing imper- fections. The dispersion coefficients used in the simulation are obtained by a polynomial fit up to six-order (red dashed line).   30 2.11 Comparison of the simulated output spectra with 13.8 mW average propagating power, obtained from Eq.(2.6) with a) all terms included, b) neglecting the self-steepening term, and c) by applying the slowly varying envelope approximation to the Raman term. Note that while the self-steepening term differentially modifies the rate of Raman shift- ing, the slowly varying envelop approximation fails to capture the proper soliton break up.   32 3.1 2D photonic crystal microcavities, a) an isolated cavity and b) adding a 1D waveguide channel to the cavity structure.   40 3.2 The model cavity can communicate with two continuum channels.^50 List of Figures^ xi 3.3 Spectrum at threshold of squeezing for the Y quadrature in a degener- ate down-conversion process. The solid line is when loss into channel two is zero (-ye = 0) and the dashed line is when rye = yi.   51 3.4 Shift of the holes next to the cavity in order to increase its Q and also its coupling efficiency to the waveguide. The x and y axes represent the electronic axes of the underlying crystal  55 3.5 a) The frequency spectrum of the microcavity fundamental mode with the excitation pulse apodized before Fourier transforming, and b) log plot of the electric field versus time  56 3.6 Intensity profile of a) X component and b) Y component of the electric field associated with the cavity in Fig.(3.4a), and c) total intensity of the pump beam in the vicinity of the cavity.   56 3.7 The spectrum of squeezing for the a) Y quadrature of the sample along [100] direction and b) X quadrature for [111] growth direction for the structure shown in Fig.(3.4), assuming 10 mW of CW excitation at 2fi . The squeezing spectrum for X and Y quadratures are expressed in Eqs.(3.34) and (3.35).   58 3.8 a) Shift of the holes next to the cavity in order to increase its Q and also its coupling efficiency to the waveguide, b) the cavity is tilted with respect to the waveguide to boost their coupling efficiencies.   59 3.9 Intensity profile of a) X component and b) Y component of the electric field associated with the cavity in Fig.(3.8a), and c) total intensity of the pump beam in the vicinity of the cavity.   60 3.10 The spectrum of squeezing for the Y quadrature of the sample in Fig (3.8) for a crystal oriented along the [100] direction for, a) CW pump and b) 500 ps pulses  60 List of Figures^ xii 4.1 Schematic diagram of Si slab photonic crystal structure in a SOI wafer. All features except the substrate (in vertical direction) are to scale. There is no oxide layer beneath the microcavity region (see text). . . 69 4.2 3-missing hole microcavity and its interface with waveguides  70 4.3 An expanded view of the cavity-waveguide coupling region that shows the connection of the microcavity to the ridge waveguide through a 6 missing-hole PhC waveguide.   71 4.4 Monitors (screenshots) of the FDTD simulation for estimating the cou- pling efficiency from a) the source to the microcavity and b) from the microcavity to the ridge waveguide. The simulation area is shown by an orange box.   72 4.5 a) The surface monitors (screenshots) on top of the grating coupler and perpendicular to the ridge waveguide axis and b) 3D view of the simulation area  74 4.6 Farfield pattern of the light diffracted from the grating coupler. Nu- merical labels indicate angles in degrees.   74 4.7 SEM image of the fabricated microcavity-waveguide structure, before the undercutting process.   76 4.8 Optical image of the structure a) after developing the photoresist and b) after wet etching. The dark region in the left side of the Fig.(a) shows the area covered by the mask.   76 4.9 The detailed SEM images of the sample after undercut process. a) shows the undercut region around the microcavity segment b) clearly shows the width of the ridge waveguide c) displays smooth walls of the taper waveguide and finally d) shows the 2D grating coupler and the termination of the taper waveguide  77 4.10 Focusing the light on the microcavity and aperturing light out of cavity or grating coupler.  ^79 List of Figures^ xiii 4.11 Aperturing the microcavity (a) and the grating coupler (b). ^ 80 4.12 Resonant scattering (a) and background-free signal emanating from the grating coupler (b).  ^81 A.1 Pulse intensity (solid line) and frequency chirp (dashed line)^ 92 A.2 Raman gain spectrum of silica, after Stolen et al [J. Opt. Soc. Am. B 6, 1159-1166 (1989)].  ^94 A.3 Self-steepening effect on a secant hyperbolic pulse. ^ 101 B.1 The FROG experiment setup. The specifications of the optical ele- ments are summarized in table (2.1) of chapter 2. ^ 109 D.1 Signal and idler field quadrature correlations in a nondegenerate para- metric down conversion. ^  127 E.1 The set up of an homodyne detection measurement. The signal and the local oscillator beams are combined in a beam splitter and the combined light is absorbed by a detector^  129 E.2 Photo detection of a nonclassical light source.  130 E.3 Experimental setup of a homodyne detection for measuring the spec- trum of squeezing. ^  132 Acknowledgements First and foremost I offer my sincerest gratitude to my supervisor, Dr. Jeff Young, who has supported me throughout my graduate study with his patience and knowl- edge whilst allowing me the room to work in my own way. I attribute the level of my PhD degree to his encouragement and effort and without him this thesis, too, would not have been completed or written. I would like to thank deeply my research supervisory committee members Dr. Tom Tiedje, Dr. Fei Zhou, and Dr. David Jones for their helpful advice and insight over several years. In my daily work I have been blessed with a friendly and cheerful group of student colleagues for providing a stimulating and fun environment in which to learn and grow. I am especially grateful to Allan Cowan, Iva Cheung, Miryam Elouneg-Jamroz, Tian Si Wang, and Charles Foell. Life blessed me with the opportunity to meet and work with Murray McCutcheon. His permanent enthusiasm for the work and his timely comments were invaluable throughout. Many thanks to the research associates members of Photonics and Nanostructures laboratory whose friendship and support have made it more than a temporary place of study. Special thanks to Dr. Georg Rieger, Dr. Andras Pattantyus, Dr. Haijun Qiao , and Dr. Mario Beaudoin for their assistance and guidance. Finally, I am very grateful for my wife Ameneh, for her love and patience dur- ing the PhD period. Her incredible support and understanding helped make the completion of my graduate work possible. xiv Co-Authorship Statement The manuscript of chapter 4, which has been published in Applied Physics Letters was co-written. The following areas shows the contribution of the first author: • Identification and design of the proposed geometry for the waveguide-microcavity structure. • Literature search and feasibility study. • Numerical simulation of the coupling efficiency and data acquisition of the op- tical characterization. • Mircofabrication of the proposed structure. • Manuscript preparation. XV 1Chapter 1 Introduction Miniaturized photonic devices, which can efficiently operate at practical input powers, will be realized only when we can manipulate photons on the wavelength scale with similar precision to what has been achieved for electrons in semiconductor devices. Photonic crystal (PhC) structures [1, 2] exploit the effects of periodic, high-refractive- index texture to achieve this objective for a wide range of applications such as lasers, detectors, filters, sensors, and optical channels [3, 4]. They offer much flexibility in the design and fabrication of microstructured optical materials, which allows us to integrate various photonic elements in a variety of configurations. In addition, the growing demand for all-optical signal processing devices [5] motivates consideration of the nonlinear optical properties of photonic crystals [6]. Two advantages of using PhC structures to harness nonlinear optical processes include the ability they offer to dramatically alter the dispersion properties of propagating guided light, and to accumulate and store electromagnetic energy in wavelength-scale volumes of nonlinear material. There are a host of nonlinear optical processes that influence the properties of ra- diation in materials, and that can allow for the control of a signal's fate all-optically, without the intervention of local electronic controls. This thesis reports thorough investigations of two such nonlinear processes; (i) the breakup and frequency shift of high-order classical solitons into a series of fundamental solitons propagating in a photonic crystal-clad silica optical fibre, and (ii) the generation of squeezed (quan- tum) light via parametric down-conversion inside a three-dimensional (3D), wave- length scale optical microcavity formed in a III-V semiconductor host. In (i), the Chapter 1. Introduction^ 2 high refractive index contrast of a porous clad silica core strongly confines light in two dimensions while the detailed shape of the porous cladding network provides an engineered group velocity dispersion for light propagating in the third dimension. In combination, this allows for the dramatic conversion of relatively low power, 130 fs long pulses injected into the fibre, into a series of fundamental solitons at the end of a 1.1 m length of fibre, the number and frequencies of which are tunable by varying the input power. In (ii), the tight confinement of electromagnetic energy for many optical cycles in a cubic wavelength scale III-V semiconductor microcavity is used to convert classical radiation incident on the cavity at twice the resonant frequency, into a squeezed (quantum) stream of light in a single channel waveguide efficiently coupled to the microcavity at the resonant mode frequency. Both of these projects compare experimental data with detailed optical models of the relevant nonlinear processes. In (ii), a novel photonic crystal structure capable of efficiently coupling radiation from wavelength scale cavities to single-mode waveguides is also designed and fabricated by the author. The thesis is divided into three main chapters. In the remainder of this introduc- tion, the outline of each chapter is described. The details of some calculations are explained in the appropriate appendixes. High-order soliton breakup in a microstructured optical fibre Conventional optical fibre is used to build the photonic highways that carry infor- mation around the world via the internet. These fibres are made from ultra-pure silica glasses that are drawn into fibres with a precisely controlled radial refractive index profile that is a maximum at the centre, along the fibre axis. A relative variation of refractive index of --,0.003 over a few microns in radius is sufficient to robustly guide the light through total internal reflection for kilometers [7]. Most optical commu- Chapter 1. Introduction^ 3 nication channels are designed to carry the maximum number of laser-based optical signals, each with as much optical power as possible. The limit is often the nonlinear response of the fibre to very high electric fields, which causes distortion of the signals. Appendix A describes a number of these nonlinear processes, which are all third-order in the electric field, due to the centro-symmetric properties of the glass. In some long-distance applications it is actually possible to take advantage of these nonlinear properties to achieve optimal signal transport efficiencies. Stolen et al. [8, 9] and others demonstrated and optimized the propagation of optical solitons through conventional single-mode telecommunication fibres in the 1980s. Optical solitons are short pulses of light that propagate without changing shape or shedding energy. This requires a nonlinear process because in a linear dispersive medium, large-bandwidth optical pulses necessarily change shape due to the different propagation velocities of their various spectral components. If the propagation wavevector of the guided mode at frequency w is k (co), then in pure silica the group velocity dispersion, 2 k2  = (dco2 dk ),,° ^ (1.1) with coo as the carrier frequency of the pulse, is "normal", which means it decreases as the frequency increases. In optical fibres, the net mode dispersion is that due to the intrinsic silica dispersion, and that influenced by the total internal reflection process that guides the light. It is therefore possible to have fibres that exhibit either "normal" (k2 > 0) or "anomalous" (k2 < 0) group velocity dispersion. For linear propagation, both types of dispersion lead to pulse distortion with propagation. However, one of the third-order nonlinear processes described in Appendix A, self-phase modulation, nonlinearly changes the spectrum of the propagating pulse due to the fact that the effective refractive index in silica actually depends on the intensity of the electric field n(I) = no + n2 /,^with n2 > 0^ (1.2) The frequency of light propagating through a region of time-dependent refractive index is shifted, or "chirped" . At the leading edge of the pulse, the frequency is Chapter 1. Introduction^ 4 decreasing as the pulse propagates, while at the trailing edge, the opposite is true. If the energy in the pulse is just right, the nonlinear chirping can exactly compensate for anomalous linear distortion, resulting in a pulse that does not change shape as it propagates. Pulses with the appropriate energy and duration to satisfy these exact cancelation conditions are referred to as fundamental solitons. When high-power optical pulses (more intense than those used in communication applications) are injected in conventional single-mode optical fibre, some researchers have reported that the centre frequency of the solitons shift over kilometers of prop- agation, and in some cases, multiple solitons that shift at different rates are observed [10]. The centre-frequency shift was explained by including another third-order non- linear property of silica, Raman scattering [11]. The generation of multiple solitons was qualitatively explained by including Raman scattering and the so called "self steepening" effect in propagation models (see Appendix A), but a detailed quantita- tive comparison of high-quality data and numerical models was lacking. Figure (1.1) shows a cross sectional scanning electron microscope image of the textured photonic crystal optical fibre used in the study reported below in Chapter 2. Figure 1.1: A scanning electron micrograph, in cross section, of the photonic crystal fibre used in this thesis. The core is a very small (--1 pm) diameter rod of silica, and the cladding is a hexago- nal network of thin silica walls. The average refractive index of the cladding material Chapter 1. Introduction^ 5 is very low, as it is mostly made up of air, which means that light can be more ef- fectively confined to the micron core region than in conventional single-mode fibre, where the cladding's refractive index is less than a percent different from that of the core. The cross sectional area of the mode supported by the photonic crystal fibre is ,--,2.6 ii,m 2 , whereas it is typically ,--, 100 ii,m 2 in conventional fibre. This means that for a given guided mode power, the peak electric field intensity is ,-,-,40 times higher in the fibre with textured cladding. The dispersion of the propagating modes supported in this structure can be en- gineered over a vast range of parameter space in comparison to conventional fibre [12, 13]. The particular fibre used in Chapter 2 has a very wide range of wavelengths over which there is nearly constant and anomalous dispersion for the fundamental mode. These two properties combined, made it possible to observe and study the frequency shifted multiple soliton process previously observed using kilometers of conventional fibre [10](1 km, and 36 mW coupled average power of 830 fs pulses at an 82 MHz repetition rate), using just 1 m of photonic crystal fibre and similar input power (pumped by 130 fs pulses at 80 MHz repetition rate and with average propagat- ing power of 13.8 mW). By comparing a comprehensive set of soliton spectra obtained for varying input powers and distances of propagation with a third-order nonlinear pulse propagation model, it was possible to explain the observations and attribute the main effects to high-order soliton breakup near the beginning of the fibre, and then the subsequent Raman shift of fundamental solitons generated in the breakup process. The relative importance that various third-order nonlinear processes play in the propagation was therefore revealed for the first time [14]. Squeezed state generation in photonic crystal mi- crocavities There has been a great deal of progress made over the past decade on the possible Chapter 1. Introduction^ 6 use of quantum mechanics to process and/or transmit information in fundamentally different ways than is possible using purely classical phenomena [15]. In the informa- tion processing domain this is often referred to as "quantum computation", and in the information transport domain, it is often referred to as "quantum communication". Algorithms exist for quantum computers that, if they can be realized in practice, could be used to solve certain types of problems that could never be solved using the largest, fastest classical computer imaginable. There are a huge variety of physical implementations of quantum computers being pursued. Some of these employ pho- tons and require non-classical sources of light to function. Similarly, the fundamental requirements of perfectly secure communication using quantum cryptography have already been demonstrated experimentally using a source of (non-classical) entan- gled photons [16], generated using parametric down-conversion (in this process, the incident photons on a nonlinear crystal will split into pairs of photons of lower en- ergy whose combined energy and momentum are equal to the energy and momentum of the original photon). For quantum optics based quantum information process- ing to be practically feasible, it is very likely that arrays of compact, non-classical light sources that require limited drive power will be required. Chapter 3 describes a model and its numerical solution, which estimates what optical power would be re- quired to realize a useful source of squeezed light in a compact, semiconductor-based device. Chapter 4 describes the design, fabrication, and optical characterization of a structure that could be used to efficiently couple such a source of squeezed light into a single-mode optical channel capable of transmitting the signal anywhere on the "optical chip" . These structures make use of an artificial class of dielectric materials known as "photonic crystals" . The concept of a "photonic bandgap" structure, simultaneously introduced by John and Yablonovich [17, 18] in the late 1980's, suggested how 3D periodic texture in a sufficiently high-refractive index host material could be used to qualitatively alter electromagnetic phenomena by completely eliminating the photonic density of Chapter 1. Introduction^ 7 states over a finite range of frequencies, without necessarily introducing any loss. A huge body of research and development has been produced over the intervening 20 years, substantiating the original claims. Semiconductors such as Si, GaAs, InP etc. have been particularly popular host materials for these studies, due to the fact that their refractive indexes in the near infrared are ,3-4, high enough to support a full 3D photonic bandgap if an appropriate 3D periodic texture is applied. The relevant property of such materials for the work described in chapters 3 and 4 has to do with the fact that defect regions specifically designed into otherwise perfectly periodic photonic crystals are ideally suited to act as high quality factor (high Q), 3D optical microcavities with mode volumes less than a half cubic wavelength of radiation in the material at the cavity resonance frequency. By using semiconductor-based microcavities with high Q factors and small mode volumes, we can hope to realize useful levels of common nonlinear phenomena at practical input optical power levels [19]. This is because if such cavities are efficiently coupled to via a single-mode, 1D waveguide channel, the internal electric field inten- sity established by an excitation field with bandwidth less than the cavity linewidth, is enhanced by a factor of Q times the incident field strength. This was recently demonstrated in our laboratory by Murray McCutcheon [20], who measured cavity- enhancement factors of over 1000X in second harmonic frequency conversion within such a microcavity, using less than 1 mW of laser diode radiation. This was achieved despite the fact that the coupling efficiency to the cavity in his experimental setup was only a few percent, and that it relied only on the intrinsic second order nonlinear susceptibility of the InP (i.e. there was no attempt to introduce electronic resonant media such as quantum dots, to resonantly enhance the susceptibility). The parametric down-conversion process is very closely related to the second har- monic generation process. Based on the harmonic generation results of McCutcheon, rough estimates suggested that it may be feasible to use similar cavities pumped from the top half space at twice the cavity frequency, to induce parametric down-conversion Chapter 1. Introduction^ 8 into cavity mode photons, which would have a squeezed spectrum. Figure 1.2: Down-conversion process in a 2D photonic crystal waveguide-microcavity structure. As shown in Figure (1.2), the microcavity is pumped from the top by a tightly focussed laser beam at two = w1 + w2. The parametric down-conversion process generates photons at w 1 and w2 in the microcavity modes, and these will have a squeezed spectrum with the degree of squeezing dependent on the net parametric down-conversion efficiency. The cavity mode photons must then be coupled as effi- ciently as possible to a single-mode ridge waveguide, which itself is ultimately con- nected to an optical fibre or an output grating coupler for testing purposes. Chapter 3 describes a quantum mechanical formalism developed by the author to quantitatively calculate the squeezed spectrum of the light generated in a multi-mode photonic crystal microcavity due to this down-conversion process. The idea is to study the dynamics of an open quantum system (in our case an open microcavity embedded in a photonic crystal structure) interacting with the reservoirs of free space and waveguide channels, which constantly perturb the quantum evolution of quantized cavity modes by letting information escape irreversibly to the environment or by Chapter 1. Introduction^ 9 injecting an exciting field into the system. The Hamiltonian describing the second-order nonlinear interaction between the quantized cavity modes with a classical pump field is consistently incorporated with the system-reservoir Hamiltonian of an open cavity proposed in Ref. [21], and the spectrum of squeezing for the field quadratures of the light guided into a single-mode waveguide channel is calculated. A 3D finite-difference time-domain (FDTD) simu- lation is used to estimate the degree of squeezing for the non-classical light generated in a microcavity-waveguide structure made of AlGaAs. All cavity losses and nonlin- ear coupling coefficient between cavity modes are calculated by assuming that the structure is excited by a coherent state. In order to maximize the coupling efficiency between the cavity modes and 1D waveguide, the structure fabricated and character- ized in chapter 4 is used in the simulation. