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Nonlinear optics of multi-mode planar photonic crystal microcavities McCutcheon, Murray William 2007

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Nonlinear Optics of Multi-mo Planar Photonic Crystal Microcavities by Murray William McCutcheon B.Sc, The University of British Columbia, 1997 M.Sc, The University of British Columbia, 1999 B.A.., The University of Oxford, 2001 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF Doctor of Philosophy in The Faculty of Graduate Studies (Physics) The University Of British Columbia June, 2007 © Murray William McCutcheon 2007 11 Abstract The nonlinear properties of multi-mode InP and Si planar photonic crystal microcav-ities are investigated in experiments relevant to integrated schemes for classical and quantum optical information processing. Normally incident, short laser pulses are used to coherently initialize the relative phase and amplitudes of two modes of a single-missing-hole InP microcavity. The two modes are orthogonally polarized, and separated by less than the bandwidth of the ~ 130 fs excitation pulses. The relative amplitudes of the two modes can be controlled by adjusting the polarization and the centre frequency of the excitation'beam. Cross-polarized detection of the resonantly scattered light reveals a well-defined relative phase between the modes that is characteristic of their coherence. When the short-pulse excitation is used to coherently excite two modes in a three-hole line-defect (L3) InP microcavity, second-order harmonic radiation is observed due to the interactions of the resonant fields with the second-order nonlinear susceptibility (x^) of the host InP slab. Second-harmonic and sum-frequency generated signals are observed due to the intra- and inter-mode nonlinear mixing of the microcavity fields. When a separate non-resonant pulse is focussed onto an InP microcavity, sum-frequency light is generated conditional to the resonant mode population of the microcavity. The conditionally generated signals can be tuned by tuning the frequency of the non-resonant pulse. Al l of the results can be explained with reference to the bulk properties of the InP slab. While the transient, multi-mode response of the microcavities is harnessed with the short-pulse technique, a continuous wave excitation laser exploits the local-field Abstract 111 enhancement intrinsic to these wavelength-scale microcavities. A single-mode InP L3-microcavity with Q = 3,800 is pumped on resonance with a C W laser, and the 2D pat-tern of far-field second-harmonic radiation is directly imaged. The second-harmonic light is enhanced by 1000 times compared to non-resonant excitation, demonstrating integrated low-power frequency generation. The spatial pattern of the radiation is consistent with simulations based on the bulk tensor, and reveals the importance' of scattering and material absorption of the harmonic light. Ultrafast, all-optical switching is demonstrated in a Si microcavity with a single Q = 35,000 resonant mode. The mode is resonantly excited with a weak probe pulse, and a non-resonant 200 pJ pump pulse with a precisely controlled time delay is used to inject free-carriers above the silicon bandgap. The free-carrier dispersion shifts the mode frequency by 9 line-widths, and broadens its width by a factor of 4. When the excited mode is perturbed while it is ringing down, coherent oscillations in the spectra are observed which can be explained in terms of a model of an instantaneously perturbed harmonic oscillator. The implications for frequency conversion and for the generation of squeezed optical states are considered. iv Table of Contents Abstract 1 1 Table of Contents iv List of Tables viii List of Figures i x Acknowledgements x u 1 Introduction 1 1.1 Overview 1 1.2 Photonic crystals 3 1.2.1 Basic concepts 3 1.2.2 Modes of 2D planar photonic crystals 5 1.2.3 Microcavity properties 7 1.3 Nonlinear effects involving optical cavities 12 1.3.1 Classical regime . 12 1.3.2 Quantum regime 14 1.4 Thesis outline 16 2 Fabrication 18 2.1 Introduction 18 2.2 Electron-beam lithography 20 Table of Contents v 2.2.1 General considerations 20 2.2.2 Specific processes 23 2.3 Etching 29 2.3.1 Resist-removal 31 2.4 Undercutting 32 2.5 Summary 32 3 Model l ing 35 3.1 Introduction 35 3.2 FDTD simulation overview 36 3.3 Extracting mode profiles with symmetry 38 3.4 Mind your V's and Q's 43 3.5 Second-order polarizations 46 3.6 Far-field projections 49 3.7 Summary 51 4 Experiment 53 4.1 Resonant scattering 53 4.1.1 Alignment 60 4.1.2 Samples 61 4.1.3 Laser sources 62 4.2 Single-mode resonant scattering 63 4.3 Photoluminescence 66 4.4 Second-order techniques 68 4.5 Pump-probe spectroscopy 71 4.6 Summary 74 5 Mul t i -mode microcavity characterization 75 5.1 Introduction 75 Table of Contents vi 5.2 Mode profiles 77 5.2.1 Localized modes 78 5.2.2 Quasi-localized modes 85 5.3 Multi-mode resonant scattering 87 5.3.1 Coherent multi-mode excitation model 89 5.4 Summary 94 6 Second-order responses . 95 6.1 Introduction 95 6.2 Bulk response 97 6.3 Microcavity impulse response 103 6.3.1 Single-mode excitation 103 6.3.2 Polarization properties of second-order mode radiation . . . . 104 6.3.3 Monitoring of multi-mode excitation 107 6.3.4 Monitoring of multi-cavity occupation 110 6.4 Frequency domain microcavity response 114 6.4.1 High-Q localized mode 114 6.4.2 Quasi-localized mode 116 6.5 Summary 118 7 Ultrafast mode switching 120 7.1 Introduction 120 7.2 Carrier-induced optical switching 122 7.3 Ultrafast perturbation of a harmonic oscillator 126 7.4 Implications for squeezed state generation . . . " 132 7.5 Summary 134 8 Conclusions 135 8.1 Implications 137 Table of Contents vii Bibliography 139 V l l l List of Tables 2.1 Typical e-beam writing parameters for the SEM 21 2.2 Original GaAs development recipe 29 2.3 Alternative GaAs development recipe 29 2.4 Silicon development recipe 29 2.5 P E C V D pre-etch "descum" step for silicon samples 30 2.6 Etch recipe for GaAs samples 31 2.7 Etch recipe for Si samples 31 5.1 Summary of experimental and simulated parameters 83 7.1 Summary of pump-probe data fit parameters 132 IX List of Figures 1.1 Schematic dispersion of a ID photonic crystal 4 1.2 Schematics of 3D and 2D photonic crystals 5 1.3 Typical bandstructure of a hexagonal planar photonic crystal 7 1.4 Schematic of a ID wavelength-scale Fabry Perot cavity 10 1.5 Channel drop filter schematic 12 2.1 Photonic crystal fabrication process 19 2.2 Defect microcavity in a GaAs hexagonal photonic crystal 22 2.3 Top and side views of the SEM sample holder 23 2.4 Intermediate etch profile of Si under a cross-linked P M M A mask . . . 25 2.5 Nearly vertical sidewall profile of Zep resist and etched Si 26 2.6 Schematic of the circular dose protocol 28 2.7 Typical 3 x 3 array of microcavities . : 28 2.8 Free-standing silicon photonic crystal microcavity 33 2.9 Profile of a free-standing silicon photonic crystal 34 2.10 A three-missing-hole silicon photonic crystal microcavity. 34 3.1 SEM image of an InP L3-cavity 36 3.2 Real-space rendering of the FDTD dielectric structure 37 3.3 Spectrum of \EX\2 from time monitor 1 39 3.4 Fundamental high Q mode of the L3-cavity 41 3.5 Fully symmetric s'-polarized mode of the L3-cavity 42 3.6 Calculating the Q from the time-dependent fields 45 List of Figures x 3.7 Simulated second-harmonic polarization distribution 48 3.8 Near and far-field patterns with and without absorption 50 4.1 Free-space resonant scattering set-up 54 4.2 Material polarization (P) of three microcavity modes 56 4.3 P • /^ in c i den t for each of the material polarizations 56 4.4 Cross-polarized detection schematic for resonant scattering 57 4.5 Evanescent coupling vs. free-space resonant scattering 58 4.6 Schematic of transmission sample mount 62 4.7 Short-pulse and C W resonant scattering spectra 65 4.8 PL spectrum from an InP L3-microcavity 67 4.9 Second-order spectroscopy set-up 69 4.10 Pump-probe spectroscopy set-up 72 5.1 Schematic photonic density of states 77 5.2 FDTD simulation results from the 5-cavity 79 5.3 F D T D simulation of an elliptical-hole S'-cavity 79 5.4 Spectra showing all localized modes of an L3-microcavity 82 5.5 SEM image of an InP L3-cavity 83 5.6 Spatial intensity profiles of all localized modes of an L3-cavity . . . . 84 5.7 Stitched sequence of broadband resonant scattering spectra 86 5.8 C W resonant scattering spectrum from an InP L3-cavity 87 5.9 Images of the electric field intensity of the QL modes 88 5.10 Four coherently-excited quasi-localized modes 89 5.11 Spectra from an InP 5-cavity with two non-degenerate dipole modes 90 5.12 Coherent fit of two-mode resonant scattering spectrum . 92 5.13 Spectra acquired as the OPO is tuned 93 6.1 Scattering spectra from the untextured InP slab 97 6.2 Non-resonant laser SHG for different incident polarizations 98 List of Figures xi 6.3 FDTD simulations of a focussed Gaussian beam 101 6.4 Far-field second-order radiation patterns from an untextured InP slab 102 6.5 Fundamental and second-order spectra from the high-Q mode . . . . 104 6.6 Relative orientation of axes 105 6.7 Polarization-resolved second-order spectra from the high-Q mode . . 106 6.8 Simulated field intensity profiles and associated PZ distributions . . . 106 6.9 Simulated far-field intensity patterns 107 6.10 Second-order impulse response of two-mode L3-microcavities 109 6.11 Schematic optical circuit for the joint state nonlinear monitor . . . . 110 6.12 Spectra given by the interaction of a non-resonant idler pulse; a reso-nant laser pulse; and a two-mode PPC microcavity 112 6.13 Spectrum of the OPO laser beam showing ps and fs pulses 113 6.14 CCD image of the measured second-order radiation pattern 115 6.15 Measured QL mode far-field second-order radiation pattern 117 6.16 Relative orientation of axes 118 7.1 Resonant scattering spectra for a wide range of pump-probe delays . 123 7.2 Example of a raw pump-probe spectrum 124 7.3 Schematic of free-carrier density as a function of time 124 7.4 Mode energy as a function of pump-probe delay time 125 7.5 Schematic of a perturbed harmonic oscillator 128 7.6 Spectra of a dynamically perturbed high-Q microcavity mode . . . . 129 X l l Acknowledgements I am very grateful to Professor Jeff Young for his many years of mentorship throughout my graduate education. He has taught me how to do "real" physics, the kind that one can bring to bear on a daily basis in the lab, and has given to me a wealth of insight both theoretically and experimentally. With his guidance and support, I have been challenged to develop independence as a researcher, and these skills will serve me well in future research, as in life. I have benefitted greatly from working with many others in the Young lab, includ-ing Allan Cowan, Iva Cheung, Haijun Qiao, Miryam Elouneg-Jamroz, Mario Beau-doin, and Alexandre Zagoskin. I would like to give special mention to Dr. Andras Pattantyus-Abraham for his many contributions to this work, most notably on the fabrication side, and to Dr. Georg Rieger, for contributing his experimental prowess to this work. It has been a privilege sharing the doctoral journey with a fellow trav-eller, Mohamad Banaee, and I have learned much from him. I have also been greatly influenced by two friends, Paul Paddon and James Analytis. I would also like to thank the members of the Williams group at the Institute for Microstructural Sciences at the National Research Council in Ottawa. In particular, I had many useful discussions with Robin Williams, and the expertise of Dan Dalacu in fabricating the InP samples was instrumental to this work. On a personal level, I am deeply indebted to my partner and teammate, Jen Capell, for her constant love, support and encouragement through the highs and lows of my PhD. And of course, I thank my parents for their unconditional love and support in this endeavour, as in all I do. Chapter 1 Introduction i 1.1 Overview This thesis concerns the trapping and manipulation of photons on a semiconduc-tor chip. Ultimately, this work is part of a research effort in the field of quantum information and quantum computation (QIQC), which is a vibrant area of interest for exploring the fundamental intricacies of quantum mechanics, the new paradigm of quantum information theory, and the practical goal of building a quantum com-puter [1]. There are many domains in which QIQC ideas are being explored, including superconductivity, nuclear magnetic resonance, and optics. Like its classical coun-terpart, an optimal quantum