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide To preserve the squeezed property of the light generated in a photonic crystal microcavity it should be efficiently coupled to a single-mode output channel [22]. For instance, if the coupling efficiency from the cavity to the waveguide channel is only 50%, the maximum degree of squeezing that can be achieved in the output channel is 50%. Chapter 4 of the thesis is devoted to the development of such a microcavity-waveguide structure using 2D photonic crystals fabricated in 200 nm thick semiconductor slab waveguides. Experience has shown that it is far easier to produce very high Q microcavities and associated 1D waveguides in a 2D photonic crystal geometry, rather than using full 3D texturing approaches [23]. The proposed waveguide-microcavity structure, which is shown in Fig.(1.2), achieves a coupling efficiency of ,56%, and the numerical model used to simulate its properties suggests that slight modifications can result in coupling efficiencies up to 90%. By using Chapter 1. Introduction^ 10 mature semiconductor fabrication technology, a scalable array of such structures can be integrated in an optical chip. To accomplish all of this, we designed a structure, Fig.(1.2), in which a cavity is connected to a single-mode ridge waveguide via a short segment of photonic crystal waveguide (in order to minimize the impact of its strong dispersion). The detailed shape of the interface between these two waveguides is adopted from Ref.[24]. A fully 3D electromagnetic simulation based on a FDTD algorithm is used to design this structure and estimate its properties [25]. To experimentally investigate the coupling efficiency of this waveguide-microcavity geometry, a structure was fabricated in SOI (Silicon On Insulator). Although Si does not have a bulk second-order nonlinearity, the basic fabrication technique could easily be adopted with a III-V based structure. Alternatively, the silicon microcavity fabricated here could be decorated with PbSe or PbS semiconductor nanocrystals 1 to enable second-order nonlinear processes to occur in the vicinity of the cavity. 2D grating coupler I single-mode ridge waveguide 3-missing hole microcavity taper waveguide Figure 1.3: SEM image of the fabricated microcavity-waveguide structure. 'substantial progress has been made in this regard already, by other members of the Young group. Chapter 1. Introduction^ 11 Figure (1.3) shows a test sample, fabricated on an SOI wafer (consisting of a 720 pm thick substrate covered by a 1.2 pm thick layer of silicon oxide which in turn supports a 196 nm silicon thin film layer). A two-dimensional grating coupler with a parabolic shaped taper waveguide was used to couple radiation off-chip from the ridge waveguide. The actual coupling efficiency of the structure was measured by using a 130 fs pulses to inject energy into the microcavity mode, and then measure the light coupled out of the 2D grating coupler, from the microcavity, via the waveguide. By comparing the measurement result with the numerical estimation of the coupling efficiency, and by comparing the cavity Q value with and without the waveguide coupler, we were able to deduce a coupling efficiency from the cavity to the waveguide of ,--, 55%. The following three chapters elaborate on these three elements of the thesis work. The results of chapters 2 and 4 are already published, while those in chapter 3 are in the process of being written into a manuscript for publication. 12 Bibliography [1] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: Princeton University Press, New Jersey (1995). [2] K. Sakoda, Optical properties of photonic crystals: Springer-Verlag, New York (2001). [3] 0. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, I. Kim, Two-dimensional photonic band-gap defect mode laser, Science 284 1819- 1821 (1999). [4] P. Bhattacharya, J. Sabarinathan, J. Topolancik, S. Chakravarty, Y. Pei-Chen, and Z. Weidong, Quantum dot photonic crystal light sources, Proceedings of the IEEE 93, 1825-1838 (2005). [5] M. Soljacic, and J. D. Joannopoulos, Enhancement of nonlinear effects using pho- tonic crystals, Nature Materials 3, 211-219 (2004). [6] R. E. Slusher, B. J. Eggleton (Eds.), Nonlinear photonic crystals: Springer-Verlag, Berlin (2003). [7] G. P. Agrawal, Nonlinear fiber optics: Academic Press, San Diego (2001). [8] L . F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett. 45, 1095-1098 (1980). Chapter 1. Introduction^ 13 [9] R. H. Stolen, L .F. Mollenauer, and W. J. Tomlinson, Observation of pulse restora- tion at the soliton period in optical fibers, Opt. Lett. 8, 186-188 (1983). [10] P. Beaud, W. Hodel, B. Zysset, H. P. Weber, Ultrashort pulse propagation, pulse break up, and fundamental soliton formation in a single-mode optical fiber, IEEE J. Quan. Elec. 23, 1938-1946 (1987). [11] J. P. Gordon, Theory of the soliton self-frequency shift, Opt. Lett. 11, 662-664 (1986). [12] M. H. Frosz, T. Sorensen, B. Ole, Nanoengineering of photonic crystal fibers for supercontinuum spectral shaping, J. Opt. Soc. Am. B 23, 1692-1699 (2006). [13] K. Nakajima, T. Matsui, Dispersion-flattened photonic crystal fiber, Review of Laser Engineering 34, 17-21 (2006). [14] M. G. Banaee, and Jeff F. Young, High-order soliton breakup and soliton self- frequency shifts in a microstructured optical fiber, J. Opt. Soc. Am. B 23, 1484-1489 (2006). [15] D. Bouwmeester, A. Ekert, and A. Zeilinger, Physics of quantum information: Springer-Verlag, Berlin (2000). [16] M. Dusek, N. Lutkenhaus, and M. Hendrych, Quantum Cryptography, Progress in Optics 49, edited by E. Wolf: Elsevier, Amsterdam (2006). [17] S. John, Strong localization of photons in certain disordered dielectric superlat- tices, Phys. Rev. Lett. 58, 2486-2489 (1987). [18] E. Yablonovitch, Inhibited Spontaneous Emission in Solid-State Physics and Electronics, Phys. Rev. Lett. 58, 2059-2062 (1987). Chapter 1. Introduction^ 14 [19] J. Vuckovic, D. Englund, D. Fattal, E. Waks, and Y. Yamamoto, Generation and manipulation of nonclassical light using photonic crystals, Physica E 32, 466-470 (2006). [20] M. W. McCutcheon, G. W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, R. L. Williams, Resonant scattering and second- harmonic spectroscopy of planar photonic crystal microcavities, Appl. Phys. Lett. 87, 221110 (2005). [21] C. Viviescas and G. Hackenbroich, Field quantization for open optical cavities, Phys. Rev. A 67, 013805 (2003). [22] H. J. Kimble, Fundamental Systems in Quantum Optics: Elsevier Science Pub- lishing, Amsterdam (1992), chapter 10. [23] E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, and T. Tanabe, Ultrahigh - Q photonic crystal nanocavities realized by the local width modulation of a line defect, Appl. Phys. Lett. 88, 041112 (2006). [24] E. Miyai, and S. Noda, Structural dependence of coupling between a two- dimensional photonic crystal waveguide and a wire waveguide, J. Opt. Soc. Am. B 21, 67-72 (2004). [25] M. G. Banaee, A. G. Pattantyus-Abraham, M. W. McCutcheon, G. W. Rieger, and Jeff F. Young, Efficient coupling of photonic crystal microcavity modes to a ridge waveguide, Appl. Phys. Lett. 90, 193106 (2007). 15 Chapter 2 High-order soliton breakup in a microstructured optical fibre Incorporating the concept of photonic crystals in optical fibre technology has opened up interesting and active research areas in optical communications and nonlinear op- tics [1, 2, 3, 4, 5, 6]. These microstructured fibres consist of a uniform core region (typically either air or silica) that is surrounded by a photonic crystal cladding layer that most commonly consists of a periodic, hexagonal array of air holes defined by silica walls. Photonic crystal fibres (PCFs) offer a particularly attractive medium in which to study guided wave nonlinear optics for two reasons'. First, the index con- trast between the silica core and the surrounding photonic crystal lattice is relatively large compared to conventional single mode fibres, thus reducing the effective mode diameter [7]. The smaller mode has a proportionally higher electric field strength in the silica core region for a given guided power level. Second, by being able to engineer the cladding crystal parameters (air hole size, pitch, lattice type) one can create fibres with anomalous group velocity dispersion (GVD) in the visible and near infrared part of the electromagnetic spectrum. The GVD of a fibre is important not only in optical communication systems, but also in the context of the nonlinear propagation of light through fibres. When a fibre is operated in the anomalous dispersion regime, it is possible to excite optical solitons lA version of this chapter has been published. Banaee, M. G. and Young, Jeff F. (2006) High- order soliton breakup and soliton self-frequency shifts in a microstructured optical fiber J. Opt. Soc. Am. B 23, 1484-1489. Chapter 2. High-order soliton breakup in a microstructured optical fibre^16 that propagate without distortion by canceling the effect of group velocity dispersion through self-phase modulation [8, 9]. Nonlinear effects in optical fibres such as Raman scattering, self-phase modulation, and self-steepening cause spectral shifts and reshaping of propagating pulses. The precise form of the nonlinearly shifted output spectrum varies from fibre to fibre, and is strongly dependent on the fibre dispersion as well as the wavelength, power, duration, and chirp of the input pulses. In some circumstances the output spectrum consists of one or more discrete peaks that can be either red shifted, or both red and blue shifted. Previous studies of ordinary silica fibres showed that a combination of four-wave mixing, stimulated Raman scattering, and self-phase modulation processes can result in either discrete or continuous spectral broadening [10, 11], depending on the particular circumstances. Already, several groups have reported discrete broadening [12, 13] and supercon- tinuum generation[14, 15, 16, 17, 18] across the visible and near infrared in short segments of PCFs pumped with laser pulses ranging in duration from 20 fs to 0.8 ns, at average power levels of only a few milliwatts. The low threshold powers make this system attractive for potential applications in dense wavelength-division-multiplexed (DWDM) systems [19] and optical metrology[20]. In the present work, the discrete broadening of 130 fs pulses passing through 1.1 m of photonic crystal fibre with 1 µm core diameter and a 1.6 ,um pitch photonic crystal cladding region is investigated for a range of input powers as a function of the propa- gation distance. For certain excitation conditions, the output spectra contain several discrete spectral components that gradually red-shift with increasing input power and propagation distance. The broadening is strictly to the red, and is similar in some respects to the Raman self-frequency shifting reported by others in regular silica fibre [21], and to spectra reported by others in PCFs[12, 13]. In previous studies that specifically address the appearance of multiple solitons in PCFs, a variety of explana- tions have been offered[12, 13, 22, 23]. While recent work quite convincingly argues Chapter 2. High-order soliton breakup in a microstructured optical fibre^17 that multiple solitons can result from the breakup of a high-order soliton launched into the PCF, quantitative comparisons between the model and experimental results are limited[24, 25]. Below we quantitatively compare numerical solutions of a gener- alized nonlinear SchrOdinger equation with a comprehensive set of measured soliton spectra whose shift can be followed accurately as a function of excitation power. This comparison allows us to convincingly demonstrate that the multi-soliton behaviour observed in these experiments is due to the breakup of high-order solitons in the presence of Raman scattering and self-steepening. We also make clear that while the self-steepening term slightly modifies the rate at which the solitons Raman shift, the slowly varying envelope approximation applied to the Raman term leads to qualita- tively different breakup characteristics that are inconsistent with the experimental results. Thus the finite time response of the material must be considered explicitly in order to explain the behaviour of the solitons. Third-order nonlinear effects in optical fibres, and a derivation of the generalized nonlinear SchrOdinger equation are provided in Appendix A. 2.1 Experimental setup Fig.(1.1) (in the introduction) shows a scanning electron micrograph of the cross section of the cobweb silica-based photonic crystal fibre (made by Crystal Fibre A/S in Denmark) used in the work described below. It has 1 core diameter and ,-- 1.6 fim pitch. The objective of this study was to characterize the spectral properties of the light emanating from the PCF when sub-picosecond pulses from a Ti-Sapphire laser oscillating at a 80 MHz repetition rate are coupled into it using a 40X microscope objective, Fig.(2.1). By using a beam splitter (BS), the laser pulse is divided into two parts. The part that is directed towards the PCF passes through a half-wave plate (HWP) and a polarizer (PL) and is coupled by the Ll lens onto the cleaved input facet of the fibre. The combination of HWP and PL allows continuous control PM Chapter 2. High-order soliton breakup in a microstructured optical fibre^18 of the power coupled into the fibre. A parabolic mirror (PM) collimates the output of the fibre. Table .(2.1) summarizes specifications of the optical elements used in the experiment. The time-integrated output spectrum of the photonic crystal fibre was measured using a Fourier transform spectrometer. The uniform spatial distribution and polarization of the fibre output confirmed that the coupled light propagates in the fundamental mode. The measured coupling efficiency was 11%. FT spectrometer 130 fs BS H P^Ll HI LL Pulses  \ S ^M2 L2 RF L3V^NLC Fl^ c"L:)L4 L5 c Delay line  Photon counter PMT Monochromator Figure 2.1: Experimental setup. A nonlinear crystal (NLC) was used to perform the frequency resolved optical gating measurements to characterize the input pulses. A CCD camera positioned to Chapter 2. High-order soliton breakup in a microstructured optical fibre^19 Table 2.1: Optical elements used in the experiment. Optical elements Specifications M1 -M2 Dielectric mirrors, 25.4 mm Diameter, 0.5-18 pm HWP Half-wave plate PL Polarizer Ll 40X Objective lens, Newport F-L4OB L2-L3 Lens with respectively 10 and 15 cm focal length L4-L5 UV lens, Focal length 7.5 cm, AR coated, 250-430 nm Fl BG40 filter BS 50/50 Beam splitter PCF Numerical aperture > 0.5 at 633 nm NLC Beta-Barium Borate(BBO) crystal, 5 x 5 x 0 5 mm 3 , cut at 38° PM Parabolic mirror, 90° off axis, 28.75 mm focal length RF Newport Gold-coated retro-reflector display the surface of the nonlinear crystal helps to overlap the two input beams in the crystal. The nonlinear crystal rests on a motorized rotation stage that has an accuracy of 0.1°. Lens L4 collimates the up-converted signal and lens L5 images it on the entrance of a monochromater (Digikrom DK242 from CVI Laser Corporation). Filter Fl prevents any IR wavelengths from entering the monochromator. A photomultiplier tube (Model 9784A made by Thorn EMI Electron Tubes Ltd.) at the exit of the monochromator is used to monitor the sum frequency signal in a photon counting mode (Hamamatsu model C3866). By using a SHG-FROG setup, the duration of transform limited pulses centred at 800 nm was measured to be 130 fs. The frequency resolved optical gating (FROG) method, which is explained in detail in Appendix B, reveals the temporal phase and the intensity of the pulse. By processing the phase retrieval data of the FROG experiment we found that the input pulses coupled into the fibre (centred at 800 nm) were chirp-free. 6.2 4.8 3.4 2.0 1050 11001000950900850 6 Chapter 2. High-order soliton breakup in a microstructured optical fibre^20 2.2 Output spectral dependence on input power and propagation distance Fig.(2.2) shows the output spectra observed from a 1.1 m length of the fibre when it was pumped at 800 nm (13.5 nm bandwidth) for input average powers ranging from 10 mW to 120 mW (labels on the right of this figure show the measured power at the output of the fibre in each case). At the highest powers used in the experiment WAVELENGTH (nm) Figure 2.2: Experimental output spectra of the photonic crystal fiber pumped at different average propagating powers (shown at right). The spectra marked with a) and b) have the most shifted and the second most shifted solitons centred at 850 nm respectively. mW propagating in the fundamental mode), the output spectrum extended as far as 1000 nm and it contained three well-separated spectral peaks. As this figure shows, the number of peaks in the output spectrum depends on the power, and the centre wavelengths of these well-defined spectral components gradually red-shift with increasing power. Square symbols in Fig.(2.3) show the frequency shifts of the three well-separated spectral components in Fig. (2.2) versus power in the fundamental •• 0 s • • 0• • ■•o^■• • ■^o ° 0o • •• • • 0 o ° 110 100 90 80 70 N 60 I- 50 c,C)^ 40 30 20 10 120 Chapter 2. High-order soliton breakup in a microstructured optical fibre^21 mode. 0 0^20^40^60^80^100 INPUT POWER (mW) Figure 2.3: Frequency shifts of the experimental peaks in Fig.(2.2), red squares, and corresponding simulation results from Fig.(2.8), blue circles, versus average propa- gating power. The development of the spectrum excited by a fixed input power was studied as a function of propagation distance in the fibre by bending the fibre at different lengths and spectrally analyzing the radiation that leaked out of the core. Fig.(2.4) shows the resulting spectra obtained at three points in the fibre when the average propagating power was 13.8 mW. The quality of the spectra suffers when having to bend the fibre to extract light scattered out of the fundamental mode, and couple this into the spectrometer, but nevertheless it is clear that the centre frequency of the well-separated peaks gradually decreases with increasing distance. The shifts of peaks in Fig.(2.4) are shown by the squares in Fig.(2.5). Qualitatively similar experimental results have been reported before in both PCF [22, 23] and standard fibre [21], but often the Stokes components are accompanied by anti-Stokes components in the spectra, or the presence of multiple distinct spectral peaks is not obvious. While the continuous shift to the red is clearly associated with Raman self-shifting solitons (RSSS) [26, 27], the origin of multiple peaks is less clear. 2.5 800^850^900^950^1000 WAVELENGTH (nm) b) a) 1050^1100 0.5 0 0 0 0 0 0 o 0 0 ^0 0 0 0 Chapter 2. High-order soliton breakup in a microstructured optical fibre^22 Figure 2.4: Spectrum of the PCF bent at a) 30 cm, b) 50 cm, c) 70 cm, and d) the whole length of the fibre with 13.8 mW of average power launched into the fundamental mode. 110 100 90 80 70N 60 50 I 40 30 20 10 0 0.2 0.4 0.6 0.8 1.2 LENGTH (m) Figure 2.5: Frequency shifts of peaks in Fig.(2.4), and Fig.(2.9) versus propagation length. Squares are the frequency shifts of spectral components observed in the experiment, and circles are the simulation values for shifts. Chapter 2. High-order soliton breakup in a microstructured optical fibre^23 At least one report attributes multiple red shifted peaks to a set of RSSS excited in different transverse spatial modes of the fibre [12]. This can not explain the results presented here, as there is no indication of high-order mode propagation in the spatial distribution of the output signal. Fig.(2.6a) and Fig.(2.6b) show the far-field mode profile obtained by putting a bandpass filter centred at 850 nm at the fibre output (after PM mirror in Fig.(2.1)), when the first most red-shifted soliton, spectrum a) in Fig. (2.2), and the second most shifted soliton, spectrum b) in Fig. (2.2), were centred at this wavelength respectively. A digital camera connected to an IR viewer was used to take the images in Fig.(2.6). Figure 2.6: Spatial profile of the fibre output when a) the most shifted soliton is centred at 850 nm and b) the second most shifted soliton is at 850 nm. These correspond to spectrum a) and b) of Fig. (2.2) respectively. Other reports [13, 22, 23, 24] assume that all observed solitons propagate in the same transverse mode of the fibre, and attribute them to a breakup of the high- order soliton launched into the fibre. This interpretation is supported by solutions of the nonlinear SchrOdinger equation, augmented by additional nonlinear terms to take into account Raman scattering and self-steepening processes. In one case, semi- quantitative agreement between the measured and the modeled spectral shift of one of the fundamental RSSS peaks as a function of incident power was reported [24]. The current experimental results, reported above, appear qualitatively consistent with the breakup of high-order solitions within a single transverse mode of the fibre. Below Chapter 2. High-order soliton breakup in a microstructured optical fibre^24 we show that simulations similar to those reported in Ref.[24]. provide quantitative agreement with the measured power and propagation distance dependences of all RSSS peaks. This removes any ambiguity about the origin of these multiple solitons, and it further shows that a full treatment of the Raman process (as opposed to the more commonly used slowly varying envelop approximation [28, 29]), is essential to obtain a good level of quantitative agreement between model and experiment. 2.3 Numerical simulation Appendix A provides a detailed derivation of the generalized nonlinear SchrOdinger (GNLS) equation that we solved numerically to compare with the experimental re- sults described above. The simplest form of the GNLS equation describes how the instantaneous Kerr effect (the refractive index of silica depends instantaneously, on the intensity of the local electric field) influences the propagation of pulses in a dis- persive medium (where dispersion terms up to 6t h order are included). This model can describe soliton propagation, but it fails to explain the continuous red shift of the solitons as they propagate. To model this shifting, which is due to stimulated Raman scattering (a third-order interaction mediated by vibrational excitations in the silica, appendix A), one has to include a non-instantaneous response term in the general third-order response function. The most common third-order nonlinear effects in optical fibre (silica does not have a second-order nonlinearity) originate from nonresonant (the carrier frequency of the pulses is far from any electronic transitions), incoherent (intensity-dependent) nonlinear effects with the following general form: P (3) (r, t) = 3—€0x (3) E(r, t) f R(T)1E(r, t — 7- )1 2dr, 4 (2.1) where x (3) is the third-order susceptibility and R(T) is the response function of the silica, which can be expressed as [29]: Chapter 2. High-order soliton breakup in a microstructured optical fibre^25 R(t) = (1 - fR) 8 (t) + fRhR(t)•^ (2.2) This explicitly separates the instantaneous electronic contribution (Kerr effect)from the vibrational Raman contributions, hR(t). The fR coefficient represents the rela- tive strength of these two contributions and for the silica fibre fR is experimentally estimated to be 0.18 [29]. The Raman response function of the silica core is usually approximated by: hR(t) =  ̂exp(-- t ) sin(—t ), 2^2 Ti T2^72 1- 2 (2.3) with T1=12.2 fs and T2=32.0 fs [29]. The electric field amplitude of a polarized pulse propagating through a single- mode fibre satisfies the following wave equation: — V2 E(r, t) + e2 1 a2E(r, t)^a2P(r, t) at2 ate ^(2.4) where P(r, t) is a combination of the linear and nonlinear polarization induced by the pulse, Eq.(2.1), in the fibre core. By considering the axial symmetry of the fibre, the electric field can be expressed (in the frequency domain) as follows: E(r, t) = Jdo) E (r , w)e - ' = f dwA(z , c.o)F (1, co)ezk(w ) z- iw t ,^(2.5) where I shows the (x,y) transverse coordinates. We suppose the transverse compo- nent of the field, F(±, w), is not altered by the nonlinearity. By substituting Eqs. (2.1) and (2.5) into Eq.(2.4) and supposing the envelope of the field, A(z, w), is a slowly varying function (its second-order derivative can be ignored) we have: ^6 ^m-1 rn^ +xDA = a A^k  — i — 0 )[A(Z, t)^R(e)0( t — t')1 2 dti]. ^ az^2 m i^a A  mml atni=1^• wo at Chapter 2. High-order soliton breakup in a microstructured optical fibre^26 (2.6) The first term in the above equation is added to allow for the linear loss, k l is the inverse of the group velocity, km (with in > 2) are higher-order dispersion coefficients, and -y is the nonlinear instantaneous Kerr effect coefficient, 3X (3)wo 7 = Q . (2.7)ocnef f Aeff Here Aeff and nen. are the effective area and the refraction index of the funda- mental spatial mode respectively. Equation (2.6) is called the generalized nonlinear SchrOdinger equation, which describes short pulse propagation in a dispersive non- linear medium, where various different third-order contributions to the nonlinear response of the silica are included. The self-steepening effect (appendix A), which is due to the intensity dependence of the group velocity, is represented by the time derivative term in Eq.(2.6). For pulses with duration longer than 100 fs an approxi- mate form of Eq.(2.6) is often used to simulate the pulse propagation in fibres [29]. The envelope function in the integration is simplified as so that where IA( z, t — t')1 2 -̂  IA( z, 0 1 2 — t' at ^t) 1 2 , f R(3)(e)0 (z, t — 01 2 de = 1A(Z 7 01 2 — TRIA(Z )01 2 ) TR = f t' R(e)de = fRfehR(e)de. (2.8) (2.9) (2.10) Using the form of the Raman response function of silica given in Eq.(2.3), the TR coefficient is found to be 1.46 fs. Therefore, with this approximation, the GNLS equation is: Chapter 2. High-order soliton breakup in a microstructured optical fibre^27 aA 6^im-1 am =^A + [E km ^]A + i-y[AIAl 2 — TRA—a 012 + —a 001 2 )]• (2.11)az^2 m! atm at^woatm=1 Equations (2.6) and (2.11) were both solved using the split-step Fourier (SSF) method [29], in which the equations for the field envelope are written in the form: OA LA+ NA, az where the linear operator L contains linear loss and dispersion terms, (2.12) a E k 6^im-1 am 2^mi m ^][ m! atm '= while the nonlinear operator N includes all of the third-order nonlinear terms, (2.13) +co N = i A(z, t)^c.o0(1 +^— a )[A(z, t) f^cleR(e)1A(z,t — t')I 2],^(2.14)at for Eq. (2.6), and a similar form applies for Eq. 2.11). With reference to Fig. (2.7), the SSF method involves discretizing the entire propagation length and in each segment we suppose the linear and nonlinear effects act independently. Therefore the field solution at the input and output ends of a segment are related by the following: A(z + h, t) e h A(z,t)J= LB(z, t). (2.15) The linear operator only contains the time derivatives a/at, which can be replaced by a simple algebraic multiplication by in frequency space. The operation of the linear operator can then be evaluated using Fourier Transforms (FT represents the Fourier transform operation and FT -1 its inverse) as: ehLB(z,t)= FT- 1 FT[e B(z,t)]= FT - l [ehL(' ) FT B(z,t)].^(2.16) Thus the nonlinear operator (evaluated at the input of the segment) is applied to the field envelope, the Fourier transform of the result is taken, and after multiplying that • • •a) z=0 z=L dz/2 dz/2 Chapter 2. High-order soliton breakup in a microstructured optical fibre^28 with ehL( 'w) (which is a number in frequency space), an inverse Fourier transform is applied. given initial envelope A(z=0,t) final envelope A(z=L, t) Figure 2.7: a) Discretizing the entire fiber length and for each segment, b) applying linear and nonlinear operators. The accuracy of this method can be improved if we symmetrize the role of the linear and nonlinear operators [29], so we end up with the following form A(z + h, t) ,., ehLI2 ehlCr ehL12 AII(Z , t) . (2.17) A parallelized C++ program was written to execute the SSF method with spatial resolution of h=200 pm and 2 14 Fourier components were used in the Fourier trans- formation at each step. By using 16 Pentium' 4 processers (2.4 GHz) it takes roughly 12 hours to run the simulation for a 1.1 m length of the microstructured fibre. The spectra obtained from the simulation for a fixed length of 1.1 m, for different propagating powers, are shown in Fig.(2.8), and the simulated spectra at different distances of propagation for a fixed propagating power of 13.8 mW are shown in Ak 0.5 2.5 1.5 1.2 Chapter 2. High-order soliton breakup in a microstructured optical fibre^29 Fig.(2.9). In the above calculations we set -y = 7.24 x 10 -8 (pm W) -1 (this value of 'y is exactly what is expected if the light propagates in the fundamental spatial mode with the effective area set to the square of the pitch in the surrounding photonic crystal cladding region [7]). The dispersion coefficients used in the simulation results are fitted to the group velocity dispersion curve t in Fig.(2.10), which was provided by the fibre manufacturer. 7 0.8^0.9^1 WAVELENGTH (µan) 13.8 mW 12.6 11.7 10.8 9.7 8.8 7.4 6.2 4.8 3.4 2.0 0.9 1.1 Figure 2.8: Simulated spectra after propagating 1.1 m at different average propagating powers. Each spectrum was calculated for the same average propagating powers as in the experimental results shown in Fig.(2). The input pulses are taken to be 130 fs in duration. The simulated results agree very favourably with those obtained experimentally. It is clear that Eq. (2.6) does describe how high-order solitons excited near the fibre input can spawn a series of fundamental Raman solitons that gradually shift to the red as they propagate. Fig.(2.3) compares the frequency shifts of spectral components in Fig.( 2.2) and Fig(2.8) versus power. Also, the shift of the centre frequencies of 2 There are two definitions for the group velocity dispersion. It can be represented by k 2 = L(,÷) or D = criA ( ) 2.2 x̂ 1.8 • 1 -6 1.4 • 1.2 0.8 0.6 0.4 0.2 0 ^ 0.7 0.8^0.9^1 WAVELENGTH (pm) 110 cm 50 cm 70 cm 30 cm 1.1^1.2 Chapter 2. High-order soliton breakup in a microstructured optical fibre^30 Figure 2.9: Numerical solution of Eq. (2.6) at four different propagation lengths with 13.8 mW average propagating power and a 130 fs pulse duration launched into the fibre. 1 ^ 1.5 WAVELENGTH (pm) Figure 2.10: Group velocity dispersion provided by the fibre manufacturer (blue solid line), which is the dispersion calculated by using a fully-vectorial beam propagation method in the absence of any manufacturing imperfections. The dispersion coeffi- cients used in the simulation are obtained by a polynomial fit up to six-order (red dashed line). Chapter 2. High-order soliton breakup in a microstructured optical fibre^31 the various discrete spectral components in Fig.(2.9) are plotted versus the length of propagation in Fig.(2.5), using circles. It should be mentioned that there was no attempt to get a better match between the observed and the simulated frequency shifts by varying the dispersion coefficients. These were fixed from fitting to the calculated dispersion offered by the fibre manufacturer, solid line in Fig. (2.10). To help interpret these results we use the well known formula[29], N2^ YP°7" 29^(2.18) jk21 to estimate the highest order of soliton that could have been launched at the input of the fibre at the highest power excitation. With k 2 = —1.625 x 10 -2 fs2 /pm at 800 nm (from the dispersion curve in Fig.(2.10)), 'y = 7.24 x 10 -8 (gm W) -1 , P0 =1.15 kW, corresponding to 13.8 mW of average power, and a 130 fs pulse duration (To = 130/1.76 fs for secant hyperbolic pulses) one gets N L.--2 5. Simulations show that a soliton of order 5 compresses to ti 10.0 fs within the first few centimeters of its propagation through the fibre[30]. This occurs in the simulation results with or without the Raman and self-steepening terms in Eq.(2.6). When the Raman term is included, there is a sudden breakup of the compressing soliton just before it first reaches its minimum pulse duration, and three distinct RSSS gradually separate from the residual, complicated pulse that continues to propagate near the launch frequency. Each of the three RSSS have distinct pulse durations and peak powers, which explains why they frequency shift at different rates. It is difficult to quantitatively interpret the evolution of the pulses when the simulation is done with up to 6th order dispersion. By only including second order dispersion (a constant GVD across the entire spectrum), it is possible to verify that the various RSSS that are generated when the high-order solitons "break up" after the initial compression, are all fundamental solitons; their durations are inversely proportional to their amplitudes, with a fixed proportionality constant for each set of solitons. For example, the most shifted soliton after 40 and 60 cm propagation through the fibre had (A 0 =1.0, To = 26.60 fs) and (Ao=0.933, 1.8 • 1.6 1.4 • 1.2 • -NW ^ 0.6 0.4 • 0.8 • 1^1.1 kir. ) 0.8^0.9 0.2 077 (a) 1.2^1.3 Chapter 2. High-order soliton breakup in a microstructured optical fibre^32 To = 30 fs) normalized amplitude (square root of the peak intensity) and duration respectively, and the proportionality between the amplitude and the inverse of the duration was N 27. For the second most shifted soliton the corresponding amplitude and the duration were (A0=0.698, T0 = 23.01 fs) and (A0 -=0.644, T0 = 26.92 fs) and the proportionality constant was 16. The self-steepening effect alone does not cause any soliton breakup in the out- put fibre spectrum but it slows the frequency shift of separate fundamental solitons [25]. Fig.(2.11) compares the simulated spectra obtained with only the Raman term (2.11b) and with both Raman and self-steepening terms included (2.11a) in the GNLS equation. (b) WAVELENGTH (pm) Figure 2.11: Comparison of the simulated output spectra with 13.8 mW average prop- agating power, obtained from Eq.(2.6) with a) all terms included, b) neglecting the self-steepening term, and c) by applying the slowly varying envelope approximation to the Raman term. Note that while the self-steepening term differentially modifies the rate of Raman shifting, the slowly varying envelop approximation fails to capture the proper soliton break up. Figure (2.11c) shows the simulated spectrum with the approximated form of Eq.(2.6) and TR = 1.46. This figure hints that the approximate form of the Raman response Chapter 2. High-order soliton breakup in a microstructured optical fibre^33 can not be used to describe the experimental results reported here. Numerous at- tempts to fit the full set of data from Figs. (2.2) and (2.4) using this approximation to the Raman contribution, and varying the other model parameters away from their nominal values, failed to come close. It is also worth mentioning that the high-order dispersion terms in Eq.(2.6), extracted from the group velocity dispersion shown in Fig.(2.10), did not cause any soliton breakup when the Raman and self-steepening nonlinear terms in Eq.(2.6) were switched off. In Appendix A it is shown that by using the inverse scattering method a high- order soliton can be expressed as a combination of several fundamental solitons. In addition it is shown that how Raman scattering and self-steepening disturb the propagation of a fundamental soliton. The process of high-order soliton breakup in the presence of the Raman effect has been noted previously [31, 32, 33], but we believe that the current data and interpretation help to provide a better understanding of this interesting and potentially useful phenomenon. In conclusion, the output spectrum of a photonic crystal fibre, pumped by 130 fs pulses centred at 800 nm, exhibits discrete broadening of the incident pulse through creation of several fundamental solitons, which continuously red-shift with increasing power and propagation length. The far-field pattern of the fibre output as well as the similar spectral intensity for the most shifted and second most shifted solitons at the same wavelength, validate the idea of single spatial mode propagation of multiple solitons through this structure. The numerical solution of a generalized nonlinear SchrOdinger equation that includes Raman scattering, self-steepening, and dispersion up to the sixth-order, provides a quantitative description of the experimental results, and it can be used to interpret the results as due to the breakup of high-order solitons into fundamental self-shifting Raman solitons. Creation of these multiple fundamen- tal Raman solitons suddenly happens when the launched pulse compresses to a critical pulse duration, close to the input of the fibre, but only when the full time-dependent Raman term is included in the generalized nonlinear SchrOdinger equation. 34 Bibliography [1] J. C. Knight, T. A. Birks, P. St. J. Russel, and D. M. Atkin, All-silica single -mode fibre with photonic crystal cladding, Opt. Lett. 21, 1547- 1549 (1996). [2] R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russel, P. J. Roberts, and D. C. 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Lett. 13, 392-394 (1988). 38 Chapter 3 Squeezed state generation in photonic crystal microcavities Deterministic and efficient nonclassical light sources play a central role in implement- ing future quantum information protocols [1, 2]. Two examples are single photon and squeezed state light sources. Optimum implementations of quantum cryptography [3] and quantum computation with linear optics [4] rely on single photon sources whereas quantum information processing with continuous variables depends on squeezed state sources [5, 6, 7], which produce two strongly correlated (entangled) light beams Semiconductor-based implementations of these non-classical light sources are partic- ularly attractive due to their small size, low power consumption and the possibility of integration with the solid-state qubits [8, 9]. The first protocol proposed for continuous variable teleportation [6] used a bulk nonlinear crystal as a source of entangled photons, which were produced via a nonde- generate parametric down-conversion process. The work described below l investigates the feasibility of miniaturizing squeezed state generation by using a photonic crystal microcavity efficiently coupled to a single-mode waveguide channel [10]. This was mo- tivated by the observation of the nonlinear sum-frequency generation in a microcavity embedded in an InP membrane[11], which suggests the possibility of generating para- metric down-converted light in the microcavity-based photonic structures. Already there have been reported attempts to use 1D photonic crystal slabs and 2D photonic 'A version of this chapter will be submitted for publication. Banaee, M. G. and Young, Jeff F. Squeezed state generation in photonic crystal microcavities. Chapter 3. Squeezed state generation in photonic crystal microcavities^39 crystal waveguides to generate squeezing [12] and entangled photon pairs [13, 14, 15], but as far as we know this is the first attempt to accomplish the squeezing process in cubic-wavelength scale, 3D photonic crystal microcavity structures. Due to the non-classical characteristics of squeezed light, a quantum mechani- cal model is needed to describe the down-conversion process in the cavity, and how it couples to various leakage channels. The overall aim of this work was to obtain a quantitatively accurate estimate for the (nonclassical) spectrum of radiation that propagates away from a 3D, wavelength-scale microcavity that supports two distinct electromagnetic modes with frequencies co l and w2 , and which is pumped/excited by a classical source at w 1 + o)2 . We adapt a cavity/reservoir quantization formalism from Ref. [16], that allows us to rigorously relate the output channels' quantum field variables to those of the input channels, and the leakage from the excited modes in the cavity (the so-called input-output relations [17, 18]). The approach of Ref. [16] was adopted because it gives an explicit mathematical expression for the damping coefficients as an overlap integral between the cavity (system) and channel (reservoir) modes at the interface that defines the cavity region, which are more usually consid- ered as phenomenological parameters when dealing with photonic crystal cavities. In the following, an interaction Hamiltonian due to the second-order nonlinear coupling between the cavity modes is added to the system-reservoir Hamiltonian of the Ref. [16] and all the appropriate approximations are explained explicitly. The crucial component of this model, that must be solved numerically, is the overlap integral describing the nonlinear interaction of the classical excitation field at c.o i +4,02, with the two quantized cavity modes. This is evaluated using an FDTD solver [19] to find the relevant field amplitudes for the 3 fields involved in the parametric down- conversion (PDC) process. Finally the squeezing spectrum of the light generated in a 3D photonic crystal microcavity-waveguide geometry is evaluated. Chapter 3. Squeezed state generation in photonic crystal microcavities^40 3.1 Field quantization in an open optical cavity The system of interest consists of a defect-state optical microcavity in a thin semicon- ductor membrane perforated by a regular 2D array of through-holes, Fig. (3.1a). The light is confined in the cavity by Bragg reflection in the plane (due to the photonic crystal bandgap) and conventional total internal reflection in the vertical direction. For appropriately designed defect states and large enough surrounding photonic crys- tal material, the intrinsic loss of the cavity is due entirely to out-of-plane scattering [20, 21]. 0000 0 0 00000 0 0000 0 0 OCOCCC000000 000 000 000000000000 0000 000 l o 00000 000 0 0 0000 0 0000COCC 00 OVIICZ %%P^00 co co 04000 00000 -^OCOu Occp cc C0000 00c 00000 Cm 00 ^Oc CCOt00000° 407 cooc 00occcoccooccatcooccocco ccoc occoo 0000cocpc c000 cco00 ----___ (a) Figure 3.1: 2D photonic crystal microcavities, a) an isolated cavity and b) adding a 1D waveguide channel to the cavity structure. When a 1D single-mode waveguide is introduced in proximity to the microcavity, as in Fig. (3.1b), it is possible to have the coupling of the cavity mode(s) to this waveguide channel dominate the total leakage. An example of how this can be realized in practice is described in Chapter 4, but for the purpose of this chapter, it is assumed Chapter 3. Squeezed state generation in photonic crystal microcavities^41 that preferential coupling to a single-mode output channel can account for at least 90% of the total leakage [22]. For modeling purposes, this system is conceptualized as consisting of two com- pletely localized cavity modes, ch i (r) and 02 (r), that can be driven by polarization induced in the microcavity by classical and/or quantum field sources incident via the continua channels (either from the top or bottom half space radiation modes, or other in-plane waveguide channel modes). Each continuum channel carries away radiation excited by the incident fields, possibly mediated by resonant scattering in the cavity. We adopt a system-reservoir formalism from Ref. [16] in which the cavity is considered as a system with a discrete set of quantized modes (associated with normal modes of an isolated cavity). The reservoir consists of a set of continuum field operators which model all the distinct channels that the cavity can leak into. Here we introduce the key elements of the system-reservoir Hamiltonian formalism from Ref.[16], which is derived using the Feshbach projection technique [23, 24]. A more complete description of this derivation is provided in Appendix C. The exact eigenmodes of the vector potential for the multi-mode photonic crystal microcavity and several continuum channels that couple to the cavity, f m, (r, w), are expressed in terms of the discrete eigenmodes of the cavity, 15), (r), and the continuum of modes, V ri (r, co'), in the channels [16, 25]: f,, (r, w) = E arAn(w)UA (r) + E f dc,/,3,„-(c,), w')vn (r, w'). (3.1) A The discrete index m labels an asymptotic solution in which an incoming wave through channel m is scattered by the structure into all other channels. Here UA (r) is a localized cavity mode with a discrete spectrum, and V, (r, co') is a continuum eigenmode of channel n. The cavity and channel modes are orthogonal and they have nonzero values only in the appropriate regions. The coefficients criAn and /3n7n(w, w') are the coupling coefficients between the cavity/channel modes and the exact eigenmode associated with excitation of the system through channel m. Chapter 3. Squeezed state generation in photonic crystal microcavities^42 The quantization of the fields may be achieved by expanding the vector potential in a complete set of mode functions and imposing canonical commutation relations for the expansion coefficients [16, 26]. By using the eigenmodes of the structure, Eq.(3.1), the field expansion for the vector potential takes the following form A(r,^= 2E0WA [aAPA , t)UA(r, WA) + a tAPA, t) 1";,(r, WA)] +^f du.) 20w ^t)V ni (r,^+ f-L(W, t)Vm* (r, co)] , (3.2) where as and ettA are the creation and annihilation operators of the discrete cavity modes, whereas f-in and f"tni are the corresponding operators of the continuum modes of channel m. By substituting the above equation into the Hamiltonian of the elec- tromagnetic field in a linear medium f d3 r[Eoc(r)( at ) 2 +^(V x A) 2 ],^(3.3) we get: E 2+ + E f dwhwv.! i (w),,,(w) +A hY-'^f dw[WAm (w)dtA i.ni (w) +147-;,(w)a),4,(w) + TAm(w)etAi'm(w) + nn,(w)aliln(w)1, T 72 (3.4) where WA,,,, and TA,,, are the coupling coefficients between the cavity and channel oper- ators at an interface that defines the cavity region[16]. Analytic expressions for these coupling coefficients are derived in Appendix C. Because a and 1- vary approximately as e' t , and at and 7-4 as CH'', Eq.