computer should be scalable. This means that the fundamental computational unit, the qubit, can be replicated (or scaled up) to cre-ate a multi-qubit device without redesigning the architecture. Semiconductor wafers are the ultimate platforms for scalable devices, as a result of the tremendous ex-pertise and investment in lithographic and etching technologies that has driven the microelectronics industry. If these techniques could be leveraged for a QIQC device, semiconductors would be an ideal platform for quantum computation. On the macro scale, photons are efficient carriers of quantum information, because they are weakly interacting (and so resistant to decoherence); easily transportable and directable; can be efficiently delayed using phase shifters; and are easily com-bined with the aid of beamsplitters. If these same properties could be achieved on the micro scale, by fabricating elements such as low-loss waveguides, beamsplitters, and filters for photons propagating on a semiconductor wafer, such "photonic chips" Chapter 1. Introduction 2 would offer an extremely attractive platform for multi-qubit quantum processors. The key requirement then would be finding mechanisms for coupling information from the qubits to the photons. Similar reasoning about the attractive properties of integrated optical networks led to the late-1990s boom in developing classical optical information processing applications. Much of the attention was focussed on a class of artificial materials with a periodically modulated dielectric constant known as pho-tonic crystals [2, 3]. In particular, a large body of work addressed the development of photonic crystals in planar semiconductor membranes, as described later in the introduction. This showed the feasibility of controlling the flow of light in compact optical circuit geometries. One of the major challenges encountered during this era was the relatively high loss in photonic crystal-based devices, in comparison to com-peting fibre optic technologies, due to unavoidable deviations from ideality in the fabricated structures. Ultimately, this loss has prevented the commercial viability of integrated optical telecommunication platforms based on photonic crystals. Although photonic crystal-based optical circuit platforms cannot currently com-pete with fibre optic, or even glass waveguide, technologies in telecommunications, they do offer unique capabilities that might suit more specialized applications. In particular, the recent demonstration that semiconductor-based photonic crystal mi-crocavities can be used to trap 1.55 micron light in volumes less than 0.2 ^ m 3 for over a million optical cycles [4], motivates research on the nonlinear interactions of pho-tons and electronic excitations within the microcavities. Low-power, nonlinear optical responses can be used to perform all-optical logic and switching operations, and if the nonlinear response can be induced with only a single photon in the microcavity, quantum optical chips are conceivable. This thesis focusses on elucidating some of the nonlinear interactions specific to light confined in photonic crystal-based microcavities.. The remainder of the introduc-tion will explain some of the background information needed to understand photonic crystal microcavities, and cavity-based nonlinear optics. Chapter 1. Introduction 3 1.2 Photonic crystals 1.2.1 Basic concepts In one dimension, a photonic crystal takes the form of a Bragg stack of alternating layers of two different dielectrics, each with a different thickness and refractive index. When normally incident on the structure, light which satisfies the condition that an integral number of half-wavelengths matches the periodicity of the structure can be diffracted such that there is complete reflection over a frequency range known as the stop-band. If the frequency of propagating modes inside the dielectric is plotted versus wavevector in a dispersion diagram, as shown schematically in Fig. 1.1, the dispersion is strongly modified from its bulk trend in the vicinity of this diffraction condition. The group velocity (du/dk) of the light goes to zero, and standing waves are formed which consist of electric field maxima in either the high-index (lower energy mode) or low-index (higher energy mode) material. There is a gap in energy between these two standing-wave conditions: light is forbidden to propagate at energies within this photonic bandgap. The position and width of the stop gap depends on the incident angle, and the width of the gap also scales with the refractive index contrast between the high- and low-index materials comprising the stack. If the periodic ID structure contains a defect, as shown in Figure 1.1(b), a sin-gle mode can be created with an energy inside the stop gap. The defect acts like a ID Fabry Perot cavity, and the dielectric stacks to either side serve as mirrors to diffract light at the resonant energy back into the cavity. The transmission spectrum in Fig. 1.1(b) exhibits a Lorentzian-shaped resonance feature when the incident fre-quency matches the energy of the mode localized between the two dielectric stacks. The width of the resonant feature is a measure of how well-confined the mode is - the higher the reflectivity of the dielectric mirrors, the narrower the feature, and the higher the quality (Q) factor of the localized mode. The Q-factor of the mode increases as the number of layers in the Bragg stacks is increased: in a defect within Chapter 1. Introduction 4 7l/A E, E 2 Wavevector Energy (a) (b) Figure 1.1: (a) Schematic dispersion of a ID photonic crystal (Bragg stack). The wavevector corresponds to the direction of propagation normal to the slab planes, (b) Transmission spectrum of a ID photonic crystal with a defect, which introduces a resonant mode into the energy gap. an infinite periodic dielectric stack, the mode would be completely localized. This concept of localizing light by diffraction is far more powerful, and more relevant to integrated optical circuits, if generalized to three dimensions. Since 1987, there has been an active research effort dedicated to fabricating structures which have a 3D periodicity, so that overlapping photonic bandgaps exist for any direction of propagation [5, 6, 7]. However, even uniform structures such as the one depicted in Figure 1.2(a) are extremely difficult to manufacture, and the introduction of tailored defects to localize light, and channels to couple that light to the outside world, is prohibitively difficult with current methods [8]. An alternative approach to photonic bandgap engineering is based on a two-dimensional (2D) texturing of a planar semiconductor wafer, as shown in Figure 1.2(b). This geometry uses total internal reflection (TIR) to confine light to the 2D slab,1 and photonic crystal concepts in 2D to control the flow of light within the slab. These 1 T h e r e is a h i g h r e f r a c t i v e i n d e x c o n t r a s t be tween a i r ( n = l ) a n d m a n y s e m i c o n d u c t o r s (e.g. S i , I n P , G a A s ) i n t he n e a r i n f r a r e d ( n ~ 3) , a n d so a s l a b o f s e m i c o n d u c t o r w i t h a i r a b o v e a n d b e l o w c a n v e r y e f fec t ive ly con f ine l i g h t t o p r o p a g a t e o n l y w i t h i n t he 2 D s l ab . Chapter 1. Introduction 5 1 i l l 1 1 1 1 1 V (a) (b) Figure 1.2: Schematics of (a) a 3D "log-pile" photonic crystal like the one fabricated in [5], and (b) a 2D photonic crystal of air holes in a dielectric slab with a defect microcavity and waveguide incorporated. 2D structures are much easier to fabricate than 3D structures, because they take ad-vantage of the highly-developed thin film growth, patterning, and etching techniques used in the microelectronics industry, and are ideal for integration with other opto-electronic devices like semiconductor lasers. Point defects are readily introduced into the periodic lattices in order to act as microcavities, and line defects can serve as waveguides. 1.2.2 Modes of 2D planar photonic crystals The bound modes of a uniform, free-standing (as often they are) 2D slab have either even (TE) or odd (TM) parity with respect to a mirror plane in the middle of the slab. These modes are completely confined to, and are free to propagate within, the slab by total internal reflection. Texturing the slab with a 2D array of air holes has a profound impact on the spectrum of electromagnetic excitations which can exist in the slab. In a similar manner to the ID example, the 2D propagating modes of the bare slab are renormalized by Bragg scattering from the air holes. Bragg scattering can open up a large 2D energy gap in the spectrum of the fundamental TE-like modes. This gap is not a complete photonic bandgap, since it only exists for light with sufficient in-plane momentum to be confined in the vertical direction. Chapter 1. Introduction 6 These concepts are made clear in Figure 1.3, which shows the bandstructure of a hexagonal photonic crystal for the high symmetry directions of the 2D Brillouin zone shown in the inset. The dashed lines depict the dispersion of light in the air cladding which bounds the slab. Modes which are everywhere below this light line, such as shown by the lower band of blue squares, are bound modes which are completely localized to the slab. Above the light line, in the lighter shaded region, there is a continuum of radiation modes. Modes which lie in between these two extremes, such as the one represented by the upper band of circles, are known as resonantly bound, or leaky, modes. Although largely confined to the slab, these modes have radiative components which lead to out-of-plane loss. The two bands shown in the diagram are known as the air and dielectric bands, and together they define the photonic bandgap, shown by the dark grey region below the light line. When considering the in-plane dispersion in 2D, the radiation zone is referred to as the light cone, since the light line sweeps out a cone in the volume defined by (kx, ky,u>), where k\\ = (kx, ky). Much of the early work on planar photonic crystals (PPCs) focussed on studying and tailoring the allowed (propagating) mode spectrum of uniform 2D crystals [9, 10]. For the present work, the properties of defect states intentionally introduced to localize light in three dimensions are the subject of interest. When a defect is placed within the periodic lattice of air holes, it can introduce a localized mode into the bandgap, as shown in the ID example above. The nature of these defects has been studied extensively by several groups [11, 12, 13], leading to a significant body of literature which focusses on the tailoring of the photonic lattice around the defect to produce favourable mode symmetries, volumes, and Q-factors. Chapter 5 deals in detail with the properties of defect modes associated with hexagonal PPCs having three missing holes (the L3-defect), as shown in the schematic in Figure 1.2(b). Chapter 1. Introduction 7 r K M r Figure 1.3: Typical bandstructure of the fundamental TE-like guided modes of a hexagonal planar photonic crystal, with a photonic band-gap shown in dark grey. Above the dashed cladding light lines are the radiation zones (light shading). The inset shows the high symmetry points T, K, and M in the first Brillouin zone. The upper and lower bands are labelled as the air band (AB) and dielectric band (DB), respectively. The photonic crystal has a pitch of 500 nm and air-hole radius 150 nm, and is etched in a slab of thickness 300 nm with index n = 3.4 (GaAs). 