(3.4) contains resonant (atr , oa) and nonresonant (ar , ato) terms. When the photonic crystal microcavity modes have high quality factors (where the bandwidth of the modes are very small compared to the mode Chapter 3. Squeezed state generation in photonic crystal microcavities^43 frequencies), a rotating-wave approximation can be used and the nonresonant terms in the Hamiltonian can be ignored. Assuming a microcavity with two localized modes at w 1 and w2 , which are coupled to several output channels, the Hamiltonian in Eq.(3.4) can be written as: + 1w2ei2a2 + E dwhcei-tm (w)1.,,(w) + hE dw[Wim(w)ati'm(w) + Wim (w)aM,i (w) + W2m(w)a2i'm(w) + W1m(w)a2 11n(w)]• Tri (3.5) 3.2 Parametric down-conversion in an open cavity The Hamiltonian in Eq. (3.5) is for the electromagnetic field within a linear material. In the system of interest here, the intent is to use the cavity modes to mix nonlinearly with a strong classical field incident on the cavity from the top half space, at the sum frequency of the cavity modes. This parametric down-conversion process introduces correlations in the cavity mode field operators, which gets transferred to the output channel field operators. To describe the nonlinear mixing in the cavity, we need to introduce higher order terms in the part of the Hamiltonian that describes the cavity mode dynamics in the presence of an external (classical) field at w 1 + w2. The Hamiltonian of the electromagnetic field inside a nonlinear dielectric medium can be written as [27, 28]: 1 H = co f d3rr 1— (E 2 + c2B2)± 2 — E.x (1) : E+ 2 — E .x (2) : EE+ - 4 E.x (3) : EEE+• • •]. (3.6) 3 The first 3 terms include all of the linear interaction with the material, while the fourth term in above equation describes the nonlinear interaction to second order Chapter 3. Squeezed state generation in photonic crystal microcavities^44 in the fields (this term, that assumes the normalized response function of the host material is instantaneous, is responsible for parametric down-conversion): -1-// = -3-26° d3rE(r, t).x (2) : E(r, t)E(r, t), (3.7) where E(r, t), the total field in the cavity region, is the superposition of the fields associated with the cavity modes and the pump field. The quantized field due to the cavity modes can be written by using Eq. (3.2) and the fact that E = —aAiat, as Ec (r, t) = i 2E0 „^ [ct i (t)U i (r) at(t)uT(r)]+ hw22€0^ (t)U2 (r) - at2 (t)U;(r)]. (3.8) The pump field, which is taken to be a classical coherent state at 2w 0 is written as: Ep (r, t) = iAp (t)[Up (r)e -2iw° t -Up*(r)e±2iwl,^(3.9) where Ap and Up (r) are the envelope function and the spatial distribution of the pump field in the cavity region respectively. By substituting the total field E(r, t) = E c (r, t) + Ep (r, , t) in the Hamiltonian of equation (3.7) and only considering the resonant terms (assuming 2w 0 = w l + W2) we have Ht = ih[gal(t)azme -2iwo — g* alma2(0e2iw0l, where (3.10) VW].g = ^3 W2 Ap (t) f d3r[Ui.x2 : U2Up +Ui.x2 : UpU2 +U2.x 2 : U lUp + U2.x2 : UpUi + Up.x 2 : U2U1 + Up .x2 : U1112]. (3.11) Here we suppose the spatial distribution of the fields in Eq.(3.11) are real valued functions and therefore g is a real number. By adding the Hamiltonian of the nonlinear interaction, Eq.(3.10), to the original system-reservoir Hamiltonian, Eq. (3.5), the Chapter 3. Squeezed state generation in photonic crystal microcavities^45 Heisenberg equations of motion for the cavity and reservoir operators will be (to simplify the notation we now omit the hat symbol on the operators): 64(0 =^+ g4e -2iw° t — iE f dwWim (w)rm (w,t),^(3.12) (12(0 = — iw2a2(t) + gat, e-2iw°t — jE f dwW2,n (w)rm (w, , t), i*in (W7 t) = —iwrm (w, t) — inn (w)ai — iW2m(w)a2, The integration of reservoir operators starting from initial time t o < t yields (3.13) (3.14) -,rrn p ,^e-iw(t-to)rm(w, f t oto to)—inĵw) dee-iw(t-e)ai(e) inn (w) f de e- iwa e)a2(t r ) (3.15) By expressing the same reservoir operators in terms of their values at t i > t, and taking the limit t 1 +oo, to —oo, the Fourier transforms of the operators can be expressed as [16] rZa (w) ern(w) = — iA/27 [Wim (w)ai (w) +^(w)a2(w)] where the input and output operators are defined as: rin(w) = lim eiw t°rm (w, to),^rmout(w)^lim eiw , r (w, t i ).rn (3.16) (3.17) The above equation is called the input-output relation [17]: it expresses the output field operators that are directly related to what is ultimately measured, to the input (homogenous) driving fields incident from outside the system, and the localized modes that leak into the output channel. After substituting the solution of the reservoir operator, Eq. (3.15), into the equation of motion for the cavity operators we have: a 1^+ gat2C2iwot^&Am/1m (w ) e -iwt rinin (w ) E fdwovim(w)12 ft t dee- (t-e) a i (e) + Wim(w)n, (w)^dee'w(t-ti) a2(t 1 )],to (3.18) Chapter 3. Squeezed state generation in photonic crystal microcavities^46 a2^gale-ziwot —^I dww2m (w )e-iwt rmin(w ) ^E f dw[w2,i(w)inn (w) j.^e-' ( '-e) (e) + I W2m (w)1 2 1'^e'w (t-ti) a2(e)] , ^ to to (3.19) By using a rotating frame in which, ai,2(t) = "Ct i ,2(t)e- iw° t , and rm (t) = an(t)e iwot and manipulating the integration [17] in the last two terms of Eqs.(3.18) and (3.19) we have: ai = —i(wi — wo + kii)Eti(t)+gart2 (t) —nliai(t)-7/12a2(0— iki2a2(t) — iE f dww,m (w)e-i-ti-irrnz(w), a2^—i(w2 —^k22)a2(0+gat(t) — n2iai(t) — 7722a2 (t)^(t) E 1 dww2m ( o) e -iwt min (w ) (3.20) where we used the following notations: ^W2m(w)W3*m(w)^Ka., =EPI dw^,^i, j = 1,2,^(3.21) rn^4,4-10 — (.4-) and 712.3 = i E wzm(wo)w,m(00),^i, j = 1 , 2 -^(3.22) The kip are responsible for frequency shifts and the 7/23 account for losses of the cavity modes. Now we make two assumptions to simplify the above equations of motion. First we suppose the cavity modes are weakly coupled to the reservoir, therefore the cavity modes are not coupled to each other due to their interaction with the reservoir: this is equivalent to neglecting the off-diagonal elements of k i3 and 7h3 . Also due to the weak coupling, the eigenfrequency of the cavity modes will not be affected by the reservoir and even diagonal elements of tc z, (frequency shift of the cavity modes caused by their interaction with the reservoir) can be ignored. Chapter 3. Squeezed state generation in photonic crystal microcavities^47 The second assumption is the Markov approximation (reservoir correlation times are assumed negligibly short compared to the characteristic time scale associated with the cavity mode dynamics [18]), in which we assume that the coupling coefficients W, (w) are independent of frequency, yielding: a l^i(wo — woai(t) + gat2(t) — 7hiai(t) — iv27 E wi9,47(t), a2^i(wo — w2)a2(t) + ga ti(t) — 122a2 (t) — ^27 E w2,01n (t). 97, Now we can separate the channels that a microcavity mode can communicate with into two parts. Channel 1 is a 1D single-mode waveguide and rm (m = 2, .) are the rest of reservoir operators which the cavity modes can couple with. The coupling coefficient between the cavity modes and different channels can be represented by: Wig^ V 71 (3.23) where -ya3 shows the loss of cavity mode i into channel j and we set the Wu to be purely imaginary. By using the equations (3.22) and (3.23) we have: 1711 =^'71m = F1,^1722 = E 72m — F2. m m (3.24) The F 1 and F2 show the total loss of each cavity mode. Finally our equations of motion become a l =^— wi)ai(t) + ga t2 (t) — 1'1E4 (t) + V271177 (t) + E^(t),2 a2^i(wo — W2)a2(t) + 0'1(0 — F2 d2 (t) + V2721 (t) + E \/272mPrin' (t).2 After Fourier transforming these field operators and setting A, = w o — wi as the offset of the cavity modes with respect to the half of the pump frequency, the field operators of the cavity modes can be expressed in terms of the input channel field operators as: (Q)^9-072171n ( — Q) + g Em=2 -\/272mP": t (—Q) [F 1 —^+ Q)] [F2 + i(02 — Q)] — g2 [F2 + i(0 2 — s2)] ^2-y11f1 (Q) + Eni=2 O'bm1zT(Q)1 [F 1 —^+ Q)] [F2 + i(A2 — Q)] — g 2 Bm (52) C(S2) Dm( 12 ) Chapter 3. Squeezed state generation in photonic crystal microcavities^48 a2 (12) ^t(—C2) + g Em=2 -V2'yi rn izZ2 t ( --C2) [F 1 +^— s2 )]  [r2 — i(A2 + s2)] — g2 ^ [F, + i(A1 — Q)]i .V2721FT(C2 ) + Em=2 N 2272mi";n1, ( Q )} •[F 1 + i(A1 — s2)] [r2 — i(02 + Q)] — g 2 (3.25) We are interested in the field at the output of channel 1 (single-mode waveguide). By using the input-output relation, Eq.(3.16), the output operator in the desired channel is: ilut (Q) = A(Q)47(Q) + E Bm (Q)P;7:(Q) + C(Q)fin t ( — Q) + E Drn(Vir9n/ t(—c2),m=2^ m=2 (3.26) where 2-yll [r2 + i(02 — (-2)] A(Q)^ 1 [r, — i(A1 + c2)] [F2 + i(A2 — Q)] g2 2-y2i [Fi + (Al — C2)] [F 1 + i(A1 — 1 )] [r2 — i(A2 + Q)] — g2 ' -V2711N/27im[r2 + i(A2 — (2)] [r, — i(A1 + 12)] [F2 + i(02 — Q)] — g 2 V2721V272,,,[ri +^— c2)] +^— (-2)] [F2 — i(02 + s2)] — g2 ' Nry1J.72]. [r, —^+ s2)] [F2 + i(A2 — (2)] — g2 29 .\/711')/21 [F, i(Al — (-2)][1'2 — i(A,2 + 52)] — g2 g1/2711,7272„, [F 1 —^+ (2)] [F2 + i(02 — Q)] — g2 91/2721N/271m [r, + i(Ai — ( -2)] [F2 — i(02 + Q)] — g2. (3.27) To measure the degree of squeezing in the output channel 1, we have to evaluate the following correlation functions [18, 26]: sx (Q) < xont (Q), xout(—Q) > —1, (3.28) Chapter 3. Squeezed state generation in photonic crystal microcavities^49 Sy(Q) < yout( Q ) ,yout(_ 1") > —1, (3.29) where Sx (C2) and Sy (Q) are called the spectrum of squeezing for the X and Y field quadratures respectively, with xout (Q)^ilut(-2)^flut t (_ c2), yout( Q )^_irt(Q) _ Tint f(_Q)] . (3.30) It is shown in appendix E that the variance of the photocurrent in a homodyne detection setup is the integration of the squeezing spectrum. Equations (3.28) and (3.29) are derived in Appendix E. By supposing the reservoirs are in the vacuum state (< t(Q) >=- 8„,i ) we get: < xout(Q) , xout (_Q ) > [A(Q) + C*(—Q)][C(—Q) + A*(C2)] + E [D7, (-C2) + Bm (Q)][B:n (Q) + Dm (—Q)], m=2 < yout (Q)yout = [A(Q) — C*(-52)][A*(Q) — C(—C2)] + E [Bm (Q) — D',,,,V—Q)][B7„(C -2) — Dm (-52)], m=2 (3.31) also (3.32) where the A, B, C, and D coefficients are given by Eq.(3.27). Equations (3.31) and (3.32) contain term such as: i hl M 1 and i V1/2m, m=2^ m=2 (3.33) which means that if we are only interested in monitoring the m=1 channel, it is not important to distinguish between the other m > 1 channels when expressing and [ (71 + -Y2) — 2g ] 2 +^' Sx (Q) =^8971 (3.34) Chapter 3. Squeezed state generation in photonic crystal microcavities^50 their contributions to the overall cavity loss/excitation. Therefore our microcavity- waveguide photonic crystal structure can be considered as an open cavity, which can leak out into just two channels, Fig.(3.2). Channel 1 is a 1D single-mode waveguide that is strongly coupled to the cavity, and through which the cavity is monitored. Channel 2 represents the total leakage into all of the other continua channels, any one of which is only weakly coupled to the cavity. In the actual structures, the m > 1 modes are the radiation modes that carry energy away from the microcavity into the top and bottom half spaces. Channel 1 Figure 3.2: The model cavity can communicate with two continuum channels. Before we show the numerical calculation of the spectrum of squeezing in a para- metric down conversion process generated in a photonic crystal microcavity, it is useful to review the general behavior of the squeezing spectrum for a degenerate pro- cess. By setting a l = h2 = a in Eq. (3.10) for the Hamiltonian of the nonlinear interaction and Eq. (3.12) of the equation of motion for the cavity operator, the spectrum of squeezing for the X and Y quadratures become: Sy(Q) = ri^—89-Y1 R71. + 72) + 2912 + Q2 • (3.35) Chapter 3. Squeezed state generation in photonic crystal microcavities^51 The threshold of this process occurs when the nonlinear gain of the cavity mode compensates the total loss, or when 2g = 7 1 + 72 , in which case 4-y? + 471 -y2 Sx(Q) =^Q2^, -4-yry2 4(71 + -y2)2 + Q2. (3.36) -15 -10 -5 0 12/71 5 10 15 Figure 3.3: Spectrum at threshold of squeezing for the Y quadrature in a degenerate down-conversion process. The solid line is when loss into channel two is zero (7 2 = 0) and the dashed line is when 7 2 = Eq.(3.36) shows that at this threshold, the X quadrature squeezing spectrum diverges and the Y quadrature spectrum becomes negative. Figure (3.3) shows the spectrum of squeezing for the Y quadrature for two different cases, when there is only one output channel, the waveguide channel (solid line), and when loss through channels 1 and 2 are equal. When there is perfect coupling of the nonlinear cavity to a single mode channel, Sy = —1 at threshold when Q = 0. In this situation there is a perfect correlation between the Y field quadratures of the degenerate photons at the waveguide output [29, 30]. A physical interpretation of the squeezed quadrature field components is discussed in Appendix D. Chapter 3. Squeezed state generation in photonic crystal microcavities^52 3.3 Numerical estimation of the squeezing spectrum The crucial steps in quantitatively estimating the degree of squeezing expected for a realistic cavity/waveguide system, excited by a realistic pump source involve ob- taining numerical estimates of g (the nonlinear coupling coefficient in Eq.(3.11)), and -yzj ,i, j = 1, 2 (cavity modes losses through channels 1 and 2). Chapter 4 describes the design, fabrication and measurements of a system with a coupling efficiency of > 50% into a single mode waveguide, and higher coupling efficiencies are certainly possible [22]. To estimate g (in Eq. (3.11)) for a particular microcavity and host material, one needs to determine the tensor x (2), and the 3D field profiles in the cavity region corresponding to the localized microcavity modes and the pump field. Here we consider a "3-missing hole" cavity [31] made in an A10.3Ga0 . 7As membrane (n=3.23 around 1.5 pm). The concentration of Al in A1 0 . 3 Ga0 .7As is chosen such that its bandgap [32] is at 1.798 ev (or 689 nm), so that the pump beam (at ,720 nm, twice the cavity mode frequency) does not generate any free carriers 2 in the cavity region (free carrier generation in the cavity region changes the refraction index of the medium and induces undesired losses due to free carrier absorption). The value of the g coefficient, Eq.(3.11), depends on the x (2) tensor symmetry of the host photonic crystal material. A10. 3 Ga0 . 7As belongs to the 43m symmetry group. If the wafer used to fabricate the microcavity is oriented in the [001] direction, then x (2) has the form / o^o^o x( 2y)x i/(2)Axyy 0 0 0 0 x (2) (^2w0, -w2) = 0^0^0 0 0 ,,()Ayyz i/(2Ayzx)  0 0 0^0^0 0 0 0 0 ,,(2)xzzy (2xzzz 2 free carriers can still be generated by two photon absorption, but that is a weaker process and can't be easily avoided using conventional III-V semiconductors. Chapter 3.^Squeezed state generation in photonic crystal microcavities^53 1 0 0 0 1 1 0 0 0 0 = 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 (3.37) where /=216 pm/V for w 1 , w2 1.5 pm [33]. The sparse nature of this tensor restricts the possible orientations of the photonic crystal with respect to the electronic axes of the host crystal. To reduce these restrictions, we also consider wafers grown along the [111] direction. The x (2) tensor for this [111] crystal orientation is: /^V-2- 5-40^0 60 2 60 2^60^20^ 1^5-0^1^0 (2) (^ 2w0, —w2) =^0 0^1^___^n ^1^1^1 ^02 20 —2 f^20 2 2/ 20^0 —2 ^0 0 ^ 2^0^0 (3.38) The nonlinear coupling coefficient also depends on the spatial distribution of the cavity modes and the pump laser, which can be determined using a finite difference time domain (FDTD) simulation [19]. The cavity field distribution, U z (r) in Eq.(3.11) are modes of a cavity without any leakage (true modes of a cavity). The mode solution determined by FDTD does not eliminate the role of the cavity intrinsic losses, but for a cavity with a very high quality factor we can approximate these as the true modes of the cavity. The following describes how the FDTD simulator is used to estimate the localized mode field distributions. First, the temporal variation of the field distribution for the two cavity modes at each point in a volume in the vicinity of the cavity region is determined by the FDTD simulation. The bandwidth of a pulsed source used to excite the cavity modes is narrow enough that it can excite each cavity mode separately, by adjusting its centre frequency. Then the Fourier transformation of these cavity field solutions are obtained, c5 i (r, w) and 02 (r, w), which depend on the strength of the source used to excite them. The spatial distribution of the localized cavity modes are these solutions Chapter 3. Squeezed state generation in photonic crystal microcavities^54 at the cavity mode frequencies. To do the Fourier transformation, an apodization of the time window should be applied which minimizes the contribution of the nonres- onant (instantaneous) fields during the excitation pulse to the Fourier transformed field. Then we evaluate the mode volume for each of the cavity modes using = f dr3c(r)Wi (r, wz )I 2 , max[f(r)10,(r, wi)12]^ i = 1 2^(3.39) If we designate max[E(r)Wi (r, cv,)1 21 = a i , Va, has the same unit as (P,(r,(.2.4). Now we define the spatial distribution of the cavity modes used in Eq.(3.11) as: 1 Oi (r, wi ) Ui (r) =^ (3.40) Vi fa7 therefore Ui (r) are normalized (independent of the exciting source strength) and they have units of Tri -3/ 2 . The units of Vh,c4.4/26 0U,(r) are V/m, the same as the units of the pump field distribution, U p (r). The normalized spatial distribution of the pump beam, incident from the top half space and centred on the cavity with a 1.25 pm beam waist, U p (r), is determined similarly by recording the field distribution from the FDTD simulation in the vicinity of the cavity. 3.3.1 Degenerate parametric down-conversion We consider two specific cavity geometries in an hexagonal 2D photonic crystal struc- ture to estimate the squeezed spectrum in the case of degenerate parametric down conversion involving a single cavity mode pumped at twice its natural frequency. The first structure is shown schematically in Fig. (3.4a), it has a lattice constant of a=420 nm, an air hole radius r=0.29a, and the semiconductor membrane thickness is h=200 nm. The cavity is a 3-missing hole cavity and the holes adjacent to the cavity were shifted by 0.15a in order to increase its Q and the coupling of the cavity mode to the ridge waveguide, Fig.(3.4b). Chapter 3. Squeezed state generation in photonic crystal microcavities^55 O 0 0 0 0 0 0 O 0 0 0 0 0 0 • 0 0 0 0 O 0 0 01'0 0 0 O 0 0 0 0 0 0 0 0 0 0 0 Oo 0 0 0 0 Oo 0 0 0 00 0 0 0 0014,, '0  0 0 0 0 0 0 0 0 O 0 0 0 0 0 0  00000000000 O 0 0 0 0 00 0 0 0 0 0 0 0 0 000000 0 0 0 0 0 0 0 0 0 02020 02020 0 0 0 00 0 0 0 0 0 0000 0000 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O 0^0 00 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O 0^0 00 0 0 •0 0 PhC waveguide ridge waveguide Figure 3.4: Shift of the holes next to the cavity in order to increase its Q and also its coupling efficiency to the waveguide. The x and y axes represent the electronic axes of the underlying crystal. The design and properties of the ridge-waveguide and its interface with the PhC waveguide is described in chapter 4. The cavity shows a fundamental mode at f1=209.32 THz (1.43 fim), Fig. (3.5a). The spatial distributions of this mode's field components are shown in Fig. (3.6a,b). The field polarization at the cavity center is in the x-direction, therefore the mode is called an x-polarized mode. The pump frequency is considered as 2f 1 =415.28 THz (--720 nm) and it is x-polarized. Figure (3.6c) shows the spatial distribution of the pump beam intensity. To estimate the cavity Q, a log plot of the electric field versus time is obtained, Fig.(3.5b). By using lEx (t)I = 1E0 le-w1t/ 2Q we have: log io (e)wi Q = 2m (3.41) (a) (b) where w 1 is the angular eigenfrequency of the cavity mode and m is the slope of loglEx I versus time. The quality factor of both an isolated cavity and one connected 31.45 1.44 co 1.43 Wx 0 CO 1.42 0 1.41 1600^700^800^900 1000 1100 1200 1300 1400 1500 TIME (fs) rum (a) (b) ID 00 0 00 -2^-1 ^ 0^1 ^ 2 ^ 0 ^ 2^ -1^0^1 ^ 2 X (µn) >cum) Mn) Chapter 3. Squeezed state generation in photonic crystal microcavities^56 1 0 1 5 • 190 195 200 205 210 215 220 225 230 235 FREQUENCY (THz) Figure 3.5: a) The frequency spectrum of the microcavity fundamental mode with the excitation pulse apodized before Fourier transforming, and b) log plot of the electric field versus time. Figure 3.6: Intensity profile of a) X component and b) Y component of the electric field associated with the cavity in Fig. (3.4a), and c) total intensity of the pump beam in the vicinity of the cavity. Chapter 3. Squeezed state generation in photonic crystal microcavities^57 to the single-mode ridge waveguide are found to be Q z =25560, and Q 7-8852, using the FDTD simulator. By using the following equation: 1^1^1=^ QT Q2 Qwg^ (3.42) we can estimate the quality factor of the cavity mode due to its leakage to the waveguide as Q w9=12852. Therefore the value of the cavity losses to be used in the squeezing simulation are y l = 27rfi /Qwg = 102 GHz and -y2 = 27/1 /Q, = 51 GHz. We now have all the fields required to evaluate the critical nonlinear coupling term, g in Eq. (3.11), except for the absolute value of the pump field intensity. If a CW excitation beam with an average power of 10 mW is assumed, Eq. (3.11) yields g[100] = 48 MHz,^g[1111 = —373 MHz, for samples made from [100] and [111] oriented wafers. Figure(3.7) shows the result of calculating the squeezed spectra for the X and Y quadratures for these two cases. For the [111] direction, g is negative and its X quadrature shows squeezing below the shot noise. The threshold of the degenerate down-conversion process is 2g = ryl + = 153 GHz, therefore, with 10 mW input CW power we are far below the threshold, hence the relatively small degree of squeezing. The g factor is proportional to the pump amplitude, therefore by using a pulsed laser with narrow linewidth (with respect to the cavity linewidth) the peak power in the down-conversion process can be increased. If instead of a CW beam, we assume an 80 MHz train of 500 ps pulses with 10 mW average power the g factor increases due to the increase in peak power to: g[loo] = 240 MHz, g[m] =- —1870 MHz The 500 ps pulses have a narrow linewidth with respect to the cavity mode (for QT = 8500, the cavity linewidth is '50 times larger than 500 ps laser linewidth) Chapter 3. Squeezed state generation in photonic crystal microcavities^58 x x 10 >- U) -10. 0 0 -10. 03-0.2^-0.1^0^0.1^0.2 (THz) -0.2^-0.1^0^0.1^0.2 0 (THz) 03 -15. -20 -15 Figure 3.7: The spectrum of squeezing for the a) Y quadrature of the sample along [100] direction and b) X quadrature for [111] growth direction for the structure shown in Fig.