1.2.3 Microcavity properties Quality factor A key parameter that characterizes a defect mode is its lifetime, r, or its quality factor, Q (= U)QT). The Q-factor determines the ratio of energy stored in the cavity to the power loss. Along with a small mode volume, a high Q-factor is a key determinant for a range of applications that require high field amplitudes, long interaction times, and narrow line-widths. In the PPC geometry, the Q-factor can be resolved into in-plane, Q\\, and out-of-plane, Q±, components as 1/Q — 1/Q\\ + 1/Q±, and if there are enough periods (~ 15) of the photonic crystal lattice surrounding the defect microcavity, the in-plane confinement is usually such that Q is limited by Q±. Because uniformly textured slabs have discrete translational symmetry in the plane, any photonic mode Chapter 1. Introduction 8 of the background photonic crystal region can be represented as a 2D Bloch state, characterized by its in-plane wavevector, k\\. In the out-of-plane direction, the lack of translational invariance means that there is a continuum of plane-wave modes which are not confined to the plane of the slab. A localized defect can mix the 2D Bloch modes from the band-edge states of the uniform photonic crystal [11] to form localized modes. The properties of the defect modes can be understood by considering the Fourier decomposition of their material polarizations into plane-wave components, each characterized by its in-plane wavevec-tor, Whether or not a given component is confined to the slab is determined by the dispersion relationship for a plane wave of frequency u> in the air (n=l) cladding: | / c n | 2 + ^ = ( W /c) 2 . (1.1) A high-Q defect mode at frequency to will be dominated by Fourier components which satisfy the waveguiding condition, given by |fc||| > w/c. • • ' ' (1.2) Such components can only couple to evanescent fields in the z direction, since k\ < 0 in eq. (1.1). However, due to the complete lack of translational symmetry, a defect mode always contains some Fourier components that fail to satisfy this condition. These components are located inside the light cone, and lead to loss due to coupling to out-of-plane radiation modes with k\ > 0. Much of the high-Q mode design literature has aimed at reducing lossy Fourier components [14, 15], and <Q-factors of ~ 1 million have now been realized in silicon photonic crystals [16, 17, 18]. While these radiative Fourier components of the in-plane polarization associated with the defect states are the dominant loss channels, they can also be exploited as in-coupling channels to excite the mode, and the free-space resonant scattering technique used in this work is premised on this fact. Chapter 1. Introduction 9 In this thesis, the term "high-Q" is often used in reference to a particular mi-crocavity mode. This refers to modes which have, or could be optimized to have, Q-factors in excess of ~ 10,000, which is the estimated threshold necessary to realize quantum interactions with excitonic two-level systems [19, 20, 21] (see cavity QED discussion below). Mode volume and multiple modes Another crucial feature of a P P C microcavity is its small mode volume, V. The ultra-small mode volumes achievable in PPC microcavities are one of the key drivers of the research. For a given Q-factor, the mode volume determines the peak electric field that can be realized inside the cavity, and high field amplitudes are important for many of the nonlinear phenomena discussed below. In a wavelength-scale micro-cavity, there is no advantage to larger volumes in terms of phasematching, since the conversion efficiency is determined by the peak field strength. In a cavity driven on resonance by a C W laser, the Q-factor determines how much energy can be stored in the cavity for a given incident power. In a high-Q cavity that also has a small V, the internal electric field can be very high, since the energy trapped in the cavity, U, is given by U ~ J \E\2dV. In fact, if the cavity dimensions are on the scale of a wavelength (V ~ A 3 ), the external (incident) field intensity can be enhanced by a factor close to Q inside the microcavity; i.e., [-£7int12 ~ Q\Eext\2, as depicted in the ID case in Figure 1.4. High local field enhancements are fundamental to the properties of wavelength-scale microcavities, and are exploited for many nonlinear phenomena in the literature and in this work, as discussed below. The relationship between V and Q is not entirely independent. Intuitively, the properties of the Fourier transform dictate that the extent of a mode in real-space is inversely proportional to its spread in A;-space. Very small real-space mode vol-umes can therefore lead to extended fc-space distributions and more lossy components within the light cone [22]. Indeed, the highest Q (and highest Q/V) microcavity de-Chapter 1. Introduction 10 d ~ X lEextNl 2 c c H |Eext(co)| p|E i n t(co) Figure 1.4: Schematic of a ID wavelength-scale Fabry Perot cavity. The incident field intensity is enhanced by a factor of Q inside the cavity. signs in the literature have volumes V ~ (A/n) 3 which, while very small compared to non-integrated cavities, are considerably larger than the fundamental limit of (A/2n) 3 (the lowest-order standing-wave condition). Microcavities designed to support higher Q modes at the expense of slightly larger mode volumes are usually multi-mode. Multi-mode cavities are of significant interest in their own right because many classical and quantum information protocols hinge on multiple modes (or basis states) to perform logical operations (or gates) [23, 24]. Based on experiments using a short-pulse laser to coherently excite multiple micro-cavity modes, the linear properties of multi-mode PPCs are explored in Chapter 5, and their nonlinear interactions are studied in Chapter 6. Applications Ten years ago, there was much optimism that the bulky macroscopic optics which per-formed such functions as multiplexing and adding or dropping channels in multi-mode optical fibre communication networks could be completely replicated in PPCs [2, 25]. PPC waveguides consisting of a row of missing holes in an otherwise uniform 2D structure, like the one sketched in Fig. 1.2(b), were designed to have tight (60°-90°), wavelength-scale bends and Y-branches, and various filtering and routing functions were proposed. Despite considerable progress in the practical implementation of many of these proposals, out-of-plane losses in these high-index-contrast structures Chapter 1. Introduction 11 due to unavoidable fabrication imperfections have, for the most part, prevented these devices from surpassing implementations based on conventional optics [26, 27, 28]. Imperfections such as sidewall roughness and variations in feature size and period-icity lead to loss due to backscattering and scattering into radiation modes, and in many cases the loss is polarization-dependent, which is another key disadvantage for commercial applications [26]. In some cases, simpler integrated solutions have been developed that are not based on photonic crystals. For example, silicon-on-insulator rib waveguides have been demonstrated to have < 0.1 dB loss (~ 2%) in a bend of radius 1 /um and 3.6 dB/cm propagation loss [29, 30], whereas the best reported photonic crystal waveguides have propagation losses of ~ 10 dB/cm [31]. However, as discussed below, devices based on PPC microcavities hold promise for new integrated optical functionality in applications where the loss tolerance is higher. Integrated, wavelength-scale optical microcavities [32] in PPCs may be useful building blocks of optical networks, because they can be naturally connected via in-tegrated waveguides to other miniature optical and electrical devices, and then chan-neled to the outside world through optical fibres or electrical wires. When designed in direct bandgap hosts, PPC microcavities are ideal for integrated lasers because they are small, their emission wavelength and radiation pattern can be tuned by tailoring the defect geometry, and they have high spontaneous emission coupling factors [33]. Microcavity 2D superlattices with flat dispersion bands have also been used to create lasers [34]. There have been a number of proposals which use microcavity filters to add or drop signals from multi-channel waveguides, which is a topic of great interest in optical signal processing. This passive functionality can be achieved in direct or indirect bandgap semiconductor hosts. As sketched in Figure 1.5, these proposals involve tunneling between two ID waveguide continua mediated by resonant localized states of the microcavities [35, 36], and transfer efficiencies of 80% have been obtained [31]. Chapter 1. Introduction 12 (Ol, C0 2 , 0 ) 3 0D2, 0 ) 3 Q Figure 1.5: Schematic of a channel drop filter based on resonant microcavity tunneling between two ID waveguides. 1.3 Nonlinear effects involving optical cavities To process, rather than just to transport and filter, light signals in these integrated optical chips, it is necessary either to incorporate electro-optic components, or to exploit the nonlinear optical properties of the host material. A nonlinear interaction is required because photons do not interact with each other in linear media. Due to the local field enhancement enabled by large quality factors and small mode volumes, microcavities are well-suited to enhancing nonlinear effects [37, 38] for classical and quantum integrated networks. 1.3.1 Classical regime In classical optical communication systems, much of the conditional switching and routing of multiple channels (each carried by a different wavelength of light) from input to output optical fibres is achieved by demultiplexing all of the input chan-nels, converting the information to the electronic domain, making decisions using purpose-built electronic hardware, and then regenerating the information in appro-priate output optical channels. It would be far more efficient if this switching could be achieved without converting the information into the electronic domain. This requires the ability to "make decisions" on the chip based on input optical signals, and then conditionally switch selected channels, or regenerate information from one Chapter 1. Introduction 13 channel (wavelength) on a different channel. The switching of one channel contingent on the presence of another channel, a process known as all-optical switching, has been demonstrated in uniform ID and 2D photonic crystals [39, 40, 41, 42, 43], and both photonic crystal [44] and ring res-onator microcavities [45] with large local field enhancements. It has also been demon-strated in a bacteriorhodopsin-coated microsphere, in which a photoinduced confor-mational change of the protein coating induces a microcavity resonance shift [46]. These schemes are all based on a control channel changing the refractive index of the host material enough to effect a significant change in the reflection/transmission prop-erties of the signal channel through the region of the optical chip where they interact. This topic is the subject of Chapter 7, which describes a pump-probe technique with sub-picosecond time resolution for the switching of a microcavity mode. A higher level of functionality would involve designing the switching function to be dependent on the simultaneous presence of two control channels in the interac-tion region. This would require exploiting some nonlinear interaction between the two control beams, such that the reflectivity/transmission of the signal beam is only changed when both controls are present. Chapter 6 explores the second-order non-linear interaction of two modes inside PPC microcavities, and identifies a mechanism that could be used for this conditional switching operation based on an optical AND operation. The regeneration of information from one channel (wavelength) to another can also be achieved using the second-order nonlinear response of the host material. Chapter 6 illustrates this principle by demonstrating the frequency-doubling of a low-power continuous-wave diode laser which is resonantly enhanced by a P P C microcavity. A more sophisticated scheme for conditionally regenerating information is also presented which shows how electromagnetic energy at a well-defiried frequency, stored in a localized microcavity mode, can be mixed with a variable-frequency control channel to produce radiation at the (continuously variable) sum frequency. Chapter 1. Introduction 14 1.3.2 Quantum regime The QIQC literature contains a vast and diverse set of strategies for implementing quantum information processing protocols [1]. This breadth of approaches stems from the many possible physical representations of a qubit. Amongst the abundant examples in nature of this fundamental carrier of quantum information are the two spin states of a spin-1/2 particle, and the two polarization states of a photon. Like a classical bit, the qubit has two states, denoted |0) and |1), but unlike its classical counterpart, the qubit can exist as a linear combination, or superposition, of the pure states, |V) = a|0)+/3|l>, (1.3) where a and B are complex amplitudes. Two QIQC protocols that are relevant for the work in this thesis are based on linear optics and cavity quantum electrodynamics (cavity QED). In 2001, a linear optics quantum computation (LOQC) scheme was put forth by Knill et al. [47]. The proposal involved only beamsplitters, phase-shifters, single-photon sources and photo-detectors. These linear elements perform only single qubit operations (rotations), but there is an effective nonlinearity