(3.4), assuming 10 mW of CW excitation at 2h. The squeezing spectrum for X and Y quadratures are expressed in Eqs.(3.34) and (3.35). which justifies scaling the CW result by the peak power in the pulses. Now the minimum in the squeezing spectrum for [111] direction increases from —1.3 x 10 -2 to —6.2 x 10 -2 . To get closer to the threshold of the down-conversion process we can use a cavity with higher quality factor. The 3-missing hole cavity shown in Fig. (3.8a) shows a better temporal confinement than the previous design with the same pitch, hole radius, and slab thickness. Three holes next to the cavity are shifted to increase the cavity Q factor [34]. The shift at the A, B, and C locations are 0.2a, 0.025a, and 0.2a respectively. The cavity is tilted with respect to the waveguide axis to increase its coupling efficiency to the waveguide [22]. The cavity shows a fundamental mode at h=207.98 THz (1.44 pm). The spatial distribution of the cavity mode and the pump beam at 2f1 =415.96 THz (,721 nm), with the same polarization as the cavity mode, are shown in Fig. (3.9). The quality factor of both an isolated cavity and one connected to the single-mode ridge waveguide as in Fig. (3.8b), are found to be Q i =79420, and QT=20940. By using Eq.(3.42) we can extract the quality factor of O 0 0 0 0 0 0 O OT00 00^0 0 0 Ot0B0 0 O 0 Ot0A0 0 0 O 0 0 0 0 0 O 0 0 0 0 0 O 0 0 0 0 0 O 0000 0 o 04- Ao o0 4.0B0 0 00 0 0006-000 X^(a) Chapter 3. Squeezed state generation in photonic crystal microcavities^59 the cavity mode due to its coupling with the waveguide, which is Q wg=28440. The values of the cavity losses for the model calculation are thus yl = W1/Q,,,9 = 46 GHz and -y2 = wi/Qz = 16 GHz. 0 0 o0 0 0 0 0 0 0 0 0 0 0 0o ^o o o o o0000000 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (b) ridge waveguide Figure 3.8: a) Shift of the holes next to the cavity in order to increase its Q and also its coupling efficiency to the waveguide, b) the cavity is tilted with respect to the waveguide to boost their coupling efficiencies. For this cavity configuration, and again assuming a 10 mW CW pump laser, the FDTD calculation of the g coefficients for the two growth direction are: g[100=0.483 GHz and g[iii]= -0.199 GHz respectively. Again by using the 500 ps pulses (the linewidth of the cavity mode is '16 larger than pump laser linewidth in this case.) with 80 MHz repetition rate and 10 mW average power, the g coefficients for the two growth directions increase to: g[100]=2.41 GHz and g[iii]= -0.996 GHz respectively. Figure (3.10) shows the result of calculation for the spectrum of squeezing of the Y quadrature for the [100] growth direction for two these two choices of the pump laser. In the case of the 500 ps pump source the squeezing is 20% below the shot noise 0 -0.005 • -0.01 - -0.015 • -0.02 • >- -0.025 - -0.03 • -0.035 • -0.04 • -0.045 • -0.06 U) -0.05 -0.1 • -0.15 • -0.2. Chapter 3. Squeezed state generation in photonic crystal microcavities^60 1 3 3 3 2 2 2 0.8 • 0.6 O . 0 ())) o 1,11‘t -1 -1 -1 r - 0.4 -2 -2 -2 0.2 -3 -3 -3 -4 -2 X (am) 2 4^-4^-2 0 X (km) 2 4^-4^-2 X(um) 2 0 4 Figure 3.9: Intensity profile of a) X component and b) Y component of the electric field associated with the cavity in Fig. (3.8a), and c) total intensity of the pump beam in the vicinity of the cavity. -0.2^-0.1^0^0.1^0.2^0 3^ -0.2^-0.1^0^0.1^0.2^0 3 (THz) i2 (THz) Figure 3.10: The spectrum of squeezing for the Y quadrature of the sample in Fig. (3.8) for a crystal oriented along the [100] direction for, a) CW pump and b) 500 ps pulses. Chapter 3. Squeezed state generation in photonic crystal microcavities^61 level. The threshold of the degenerate down-conversion process for this structure is at 2g = 7 1 +72 = 62 GHz, therefore, with the above chosen pump beam parameters we are still below the threshold. The optimum squeezing degree for the structure at the threshold is ,---, 70%. One of the squeezed state applications is in continuous variable quantum telepor- tation [6]. If there is any bipartite entanglement, the fidelity of the teleportation (the overlap between the initial state and the constructed state in the receiver station) becomes higher than 0.5 [7]. It can be shown that [35] by having 20% squeezing in a degenerate down-conversion process the fidelity becomes 0.54 and for the opti- mum squeezing degree of '70% for the structure in Fig. (3.8), the fidelity reaches 0.62. It should be mentioned that in the first experimental demonstration of quantum teleportation with squeezed states in 1998, the measured fidelity was 0.58 [6]. In summary, a formalism for field quantization of open optical cavities was in- tegrated with a quantum treatment of second-order parametric down-conversion in a realistic 3D microcavity structure and finite-difference time-domain simulations to estimate the squeezing spectrum of radiation coupled out of the microcavity mode into a single channel waveguide. The cavity intrinsic loss and its leakage to a single- mode waveguide were treated as two distinct output channels respectively. In order to increase the coupling efficiency between the cavity and the single-mode channel, the cavity was tilted 60 degrees with respect to the waveguide axis on a [100] ori- ented A1GaAs crystal. The Q of the cavity without the waveguide was 80000. The excitation beam was considered as 500 ps pulses at 80 MHz repetition rate with 10 mW average power. The spectrum of squeezing at the waveguide output for a degen- erate down-conversion process showed a promising " 20% squeezing. 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Feshbach, A unified theory of nuclear reactions. II, Ann. Phys. 19, 287-313 (1962). [24] S. Nordholm and S. A. Rice, A quantum ergodic theory approach to unimolecular fragmentation, J. Chem. Phys. 62, 157- 168 (1975). [25] M. Shapiro, P. Brumer, Quantum control of bound and continuum state dynam- ics, Phys. Rep. 425, 195-264 (2006). [26] L. Mandel, E. Wolf, Optical coherence and quantum optics: Cambridge Univer- sity Press, New York (1995). [27] M. Hillery, L. Mlodinow, Quantized fields in a nonlinear dielectric medium: a microscopic approach, Phys. Rev. A 55, 678-689 (1997). [28] P. D. Drummond, Z. Ficek (Eds.), Quantum Squeezing: Springer-Verlag, Berlin (2004). [29] Z. Y. Ou, S. F. Pereira, H. J. Kimble, Realization of the Einstein-Podolsky-Rosen paradox for continuous variables in nondegenerate parametric amplification, Appl. Phys. B 55, 265-278 (1992). [30] M. D. Reid, P.D. Drummond, Quantum correlations of phase in nondegenerate parametric oscillation, Phys. Rev. Lett, 60, 2731 (1988); Correlations in nondegen- Chapter 3. Squeezed state generation in photonic crystal microcavities^66 crate parametric oscillation II, below threshold results, Phys. Rev. A 41, 3930-3949 (1990). [31] Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, Design of a channel drop filter by using a donor-type cavity with high-quality factor in a two- dimensional photonic crystal slab, Appl. Phys. Lett. 82, 1341-1343 (2003). [32] S. Adachi, GaAs and Related Materials: Bulk Semiconducting and Superlattice Properties: World Scientific Publishing Company, Singapore (1994). [33] Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Meriadec, and A. Levenson, Phase-matched frequency doubling at photonic band edges: efficiency scaling as the fifth power of the length, Phys. Rev. Lett. 89, 043901 (2002). [34] T. Asano, Bong-Shik Song, Y. Akahane, and S. Noda, Ultrahigh-Q Nanocavities in Two-Dimensional Photonic Crystal Slabs, IEEE J. of Selec. Top. in Quant. Eelec. 12, 1123-1134 (2006). [35] P. van Loock, and S. L. Braunstein, Multipartite Entanglement for Continuous Variables: A Quantum Teleportation Network, Phys. Rev. Lett. 84, 3482-3485 (2000). 67 Chapter 4 Efficient coupling of photonic crystal microcavity modes to a ridge waveguide The utility of the microcavity-based sources strongly depends on the efficiency with which electromagnetic energy can be coupled from the microcavity to single channel waveguides connected to other circuit elements'. This feature is especially important for quantum optical applications since the statistics or reduced quantum noise of the non-classical light generated in microcavities is not well-preserved when there is more than one channel in communication with the microcavity [1, 2]. A figure of merit to characterize the efficiency of a cavity for quantum optical applications is the ratio of cavity loss through a desired output channel to the total loss [1]. To have optimum squeezing in the field quadratures induced in a nonlinear cavity, or to preserve the photon distribution of the non-classical light generated in these structures, this ratio should be close to one. In other words, the desired channel should be the main channel that contributes to the cavity loss. As was shown in Chapter 3, Fig. (3.3), the relative coupling efficiency of a cavity mode to a single channel output, versus all possible continua, has a strong influence on the degree to which the radiation coupled to the single channel guide has a non-classical nature. ' A version of this chapter has been published. Banaee, M. G., Pattantyus-Abraham, A. G., McCutcheon, M. W., Rieger, G. W. , and Young, Jeff F. (2007) Efficient coupling of photonic crystal microcavity modes to a ridge waveguide, Appl. Phys. Lett. 90, 193106. Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 68 In this chapter, a new geometry is introduced to outcouple a photonic crystal microcavity mode through a single-mode ridge waveguide in a silicon on insulator (SOI) wafer. The microcavity is free standing (to increase its quality factor due to better vertical confinement), but the rest of the structure is supported on a SiO 2 layer, which makes it robust. The ridge waveguide is connected by a short segment of a PhC waveguide to the microcavity. The length of the PhC waveguide is kept to a minimum to reduce the impact of the strong dispersion that it introduces. Once the light is coupled from the microcavity to the ridge waveguide efficiently, it can be guided anywhere on the integrated photonic chip. In this experiment, a 2D PhC grating coupler was added to the structure to diffract the light off the chip for monitoring purposes. The key component of this project though is the demonstration of an ,--56% coupling from the microcavity to the ridge waveguide, which is independent of the excitation source and the grating output coupler. To our knowledge, there are just three other experimental reports [3, 4, 5] of a structure for the unidirectional out-coupling of light from a PhC microcavity. Side- and unidirectional coupling between multi-mode PhC waveguides and microcavities was studied qualitatively in Ref. [3]. The authors in Ref. [4] coupled the microcavity to a 1D PhC waveguide, with the entire structure made free-standing. They deduced a coupling efficiency of 13% from the cavity, via the PhC waveguide, to a tapered optical fiber coupler. Very recently, a coupling efficiency of 40% from a microcavity to a PhC waveguide (with two-hole separation) on a GaAs membrane was reported, Ref. [5], based on a comparison of simulated and measured cavity Q values. They estimated a 90% coupling efficiency when the cavity is tilted with respect to the waveguide axis. The work reported below emphasizes coupling from a free-standing microcavity to a simple, single-mode silicon ridge waveguide supported on SiO 2 : this offers scalability and relatively low loss connectivity over long distances on a SOI chip, with a coupling efficiency of ,60%. Although this work is done on a SOI structure, which does not have a second-order nonlinearity (to generate the down- taper waveguide ridge waveguide microcavity Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 69 conversion process discussed in chapter 3) the current design could be realized in III-V semiconductor materials using a slightly different process. The choice of SOI material for this project was based on several years of experience in the Young lab processing both SOI and A1GaAs. SOI is much easier to reproducibly process, which significantly reduces the overall development time for a new structure such as this. 4.1 Design and numerical simulation of the waveguide-microcavity structure Fig.(4.1) shows the geometry of the SOI structure designed and fabricated for the work described below. Out^grating coupler Figure 4.1: Schematic diagram of Si slab photonic crystal structure in a SOI wafer. All features except the substrate (in vertical direction) are to scale. There is no oxide layer beneath the microcavity region (see text). 0 0 0 0 0 0 000°o°o°o°oo° °o°°o 0o°o°o°o°oo0o°o o°o°od00% o°o°odoo% o°o°ooo0o°o°o°oo0o°°o°o°odoo°o o°o°od00% o°o°oo0o% o°o°ooqa(kr-rPo%ocio°o °0%ogo% o°q°0 1.152a 0 0.  00 0 0 00 000 0 ^ 0 O °  0 ° 040.5a t\.l Zia 4-0 0.974a (a) ^ (b) Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 70 It consists of a microcavity, a short segment of PhC waveguide, a single-mode ridge waveguide, a tapered waveguide, and a 2D grating coupler. The microcavity is a 3-missing hole cavity [6] in a triangular lattice, with nominal parameters of lattice constant a=420 nrn and air hole radius r=0.29a, Fig.(4.2a). The Si slab PhC structure is 196 nrn thick and lies on top of a 1.2 pm Si0 2 layer and a 720 pm Si substrate. The PhC waveguide is made by removing six air holes along the F-K direction, and it is separated by two holes from the microcavity. The geometry of the interface between the PhC and ridge waveguides is shown in Fig (4.2b). The connection region of the ridge waveguide was tapered to maximize its coupling efficiency to the PhC waveguide segment [7]. The width of the triangular taper in Fig. (4.2b) is t=0.54a. To ease the coupling of light out of the 10 pm long ridge waveguide, it is connected to a 30 long parabolic tapered waveguide (with start and end widths of 400 nm and 14 respectively) and a 2D rectangular PhC grating coupler [8, 9] with pitches of a11=795 nm (parallel to the waveguide axis), a1 =750 nm and hole radius of r g=244 nm. Figure 4.2: 3-missing hole microcavity and its interface with waveguides. A three-dimensional finite-difference time-domain (FDTD) simulation [10] is used Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 71 to characterize the structure and estimate its coupling efficiency. To calculate the overall coupling of an excitation pulse incident on the microcavity, to the signal diffracted off the wafer by the 2D grating coupler (see Fig. (4.1)) we separately determine the coupling from a Gaussian source to the cavity, from the cavity mode to the ridge waveguide, from the waveguide to the grating coupler, and finally from the grating coupler to the far field. The overall coupling efficiency is compared to the experiment used to verify the calculation, but in an integrated quantum optical chip, only the coupling efficiency from the cavity mode to the ridge waveguide is important. First the quality factor of an isolated cavity (cavity without any waveguides) with its fundamental mode centred at A=1.520 Am (197 THz) is estimated as Q=4910. After adding the waveguides (PhC and the ridge waveguides) to the microcavity, Fig.(4.3), its Q drops from 4910 to 2650, and the frequency of the cavity mode changes by less than 0.1%. single-mode ridge waveguide PhC microcavity 6 missing-hole PhC waveguide Figure 4.3: An expanded view of the cavity-waveguide coupling region that shows the connection of the microcavity to the ridge waveguide through a 6 missing-hole PhC waveguide. The simulation showed that by keeping the ridge waveguide and adding back all 6 volume monitor •■■•11. .1111■11. Si substrate Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 72 air holes along the PhC waveguide, the Q of the cavity reaches the isolated cavity value. Therefore, removing the air holes of the PhC waveguide allows the waveguide to serve as the main channel of the cavity mode leakage to the environment. To calculate the coupling efficiency from the microcavity to the ridge waveguide, a 130 fs Gaussian pulse is used to excite the microcavity from the top (see Fig. (4.4a)), and the energy stored in the microcavity right after the pulsed excitation, and the energy that ultimately leaks out to the ridge waveguide, Fig.(4.4b) are calculated. \ / surface monitors Figure 4.4: Monitors (screenshots) of the FDTD simulation for estimating the cou- pling efficiency from a) the source to the microcavity and b) from the microcavity to the ridge waveguide. The simulation area is shown by an orange box. The beam waist of the Gaussian source that resonantly deposits energy in the cavity mode is set to 1.25 pm. The source has its center frequency at 197 THz with a 130 fs pulse duration and an 550 fs offset: it is polarized along the x-direction (the polarization direction of the cavity mode). The simulation area in this case, Fig.(4.4a) only contains the source and free space. Then the Poynting vector at each point in a surface at z=1.0 iim is obtained, which after a surface and time integration, gives the energy of the source incident on the microcavity structure. The area of this Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 73 surface is S = Ax x Ay = 9.4 x 9.4 itm2 . Then the full structure is included and the energy contained in the microcavity is calculated in a volume V = Ax x Ay x Az = 130a x 36a x 2.4a at the time when the field inside the cavity reaches its maximum. The ratio of the energy contained in the cavity to the energy of the Gaussian source is 0.02, which is the coupling efficiency from the incident source to the cavity mode. To calculate the power passing through the waveguide, the Poynting vector along the waveguide axis at a surface 5.70 Lim away from the interface of the two waveg- uides with area=-Ay x Az = 2 x 1 ,u,m2 is considered, Fig.(4.4b). By comparing the energy passing through the ridge waveguide to the energy scattered into the cavity by the pump beam, the coupling efficiency from the microcavity to the waveguide is calculated to be 0.56. 4.1.1 Power coupled out of the grating coupler As mentioned in the introduction, the tapered waveguide and 2D grating coupler are added to the microcavity-waveguide geometry to facilitate the first coupling efficiency measurement of the proposed structure. These elements and their coupling efficiencies are not crucial to the main result of getting 56% coupling efficiency from the cavity to the single-mode waveguide, as estimated above, but they are crucial in order to compare the simulation result with the experiment, which measures the light coupled out of the grating. In the following, the key parameters of these components are estimated by FDTD simulation. The ridge waveguide is connected to a L=30 urn long parabolic tapered waveguide with the following profile y2 = C2 - 2cx^ (4.1) where c =^— w2)/8L and the start and end widths of the taper are w, =400 nm and wf =14 um respectively. Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 74 surface monitor on top of the grating coupler surface monitor perpendicular to the ridge waveguide axis Figure 4.5: a) The surface monitors (screenshots) on top of the grating coupler and perpendicular to the ridge waveguide axis and b) 3D view of the simulation area. xi 012 Figure 4.6: Farfield pattern of the light diffracted from the grating coupler. Numerical labels indicate angles in degrees. Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 75 Fig.(4.5) shows the geometry of the monitors that were used to estimate the coupling efficiency from the ridge waveguide to the top of the grating coupler. The distance between the surface monitor in Fig.(4.5a) on the top of the grating coupler and the wafer is 0.5 pm and it has S = Ax x Ay = 18 x 18 pm'. The simulation shows a 0.17 coupling efficiency from the ridge waveguide to the top of the grating coupler, and the light comes out of the grating coupler at 17° with respect to the normal, Fig.(4.6). Considering all these factors together, the simulation suggests that ,0.22% of the incident Gaussian pulse is coupled off the chip via the microcavity mode, waveguides, and grating coupler. This is the quantity that can be compared to the experimental results below. 4.2 Fabrication procedure A scanning electron microscope (SEM), operating at 30 kV, was used to write the pattern in a 500 nm thick ZEP-520A (Zeon Corp.) e-beam resist layer. The pattern was transferred to the Si slab by dry etching with C1 2 2 . Fig.(4.7) shows the fabricated structure. The trenches, which define the walls of the taper and the ridge waveguide should be defined as a filled polygon (line type 5) in the nanometer pattern generation software (NPGS) of the SEM machine: an 0.3 nC/cm dose was used to write these trenches and the dose for writing the holes of the cavity and the grating coupler was 0.26 nC/cm. In order to maximize the Q of the localized mode, the Si02 layer under the microcavity membrane was removed by an undercutting process. In this procedure, the already patterned structure is spin coated with HMDS (Hexamethyldisalizane, a chemical primer used to remove surface moisture and improve photoresist adhesion), then the photoresist (AZP 4110) is spun on at 5000 rpm for 40 sec. The thickness of 2 The recipe for e-beam writing and plasma etching of SOI wafer was developed by Andras Pattantyus-Abraham, which is summarized in Murray McCutcheon's PhD thesis (Nonlinear Optics of Multi-mode Planar Photonic Crystal Microcavities. submitted to UBC library in July 2007). Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 76 Figure 4.7: SEM image of the fabricated microcavity-waveguide structure, before the undercutting process. area covered by mask (a)^ (b) Figure 4.8: Optical image of the structure a) after developing the photoresist and b) after wet etching. The dark region in the left side of the Fig.(a) shows the area covered by the mask. •• • 04•c C • C C C. C. • CCO'• CC CC C.' • • • 411, • •• • or■ - 4; 0000 C000 0000 30000000000000000000 L 000000 0000000000000000'00 000000 0000000000 00 000000000 00 OCC 0000000000090 000 OCCC0 0000000000000000 0 0000000 0 000 000000cco CO CJC: 00000000000 ono^0,00 0.000 0 000000000000 G C °C C, 000000000000000 00000000000000 CC COCC0 000000000CCGC0CGC0CTOOCC Coo0000000000.(mcocacco,:c 0390000000 000000000007 - 000 ,200000,0000000000000 ,000010(i0000000000 , r' 00000000000000 '" ,0000( , 000000 00000000030 CO00t000000 00000(0000000 0300000000000300 ^000 •^00000000, 0000000co.:, C 00 00400000, ^ 00000^1100=00000: u0000^cr000maraccc 10 im tapered waveguide ^2D grating coupler Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 77 halo area shows the undercut region ridge waveguide ' Figure 4.9: The detailed SEM images of the sample after undercut process. a) shows the undercut region around the microcavity segment b) clearly shows the width of the ridge waveguide c) displays smooth walls of the taper waveguide and finally d) shows the 2D grating coupler and the termination of the taper waveguide. Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 78 the coat was measured by an alpha-step 200 machine (Tencor Instruments) as 1 pm. Before we put the wafer in the mask aligner, it was soft baked at 90° for 10 minutes. Small chromium squares (200 x 200 inn') on a 4 inch optical mask (made of soda lime with 3 mm thickness) were used to cover the waveguides and the grating coupler. After exposing the photoresist under a mask aligner (Canon PLA-501F double-side 100mm mask aligner) for 15 sec (current passing through the Hg lamp was set to 5A), the photoresist was developed in a solution (4:1, DI water:AZP4110 developer) for 60-120 sec. Then by looking at the sample under an optical microscope, Fig. (4.8a), we make sure the grating coupler and waveguides are properly covered by the photoresist. After hard baking (120° for 10 minutes) we let the sample cool down for ten minutes. Finally the sample is dipped in a HF solution (10 parts 40% NH 4 F to 1 part 49% HF) for 15-20 minutes and rinsed with DI water, Fig. (4.8b). Fig.(4.9)shows the SEM images of the final fabricated structure. Fig.(4.9a) shows the undercut microcavity and its interface to the ridge waveguide. Although the end of the ridge waveguide is also undercut (which is unavoidable) it remains single mode throughout its length. 4.3 Optical Characterization The microcavity mode frequency and quality factor are measured by a resonant scat- tering experiment. This technique was developed by Iva Cheung, and then refined by Murray McCutcheon [11]. A 130 fs pulse train of a Spectra-Physics OPO laser system is focused on the microcavity from the top half-space by a 100X microscope objective3 (the beam waist radius of the focused beam was measured by a knife-edge method [12] as 1.25 pm) and in the first instance, the light scattered from the micro- cavity back through the 100X objective is detected with a Bomem Fourier transform 3 Mitutoyo objective, M plan NIR 100, numerical aperture 0.5, working distance 12.0 mm, and focal length 2.0 mm. Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 79 spectrometer, Fig.(4.10). Figure 4.10: Focusing the light on the microcavity and aperturing light out of cavity or grating coupler. In this case the aperture between two lenses is aligned such that only scattered light from the cavity reaches the spectrometer, Fig.(4.11a). The adjustable rectangular aperture is at the common focal point of lens 1 (diameter 2.5 cm and focal length 6 cm) and lens 2 (diameter 2.5 cm and focal length 9 cm). A CCD camera (Electrophysics, Micron viewer 7290A) is used to view the whole image and align the aperture. It should be mentioned that the image in Fig. (4.10) is not to scale. The diameter of the 100X lens is 2 cm and the sample is just 80 pm long, which allows the image of the whole structure to be viewed at once on the CCD camera. To get rid of most of the pump laser in the scattered spectrum from the cavity, two cross polarizers are used in the experiment. First the horizontal polarization of the pump laser is rotated by 45° by using a half-wave plate (HWP) and a polarizer. The vertical component of this excitation beam can thus excite the cavity mode, which is oriented so it's polarization axis is in the vertical direction. The analyzer before the spectrometer, oriented at 90 degrees with respect to the incident polarization, thus Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 80 Figure 4.11: Aperturing the microcavity (a) and the grating coupler (b). largely blocks the non-resonantly scattered pump laser, but allows roughly 50% of the resonantly scattered cavity mode signal to reach the spectrometer. Fig.(4.12a) shows the resonant scattering spectrum of the cavity mode centered at 1.510 ium with Q=2120, not far from that predicted by simulation (Q=2650). The Q measured on isolated cavities of the same design was Q=4650, also close to that predicted by simulation (Q=4910). To quantitatively compare with the simulation described above, an image of the grating coupler was apertured and coupled into the spectrometer. The aperture is aligned such that it only allows the light scattered out of the grating coupler to reach the spectrometer, Fig.(4.11b). Fig.(4.12) shows the signal spectrum from the grating coupler, and compares it to that obtained using the resonant scattering geometry from the cavity itself. In this experiment, 398 [tW average OPO power focused on the top of the microcavity gives a signal with 600 nW average power out of the grating coupler, giving a measured efficiency, from the excitation pulse, to that detected from the grating, of 0.15%. The calculated total coupling efficiency from the simulation (0.22%) agrees well with the measurement (0.15%). This level of agreement, given the uncertainty in the 10 9 8 (b) 6 5 4 • 3 2 Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 81 (a) 1- z z (7) 1.48^1.49 1.5^1.51^1.52 WAVELENGTH (pm) 1.53^1.54 Figure 4.12: Resonant scattering (a) and background-free signal emanating from the grating coupler (b). actual Gaussian spot-size, offers confidence that the all-important coupling efficiency from the microcavity to the ridge waveguide is close to that simulated using FDTD (,56%). This is further supported by the measured change in the Q of the waveguide- coupled microcavity (Q=2100), as compared to the isolated cavity (Q=4650). By using Eq.(3.42) of chapter 3, the Q of the waveguide is C2,,g = 3830, therefore the loss due to the waveguide is 2100/3830=55% of the total loss. Additional simulations suggest that this can be increased by simply shifting the two holes on either end of the cavity (such shifts also affect the Q of an isolated microcavity[13]). With a 10% lateral (outward) shift of the holes, the efficiency is increased to 68%. This means that the main channel of cavity loss is through the 1D ridge waveguide and shows a promising geometry to do quantum optical processing. Even more improvement in the coupling efficiency can be achieved by tilting the cavity with respect to the waveguide axis [5] where coupling efficiencies up to 90% are possible. The simulation in chapter 3 was based on a tilted cavity with 74% coupling efficiency. In summary, a three missing hole defect cavity in a 2D hexagonal silicon photonic Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 82 crystal membrane structure was made free standing, and coupled to a single-mode silicon ridge waveguide through a short segment of a 1D photonic crystal waveguide. The ridge waveguide was connected to an integrated grating output coupler through a tapered waveguide region in order to measure the light coupled from the cavity mode to the waveguide. The full structure was simulated using FDTD to mimic the experiment that involved exciting the microcavity from the top half space using a 130 fs pulse centred at the cavity mode frequency, and collecting the light diffracted off the sample by the grating output coupler. The overall efficiency from the excitation pulse to the detected signal was estimated by FDTD to be 0.22%, and measured to be 0.15%. This suggests that the actual structure mimics well the simulations of the ideal structure, and that the critical coupling efficiency from the cavity to the single- mode waveguide is ,55%. Slight modifications of the cavity design can push the ideal coupling efficiency above 90%. Although this was all demonstrated in silicon, a slightly modified process could be easily applied to III-V wafers in the AlGaAs system assumed for the quantum optical simulations described in chapter 3. 83 Bibliography [1] H. J. Kimble, Fundamental Systems in Quantum Optics: Elsevier Science Pub- lishing, Amsterdam (1992), chapter 10. [2] A. M. Fox, Quantum Optics: Oxford University Press, New York (2006). [3] S. N. Olivier, C. J. M. Smith, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, and U. Oesterle, Cascaded photonic crystal guides and cavities: spectral studies and their impact on integrated optics design, IEEE J. of Quant. Elec. 38, 816-824 (2002). [4] P. E. Barclay, K. Srinivasan, 0. Painter, Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper, Opt. Exp. 13, 801-820 (2005). [5] A. Faraon, E. Waks, D. Englund, I. Fushman, and J. Vuckovic, Efficient photonic crystal cavity -waveguide couplers, Appl. Phys. Lett. 90, 073102 (2007). [6] Y. Akahane, M. Mochizuki, T. Asano, Y. Tanaka, and S. Noda, Design of a channel drop filter by using a donor - type cavity with high -quality factor in a two- dimensional photonic crystal slab, Appl. Phys. Lett. 82, 1341 - 1343 (2003). [7] E. Miyai, and S. Noda, Structural dependence of coupling between a two- dimensional photonic crystal waveguide and a wire waveguide, J. Opt. Soc. Am. B 21, 67-72 (2004). Chapter 4. Efficient coupling of photonic crystal microcavity modes to a ridge waveguide 84 [8] A. Mekis, A. Dodabalapur, R. E. Slusher, J. D. Joannopoulos, Two-dimensional photonic crystal couplers for unidirectional light output, Opt. Lett. 25, 942-944 (2000). [9] G. W. Rieger, K. S. Virk, J. F. Young, Nonlinear propagation of ultrafast 1.5 ,um pulses in high-index-contrast silicon-on-insulator waveguides, Appl. Phys. Lett. 84, 900-902 (2004). [10] The FDTD code is provided by Lumerical Solutions Inc. [11] M. W. McCutcheon, G. W. Rieger, I. W. Cheung, J. F. Young, D. Dalacu, S. Frederick, P. J. Poole, G. C. Aers, R. L. Williams, Resonant scattering and second- harmonic spectroscopy of planar photonic crystal microcavities, Appl. Phys. Lett. 87, 221110 (2005). [12] L. Bachmann, D. M. Zezell, E. P. Maldonado, Determination of beam width and quality for pulsed lasers using the knife-edge method, Inst. Sci. Tech. 31, 47-52 (2003). [13] Y. Akahane, T. Asano, B. Song, and S. Noda, Fine-tuned high-Q photonic-crystal nanocavity, Opt. Exp. 13, 1202-1214 (2005). 85 Chapter 5 Summary and conclusions Two distinct nonlinear optical processes in periodically textured, high index contrast dielectric media were investigated both experimentally and theoretically in this thesis. The first work involved explaining the discrete set of peaks observed in the output spectrum of a 1.1 m length of silica-based photonic crystal fibre, whose number and redshifts with respect to the 130 fs excitation pulses increased with increasing launch power. With ,s, 13 mW average power from an 80 MHz train of 130 fs pulses launched into the fibre, three distinct peaks are observed at the output, red shifted up to 200 nm from the 800 nm excitation wavelength. The spectra obtained at a fixed propagating power, at different lengths of propagation, revealed that the red shift of each peak in the spectrum increased monotonically with propagation distance. Measurements of the far field radiation pattern associated with separate peaks in the output spectrum confirmed that all radiation was propagating in the fundamental transverse mode of the fibre. The power dependent and propagation length dependent shifts were successfully explained by solving a generalized nonlinear Schrodinger equation that included instantaneous (Kerr-like), and non-instantaneous (Raman-like) third-order nonlinear response terms, and linear dispersion terms up to sixth order. A detailed investigation of the numerical simulation predictions confirmed that the discrete peaks are the result of the breakup of a high-order soliton launched at the input of the fibre, into several fundamental solitons, which then redshift continuously at different rates, due to the Raman self-shifting phenomenon. The breakup process occurs just before the high-order soliton first approaches its minimum pulse duration within a few centimeters of the input, and this breakup is only observed when the Raman term Chapter 5. Summary and conclusions^86 is included in the equations of motion. Although the "self-steepening" process plays a qualitative roll in the spectral evolution, the key requirement to obtain a satisfactory quantitative description of the observations, was to include the full time-dependence of the non-instantaneous Raman response in the simulation, rather than using the common slowly varying envelope approximation. The parametric down-conversion process in a 3D photonic crystal microcavity- waveguide structure was studied next. Small mode volume and high quality factor photonic crystal microcavities provide a suitable structure to generate squeezed states of light in the cavity, which can be efficiently coupled into a single-mode waveguide. Such a structure was designed and fabricated in a 200 nm thick silicon membrane, on a silicon-on-insulator wafer. A combination of optical measurements and finite- difference time-domain simulations confirmed that the coupling efficiency of classical light from the microcavity mode to the single mode ridge waveguide channel in this structure was The simulations suggest modified designs could achieve classical coupling efficiencies of over 90%. A quantum mechanical model was established to quantitatively describe the para- metric down-conversion process in a nonlinear multi-mode microcavity coupled to one single-mode waveguide channel (the one used to extract the squeezed light signal), and a second, multi-mode channel that represents all of the other leakage channels from the cavity. The squeezing degree in the waveguide output in an A1 0 . 3 Ga0 . 7As membrane was estimated to be ,20% below the shot noise level when the cavity was pumped with 500 ps pulses at an 80 MHz repetition rate and 10 mW average power. Conclusions This work has demonstrated two ways in which periodic dielectric texture with relatively high refractive index contrast can be used to realize interesting and poten- tially useful nonlinear optical processes using unamplified laser sources, and far less Chapter 5. Summary and conclusions^ 87 nonlinear material than is required to observe the same effects in the bulk. In the case of the photonic crystal fibre, the main feature of the dielectric texture, relevant to the generation of Raman self-shifted solitons that redshift more than 200 nm in just a meter of propagation, is its impact on the dispersion of guided mode propagation in the core of the fibre. In the second example, where a 3D cavity on the scale of a cubic wavelength in dimension, can be used as a source of (quantum) squeezed light, it is the full photonic bandgap property of the photonic crystal material that surrounds the cavity that is crucial. This allows light to be localized in a tight volume, for many optical cycles, thus effectively enhancing the nonlinear interaction strength that would otherwise have to accumulate over much larger propagation distances. The demonstration that coupling efficiencies of at least 55% from the microcavity modes, to single channel waveguides on robust substrates is very encouraging for the future potential of these type of structures in integrated quantum optical applications. The main path to further reducing the optical power requirements for such quan- tum optical devices involves designing slightly more sophisticated cavities coupled to two waveguides (one unidirectional channel through which to extract the fundamental modes as shown here, and a second one designed to resonantly excite a higher order cavity mode at the pump frequency). The second stage of further development should incorporate resonant media in the cavity to enhance the second order susceptibility, without introducing too much loss. 88 Appendix A Third-order nonlinear effects in optical fibres The high-purity silica used to fabricate optical fibres for telecommunication appli- cations, and the photonic crystal fibre studied here, is an amorphous material and therefore there are no second-order nonlinear effects in the optical fibres. The most relevant nonlinear processes in optical fibres thus originate from the third-order sus- ceptibility, which yields a nonlinear polarization: P (3) (r, t) = E0 R (3) (t — t2,t — t3 ) : E(r, t i )E(r, t2 )E(r, t3)dt i dt2 dt3, (A.1) where R represents the full third-order response of the fibre material in the time domain'. Ignoring processes like third harmonic generation, and focussing on pro- cesses that involve an intensity-dependent refractive index (self-phase modulation) or loss/gain (Raman scattering), allows a simplification of Eq.(A.1) to the following form: P(3)(r, t) = —34 cox (3) E(r, t) R(7-)1E(r,t — T)1 2dT, (A.2) where x (3) represents the magnitude of the third-order response of silica, R(T) denotes the normalized response function, and we suppose the propagating light through the fibre is polarized. The normalized response function, R(T) in silica optical fibres has 'Primary reference for the following derivations was from: Govind P. Agrawal, Nonlinear Fiber Optics, Academic Press (2001) Appendix A. Third-order nonlinear effects in optical fibres^89 two dominant time scales. The nonresonant, virtual electronic transitions occur on a very short time scale, which can be modeled by an instantaneous delta function. The interaction of light with the ions in the silica (phonons) results in Raman scattering, which has a characteristic time scale associated with a phonon vibration frequency fs) and is modeled by a Lorentzian line shape 2 . The third-order response function that includes these two distinct processes is typically expressed as R(7-) = (1 — fR)6 (7) + fRhR(T),^ (A.3) where 2 + 2 hR(T) \ Ti 72 =- 2 eXp( — 7172) sin(7/71). (A.4) The coefficient fR in Eq.(A.3) parameterizes the relative contribution of the delayed Raman response to the instantaneous third-order nonlinear polarization. The param- eters 7-1 and 7-2 in the above equation are chosen to yield a good fit to the measured Raman-gain spectrum in silica fibre. Their appropriate values are T i = 12.2 fs and 72 = 32 fs and fR is estimated3 to be about 0.18. The third-order nonlinear polarization defined in Eq.(A.2) introduces an intensity dependent refraction index and absorption in the fibre. To show this, we write this polarization as: P(3) (r, t) = E0 6NL(t)E(r , t),^ (A.5) where the nonlinear dielectric function is: 3^ 3^2 ENL (t) = — 4 X(3) J R(T)1E(r, t — 7)1 2 d7- = 4x(3) E n c I R(r)I (r, t — 7)(17^(A.6) oo 2D. Hollenbeck and C. D. Cantrell, "Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function", J. Opt. Soc. Am B 19, 2886-2892 (2002). 3 K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers", IEEE J. Quant. Elec. 25, 2665-2673 (1989). Appendix A. Third-order nonlinear effects in optical fibres^90 where /(r, t) is the pulse intensity. In frequency space we have 3^ , ENL(w) = -4X (3) R(w) 2( ^(A.7) The overall dielectric function of the medium i Es na °combination of the linear and nonlinear responses: E (w) = 1 + xm (w) + ENL(w), which is also related to the refraction index and the absorption coefficients as4 : (w .E(w) = (n(w) + ia 2w )c ) 2 By substituting Eq. (A.6) in Eq. (A.9) we have: n(w) = no + n2 IE(r, w)I 2 , a(w) = a() + a21E(r,w)1 2 , where the nonlinear refraction index and absorption are: n2 = 8n 3 ^X (3) Re[R(w)], o 3wo a2 = ^x(3)Im[R(w)], 4cno and (A.10) (A.11) (A.12) (A.13) no = 1 + 2-1 Re[X (1) (w)i,^ (A.14) ao = w0^ im[X (1) (w)i. (A.15)no c In the following sections these intensity dependent phenomena are described in more detail. 4 G. P. Agrawal,"Nonlinear fiber optics," (San Diego: Academic Press, 2001). (A.8) (A.9) n = no + n2 1E1 2 , (A.16) ONL = won21A(0 , t) 1 2L c (A.18) Appendix A. Third-order nonlinear effects in optical fibres^91 A.1 Self-phase modulation An optical soliton represents a pulse with an envelope that is preserved during the propagation. In the field of nonlinear optics, solitons are classified as temporal or spatial, which respectively accounts for whether the confinement of the light occurs in time or space when an optical pulse propagates in a third-order nonlinear medium. Both cases are caused by the optical "Kerr effect" wherein the refractive index of the medium depends on the intensity of the light, leading to self-phase modulation or self-focusing of a shaped optical pulse. This can be understood by expressing the refraction index dependence on the local intensity of light, Eq. (A.10) where the nonlinear refraction index, n2 , is proportional to the real part of the third- order susceptibility, Eq.(A.12). For silica fibre n 2 is positive. When an intense pulse propagates through an optical fibre, Fig. (A.1) , its envelope therefore gains an addi- tional phase due to this nonlinear refraction index (this is the solution of the gener- alized nonlinear SchrOdinger equation, which is derived later, without any dispersion and Raman scattering effect) A(z , t) = A(0, t) exp[i b±,'- (no + n2 1A(0, t)1 2 )z],^(A.17) or where wo is the carrier frequency of the pulse and L is the propagation length. The phenomenon of inducing the nonlinear phase shift by the pulse itself is called self- phase modulation (SPM). More quantitatively, suppose the incident pulse at the input of the fibre is chirp-free (constant phase) with a secant hyperbolic profile, A(0, t) = A, sech(t/T), then after a distance L, the spectrum of the pulse is altered Appendix A. Third-order nonlinear effects in optical fibres^92 due to this nonlinear phase accumulation, which is different for different parts of the pulse: dO dt^° dt NL ^w e Th2 6w =  A2 L—d sech2 (t/r).^(A.19) Fig.(A.1) shows the pulse intensity and the frequency chirp caused by this SPM. The frequency shift near the peak of the pulse is zero, but the frequency of the leading edge (side with negative time) is lowered (leading edge is red-shifted), whereas those in the trailing edge are raised (trailing edge is blue-shifted). Due to the opposite frequency shift at two edges of the pulse, this SPM leads to spectral broadening of the optical pulses. — 1 • - - • -dl/dt 0.8 • 0.6 • 0.4 • 0.2 • 0 - 0.2 • - 0.4 • I - 0.6^ , - 0.8 • 0 tit Figure A.1: Pulse intensity (solid line) and frequency chirp (dashed line). A chirped pulse propagating in a non-dispersive medium is not transform limited, and the envelop gradually develops a complex shape. However, if the linear disper- sion in the medium exhibits negative group velocity dispersion (GVD), then it is possible under special circumstances for the dispersion to correct the nonlinear phase accumulation exactly, allowing the pulse to propagate with a fixed shape. This is the definition of a fundamental soliton. Higher order solitons are also possible, where the negative GVD compensation leads to a periodic (not stationary) pulse shape. .9R(C2) = cn ^X(3)fRim[hn(Q)i, o (A.20) Appendix A. Third-order nonlinear effects in optical fibres^93 A.2 Raman scattering Another important nonlinear effect in optical fibres is Raman scattering. Quantum mechanically the Raman effect is the inelastic scattering of photons by phonons (quan- tized vibrational states of the ions). The scattered photons may lose energy (Stokes components) or gain energy (anti-Stokes components). When a monochromatic light beam propagates through an optical fibre, due to the spontaneous Raman scattering, a portion of photons in the beam shift in frequency due to emission of phonons. The probability of the transformation depends on the Raman response function, Eq.(A.4), as it enters in the Raman gain spectrum, Eq.(A.13), where Q represents the frequency difference between the pump and Stokes waves. The Raman gain spectrum depends on the fibre's core composition. In silica fibres g R (C2) extends over 40 THz and has a broad peak around 13 THz, which is due to the noncrystalline nature of silica', Fig.