introduced through feedback from the detectors. In the proposal, the qubit states are represented by two optical modes,2 |0) and |1), which could be the two polarization states of propagating photons, or two modes of a cavity. In Chapter 5, a method is shown for coherently populating two modes of a PPC microcavity with a short-pulse laser. At the single photon level, this may have relevance for LOQC schemes. Knill et al.'s scheme also relies on efficient non-classical sources capable of pro-ducing single photons (on demand) to inject into the quantum optical processor [48]. Chapter 7 describes a novel way of possibly generating non-classical light from P P C microcavities, and this might present a path for the entanglement of photons in an 2 The authors actually use a variation on the dual-rail representation, and the logical states of their qubit are |0) L = |00) + |11) and \1)L = |01) + |10) Chapter 1. Introduction 15 integrated optical circuit [49]. The other QIQC implementation relevant to this thesis is cavity QED [50], which is based on the nonlinearity of a two-level system coupled to the optical modes of a cavity. In a seminal paper, Turchette et al. demonstrate the conditional coupling of two photons in a probe beam whose polarization state is mediated by a single (two-level) atom in a cavity [51]. With the development of narrow-linewidth quantum dot emitters, which are nanometre-sized islands of semiconductor in which the quantum confinement of electrons causes discrete energy levels and atom-like emission spectra, the single atoms in traditional cavity QED experiments can be replaced by tailorable semiconductor quantum dots, which can be grown epitaxially on a semiconductor wafer [52] or chemically nucleated in solution [53]. Theoretical schemes have been proposed to perform conditional quantum logic using a single quantum dot coupled to a two-mode microcavity [23], and experimental progress has been made in realiz-ing quantum interactions between single quantum dots and high-Q P P C microcavity modes [19, 20, 21]. The Young group at UBC is also active'in this area, and is de-veloping methods to couple single PbSe nanocrystals to PPC microcavities [54]. The recent advances in guiding and trapping light in microphotonic devices are now being integrated with atom chip technology [55], which enables the cooling and trapping of atoms to make chip-scale Bose-Einstein condensates. These novel hybrid photon-atom chips [56] will allow further studies of strong light-matter interactions. While these potential applications help to motivate the research described below, it is worth noting that both classical and quantum chip-based optical processors may never become commercially feasible. The true value in the following results lies in the elucidation of interesting, and new, nonlinear optical responses that can be engineered in semiconductor membranes using the concept of planar photonic crystal microcavities. Chapter 1. Introduction 16 1.4 Thesis outline This thesis describes an experimental investigation of PPC-based microcavities. The goal is to demonstrate how the multi-mode and nonlinear properties of PPC microcav-ities make them attractive building blocks for both classical optical signal processing schemes, and cavity QED implementations for QIQC. The first three chapters of this thesis describe the fabrication, modelling, and opti-cal techniques utilized in this work. Chapter 2 describes the multi-step processes that were developed to fabricate the structures using electron beam lithography. Chapter 3 describes the methods used to numerically model both the linear and nonlinear elec-tromagnetic properties of the microcavities, and Chapter 4 discusses the short-pulse laser resonant scattering technique which is the basis of all the optical experiments. The multi-mode properties of two different types of microcavities are considered in Chapter 5. The broadband spectrum of the pulsed excitation beam allows simul-taneous excitation of more than one mode of a microcavity. Simulations are used to characterize the different mode structures, and a model is developed to explain the well-defined coherence observed in the spectra. In Chapter 6, the second-order nonlinear properties of InP-based microcavities are investigated. A microcavity that is excited into a multi-mode state exhibits both second-harmonic and sum-frequency radiation from the second-order interactions of the various resonant fields. The im-plications of these nonlinear signals for conditional mode monitoring schemes are considered. When a continuous-wave laser is used in place of the pulsed source, the local field enhancement of a single mode microcavity is exploited to generate second-harmonic generation at the exceedingly low input power of 300 uW. The final chapter of the thesis provides a demonstration of dynamic mode switch-ing in a silicon microcavity. A high-Q mode is excited with a resonant laser pulse, and a second laser pulse at nearly twice the energy is used to inject free-carriers into the microcavity. The free-carrier population changes the refractive index and absorption of the cavity, leading to a very rapid switching of the frequency and lifetime of the Chapter 1. Introduction 17 resonant mode. The results depend on the relative time delay between the two pulses, and the implications for optical switching, frequency conversion, and the generation of squeezed optical states are discussed. 18 Chapter 2 Fabrication 2.1 Introduction The planar photonic crystals studied in this thesis consist of a hexagonal lattice of air-holes etched into a free-standing semiconductor membrane. Fabrication of the planar structures takes advantage of the mature microfabrication techniques that provide the foundation of the modern microelectronics industry. Remarkable techno-logical advances in this discipline have been responsible for the dramatic decrease in minimum feature size, from the centimetre length-scales of the first transistor to the current state-of-the-art 45 nm process recently perfected by Intel [57]. In this thesis, optical spectroscopy data are presented from both indium phosphide and silicon planar photonic crystals (PPCs). The InP-based PPCs, which contain a layer of In As quantum dots, were fabricated at the Institute for Microstructural Sciences at the National Research Council in Ottawa. The silicon structures were fabricated by the author in the UBC nanofabrication facility based on extensive proof-of-principle work with gallium arsenide wafers. A schematic of the full fabrication process used to make the samples studied in this thesis is shown in Figure 2.1. The process is based on three main techniques: electron-beam lithography (EBL), reactive ion etching, and wet chemical etching. E B L is a method for writing sub-micron scale patterns into planar semiconductor wafers. A thin polymer layer deposited on the sample surface is chemically altered by the focussed electron beam in a scanning electron microscope (SEM). Computer control of the beam scan allows almost arbitrary patterns to be defined. The pattern Chapter 2. Fabrication 19 1. Pattern resist 2. Develop resist 3. E C R etch 4. B O E etch with e-beam Figure 2.1: Photonic crystal fabrication process. (1) A silicon-on-insulator wafer coated with ZEP-520A resist is first patterned using electron beam lithography (EBL); (2) The resist is chemically developed; (3) The pattern is transferred into the sili-con membrane using an electron cyclotron resonance (ECR) etcher. The membrane is rendered free-standing in the patterned region by undercutting the SiC>2 with a buffered oxide etch (BOE). is then developed in a solvent that dissolves the exposed regions preferentially, leaving a polymer mask. Plasma etching is used to transfer the pattern of this polymer mask into the semiconductor layer. After the polymer has been stripped from the surface, a sacrificial sub-layer of the wafer is locally removed with a liquid acid etch to leave the final structure, which is a free-standing membrane textured with a photonic crystal. The last "undercutting" step improves the waveguiding character of the PPC by increasing the index contrast between the membrane and the cladding (air), which ultimately improves the vertical confinement of the microcavity modes. The E B L process was optimized for microcavity fabrication by the author based on documentation and training provided by Murray Thom, and the Master's thesis of Alex Busch, who implemented the e-beam writing system [58]. Initially, a number of samples were fabricated in the GaAs/AlAs material system, which formed the basis of previous work on PPCs in the Young group [59, 9]. Although no experimental results from these samples are presented in this thesis, the fabrication methods are included because they represent a significant improvement of the E B L process over Chapter 2. Fabrication 20 what was previously used for processing GaAs in the group. Although there are important details which are different, the general approach for GaAs is the same as that shown in Fig. 2.1 for Si. The Si process - in particular, the etching conditions - was developed concurrently to this work by Dr. Andras Pattantyus-Abraham, and was subsequently used by the author to fabricate the Si samples studied in this thesis. This chapter describes the methods of e-beam lithography, plasma etching, and undercutting to fabricate free-standing planar photonic crystals, and provides specific details for the GaAs and Si processes. 2.2 Electron-beam lithography 2.2.1 General considerations The scanning, electron microscope (SEM) used to pattern the samples is a JEOL JSM-840A coupled with a Nabity Pattern Generating System™ (NPGS) [60]. The typical SEM settings for this work are listed in Table 2.1. The dwell time of the beam at a given point controls the size of the exposed region, and the beam is rapidly scanned from point to point to map out the desired photonic crystal pattern. The ~ 300 nm diameter circular holes of a photonic crystal can be defined with one of two beam-writing protocols: point dose and circular dose. The first is simpler, and involves delivering a set dose to a single point, with the scattered electrons (both forward-scattered and back-scattered from the sample surface) exposing the resist in a circle centred on the point. The second involves the beam tracing out the circular profile of each hole. The protocol of choice depends on the sample material, as discussed in Section 2.2.2. The patterns are generated with a simple Matlab™ code, which outputs a text file in the .DC2 DesignCAD™ format for importing into the NPGS software interface. Individual photonic crystal defect patterns are grouped in 3 x 3 arrays (see Fig. 2.7). These groups of 9 photonic crystals are spaced by about 300-500 pm, and about Chapter 2. Fabrication 21 Table 2.1: Typical e-beam writing parameters for the SEM. Accel, voltage 30 kV (Si), 20 kV (GaAs) Beam current 5-8 pA Magnification 1000 x Working distance 15 mm 10 such groups are written in a row across the sample. A systematic variation of parameters, such as hole size and pitch, across the different groups allows the optimal parameters to be bracketed. Typically, 2-4 rows of patterns are written per sample, and after development, the sample is cleaved to give one row per sample. Generating several smaller samples in this fashion allows the etching conditions to be iterated and optimized. Also, because the SEM patterning process is time consuming, this increases the throughput of samples generated from a single EBL run. The large number of individual photonic crystals which are written allows a systematic variation of the parameters to account for uncertainties in material parameters and fluctuations in EBL conditions, thereby increasing the chances of generating optimal structures. When a particular pattern element is written into the resist, it receives a direct dose of electrons when the beam is focussed directly onto it, and an indirect dose from electrons back-scattered from the substrate. The indirect exposure due to scattering is called the proximity effect, and can be an important consideration when setting the dose of a feature, because it renders the size of features dependent on the pattern. An example of this effect is illustrated in Figure 2.2, which shows a microcavity consisting of a single enlarged hole in an otherwise uniform lattice. The defect hole is about twice the diameter of the other holes, but required four times the direct dose. The six nearest neighbour holes are