(A.2). In amorphous materials the molecular vibrational frequencies merge to form bands that overlap and make a continuum. In the next section we see how the combination of the Raman scattering, SPM, and dispersion influence pulse propagation through the optical fibres. A.3 Generalized nonlinear Schr8dinger equation This section explains the derivation of the "Generalized Nonlinear SchrOdinger equa- tion" which incorporates all of the effects described above, and is able to reproduce the experimental soliton propagation spectra reported in chapter 2. The general- ized nonlinear SchrOdinger (GNLS) equation is an equation of motion for the electric field of the propagating pulse in the presence of linear dispersion and the various 5 G. P. Agrawal."Nonlinear fiber optics," (San Diego: Academic Press, 2001). Appendix A. Third-order nonlinear effects in optical fibres^94 6 0 ^ 0 5^10^15^20^25 frequency offset (THz) 30 ^ 35 ^ 40 Figure A.2: Raman gain spectrum of silica, after Stolen et al [J. Opt. Soc. Am. B 6, 1159-1166 (1989)]. third-order nonlinear processes identified above. The electric field of a pulse propagating through a single-mode fibre satisfies the following wave equation: or 1 02 E(r, t)^02P(r, t)VxVx E(r, t) = c2 at2^tio ate 2^ 1 32E(r,t)^a2P(r, t) _^E(r, t) c2^at2 at2 (A.21) (A.22) where V. E(r, t) = 0 was used, and we supposed the field is polarized (fibre is polarization maintaining). P in the above equation can be expressed as the sum of the linear polarization P(1)(r, t) = €0x (1) E(r , t),^ (A.23) and the nonlinear polarization, Eq.(A.2). The electric field can be expressed in the frequency domain as follows: Appendix A. Third-order nonlinear effects in optical fibres^95 E(r, t) = f db.) E (r , co)e -i' t = f A(z, co)F (1, cv)e ik(w ) z-iw (A.24) where 1 denotes transverse coordinates (x,y). In the above equation it was supposed that because of the axial symmetry of the fibre, we can separate the longitudinal part of electric field, A(z, w), from the transverse part, F(_L, w). By writing V 2 = 02 /az2 + 01, and substituting Eqs.(A.2) and (A.24) in Eq.(A.22) and some simplifications we have: (9, A(z, w)[ViF (1, w) + (^— k2 (u.)))F(±, w)] + F^w)[ ẑ2 A(z, w) + 2ik(w)- a a A(z,w)] = 4 Q^w 2 f &di f dw2R(w — wi)A(z , w i )F (1, wi)A(z , w2)F (_L , w2)./1*(z,wi + w2 — w) xc F* (_L,^— w)ei[k(wi)+k(w2)-k(wi+w2-w)-k(,,,Az (A.25) where R(w — w1 ) = f d7R(T)ez (w-wi ), ^ (A.26) and we used c(w) = 1 + x( 1) (—w; w). Assuming the nonlinearity does not change the transverse mode profile significantly, the first bracket in Eq. (A.25) is zero. Also suppose the envelope function, A(z, w), is a slowly varying function (with respect to the carrier frequency) so that the second derivative term involving A can be neglected, then 3^w 2 2ik(w)F(_L,w)A(z,w) = — Ti x (3)^f dw i f dw2 R(c,,, — w i )A(z, w i )F^w i )A(z, w2 ) x W2)A * (Z, Wl +^— w)F*(i,^+W2 w)ei[k(coi)-1-k(w2)—k(w1+0)2—w)—k(4z. (A.27) Next we multiply both sides of the above equation by F* (1, w) and integrate over the 1 coordinates, a^3x(3)(2 f A(z , w)^ 8ik(w)c2 = ^ dwi f dw2 R(w — w i )G(w ,w i ,w2 )A(z ,w1)A(z,w2)A* (z , wi + b.)2 — w)x Appendix A. Third-order nonlinear effects in optical fibres^96 e i[k(wi.)+k(w2)—k(w].-102—w)—k(w)1z , (A.28) where G (u) , w i , w2 ) = f F*(i, w)F(_1_, w i )F(1, w2 )F* (1 wi +W2 - w)d2 r± f IF(.1 ,0))1 2 d2ri For simplicity suppose the mode area doesn't depend on frequency, then G in the above equation is roughly equal to the inverse of the effective mode area of the fundamental mode (f IF(±,w)1 2 d2 ri_) 2 . Aeff^f IF(1 , w)1 4d2 r-L By defining a new envelope function as: (A.29) 14(z, w) = A(z, c4))e 2 rk(w )-k01 z,^(A.30) Eq.(A.28) becomes: 7Tz A(z, - i(k(w) — ko )A(z , co) = i-y(1 w w:° ) f dw y I c/w2R(co — w i )A(z, co y );1(z, w2 ) x A* (z, wi + w2 w) (A.31) where ko = k(wo ) is wave vector at carrier frequency of the pulse and = 3X(3)wo  Q^ (A.32)ocneffAeff 7 is called the nonlinear coefficient of the fibre. By expanding k(w) around wo (centre frequency of input pulse) we have: Appendix A. Third-order nonlinear effects in optical fibres^97 1k(w) = ko + (co — wo)ki + 1 (w — wo ) 2 k2 + (w — cvo ) 3 k3 + • • + 777-2i (w — wo )"2 + • • • , (A.33) which describes the linear dispersion of the material with = ( dmk(w) ),, dwm (A.34) Finally by transferring Eq.(A.31) to the time domain and using the fact that w — w o transfers to is/at, the envelope function of a pulse inside a single-mode fibre should approximately satisfy the following equation: aA^a ^im-1 a am^ co ^= 2 A + [ rnl km  m! tm ]A + i-y(1+ wo at^deR(e)0(z, t t')1 2 ],z (A.35) where the first term in the right-hand side of the above equation was added to consider the linear loss through the fibre. This equation can be further simplified if we go to a coordinate system that moves with the group velocity of the pulse, then by using T = t — k i z we have: A^6^.m-1 am^i a^+0. 57 k[ E m i^m=2^rn! aTm iA i7(1^[A(z, T) f^dT'R(T')1.24(z,T —T')1 2 ]. (A.36) This equation is called the generalized nonlinear SchrOdinger equation. It is the equation that was solved numerically as described in Chapter 2, to simulate the experimental results reported in that chapter. For pulses with duration longer than 100 fs a further approximate of Eq.(A.36) is often used to describe the evolution of the pulse, in which the finite time response of the medium is ignored. This approximate form is usually used to obtain the solitonic Appendix A. Third-order nonlinear effects in optical fibres^98 solution of the GNLS equation using the inverse scattering method, which will be explained later. In this approximation, the envelope function in the integration is simplified as so that 1A(z,T — T')I 2^IA(z,T)I 2^IA(z,T)I2, (A.37) f R(T')IA(z,T — T')I 2 dT'^IA(z,T)1 2 — TR aT IA(z,T)1 2 .^(A.38) By using the fact that the response function R is normalized f -HZ dT'R(T') = 1 and defining +00 TR =- f dT'T'R(T'), -00 then the nonlinear term in Eq.(A.36) becomes: (A.39) WO ^i ^a^, i-y(1+ —OT )o(z,T)^R(11)1A(z,T —14 )1 2] = ilf(1 w-0 —OT )VIIAl2 TRA—aT IA11, or + 0c cc^ a^i a i'y(1+ wo OT^— —a )[A(,,,T) f^d7'' 11(711A(z,T T')1 2] = j'Y[AIAl 2 — TRA aT 01 2 + w0OT (AIAl2)], where we ignored the term with the second-order derivative. Finally the GNLS equation, in this limit takes the following form: DA a^6^im-1 am^a^i a^ =^A-F{E  ^ ]A+i-y[AIAI2 TRA aT IAI 2 + w 0 ^0012)]. (A.40)maz^2 aT rn^m= aT The first, second, and the third terms in the above equation are responsible for self-phase modulation, Raman scattering, and self-steepening effects (which will be N= 'Y Pon Ik21^• (A.43) Appendix A. Third-order nonlinear effects in optical fibres^99 described in the next section) respectively. Usually Eq.(A.40) is expressed in a di- mensionless form by introducing A^ik2lz^T ,^(A.41)U =^,^=^, T = —ToPo n where To and Po are the duration and peak power of the incident pulse, then we have au^aN2^1 1^em" N2 TR a^i  a^ u + [^km m! ( I, )rn arm ] poli u + iN2[002^u^ 02 +^ ,T (ului2)].2P0-y To ar^Tow() 0m=2 (A.42) In above equation the parameter N is By introducing u = NU we obtain au aN2^6^i m-1 1 a- N2^a^a= 21,07 71+ {mE2 k,^ (TT arm ] po7 u + iujul 2 — irRu w7dul 2 — s 5,7: (ului 2 ), (A.44) where we defined TR^1== ^ (A.45) To^woTo The inverse scattering method will be used later to find solitonic solutions of Eq.(A.44), but before that, the self-steepening effect will be described in the next section. A.4 Self-steepening The self-steepening effect is due to the intensity dependence of the group velocity, which finally leads to an asymmetry in the envelope of ultrashort pulses. To see this, Appendix A. Third-order nonlinear effects in optical fibres^100 we ignore linear loss, dispersion and the Raman term in Eq.(A.44) to obtain au^aae + s—aT aul 2u) = ilul 2 u.^ (A.46) Now by using u =^exp(icb) in Eq.(A.46) we get two equations for the real and imaginary parts + 38/—DI = 0,^ (A.47)ae + si a° =ae^ar (A.48) The intensity equation, Eq. (A.47), can be solved using the method of characteristics 6 , which gives us (e, T) = f (T — 3sIe),^ (A.49) where f (T) = I (0, T) is the incident pulse. For example, suppose the incident pulse is a secant hyperbolic pulse f (T)^sech2 (1.76r), then the pulse at e along the fibre becomes /(e, y) = sech2 [1.76(r — 3sIe)]. (A.50) (A.51) Fig.( A.3) shows the pulse at es = 0.0, 0.1 and 0.2; the pulse becomes asymmetric as it propagates through the fibre in such a way that its trailing edge becomes steeper with increasing distance. 'N. Tzoar and M. Jain, Phys, Rev. A 23, 1266 (1981). 0.8 'ZZ 0.6 C 0.4 0.2 0 0T-1-2 2 Appendix A. Third-order nonlinear effects in optical fibres^101 Figure A.3: Self-steepening effect on a secant hyperbolic pulse. A.5 Solitonic solutions of the GNLS equation This section describes the nature of solitons, which are a set of basis functions in terms of which any solution of the basic nonlinear SchrOdinger equation can be expanded. The Simplest nontrivial form of the GNLS equation, derived from Eq. (A.44), is when we only consider the group velocity dispersion and self-phase modulation. In this limit Eq. (A.44) can be written as . Ou 1 02 u + 2 aT2 + luru = 0,^ (A.52) which is called the nonlinear SchrOdinger (NLS) equation. This equation belongs to a class of equations that can be solved by the inverse scattering method'. The NLS equation for T) is formally equivalent to the following eigenvalue problem for 71) of a direct scattering problem' in which u( = 0, T) is considered as an effective potential 7 M. J. Ablowitz, and P.A.Clarkson, "Solitons, nonlinear evolution equations and inverse scatter- ing," (Cambridge University Press, Cambridge, 1991). 8 H. Hasegawa and Y. Kodama, " Solitons in Optical Communications" (Oxford University Press, New York, 1995). a u L = —a* (A.56) =  + 2 (A.59) Appendix A. Third-order nonlinear effects in optical fibres^102 =^ (A.53) = (A.54) where 1.^ (A.55) The L and M operators, which are called the Lax pair, are differential operators in the 7-derivative and involve u( - , T), and for the NLS equation they are defined as 9 : (A.57) Now we impose an invariant condition on the eigenvalue A (which corresponds to having a soliton solution for the NLS equation with a preserved shape), versus the distance of propagation, OA 0 . (A.58) In the inverse scattering method, the eigenvalues are used to solve for the u() po- tential. Each discrete eigenvalue of Eq.(A.53) ( i-12,, + 2Itil 2M = ) _u . aar _ aau; _ i :7_22^1 u 1 2 , a _L au a,^2 a, characterizes a soliton solution' u(e, 7) = ri sech[n(T — To + 6-0] exP[7(772 (52 )V2 — iS7 + 100] • (A.60) 9 V. E. Zakharov and A. B. Shabat, sov. Phys. JETP 34, 62 (1972). 10A. Hasegawa, M. Matsumoto, "Optical solitons in fibers" (Springer-Verlag, Berlin,2003). Appendix A. Third-order nonlinear effects in optical fibres^103 The imaginary part of the eigenvalue , n , specifies the required relationship between the amplitude (and the inverse of the pulse width), while the real part, 6, specifies the soliton frequency (and the velocity). The To and 00 determine the position and the phase of the soliton. Now if the input pulse is approximated by a secant hyperbolic function as u( = 0,7) = A sech(r),^ (A.61) the number of eigenvalues N is given by A- 1 < N < A ± —2 . (A.62) With this form of the input pulse the corresponding eigenvalues are imaginary 1 = 2 = i(A — n + — 2 ),^(n = 1,2, • (A.63) If A is exactly equal to N, it can be shown that the envelope function of the pulse is a linear combination of N fundamental solitons with different parameters' T) = E raj sech[rij (7 — T,9 • + 6 1 e)] exp [i (7/3? — j=1 and the amplitudes of these N fundamental solitons satisfy (A.64) = 2(N — n) + 1 = 1,3,5, • ••, (2N — 1). (A.65) In addition, with the particular form of the initial pulse given by Eq.(A.61), the eigenvalues are purely imaginary. Therefore, all solitons propagates at the same speed. The overall pulse has an oscillatory behavior due to interference among different solitons. If A is not an integer, the initial pulse evolves to N solitons and a dispersive 'H. Hasegawa and Y. Kodama, " Solitons in Optical Communications" (Oxford University Press, New York, 1995). . au 1 a2u ± 2 aT2^IU1211.= if(U)' (A.66) or Appendix A. Third-order nonlinear effects in optical fibres^104 wave. In another words, any input beam in the optical fibre can be transformed into a dispersive wave and a set of fundamental solitons corresponding to the discrete eigenvalues of the inverse scattering problem. A.6 Perturbation of solitons Now we can consider other nonlinearities that appear in the GNLS equation, as perturbations of the NLS equation. We therefore write the GNLS equation in the form where €(u) represents a small perturbation, which depends on u and its derivatives. When there is no perturbation, c(u) -= 0, the solitonic solution of the NLS equation is given by Eq.(A.60). There are several techniques to examine the effect of the perturbation on the solitons, and all assume that the functional form of the soliton remains the same but the four parameters in Eq. (A.60) change versus e as the solitons propagate through the fibre. Therefore, the solution of the perturbed NLS equation will be of the form: T) = 71(0 sech[n(e)(7- — q(0)] exp[icb(e) — i8(e)71.^(A.67) For a zero perturbation, E = 0, we have: q(e) = is — 8^and^0() = (7/2 — 62 )e/2 +00, (A.68) d dq —^ and ck . dq5^(712_ 82)l2. (A.69) For a specific perturbation there is a corresponding functional form of ri, 8, q, and 0. The common techniques to find the effect of a perturbation on solitons are the Appendix A. Third-order nonlinear effects in optical fibres^105 adiabatic perturbation method, the perturbed inverse scattering method, the Lie- transform method, and the variational method, which all give the following differential equations for the four soliton parameters' A Re foo e(u)u* (r)dr, d - co = —Im^€(0 tanh^— q))u*(r)dr,da dq = —8 + 2  Re fr € (u)(7- - q)u*(T)dT, (A.70) (A.71) (A.72) Imf c(u)11/7/ — (7- — q) tanh[n(r — q)]}u*(7)dr + (772 — 82) + q—(16 (A.73) d - 00^ 2 In order to see the effect of each specific perturbation on the soliton parameters we use the approximation form of the GNLS equation, Eq.(A.44), which gives us: au 1 02 ua^alu12 + 2U = — is ar (12/1 2 U) + TRUae ± 2 ay2^ ar where the linear loss of the fibre, and dispersion higher than the second order are ig- nored. The first and the second terms on the right hand side of Eq. (A.74) represent the perturbation induced by the self-steepening and the Raman scattering respec- tively, which are discussed separately in the following. For short pulses the role of the Raman scattering is more significant. For example, with 130 fs pulses (T0 = 74 fs) centred at 800 nm, s ti 0.006, and TR 0.04. The Raman scattering coefficient is one order of magnitude higher. ' 2H. Hasegawa and Y. Kodama, " Solitons in Optical Communications" (Oxford University Press, New York, 1995). (A.74) Appendix A. Third-order nonlinear effects in optical fibres^106 A.6.1 Soliton self-frequency shift Raman scattering is one of the most important nonlinear effects in optical fibres. It is responsible for a new phenomenon, called the soliton self-frequency shift. To clarify this effect, we set s = 0 in Eq.(A.74), then 411- ± 21 '902:2 + ittru — 7 u 31u12ar • (A.75) Now by using Eq.(A.70)-(A.73) and €(u) = —ii -Ru 0; 2 as our perturbation, it can be shown that the soliton amplitude, 7i, is invariant under the Raman effect A^.,... , .^foo^ 2 01u 2= Re[—ITR,^lui  , an= 0, ck -00^UT 1 a but the soliton frequency, 6, changes as (16^cc^ 11= —I111[ — iTR i lull 31, UT 12 tall.h[n(T — q)]c17], ck -00 by using 17./1 2 = n2sech2[n (y — q)] from Eq.(A.67), we have: 00 8cir5 = — 2TR7/ 5 f sech2 [n o- — q)] tanh2 [n o- — q)]dr = m— l-Rn 4  , CO which can be integrated to yield 6 (e) = —(8TR/15)17 4e. By setting 71 = 1, the frequency shift in original units will be (A.76) (A.77) (A.78) (A.79) AwR(z) = —8 Ik2ITRz/( 15To4 )-^(A.80) The above equation shows that the frequency shift increases linearly with propagation length and it is inversely proportional to n , therefore the shift can be very significant for short pulses. This red shift in the soliton frequency can be interpreted in terms Appendix A. Third-order nonlinear effects in optical fibres^107 of stimulated Raman scattering. When the pulse bandwidth is large enough, due to the Raman effect the low frequency components of the pulse can be amplified with the high frequency components acting as a pump. This frequency shift is called the soliton self-frequency shift (SSFS). 108 Appendix B Frequency resolved optical gating The frequency resolved optical gating' setup used in chapter 2, helped to fully char- acterize the input pulse that was coupled into the photonic crystal fibre. Here we explain the method and show that how the phase and intensity of the pulse can be determined in a FROG experiment in either the time or the frequency domain. An optical pulse is defined by its electric field as a function of time. To simplify the discussion, suppose the electric field of a linearly polarized pulse can be separated into a product of its spatial and temporal components and we just focus on the temporal part, which can be written as: E(t) = Re { \ II (t) exp(iwot — OM)}, (B.1) where wo shows the carrier frequency and I(t) and OM show the temporal intensity and phase of the pulse respectively. The temporal phase reveals the pulse's instanta- neous frequency w(t) = wo — --: . (B.2) If the phase is independent of time, one has a transform limited pulse. If the phase depends linearly on time we just have a frequency shift. A quadratic temporal phase creates a linear variation of the frequency. When the frequency increases (decreases) versus time, the pulse is said to be positively (negatively) chirped. A higher order variation of the frequency is referred to as nonlinear chirp. 'R. Trebino, et al., "Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating", Rev. Sci. Instrum. 68, 3277-3295 (1997). •Spectrometer E(t) NLCE(t-t) variable delay Esig(t,t) Appendix B. Frequency resolved optical gating^109 We can also express the pulse in the frequency domain E(w) =- I(w — coo ) exp(i0(c.0 — wo))•^(B.3) To completely characterize the pulse we need to specify the intensity and phase in either domain. A conventional spectrometer measures the spectrum I(w — coo ), but it can not reveal the spectral phase. It is also not easy to measure either I(t) or OW directly for short pulses mainly because of the limited temporal resolution of measurement devices. The main technique used to characterize ultra short pulses in the time domain, is nonlinear autocorrelation, which effectively uses the pulse to measure itself. In an autocorrelator a pulse is split into two parts and one is delayed with respect to the other but they are spatially overlapped inside a second-order nonlinear crystal, as shown in Fig. (B.1). input beam Figure B.1: The FROG experiment setup. The specifications of the optical elements are summarized in table (2.1) of chapter 2. Appendix B. Frequency resolved optical gating^110 The most common nonlinear crystals are chosen to allow efficient, phase-matched second harmonic generation (SHG) at the frequency of the pulses being measured. The intensity of the SHG signal is proportional to the product of the intensities of the two input pulses IsHG(T) a f +00  IE(t)E(t — T)1 2 dt x J +00  I(t)gt — T)dt . (B.4) 0,0 pc, The pulse duration can therefore be measured by varying the delay between the two input pulses, which generate a maximum SHG signal when perfectly overlapped, and no signal when there is no temporal overlap in the nonlinear crystal. In this way, one obtains the autocorrelation of the pulse intensity envelope. As Eq.(B.4) shows, the SHG autocorrelation technique is not a useful technique to characterize the pulse phase. The frequency resolved optical gating (FROG) method addresses this problem, by spectrally resolving the SHG signal generated in the autocorrelator, at each relative pulse delay. The set of SHG spectra, for various pulse delays, obtained at the output of the spectrometer in Fig. (B.1) is called the FROG trace. By using this measured data and the generalized projection algorithm 2 (next section) we can retrieve the temporal phase and intensity of the input pulse. The FROG method can also be used for measuring properties of an unknown pulse by mixing it with a known pulse (gate) in a cross-correlation geometry. In this case the setup is called XFROG. 2 R. Trebino, et al., "Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating", Rev. Sci. Instrum. 68, 3277-3295 (1997). \ I F ROG (U) 7 ) r_oc ) I E ikg)^7j )I^9 (B.9)eskg (w i> T7 Appendix B. Frequency resolved optical gating^111 B.1 Generalized projection algorithm for pulse retrieval in the FROG method Here we only consider the second harmonic generation (SHG) geometry in an auto- correlation setup; the algorithm can be extended to other geometries and cross- correlation easily. The goal of the FROG method is to find the complete E(t) of the input pulse. This must be extracted from the SHG signal, which is related to the input pulse, and the delayed input pulse, by Esig (t T): Esig (t,7-) oc E(t)E(t^ (B.5) )The measured FROG trace by the spectrometer F ROG GO 7 ),^is the squared magni- tude of the 1D Fourier transform of the Esig (t, 7- ) with respect to time: -Foo FROG( U) 7) = I f^Esig(t, T) exp(—iwt)dt1 2 (B.6) In the FROG algorithm we begin with an initial guess of E(t) (usually a field consisting entirely of random numbers). The field in kth step of the iteration will be: ELk(ti, r) = E (k) (t i )E (k) (ti —^ (B.7) then we Fourier transform the above signal field with respect to time: Es( ikg) (wi ,7-i )= FT[ELkg) (t i ,7-i )].^ (B.8) A properly normalized estimate of this trial function, that can be compared with the actually measured spectrum, is thus, which in the time domain is Appendix B. Frequency resolved optical gating^112 est) (ti rj) = FT-1 [est (wi, Ti)] • (B.10) An error metric, Z, is then calculated and used to derive the (k + 1)th estimate of the trial field: N Z = E iest(ti, Ti) — E(k+1 )(ti)E(k+1 )(^j)1 2 . 