also visibly enlarged due to their proximity to the large defect hole. One challenge is to preserve the beam focus across the expanse of the sample plane. Patterns are typically written across several millimetres of the sample, and if the sample moves out of the beam focus when it is translated, non-uniform and Chapter 2. Fabrication 2 2 I i i i i i i i i I I I fry 6.4mm x20.0k SE(U) 1/8/04 2.00um Figure 2.2: Defect microcavity consisting of a single enlarged hole in a GaAs hexag-onal photonic crystal. poorly defined patterns will result. The typical mounted sample has a Az ~ 50/im difference between the height of its two edges with respect to the stage, as shown in Fig. 2.3. This tilt cannot be completely compensated using the mechanical stage controls. The solution is to image both edges of the sample, and use the 2-stage fine focus adjust knob to measure the differential height Az between the two edges. It is customary to write groups of patterns across the width of the sample at a single y-setting. Knowing the heights of the near and far edges, the sample tilt can be overcome by linear interpolation. Care must be taken to adopt a uniform backlash convention for the z fine focus knob - either clockwise or counter-clockwise, whichever monotonically corrects for the change in sample height as the sample is translated during the writing. Another important technical point relates to the astigmatism of the beam. The apertures and astigmatism are both adjusted while imaging the reference gold stan-dard (labelled 1 in Fig. 2.3), which is not height-matched to the sample. After imaging the gold, it is crucial not to adjust the coarse focus to bring the sample into focus, because this destroys the established image plane. Instead, the sample should be brought into the image plane using either the z working distance knob or the z fine focus knob. The focus can then be sharpened using the fine focus knob with a Chapter 2. Fabrication 23 60 urn ooo o o o ooo r \ -*• X Top view Profile Figure 2.3: Top and side views of the SEM sample holder. The top view shows the gold reference standard (1), the sample (2), and the Faraday cup ammeter (3), which also serves to support the sample via a piece of semi-sticky conductive tape. The dots marked on the sample represent 4 rows of 4 photonic crystal arrays, each of which consists of a 3 x 3 array of hexagonal photonic crystal defect patterns (as shown in the expanded view). During patterning, the sample is translated in the x direction to write successive patterns. The sample is typically tilted with respect to the holder by ~ 50^m from edge to edge (as exaggerated in the profile sketch), and so translation by dx moves the surface of the sample out of the plane of the beam focus. Adjustment of the z control can be used to compensate. negligible effect on the astigmatism. 2.2.2 Specific processes Resist and beam protocols The point dose e-beam protocol was used for the GaAs process. It is the simpler of the two protocols and worked well for GaAs samples, although it results in sub-optimal sidewall profiles in the resist layer. A large differential etching rate between the GaAs (~ 3.2 nm/s) and the polymethyl methacrylate (PMMA) resist (~ 1 nm/s) used in the BCI3/CI2 etch process renders the verticality of the etched air holes insensitive to Chapter 2. Fabrication 24 the etch resist profile because the resist does not significantly erode during the etch. The initial silicon lithography procedure also used the point dose method, but yielded samples with significantly sloped sidewalls. The optimized process was devel-oped by Andras Pattantyus-Abraham (AP), and involved extensive characterization of e-beam and mask conditions. In comparison to GaAs, silicon is more difficult to etch in the chlorine-based etches available in the UBC nanofabrication facility. The strong etching conditions required (high flow-rate, pure CI2 plasma) can lead to significant mask erosion. This would not be a problem with an etch resistant hard mask such as Si02, but the gases required to selectively etch the mask (CF4 and H 2 ) were not available. In order to obtain high quality photonic crystals with a softer polymer-based mask, two criteria must be satisfied. The etch selectivity must be relatively high so that the mask survives the duration of the etch, and the mask must retain a relatively uniform profile as it erodes. Any non-uniformity in the vertical profile is amplified by a poorly selective etch. Neither of these criteria are satisfied by P M M A . Because the P M M A etching rate (~ 9 nm/s) in the strong chlorine plasma was 3 times greater than the silicon etch rate (~ 3 nm/s), even 450 nm thick P M M A layers completely erode during the etch. The P M M A can be hardened after deposition on the sample by exposing it to a high electron beam flux to cross-link the polymer chains (~ 3.5 nm/s etch rate), but the resulting mask has a poor cross-sectional profile. The non-verticality in the P M M A sidewalls is thus transferred into the sample as a monotonically increasing hole size, yielding sloped sidewalls, as shown in Fig. 2.4. To produce more vertical sidewalls in silicon, two .changes to the beam conditions were made in combination with a different electron beam resist. The first beam change consisted of writing at a higher acceleration voltage. Forward scattering of the electron beam leads to a downward taper of the hole profiles, as shown in Fig. 2.4. The effective increase in beam diameter from the top to the bottom of the resist due Chapter 2. Fabrication 25 Figure 2.4: Intermediate etch profile of Si underlying a cross-linked P M M A mask. As the undercut resist is eroded during the etch, the hole widens, resulting in sloped sidewalls. Image courtesy Andras Pattantyus-Abraham. to forward scattering is given empirically by [60] where df is the diameter in nm, Fit is the resist thickness in nm, and V is the e-beam voltage in kilovolts. Writing at 30 kV rather than 20 kV reduces the diameter increase by nearly a factor of 2, and therefore decreases the hole tapers. The second change in e-beam conditions was to use the circular-dose protocol. The point-dose approach delivers a highly localized dose of charge, and then relies on scattering to spread the dose over the desired circular area. A P found that the larger the point dose, the larger the sidewall taper. The taper could be mitigated by using a more distributed dose for each hole, and tracing out a series of concentric circles of increasing radius to deliver this dose. (Writing circles rather than points was the same approach used by the NRC group to produce the InP samples studied in this work.) (2.1) Chapter 2. Fabrication 26 (a) (b) Figure 2.5: a) Nearly vertical sidewall profile of Zep resist and (b) etched silicon. The roughness is due to the Au/Pd coating applied to mitigate the effects of sample charging in the SEM. Images courtesy Andras Pattantyus-Abraham. Finally, a more chlorine-resistant resist than P M M A was found to be the ZEP-520A resist (Zeon chemicals). Its etch rate (~ 4 nm/s) is still greater than the etch rate (~ 3 nm/s) of silicon, however, making the two changes in beam conditions to improve the verticality of the sidewalls all the more critical. The combined effect of these changes produces nearly vertical sidewall profiles (about 85°) in both the resist and the etched silicon sample, as shown in Figure 2.5. S a m p l e p r e p a r a t i o n The first step is to cleave a ~ 1 cm 2 piece from a wafer and remove any organic residues using the RCA clean procedure. A 4:1 mixture of H 2 0 : N H 4 O H (40%) is heated to 70°C, and then 1 part 30% H 2 0 2 is added. The sample is immersed in the bubbling liquid for 15 minutes, rinsed in de-ionized (DI) water, and then blown dry with nitrogen gas. For GaAs, a 230 nm layer of the electron beam resist P M M A (4% by weight of 950K poly(methyl methacrylate) dissolved in chlorobenzene) is deposited onto the surface of the sample by spin-coating at 8000 rpm for 40 seconds. For silicon samples, the Zep 520A resist is deposited onto the surface by spin-coating at 2000 rpm for 60 seconds to give a 480-500 nm thick layer. Chapter 2. Fabrication 27 The coated sample is baked on a hotplate at 180° C for 2 minutes. The baking removes strain from the polymer layer by bringing it above its glass transition tem-perature, and also promotes solvent evaporation. The sample is then cleaved into 2 pieces, each measuring 5 x 10 mm, the optimal size for the pattern writing process. The edge of a graphite pencil is lightly brushed along the two long edges of the sam-ple to provide focussing marks which aid the differential edge-height determination outlined previously. E-beam writing The GaAs photonic crystals are patterned in the SEM with a point dose of 30-50 fC. Depending on the measured beam current (5-8 pA), the NPGS program automatically adjusts the beam dwell time to deliver this dose to a given point (i.e. air hole site in the photonic crystal lattice) before scanning to the next point. For silicon, the circular dose is delivered to each hole comprising the pattern according to the schematic shown in Fig. 2.6. In the NPGS program, the line-width is typically fixed at 140 nm, and the line-spacing at 40 nm. To fill in the area spanned by the line-width, the beam traces n concentric circles with radius increasing in units of the line-spacing, where n is given by the line-width/line-spacing. The radius of the circle traced by the beam is bracketed using three settings: 85 nm, 90 nm, and 95 nm. The radius of each hole in the developed resist is determined by both the circle radius and by the line dose. These two parameters are typically bracketed in each 3 x 3 array by varying the radius by column and the dose by row. An SEM image of a typical array of photonic crystals is shown in Fig. 2.7. The location of the beam dump, positioned at the top right corner of the group of patterns, is often useful as a guide to the orientation of a sample, which can become confusing if the sample is rotated and the images are inverted in the optical set-up; leaving a corner site of the photonic crystal array unpatterned can also serve the same purpose. Chapter 2. Fabrication 28 Figure 2.6: Schematic of the circular dose pattern for a hole of radius, r, and line-width, lw. 30 \xm Figure 2.7: Typical 3 x 3 array of microcavities designed to bracket an ideal structure. Each bright patch consists of a photonic crystal pattern like that shown in Fig. 2.8, with the defect region at its centre. The beam dump is visible in the top right corner. Development The main chemical agent used to develop the patterned e-beam resist is methyl isobutyl ketone (MIBK) for P M M A , and o-xylene for Zep 520A. The P M M A de-velopment procedure used by the author for GaAs samples is outlined in Table 2.2. More recently, the simpler and equally effective process outlined in Table 2.3 has been used.3 The patterned silicon samples are developed according to Table 2.4. During each development step, the sample is held with tweezers and dipped in the solvent beaker, which is gently swirled to ensure the solvent remains well mixed. After the 3 N o t e t h a t t he e - b e a m dose m a y need t o be a d j u s t e d i f t h e d e v e l o p m e n t p rocess is c h a n g e d , because the s ize o f t h e d e v e l o p e d holes d e p e n d s o n b o t h t h e r e c i p e a n d t h e d e v e l o p m e n t t i m e . Chapter 2. Fabrication 29 Table 2.2: Original GaAs development recipe. (MIBK = methyl isobutyl ketone, IPA " = isopropyl alcohol, DI = de-ionized, E E M = ethoxy-ethanokmethanol (3:7).) Chemical Time (s) MIBK/ IPA (3:1) 90 IPA 30 DI water 30 E E M 15 Methanol 30 Table 2.3: Alternative GaAs development recipe. Chemical Time (s) MIBK/ IPA (1:3) 60 IPA 30 Table 2.4: Silicon development recipe. Chemical Time (s) o-xylene 60 IPA 30 final rinse, the evaporating solvent film is blown from the surface of a sample with clean, dry nitrogen gas to avoid leaving residues. 2.3 Etching The samples are etched in a Plasmaquest electron cyclotron resonance (ECR) plasma etcher. A microwave field generates a plasma in the ECR chamber, and a radio-frequency (RF) field biases the sample and drives the ions towards the substrate. The exposed regions of the sample are then etched away by a combination of chem-ical and physical effects. A small amount of vacuum grease is used to ensure good thermal contact between the sample and the sample chuck, and the samples are left to equilibrate with the cooled chuck for about 20 minutes prior to the etch. The vacuum grease should form a layer thin enough so that it doesn't squeeze out when the sample is pressed onto it, potentially contaminating the resist layer of the sample. Chapter 2. Fabrication 30 Table 2.5: P E C V D pre-etch "descum" step designed to remove residual resist and improve vertically of side-walls in the developed resist. . Parameter Setting O2 (seem) 30 Pressure (mTorr) 200 RF power (W) 200 Step time (s) 3.5 Before the E C R etch, a preparatory "descum" step is performed with an oxygen plasma in the Trion plasma-enhanced chemical vapour deposition (PECVD) system, as summarized in Table 2.5. This helps remove any residual resist from exposed areas to provide a clean etch. Also, the high RF power creates a large anisotropy in the descum etch process which may help create more vertical side-walls in the developed resist. Over-etching into the sacrificial layer is avoided, because erosion of the resist beyond what is necessary to etch through the sample membrane adversely affects the side-wall slopes, as discussed previously. Therefore, it is important to have an accurate knowledge of the etch rate. As a result of the day-to-day variability of etch rates, due possibly to different etch processes being carried out in the same etch chamber, the etch rate must be characterized by etching a test sample. A thin scratch is made on the resist coating of the test sample with plastic tweezers before and after the etch. Using a Tencor profilometer to measure the depths of the scratches (before and after the etch), the etch rates of both the resist and the sample can be easily determined. The etch atmosphere for GaAs is comprised of a combination of Ar, CI2 and BCI3 gases. For Si, only CI2 is used. The pressures, flow-rates, and other etch parameters for both materials are detailed in Tables 2.6 and 2.7. Chapter 2. Fabrication 31 Table 2.6: Etch recipe for GaAs samples. In step 1, the microwave and RF powers are off, so there is no plasma. This step creates a uniform atmosphere in preparation for the etching step 2. The etch time in step 2 must be determined from a test sample. Parameter Step 1 Step 2 Ar (seem) 20.5 20.5 C l 2 (seem) 1.5 1.5 BC1 3 (seem) 2.1 2.1 Process Pressure (mtorr) 10 10 Microwave (W) 0 100 Backside He (torr) 5 5 Step processing time (s) 45 test (~ 3 nm/s) DC bias (V) 0 -100 RF power (W) 0 ~ 27 Chiller temperature (°C) 10 10 Table 2.7: Etch recipe for Si samples. Step 1 consists of a backside He flow to promote temperature stability. Step 2 stabilizes the etching gas atmosphere before the action happens in Step 3. The etch time in step 3 must be determined from a test sample. Parameter Step 1 Step 2 Step 3 CI2 (seem) 0 30 30 Process Pressure (mtorr) 0 5 5 Microwave (W) 0 0 200 Backside He (torr) 5 5 5 Step processing time (s) 120 30 test (~ 3 nm/s) DC bias (V) 0 0 -210 RF power (W) 0 0 ~ 27 Chiller temperature (°C) 0 0 0 2.3.1 Resist-removal After etching, the resist is removed from the surface of the sample with a chemical solvent. P M M A can be stripped in acetone that is sonicated (agitated) for 5 min. in an ultrasonic bath. The Zep 520A resist requires a 10 min. exposure to U V light before a 60 s sonication in acetone. After removing the resist, the sample is sprayed with methanol (or hexane) and blown dry with N 2 . Chapter 2. Fabrication 32 2.4 Undercutting A buffered oxide etch (BOE) is used to undercut the sacrificial layers from beneath the etched photonic crystal membranes. BOE solutions are typical in silicon micro-fabrication processes to remove sacrificial Si02 layers. They are mixtures of N H 4 F and HF in which the addition of the buffer N H 4 F to the HF acid prevents depletion of the F~ ions and produces quicker etch rates than stand-alone HF. In the GaAs process, the wafers have sacrificial layers composed of nearly pure AlAs, which is highly reactive, and so a very dilute BOE solution of 1% N H 4 F : 0.025% HF was used [61]. Samples were etched in 3 steps of a 60 s BOE dip followed by a 30 s DI water rinse. In the silicon samples, the sacrificial Si02 layer is not as volatile as AlAs in the BOE acid, and so a stronger commercial solution of 10:1 BOE (10 parts 40% N H 4 F to 1 part 49% HF) is used. The sample is left in the B O E bath for 20-25 minutes, which yields the ideal scenario of a full undercut of each photonic crystal defect pattern, without causing nearest-neighbours to join and risking collapse of the membrane. An optical image of an ideally undercut silicon structure is shown in Figure 2.8. The lighter regions surrounding the photonic crystal pattern show the full extent of the removed Si02 layer. An image of the side profile of a cleaved structure is shown in Fig. 2.9, clearly showing the free-standing membrane. Some remaining Si02 is visible above the silicon substrate. A close-up image of a typical three-missing-hole silicon microcavity fabricated with these techniques is shown in Fig. 2.10. 2.5 Summary In this chapter, the electron-beam lithography process for making GaAs and Si pho-tonic crystal microcavities was described in detail. The factors important to success-ful e-beam writing were described, including how to maintain a focussed beam across Chapter 2. Fabrication 33 S4700 10.OkV 11 9mm x3.00k SE(U) 7/22/05 lOOum Figure 2.8: Free-standing silicon photonic crystal microcavity. The undercut region is clearly visible. the sample by compensating for the change in the sample height when the sample is translated. Two different beam writing protocols were discussed, namely point dose and circular dose, and it was shown how the latter method, in conjunction with a different resist (ZEP-520A) and a more energetic electron beam (30 kV), improved the side-wall profiles in the silicon process. The development and etching recipes for GaAs and Si were tabulated and explained, and a number of S E M images of sample cross-sections and top views were presented. Chapter 2. Fabrication 34 Figure 2.9: Profile of a free-standing silicon photonic crystal. Image courtesy Andras Pattantyus-Abraham. 4fc Jfc Jfc A A Jfc A £ A A A gfe J f e j Figure 2.10: A three-missing-hole silicon photonic crystal microcavity. Chapter 3 35 Modelling 3.1 Introduction The underlying physics of photonic crystals is not new. It is essentially completely described by Maxwell's 1865 formulation of the laws of electrodynamics. However, simply because the underlying physical laws have been known for literally an age, this does not dictate that new emergent phenomena will not be discovered as a result of a higher conceptualization and understanding of these laws. After Maxwell, it took 122 years for a serious consideration of how electromagnetic radiation propagates in periodically textured dielectrics, which led to a formulation of the concept of a photonic bandgap [62, 63]. What is new, and acts as a powerful driver for research in photonic crystals, is the combination of a powerful numerical algorithm called the finite-difference time-domain (FDTD) method [64] to model Maxwell's equations on almost arbitrary di-electric landscapes, and the advent of vast computational power as an "experimental" research tool. Together, these factors have been highly successful in modelling the electromagnetic properties of photonic crystal slabs, waveguides, and microcavities, and virtual experiments are usually performed on a computer before real experiments are done in the lab. FDTD is particularly well-suited for analyzing non-periodic struc-tures, such as defects in photonic crystal lattices, which cannot easily be modelled by analytic techniques. This chapter will describe some important FDTD techniques relevant to the work in this thesis. The use of symmetry to extract pure mode profiles is first described. Chapter 3. Modelling 36 This is important to analyze the experimental spectra from the three-missing-hole cavities that are presented in Chapter 5. The methods for calculating the Q and the mode volume of a given mode will then be discussed. Although the FDTD code is always run using purely linear material properties, a process to construct the second-order polarizations manually from the linear field information will be outlined. Finally, the method of far-field projection of a field distribution will be described. The far-fields are the experimental observables, and so this step provides a crucial link between simulation and experiment. 3.2 F D T D simulation overview Most of the microcavities investigated in this thesis are based on three-missing-hole (L3) defects in hexagonal PPCs, such as shown in Fig. 3.1. In [12], Akahane et al. show how a lateral shift of the air holes at both sides of the cavity, as indicated by the arrows in the figure, can lead to a substantial increase in the Q of the fundamental cavity mode. In their particular structure, the authors find the optimal side-hole shift to be 15% of the lattice pitch in order to maximize the mode Q factor at ~ 45,000. Although microcavity designs with much higher Q factors have since been published [16, 18], the L3-cavity has the advantage of being relatively easy to fabricate, while still yielding a high-Q mode with volume less than (A/n) 3 . \ m m mmmi 500 nm Figure 3.1: SEM image of an InP L3-cavity. The two air holes on either side of the microcavity are shifted outwards to enhance the Q. Chapter 3. Modelling 37 Figure 3.2: Top and isometric views of the real-space rendering of a dielectric structure simulated with the Lumerical Solutions FDTD code. The point time monitors are labelled 1 and 2. The (x',y') axes denote the frame of the photonic crystal, and are used to describe the polarization of the localized modes. The simulations for this work employ the FDTD code from Lumerical Solutions, Inc., an offshoot of a PPC start-up company (Galian Photonics) which was founded during the late 1990s boom to commercialize photonic crystal technology for optical signal processing applications. An image of the FDTD simulation environment is shown in Figure 3.2, with the photonic crystal axes {x',y') indicated. Clearly visible are the simulation boundaries (orange), the free-standing slab perforated by a lattice of air-holes, a Gaussian beam source, point time monitors (1 and 2), and the 2D field and index monitors (yellow square) [65]. The simulation is initialized by the injection of a pulse of electromagnetic (EM) fields from a given source, such as an electric dipole or a Gaussian beam. The FDTD method works by setting up a computational mesh and discretizing the dielectric structure at the mesh points (finite-difference), and then simulating the propagation of E M fields across the mesh in a series of time steps (time-domain). The results are captured at user-defined monitors which can output time- or frequency-domain data. The information recorded at the time monitors is the essential resource of the FDTD method. Frequency-domain information is always obtained by Fourier trans-forming the results of simulations done in the time domain. As the simulation pro-ceeds, the six fields (three electric and three magnetic) are calculated at every spatial Chapter 3. Modelling 38 point for every time step, but only specified sets of fields (at defined points, planes, or volumes) are monitored (i.e. saved) for analysis. Saving all fields at all times is unnecessary, and would produce an intractable volume of data. Although the code works in the time domain, frequency information is easily and powerfully obtained. To calculate frequency-domain data from the saved time fields at a single point, a simple fast Fourier transform (FFT) calculation can be performed after the simulation has finished (i.e. frequency data are post-selected). To generate the frequency-domain electric field profiles E(LO) in a larger domain, such as a plane, without saving all the time domain data for that domain, a discrete Fourier transform (DFT) monitor can be used to preselect the desired frequency information. The monitor works by calculating a running F F T of the fields as the simulation proceeds, which vastly reduces the memory storage requirements of a post-selected F F T calculation. Typically, when investigating a new cavity design, exploratory simulations are first run with computationally inexpensive time monitors distributed around the volume of interest, and short-pulse sources, which are positioned so as to avoid high symmetry points, are used to excite a broad spectral region. From the F F T spectra of the time-domain data, such as shown in Fig. 3.3 for the L3-cavity, modes of interest can be identified, and more targetted simulations can then be run with DFT monitors placed to capture the frequency-domain fields from a spatial region of interest. 