2 ,j (B.11) To minimize Z with respect to ,E (k+ 1 )(t,), we take the derivative of Z versus real and imaginary parts of E (k+ 1) (t) at t =- tm , therefore: az NEHE(k+1 )(trn^,Tj ) 12 (E(k+1) ( tm )^E(k+1)*( trn )) j=1 1E (k+1) (tm^Tj )1 2 (E (k+1) (trn ) + E (k+1)* (tm )) aRef E(k+ 1 )(tm )} E(k+1)(t rn^7.7 )e (sVg* ( trio )^E(k+1)( tm^T \ e(91cz) g* ^ (tm(tni^Ti, 7j) - E(k+1)* ( tm^73)e (sizg) (trn^)^E (k+1)* (tm^)es(kig)(tm^Tj Tj )1, and az V•FiE(k+1) (tm^73)12 ( E^(tm) (tm )^E (k+1)* (tm)) alm{E(k±') (tm )}^t1L • IE(k+1)( trn^73 )12 (E(k+1) ( tm )^E(k+1)* ( tm )) — E (kH-1) ( tm _ .7-3 ) est* (tm , 73 ) — E (k+1) (tm + 7,)est)*(tm + + E (k+ 1)*(tm — 7-j )est(tm ,7-.) ) + E (k±' )*(tm + 7-3 )est(tm + 73 , where we used the following relations a E(k+ 1 ) (t,) _ (ti^tm), aRefE (k+1) (tm)} a E(k+ 1 )* (ti) ^_ iS (t, — trn ).aiM{E (k+1) (tm )1 (B.12) The new, iterated field is then expressed in terms of the previous trial as: Appendix B. Frequency resolved optical gating^113 az(k), ^ 1E(k)(tE ( k±' ) (tm) = E‘ trn' ) x aE(k±o(tm) aaz^ z 1E(k)(t,,,),.) ix aIm{E(k+ 1 )(tm )} E (k) (trn)^X aRefE(k+i)(tm)f IE(k)(t (B.13) then Z becomes Z = E^- [E(tm ) xC(tm )][E(tm — Tn,) + xH(tm — 7,,)]1 2 ,^(B.14) m,n with aaz^ z c(tm)= aRefE(k±')(tm)} El (k) (trn)^ainifE(k+1) (tm ) az^ az H(tm — Tn ) = aRe fok+i) (t. — alinf ok+i)(t. — To} I E(k) (t rn -Tro - In a compact form Z can be written as: Z = E`4mn - (Em xCm)(Bmn + Dmn)1 2 1 m,n where (B.15) Amr, = est) (tm ,^Em = E(tm), Cm = c(tm), Bmn = E(tm — Tn), Dmn = H(tm — Tn), therefore, Z is a polynomial of x Z = a4x4 + a3x3 + a2x2 + aix + a0 , with the following coefficients: (B.16) Appendix B. Frequency resolved optical gating^114 a4^E m,n a3^Eipmn i2(Emc:n + E:„Cm ) + ICm 1 2 (BmnDL, + m,n a2^E lErri i21,0,2n 12 + icm i2iBm,n 12 + (Emq + .E:„Cm )(BmnDL, + rn,. Am„C,',̀ 14,7, — A nnCmDmn, N a l^EiE,12(BmnDL, B 2nD m) 113.1 2 (Emqn m,n Amn (E:n D,„ + C,B7,,n) A,n (EmDmn CmDmn) ao^E IA.1 2 — Amn E ri B,„ — Amn EmBmn + I Emi 2 1B mn1 2 m,n Therefore, in each step we find the x value that minimizes the Z function, then we evaluate the new fields for the next step by using Eq.(B.13), and these steps are repeated until the error becomes minimum. The error function in the FROG method is defined as: N C = [^ E^— _TFRoG(co,73 )1 2 1 1,2 ^(B.17)N2 /0=1 where N is the number of frequency or time data points. The whole process was written in Matlab (file shg-frog.m at nanolab directory in berserk server) and it takes roughly 3 minutes to do 200 iterations and reach an error of G 10-2 when N=64. 115 Appendix C Field quantization in open optical cavities The theory of squeezed state generation in photonic crystal microcavities of chapter 3 was based on a system-reservoir Hamiltonian, Eq.(3.4), which is obtained by ex- pressing the Hamiltonian of the electromagnetic field in terms of the quantized vector potential of the structure. In this appendix we derive this Hamiltonian and express the explicit equations of the coupling coefficients between the cavity and reservoir operators' . In the following, we suppose the fields oscillate at frequencies far from any elec- tronic resonances in the medium, which is also assumed to be isotropic, hence E(r) is a real scalar. In order to express the properties of the squeezed light generated in the photonic crystal microcavity structure, the field in the cavity must be properly quan- tized. The dynamics of the microcavity and its coupling to various continua channels can be expressed by the appropriate electromagnetic Hamiltonian. The common choice for expressing the Hamiltonian is the field vector potential. To quantize the vector potential we expand it in terms of the eigenmodes of the structure and impose the appropriate commutation relations on the coefficients of the expansion. Here the whole structure consists of a multi-mode photonic crystal microcavity and several continua channels that couple to the microcavity. The vector potential satisfies the following wave equation 1 The primary reference for the material of this appendix was: C. Viviescas and G. Hackenbroich, Field quantization for open optical cavities, Phys. Rev. A 67, 013805 (2003). w2 C2 im w), (C.4) Appendix C. Field quantization in open optical cavities^116 V x [V x A(r,w)] e(r)w2 A(r,w) = 0. c2 ^ (C.1) Now we suppose the vector potential can be expressed in terms of eigenmodes of the structure V x [V x gm (r, w)] E(c2 )w2^ (r, w) = 0,^(C.2) where c(r) is the dielectric function of the whole structure (cavity plus all leaky chan- nels). We can write Eq.(C.2) as an eigenvalue equation by the following substitution 1 gm (r, w) = ^ (r) f in (r, w), \A (C.3) then the eigenvalue equation becomes: 1 r) Lf,i (r, w) = ^V x [V x   1 r) f m (r, w)] = M VE( and the orthonormality condition is f d3rf;,',(r, co) • fn (r, co') = an,„8(co — w'). (C.5) In order to obtain the Hamiltonian of the system-reservoir we should express the eigenmodes of the structure in terms of the cavity and channel eigenmodes. The Feshbach projection technique borrowed from quantum mechanics is the right tool for this purpose. C.1 Feshbach projection To introduce cavity modes and their dynamics, we separate the electromagnetic field into two parts, the field inside and outside the cavity. The separation of space is achieved by using the projection operators Appendix C. Field quantization in open optical cavities^117 Q =f d3rir >< ri,EC^ (C.6) P = f d3rir ><^ (C.7)111C C denotes the space occupied by the cavity 2 . These operators are orthogonal, QP = PQ = 0, and complete, Q + P = I, with the following properties Q Qf^Q2 Q^p pt^P2 = P.^(C.8) Now we want to express the eigenmode of the whole structure in terms of the cavity and channel eigenmodes. A general solution of the field for the whole structure satisfies: or 2 LIS(w) >_ - C2 I S^>, w2 - iri - 41S(w) >-= 0, (C.9) (C.10) where operator L is defined in Eq. (C.4) and 7) is an infinitesimal number introduced to take care of the singular nature of the operator 1/(5 — L). First we use the completeness of projection operators (P Q = I) w2 i71)(P^(P Q)L(P Q))IS(W) >= 0, (C.11) to get two coupled equations:  u4[  — irk — QLQ]Qis(0)) >= QLPIs(w) >, (C.12) 2 The boundary between the cavity and the rest is arbitrary in principle, but obviously would surround the volume over which the localized cavity mode has most of its energy — QLQ] 14(w) >= 0, — iri — PLP]lp(w) >= 0. Appendix C. Field quantization in open optical cavities^118 w2 —i^PLP]Pls(w) >= PLQ1s(w) >,^(C.13) where we used P 2 = P, and Q2 -= Q. The Pls(w) > is a field outside of the cavity with zero values inside the cavity and vice versa for Qls(w) >. We also define eigenvalue equations for the cavity and channel regions separately, The general solution of equation (C.13) is the homogenous solution from equation (C.15) plus a particular solution obtained from equation (C.13) (we suppose the incident field is a solution in the channel region) ^P I S (w ) >= IP(w) > ^PLQIS(W) > •5 1  — — PLP By substitution of the above equation into (C.12) we get 1 — ill — QLQ^L'e+2- — — QLQ QLP Pls(w) > . (C.17)Qis(w) >= 2^QLPIs(w) >= 2 After simplifying the above equation we have: ^ Qls(w) >= „,2^QLPIp(w) >,^(C.18) ^c2 .1 — Leff where (C.16) Leff = QLQ + QLP^PLQ. C52 — 1 PLP By combining Eqs.(C.16) and (C.17) we get: (C.19) I s( ) > = QI s (w) > + PI s(w) > GQQQLPIP(w) > +[ 1 + W2 - 1 PLP PLQGQQ QLP]lp(w) >, Appendix C. Field quantization in open optical cavities^119 ( C.20) where 1 GQQ — ^ (C.21) i 7/ — Leff The first term in Eq.(C.20) belongs to the cavity space and the second term to the channel space, therefore the complete solution of the field in the entire structure can be written as: s(w) >= E aA (w)jUA > E f dcZi@n(b), w')1 17. (o') >,^(C.22) A where I UA > and I Vn (w') > are eigenmodes of the cavity and channel regions (solutions of Eqs.(C.14) and (C.15) respectively) and the expansion coefficients are given by ceA (b.)) =< UA IGQQ QLPIp(w) >,^(C.23) On (w, L.0') =< Vn(w i )1[1 + w2 1^ PLQGQQQLP]1p(w) > .^(C.24) — PLP So finally we get s(r, w) = E ceA (w)U),(r) E I clu/On (w, ul)Vn (r, w').^(C.25) A There is a corresponding linear expansion for the eigenmodes of the whole structure fm (r, w) = E a rAn (W)UA (r) E f dc.,/,37 i (b.), wwn (r, w'). A (C.26) C.2 Hamiltonian of the system-reservoir Since the modes of the whole structure can be expressed in terms of the cavity and channel modes, Eq. (C.26), the quantized vector potential of the whole structure can be written A(r, t) = E A  [aA (wA , t)U A (r, w),)^a tA (w)„ t)U*A ) 1 2€0WA + E ./du) 2cow [rn,(w, t)V,,(r, co) + rtm (w, t)V,,* (r, w)1, Appendix C. Field quantization in open optical cavities^120 (C.27) and the Hamiltonian of the quantized field in a linear medium can be written as H = —2 f d3r[coe(r)k2 + 1 (V x A) 2 ]. ^(C.28) We calculate each term in the above equation separately. Since the eigenmodes of the cavity are orthogonal to the eigenmodes of channels, we get: 2 d3 reoc (r)A2 _^hwA r_t^1-1 ^f^hwr +a,,̂ Au), + aAaAi + aw— LT^t Tre^4^rnr ^rTrt^Trtr rn • (C.29) For the second term of the Hamiltonian in Eq. (C.28) we have:  22/20 jcPr(V x A) 2 = EE  hc ^f d3 r[aA V x LTA + atA V x WA ] • [aA , V x UA‘ + atA,V x IJI,] 4WAWA,A A , + EE/dWfdWi  tic  f d3r[r,,V x Vm + r"ry x V:;.„] • [rm‘V x Vm , + rynt ,V x V:,̀ ,,,,] 4-Vcou.,'m m , 2+ EE f dw 2ĥc ^ f d3 r[aAV x UA + atA V x U*A ] • [7.,,,V x Vm + rin.,V x V:,̀ ,].f^A/wc,,A By using the following identity, VxS•VxT=V•(SxVxT)+S•VxVxT,^(C.30) we have: 2/-to^ hw^ f d3 r (V x A) 2 = E hw4 A [aAt aA + aAa tA ] + E I dch)—4 m[rt r m + rm r.;„]A^m + h E E f dW[WAni (w)aA rm (w) + 417),,,(w)aAr.,,(w)] + [L in (w)aArni (w) + Tj m (w)a A rrn (w)]. A m A m Appendix C. Field quantization in open optical cavities^121 (C.31) (C.32) (C.33) where C2 Wm 2\/ww^ f d3 r U*A .V x V x V,„), and 2.0 c2TAam= ^ f d3r UA .V x V x Vm •4.)(..oA Finally the Hamiltonian becomes: H = E hwA(atAaA + 1 + E f dwhp)(4,2 (w)rm (w) + hEEf dW[WA rn (w)a tA rm (w) + nn (w)aArL(b.))] + [TAm (w)aArm (w) + T;:m (w)ajArL(w)]. m (C.34) 122 Appendix D Quadrature correlations in optical parametric down-conversion In the following we first define squeezed states, then it is shown that the fields generated in a nondegenerate parametric down-conversion (NPDC) process can be squeezed, owing to the strong correlation between the field quadratures. The degen- erate PDC process can be treated as a special case of the NPDC process for a single mode cavity. D.1 Squeezing of the quantum fluctuations The electric field for a quantized, single-mode, and linearly polarized field' can be written as: 2E0 hw pe-i(wt-k.r)^ate i(wt-k.r)] , where the field is polarized along e2 . By defining the Hermitian operators X = a + at,^Y = — i(a — at), the electric field can be written 'M. 0. Scully and M. S. Zubairy, Quantum Optics: Cambridge University Press (1997). 2 Note that this expression for the electric field uses a slightly different convention than what was used in Eqn.(3.8). = eE0 (D.1) (D.2) Appendix D. Quadrature correlations in optical parametric down-conversion 123 E = eE0 2E0 ^cos(wt — k • r) + .17- sin(wt — k r)].^(D.3) X, Y are called the field quadratures. The commutation relation between the creation and annihilation field operators is: [et, at] _^ (D.4) then the commutation relation for the field quadratures becomes [k, Y] = 2i,^ (D.5) and the uncertainty relation for the two quadratures is AfCAY > 2—1 I <^Y] >^or AkAT > 1,^(D.6) where the uncertainty of an operator, A, is defined as AA = \/< A2 > - < A >2. A squeezed state of the radiation field is obtained if OX <1,1,^or^A)L7 < 1.^ (D.7) An ideal squeezed state is obtained if in addition to the above relation, the following relation also holds Ak‘ Ai> = 1.^ (D.8) Coherent states3 and the vacuum states are not squeezed states and they satisfy Ak‘ = AY = 1.^ (D.9) Therefore, squeezing refers to the reduction of quantum fluctuations in one field quadrature below the minimum noise of the vacuum state (or a coherent state) at the 3 A coherent state is an eigenstate of the annihilation operator, ilia >= crict >. It can be shown that light emitted by a stabilized single-mode laser operating above the threshold is a coherent state. Appendix D. Quadrature correlations in optical parametric down-conversion 124 expense of an increased uncertainty of the conjugate quadrature. A squeezed vacuum state 177 > can be obtained by the operation of the squeezing operator S on a vacuum state Iri >=^>= expEnat 2 —^>, 71 = re i2 c5 .^(D.10) With respect to each mode of the vacuum, the action of (77) is to annihilate and create correlated pairs of photons, which suggests frequency conversion as a path to squeezing. A squeezed coherent state la, i > is obtained by applying the same squeezing operator on a coherent state la, n >=^>= exp[nat2 —^]Ict > .^(D.11) By using the following formula -e B e -a a2^CX3= + [A, n] + -2T [A, [A, li]] + y [A, [A, [A, /4] + • • • ,^(D.12) it can be easily shown that ,.'t(7.1 ) a ,.(77) = a cosh(2r) + ate 22° sinh(2r),^(D.13) = at cosh(2r) + ae -12 cb sinh(2r).^(D.14) We define the quadrature-phase amplitude by: Ze = ete- z6 + ate, (D.15) by setting 0 = 0 and 2we get the original definition, Eq. (D.2), of X and Y quadratures respectively. The uncertainty of the field quadrature-phase amplitude of a squeezed vacuum state is thus Ak6, = 1/< nI4In > — < n120In > 2 ,^(D.16) Appendix D. Quadrature correlations in optical parametric down-conversion 125 after a little algebra we get: A49 = Vcosh(4r) + sinh(2r) cosh(20e 2z( 9-0) + sinh(2r) cosh(2r)e -2'(9-0). (D.17) Now if = 61 then AZT = e2r,^ (D.18) and when 0 = + z we have A,k0+^e  2r^ (D.19) Therefore the quadrature that has a "L` phase shift with respect to the phase of the2 squeezing operator has an uncertainty less than the vacuum state. The phase of the squeezing operator is determined by the phase of the pump beam that generates the down-conversion process. The squeezed coherent state has the same quadrature squeezing as the squeezed vacuum state. As we pointed out in the above, squeezing is caused by a frequency conversion process. In fact, the squeezing operator defined in Eq. (D.10) is the time evolution operator for the interaction Hamiltonian of a parametric down-conversion process. To show this, consider the Hamiltonian of the interaction in a nondegenerate down- conversion process (equation (3.10), which was derived in chapter 3) = ihrg(t)a rtat2 — g*(t)a i a2 ],^(D.20) where the cavity operators were written in the rotating frame. When the Hamiltonian operator is time-dependent but the H's at different times commute, the time evolution operator becomes i f t 0(t, to ) = exp[— fii to dt'H(t')] = exp[ria tia2 — (D.21) where ri = ftto dt'g(t'). The squeezing operator defined in Eq. (D.10) is the special case of the above time evolution operator for the degenerate down-conversion process. Appendix D. Quadrature correlations in optical parametric down-conversion 126 D.2 Strong correlation with squeezed states In the following we discuss the correlation between the signal and idler beam in a NPDC process in which photons are generated in pairs with one photon at frequency coo +w (signal) and the other at coo —w (idler), where c.4.)0 is half of the pump frequency. The time evolution of each cavity mode operator is: a, (t) = e- r(ate2-aia2) a i er(ata2-a1et2) ,^i = 1,2,^(D.22) where we assume the r coefficient is real (the phase of the pump beam is set to zero). Again by using the operator identity in Eq.(D.12) and a little algebra we have: al (t) = al (0)cosh(r) + a2(0) sinh(r), (t) = a2(0) cosh(r) + at (0) sinh(r). The quadrature-phase amplitudes of the two cavity modes at time t are: ZB (t) = a i (t)e-i° + ati mei°, (t)^a2 (t)e-icb + a2(t)eio. (D.23) (D.24) (D.25) (D.26) The quantum mechanical Cauchy-Schwartz inequality for the quadrature-phase am- plitudes is: < 4(t)4 (t) > 1 2^< [2t19(t)] 2 >< [4(0] 2 >,^(D.27) which motivates defining the associated quantum-mechanical correlation coefficient as4 < 4(04(0 > Coo V< [4(t)] 2 >< [4(t)] 2 > 4 M. D. Reid, P.D. Drummond, Phys. Rev. A, 41, 3930 (1990). (D.28) MAY*0411 'wkdit\ Appendix D. Quadrature correlations in optical parametric down-conversion 127 For an initial vacuum state this becomes C0  = tanh(2r) cos(0 + 0),^ (D.29) which shows that when r —> oo Ck 1 (t), fC2 (t)) and (11. (t), —Y2(t)) are correlated. The schematic diagram of a NPDC process is shown in Fig.(D.1a) and we suppose the idler and signal beams can be spatially separated . (2) field quadratures A A Y i A A I (1)2^XV Y2 (a)^down-conversion (b) ^ Time yi y2  '4114'V/141MIAN (c) ^ Time x l x2 Figure D.1: Signal and idler field quadrature correlations in a nondegenerate para- metric down conversion. Initially the signal and idler beams are in vacuum states, but the down-conversion process amplifies the incoming vacuum noise and produces two outgoing signal and idler beams with fluctuations in quadrature-phase amplitudes. However, the fluctu- ations of the quadratures for the two beams are strongly correlated and for the ideal condition of very large amplification they become "quantum copies" of each other s , as in Figs. (D.lb) and (D.lc). The nonclassical correlation of quadrature-phase am- 5 Z. Y. Ou, S. F. Pereira, H. J. Kimble, Appl. Phys. B 55, 265-278 (1992). Appendix D. Quadrature correlations in optical parametric down-conversion 128 plitudes is similar to Einstein-Podolsky-Rosen (EPR) pair generation' for continuous variables'. If the signal and idler modes combined together in a beam splitter, the new modes are squeezed and exhibit noise reduction below the vacuum noise level. 6 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). 7M. D. Reid, P.D. Drummond, Phys. Rev. A, 41, 3930 (1990). 129 Appendix E Spectrum of squeezing In chapter 3 it was mentioned that in order to calculate the degree of squeezing present in the output waveguide we should find the correlation between the field quadratures, Eqs. (3.28) and (3.29). In the following we derive these two equations'. Since squeezed state generation is a phase-dependent phenomenon, a homodyne detection system is usually used to observe squeezing. In a homodyne measurement, Fig.(E.1), a strong local oscillator beam is added to the signal (with the same fre- quency as the local oscillator) to be measured , and by acquiring the spectral density of the current fluctuations in a detector that absorbs this combined signal, we can estimate the spectrum of squeezing. Before discussing the homodyne detection pro- cess, we should show that how the variance of the photo-current at the detector is related to the correlation of intensity operator of the incident field. spectrum analyzer Figure E.1: The set up of an homodyne detection measurement. The signal and the local oscillator beams are combined in a beam splitter and the combined light is absorbed by a detector. 'The primary reference for this appendix was: H. J. Kimble, Fundamental Systems in Quantum Optics (Elsevier Science Publishing, Amsterdam, 1992), chapter 10.  ••• At ^t Appendix E. Spectrum of squeezing^ 130 Consider the situation depicted in Fig.(E.2), where a quantized field a(t) creates a sequence of photo-electric pulses in the detector. The photo-current i(t) can be written as: i(t) = E Q(t — tk )pk, (E.1) k where the time axis is broken into a series of small intervals At and pk is a random variable, with pk = 0 for no photo-ionization occurring at time tk, and pk = 1 for the occurrence of an event at tk within At. Q(t) specifies the shape of the current pulse. Q(t) Figure E.2: Photo detection of a nonclassical light source. The autocorrelation of the current becomes: < i(t)i(t +7) > =^Q(t — tk)pk^Q(t + r - t3 )p > k^.1 Q(t tk)CM T - tk) < Pk > ^ Q(t tk)Q(t + r - ) < PkP3 > , k^ kj where we used the fact that /4 = pk (we assume the time interval AT is so short that the events in which two or more photo-electrons are emitted at the same time are eliminated). The expectation value of < pk >, the probability for having a current pulse, is proportional to the photon flux illuminating the detector, ^< pk >= <: i(tk ) : > A tk,^ (E.2) where 7) is the quantum efficiency of the detector and I is the intensity operator kt) Appendix E. Spectrum of squeezing^131 (I a ata). Here colons denotes the normal and time ordering'. We also have: < pkp3^97 2 <: /(4)/(t3 ) :> AtkAt3 .^ (E.3) Hence, passing to the continuous limit and assuming that Q(t — t') = Q 06 (t — t'),^ (E.4) we find the following result for the autocorrelation function of i(t) < i(t)i(t + 7-) > = f den <:^Q(t — t')Q(t + - t ' ) + f de I dt"712 <: i(e)i(t") :> Q(t — t')Q(t + T - t" ) , Or < i(t)i(t + 7) >= Q,111 <: I(t) :> 47) + Q (2) 112 <: I(t)I(t + 7) :> .^(E.5) Thus < Ai(t)Ai(t + 7) >=-< i(t)i(t + 7) > — < i > 2= 071 <: At) :> 6(7) + O7i 2C(7), where C(T)^i(t)/ (t T) :> - <: I :> 2 .^(E.6) Now we apply the above result to the situation of a homodyne detection of squeezed light in a cavity3 . Fig. (E.3) shows the setup of the experiment. First, 2The appropriate operator order in the photoelectron counting distribution is normal and time ordering. For example, the intensity correlation function for t i > to is written as: <: i(t0 )1(t i ):>c« Et(tott(tot(tok(to )> all creation operators are to the left (normal ordering) and time arguments increases from the left and right to take their largest values in the center. 3H. J. Kimble, Fundamental Systems in Quantum Optics (Elsevier Science Publishing, Amster- dam, 1992), chapter 10. Appendix E. Spectrum of squeezing^132 the beam out of a local oscillator at co o (eigenfrequency of the cavity mode) is fre- quency doubled by a nonlinear crystal via second harmonic generation. Light at 2w o shines on the nonlinear cavity (PhC microcavity) and the degenerate down-conversion process inside the cavity generates twin photons at co o . Then the signal field out of the cavity, as , is combined with the local oscillator field, ah,, by a beam splitter with high transmission T for the signal field.  nonlinear^A cavity^CI() as BS local oscillator SHG spectrum .f analyzer Figure E.3: Experimental setup of a homodyne detection for measuring the spectrum of squeezing. A small fraction R of the local oscillator field is combined with the signal field and the output field a = Ra io + Tas is sent to a detector with a high quantum efficiency. We assume the local oscillator is a strong coherent state with < d 1 , >= ao e'e . Although the reflection coefficient of the final beam splitter for the local oscillator is very small, we assume Rao is still large compared to the mean value of the signal field and its noise fluctuations. After a bit of algebra and keeping the leading terms in a o we have: = R2T24ic-2ie^> ezieC(7)^< ets (t + 7), ?LA +^< ats (t), ts (t + T) > < ats (t),a5 (t + 7) > + < atjt + T), a s (t) >], Appendix E. Spectrum of squeezing^133 where we used < a, b ›E_--< ab > — < a >< b > as the variance between two operators a and b. Now by using the definition of the quadrature-phase amplitude (Appendix D) for the signal field 2e(t) -= C ie âs (t) + 61(0,^(E.7) we have: C(7) -= R2 T2 <: Ze (t), 2e (t + 7) :> .^(E.8) Finally our photo-current fluctuations becomes: < Ai(t)Ai(t + r) >= OR2 a47/[8(7) + nT2 <:^29 (t + T) :>],^(E.9) where <: I (t) :>= R2a6 was used as the average intensity of the light shining the detector. The spectral density of the photo-current fluctuation will be: cb (C2, 9) = f < i(t)Ai(t + 7) > e- zffi d7 = OR2 471[1 + qT2 S (C2, 9)],^(E.10) where S(52, 9) is called the spectrum of squeezing and is defined by: S(Q, 9) = f <: 20 (t), 29 (t + r) :> e- i c21- dr.^(E.11) For a stationary process (Wiener-Khintchine theorem') we have: <: 29(C2), 20(—Q) :>= f <:^(t), 20 (t + T) :> e- z c2T dr, and therefore S (C-2 , 0) =<: 20(Q), 20(—C2) :> . By expanding the time and normal ordering expectation value we have: 4 C. W. Gardiner, Handbook of stochastic methods (Springer-Verlag, Berlin (1985). Appendix E. Spectrum of squeezing^134 S(52,0) =--< 4(C2), 4(—C2) > —1.^(E.14) The above equation was used in chapter 3 to calculate the spectrum of squeezing.

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