3.3 Extracting mode profiles with symmetry Fig. 3.3 shows the \EX>\2 intensity spectrum obtained by Fourier transforming the time data from monitor 1 in Fig. 3.2. The sharp feature at 6400 c m - 1 is the fundamental high-Q mode of the microcavity. It is well-isolated spectrally, and so the planar field profile can be easily calculated by setting a planar DFT monitor in the middle of the slab to monitor the frequency 6400 c m - 1 . When two modes are degenerate or Chapter 3. Modelling 39 3 - J O -1 ' w / -C D 4—" c y l ) 6400 6600 6800 7000 7200 7400 Wavenumber (cm-1) Figure 3.3: Spectrum of \EX'\2 from time monitor 1 in Figure 3.2. The simulation was excited with two point dipole sources that inject Gaussian pulses with a centre frequency of 6500 c m - 1 and a full-width half-maximum of 1500 c m - 1 . spectrally overlapping, it can be difficult to isolate their pure spatial profiles. Between 6900 and 7100 c m - 1 , there are three different x'-polarized overlapping modes, two of which are nearly degenerate, and so it is not possible to extract pure mode profiles from this simulation because the modes are separated by less than their line-widths. Knowledge of the symmetry of the microcavity structure can be used to filter out modes of interest, and also to greatly reduce the running time of simulations. A powerful result when group theory is applied to Maxwell's equations reveals that the point group symmetry of the dielectric constant e(f) (i.e. the photonic crystal lattice) at a given point determines the symmetry of the photonic crystal modes [66]. The defect modes of interest in these hexagonal PPCs are TE-like, and so they have symmetric electric field distributions about the middle (z = 0) plane of the slab [67]. A symmetric boundary condition (BC) can be introduced at z = 0 to take advantage of this symmetry and halve the simulation volume, shortening the simulation time by a factor of 2. It is important to note, however, that although this technique is useful Chapter 3. Modelling 40 for mapping the TE-like mode spectra of a microcavity, it is not generally applicable when modelling experimental scattering spectra, for which the full dielectric structure should be simulated. To isolate a particular mode of interest, symmetry planes can be introduced at x' = 0 and/or y' = 0. In combination with a z-symmetry plane, these mirror planes reduce the simulation area to 1/4 or 1/8 of the full volume. Setting the reflection symmetry of the x'/y' mirror planes to be either symmetric or anti-symmetric filters out modes with the corresponding symmetry. The results of a simulation with symmetric plane BCs at z = 0 and x' = 0 and an anti-symmetric BC at y' = 0 are shown in Figure 3.4. The figure shows the spectrum of the fundamental mode of the L3-microcavity, and its associated field intensity distribution and vector plot. The orientation of the vector fields reveals that the mode is y'-polarized, as per the specified BCs. This type of mode, which has one anti-symmetric boundary and one symmetric boundary, is referred to in this thesis as a dipole-like mode because of its similarity to the 2D electric field pattern of an electric dipole. Experimentally, dipole-like modes are highly relevant because they can be excited by polarized free-space excitation fields, as discussed in Chapter 4. When all three symmetry planes are set to symmetric BCs, only a single mode is excited, as shown in Fig. 3.5. This mode is extracted from the x'-polarized triplet near 7000 c m - 1 . The other two modes in this triplet are distinguished by being anti-symmetric about the y'-axis, and so they are filtered out. Note that, in contrast to the dipole-like mode above, there are equal parallel and anti-parallel vector field lines along any line drawn through the origin. This fact is important experimen-tally, because it prevents modes of this symmetry from being excited in the resonant scattering geometry described in Chapter 4. When symmetry planes are used, care must be taken to ensure that the micro-cavity is positioned properly with respect to the symmetry planes so that meshing errors do not lead to spurious results. The simulation boundaries can shift by one Chapter 3. Modelling 41 =3 CO 6400 6600 6800 7000 7200 7400 Wavenumber (cm"1) Figure 3.4: Fundamental high Q mode of the L3-cavity excited by using symmetry and a narrow-band source. Plots of the field intensity \Ex'(u>)\2 + \Ey/(u>)\2 and vector distributions are shown. The (x', y') axes denote the photonic axes, and (x, y) the InP crystallographic axes, the orientation of which is important for the second-order nonlinear results of Chapter 6. The symmetric (S) and anti-symmetric (AS) mirror planes are shown. Chapter 3. Modelling 42 6400 6600 6800 7000 7200 7400 Wavenumber (cm 1) —: T>—"——Z——; z,—r • -- - ^ - ' - x ^ - ! - - - • - Y ; - • : i • • / i • • - » > - . — • • - » ' -i • — ! • 1 1 - i. e. , / / • : S ' ' ' ^ ** z ' ' I t : Figure 3.5: Low <3 , x'-polarized mode of the L3-cavity which is symmetric (S) about both x* and y' axes. Since it is the only mode with this symmetry, the simulation does not require a spectrally narrow excitation pulse. Plots of the field intensity \Ex'(ui)\2 + |£y(a;) | 2 and vector distributions are shown. The axes are described in Fig. 3.4. Chapter 3. Modelling 43 mesh point when the mesh is calculated, leading to significant errors [68].4 Although symmetric BCs can greatly aid efficiency if a number of similar simulations are be-ing investigated, the full structure should always be run at least once to verify the symmetry conditions are working as expected. 3.4 Mind your V's and Q's There are several different ways to calculate the Q of a mode. The Q is defined as [69] Stored energy _ U Q ~ U J 0 Power loss ~ ~^WfdV ^ but a calculation of the energy and power loss in a cavity requires lengthy, memory-intensive simulations to capture the field profiles from the full mode volume. This is unnecessary in all but the highest Q cavities, for which the other methods break down. The most common experimental method to measure Q is to take a ratio of the mode frequency to line-width, which follows directly from the solution (Lorentzian function) to the equation of motion of a damped oscillator. However, the simulation spectra are usually resolution limited, as evident in the above figures, and it is impractical to run simulations for the full lifetime of the mode (i.e. until the mode has completely decayed) in order to maximize the spectral resolution. The method used in this work is to extract the Q from the time series plot of the electric field. For this calculation to be accurate, there should only be a single mode excited in the cavity so that there are no interference beats. It is sometimes the case that there are multiple modes in a microcavity which have the same symmetry, and so symmetric filtering cannot be used to ensure the single mode condition. If the modes are separated by more than their line-widths, then a narrow line source (i.e. a relatively long excitation pulse duration of ~ 400 ps) can be used to isolate a particular mode. ^Saving the simulation always updates the mesh and can be used as a check. Chapter 3. Modelling 44 The cavity energy, U(t), can be derived from eq.( 3.1) to give U(t) = U0e-Uot/Q, (3.2) which implies that the mode fields oscillate according to E(t) = EQe-wotl2Qe-iwot. (3.3) Therefore, by plotting 'log e E'1 vs '£' from a time monitor in the FDTD code and calculating the slope, the Q can easily be found. Practically, this can be quickly done through the Analysis window of the FDTD program, but there the plot is of ' log 1 0 E"1 vs Letting s denote the slope of this plot, and / the mode frequency, the Q is given by Q = - ( log 1 0 e)irf/s = - 0 .434TT/ / 5 . (3.4) To give a concrete example, a plot of Ey> vs. t for a monitor positioned in the centre of the cavity is shown in Fig. 3.6. The mode was excited with a narrow-band pulse 'of length 400 fs which was offset so as to peak at 1000 fs. The cavity field peaks at 1500 fs and then decays with exponential dependence. In this particular sample, the mode Q is calculated from eq. 3.4 to be 3,600, which is remarkably close to the experimental value of 3,800 (see Figure 4.7). In addition to the Q factor, the mode volume, V, is also of great interest. The ratio Q/V is the figure of merit which describes the Purcell enhancement of spontaneous emission (Section 4.3), and other nonlinear and quantum optical cavity-based schemes are also based on high Q's and small V s . To calculate the mode volume accurately requires 3D field information, which is beyond the planar monitors which have been discussed thus far. An order-of-magnitude approximation can be made by assuming the field is constant in z within the slab, and extracting V from the planar field data. However, the mode intensity is ~ 2x weaker at the surfaces compared to the middle Chapter 3. Modelling 45 1 2 Time (ps) (a) 2 3 Time (ps) Figure 3.6: (a) Plot of lEy>(ty vs. T for the fundamental mode of an L3-cavity. (b) Plot of 'log10 EyS vs. '£' for times greater than lps. The linear slope of the decay gives s in eq. (3.4). of the slab, and so this approximation will overestimate V by a factor of a few times. The mode volume is denned as [70] VPmax = J P(r)d3r, (3.5) where P(r) is the total electromagnetic energy density of the mode, and Pmax is its peak value. This equation can be written in terms of the electrical field energy density m o d e max[€(r)\E(r)\*}' ' To calculate V from FDTD, 3D field and index monitors are used to capture the full mode volume. The integral can be calculated by direct summation of the field energy at every lattice point, or by using the slightly more sophisticated trapezoid rule. However, with a mesh size of 20 points/lattice, the cruder direct sum gives the same answer to within 0.5%. For the fundamental mode (Fig. 3.4), V is calculated to be 8.7 x l 0 ~ 2 0 m 3 = 0.73(A/n)3. (A simulation of the slightly different structure reported in [12] gives V = 7 x 1 0 - 2 0 m 3 , in exact agreement with the value quoted Chapter 3. Modelling 46 by the paper's authors.) This small value, less than a cubic wavelength in material, is one of the intrinsic advantages of photonic crystal microcavities over macroscopic Fabry Perot cavities, and leads to extremely large local field enhancements. 3.5 Second-order polarizations The FDTD code simulates a purely linear susceptibility, but knowledge of the second-order nonlinear susceptibility tensor allows the second-order polarization distribu-tions to be calculated from the linear fields. In Chapter 6, cavity-enhanced second-harmonic generation is observed from the fundamental mode of the L3-cavity, and the data are completely consistent with the bulk second-order nonlinear suscepti-bility of the InP. There are only three non-zero elements in this tensor, given by Xijlii J k)- The analysis here describes how to simulate this experiment with FDTD. In order to calculate the bulk x^ response of the InP host material when only a single mode is present in the microcavity, the orientation of the InP crystallographic axes with respect to the mode polarization must be known. The localized modes of interest are all TE-like, and hence are dominated by in-plane field components (Ez ~ 0). Therefore, the bulk second-order response is determined by the tensor (2) component which couples Ex and Ey fields, namely Xzxy- For the fundamental mode of the L3-cavity, the photonic and electronic crystal reference frames are shown in Figure 3.4. At each of the seven principal anti-nodes of the mode intensity, the field vectors have equal components along x and y, and therefore the mode drives considerable second-order polarizations in the ^-direction via x^x\- This analysis is valid in the low-pump-depletion limit, which is appropriate for all the work in this thesis. The second-order nonlinear polarization P^(t) can be expressed in terms of its Chapter 3. Modelling 47 Fourier components as /oo dcuPW(x,y;uj)e-^. (3.7) •oo From quadratic nonlinear response theory, the Fourier components of the second-order polarization are given by [71] /OO roo dwx j du2X('2)(x,y;u1,u;2) : oo J—oo E(x,y;ui)E(x,y]U!2)5(u - ux - u>2), (3.8) where the E(x,y;u>j) represent Fourier components of the field at frequency u>j, and the colon denotes a tensor product. Substituting eq. (3.8) into (3.7) gives /oo poo deux / dto2X{2){x,y\Ui,uj2) : •oo J —oo l ( 3 ; ,y ;a ; 1 )7 l ( a ; ,y ;a ; 2 )e -^ 1 + ^ t . (3.9) The physical field in the cavity is given by the superposition of the mode fields, E(x,y;t), each with discrete centre (carrier) frequency cD :^5 E(x,y;t) = ^ ^ ( x ^ c o s ^ t ) , (3.10) where E^ (x, y; t) are (real) envelope functions of the modes, which are slowly-varying on the time scale of 1 / W The Fourier components of the field are then simply r+OO E(x,y;uj)= / drE(x, y-r)e^. (3.11) J —oo 5 The notation Q denotes a discretely-valued u>. Chapter 3. Modelling 48 0.5 -0.5 -0.5 0.5 Figure 3.7: Simulated second-harmonic polarization distribution for the fundamental mode of the L3-cavity, given by Re{Pz2\x,y,2uJi)} ~ XzxyRe{Ex(x,y\LUi)Ey(x,y-,LUi)}. The intensity scale has arbitrary units. These components are the driving terms for the second-order radiation from the mi-crocavity. The key point is that these Fourier components can be extracted from frequency-domain field monitors in FDTD, and so the second-order nonlinear polar-ization of eq. (3.9) can be determined. Because the microcavity modes are not monochromatic, the second-order polar-ization has a finite line-width. When it is evaluated at its peak, corresponding to frequency u = &i + £o2, eq. (3.8) becomes P(2){x,y;uj1+uj2) oc x ( 2 ) ( x , y ; L O U L O 2 ) : E(x,y;UJI)E(X,y;LL>2) (3.12) In the case of second-harmonic generation, for which B\ = u>2: and with the bulk InP tensor substituted, this equation becomes P f O r . y ^ ) oc X%&(x,V;vi)E9(x,yiui), (3-13) where the arguments of the tensor have been suppressed for clarity. For the funda-mental mode shown in Fig. 3.4, this complex-valued polarization amplitude is easily calculated, and the real part is plotted in Figure 3.7. In the slightly more complicated case of sum-frequency generation of two modes, Chapter 3. Modelling 49 the peak value of the second-order polarization is given by P^{x,y;co1+Cu2) C*- Xzxy yEx(x,y;ul)Ey(x,y\u2) + Ey(x,y;u1)Es(x,y,(j2)^ , (3.14) using the fact that x^x\ = X^yx-3.6 Far-field projections The comparison between experiment and simulation must be made in the far-field, where the electric fields are measured experimentally. Ideally, the Pz2\x,y; 2u>) dis-tribution in Fig. 3.7 would be inserted as the driving polarization in the FDTD code. However, the FDTD tools don't allow arbitrary polarization distributions as source terms. Instead, the real polarization pattern is approximated by a sum of discrete dipoles driven at the peak of the harmonic radiation spectrum, u)\ + u>2, with am-plitudes and phases (+ or —) scaled to match Fig. 3.7. The near-fields generated by this second-order polarization distribution are monitored just above the slab, and the built-in near-field to far-field transformation is used to generate the far-field har-monic radiation pattern. The far-field distributions are determined at the peak value of Pz2\x,y; 2a;), and they change negligibly over its line-width. In the case of the experiments considered in Chapter 6, the second-order frequen-cies are above the electronic bandgap of the InP, and so the radiation is efficiently absorbed. The multipolar second-order ^-polarization distribution radiates strongly in the plane of the slab, and this radiation is scattered out of the plane into the far-field by the air holes surrounding the microcavity. The nature of this scattering strongly depends on the material absorption, as shown by Figure 3.8. Rows (2) and (3) in the figure show the results from simulations with absorption (n — 3.47, k = 0.22) and without absorption (n = 3.47, k = 0), respectively [72]. The images in column (a) Chapter 3. Modelling 50 Figure 3.8: FDTD simulation results when the second-order polarization distribution in Fig. 3.7 is modelled by a sum of point dipoles radiating at the second-harmonic frequency of mode A from the microcavity shown in (la). Row 2 (3) shows the results without (with) material absorption included. Column (a) shows the near-field plots of \EX>\2 + l-Ey'l2, and columns (b) and (c) show the far-field patterns of \Exi\2 and \Ey>\2i respectively. Chapter 3. Modelling 51 show the near-field distributions of \EX>\2 + \Eyi\2 monitored at a plane 65 nm above the slab. The symmetry of both distributions is similar, but with absorption, the fields are limited to a circle of radius ~ 300 nm from the centre, and have about half the peak intensity. The far-field intensity distributions of \EX>\2 and | £ y | 2 are shown in columns (b) and (c), respectively, and reveal the dramatic effect of absorption on the intensity, the spatial pattern, and the relative polarizations of the fields. Without absorption, the far-fields are not clearly polarized, and the majority of the light is radiated at high numerical apertures. With absorption, the overall intensity is about 27 times weaker, but the light is strongly polarized in the y'-direction, and there are large intensity lobes within a 40° cone of the vertical direction. These features have important ramifications for the experimental results presented in Chapter 6, and reveal the im-portance of both the real and imaginary parts of the refractive index for determining the properties of the far-field radiation. 3.7 Summary Finite-difference time-domain (FDTD) codes are essential companions to wavelength-scale photonics research. In the design and characterization of photonic crystal mi-crocavities, they enable accurate determinations of mode energies, quality factors, volumes, and profiles. For both integrated microphotonics and cavity QED motiva-tions, all of these are essential figures of merit which determine the spectral, spatial, polarization, and local field enhancement properties of a cavity. In this chapter, a selection of modelling topics relevant to this thesis work were described, namely symmetry methods to extract pure mode profiles and reduce run-times; the physics and numerics of calculating Q and V; and calculation of the second-order material polarization from the linear mode data in order to determine the far-field radiation pattern. The second-order polarization is calculated at its peak value Chapter 3. Modelling 52 by multiplying the requisite Fourier component field distributions at their line centres. The resulting distribution is modelled by a series of point dipole sources radiating at the harmonic frequency, and a near-field to far-field transformation is applied to generate the far-field distributions for comparison with the second-order nonlinear experimental results. 53 Chapter 4 Experiment 4.1 Resonant scattering Resonant scattering is a relatively simple optical technique that is adopted in this work to couple free-space optical radiation into the three-dimensionally confined fields of planar photonic crystal microcavities. It forms the basis of all the optical set-ups used in the linear, second-order, and pump-probe experiments described in this thesis. After motivating the resonant scattering technique from a physical basis and com-paring it to other approaches in the literature, the optical apparatus and alignment methods are discussed. The variations on the free-space resonant scattering set-up implemented to acquire photoluminescence, second-order, and pump-probe data are then described in turn. The basic optical set-up is shown in Figure 4.1. The incident laser beam, with a polarization controlled by a half-waveplate and linear polarizer combination, is tightly focussed with a 100 x microscope objective lens to excite the microcavity from a direction normal to the plane of the sample. At 1550 nm, the spot-size of the focussed laser beam is about 2 pm in diameter (for comparison, localized mode areas are typically about 500 nm across). The scattered radiation is collected in reflection through the same lens, analyzed in the cross-polarization, and deflected towards the spectrometer by a beamsplitter. Transmission experiments can be performed by using a different microscope objective lens to collect the transmitted light. For this technique to be a useful probe of a microcavity mode, two conditions must be met. First, the mode must be excited by a linearly polarized Gaussian beam Chapter 4. Experiment 54 imaging lens Laser Pol. plate Pol. camera FTIR 0 100x sample Figure 4.1: Free-space resonant scattering set-up. The resonant probe laser is strongly focussed onto the microcavity with a 100 x microscope objective. The reflected scat-tered radiation is analyzed in the cross-polarization, and measured with a Bomem Fourier transform infrared (FTIR) spectrometer and InGaAs detector. incident from the top half-space. Second, the detected radiation, which propagates backwards in the direction of the Gaussian excitation beam as the mode rings down, must have a non-zero component polarized perpendicular to the exciting field. To address the in-coupling, if the non-zero extent of the membrane thickness is ignored, the material polarization associated with any microcavity mode can be concentrated in a plane located at the centre of the membrane. This material po-larization will in general have a complicated, but symmetric, circulating vector field pattern, like the field vector plots shown in Section 3.3. In the paraxial limit, the driving Gaussian field is uniformly polarized (in the plane) over the mode area, and will therefore drive the electrons in the slab in the direction of the field polarization. If the mode's material polarization pattern (P) experiences a net driving force due to the incident field, some energy will be left in that mode after the short laser ex-citation pulse passes through the membrane. The paraxial assumption here reflects the dominant role of the transverse fields in the linear scattering experiments, but Chapter 4. Experiment 55 ignores the fact that the strongly-focussed Gaussian beam gives rise to longitudinal field components (which are an important element in the nonlinear experiments of Chapter 6). To lowest order, the coupling of the incident field, Incident, to the mode is pro-portional to the area integral of P • incident- Figure 4.2 shows the vector material polarization of three different types of modes (a)-(c), and the transverse vector elec-tric field (d) of the incident Gaussian beam (with spot diameter 2 /mi). The L3-cavity modes in (a) and (b) were discussed previously. The whispering-gallery mode6 in (c) is anti-symmetric about both x- and y-axis mirror planes. When the incident field in Fig. 4.2(d) is dotted with each of the material polar-izations in (a)-(c), the distributions in Fig. 4.3 are generated. The integral of these distributions, f dA P • £ j n c i d e n t , is a measure of the coupling. From the symmetry evident in these plots, it should be clear that it is only non-zero for the dipole-like mode shown in (a), and then only if the exciting field is not polarized orthogonal to the anti-symmetric component of the mode (i.e. along the x'-axis). These simu-lations demonstrate that only dipole-like modes can be efficiently excited using the free-space resonant scattering technique. It should be noted that the non-dipole-like modes in (b) and (c) could, in principle, be weakly excited if the excitation symmetry were broken. This could be achieved by laterally displacing the actual excitation beam (with its Gaussian radial intensity distribution) to excite the mode off-centre. Once a dipole-like mode is excited, the material polarization associated with the mode radiates into the half-spaces above and below the membrane (this is the leakage process that determines the lifetime of the mode). For this radiation to be captured by the spectrometer, the fraction that is radiated within the solid angle of collection must have a non-zero component polarized perpendicular to the Gaussian excitation beam. This is because the polarization analyzer must be crossed with respect to the 6 This is a mode of a cavity with a single-enlarged-hole in a hexagonal lattice; it is not a mode of the L3-cavity. Chapter 4. Experiment 56 0.5 AS o -0.5 (b). -0.5 0.5 0.51 -0.5 (C) -AS--1 -0.5 AS 0.5 1 0.5 -0.5 , / / / / / / / S S s -0.5 0.5 Figure 4.2: Material polarization (P) vector plots for (a) a y'-polarized dipole-like mode of the L3-cavity; (b) an rr'-polarized mode of the L3-cavity; (c) a whispering-gallery-mode of a single-enlarged-hole cavity. The x' and y' axis mirror planes are marked as either symmetric (S) or anti-symmetric (AS), (d) In-plane electric field, i n c i d e n t ; of a focussed Gaussian beam with a spot diameter of 2 /im. is •1 -0.5 0 0.5 1 Figure 4.3: P • i n c i d e n t for each of the material polarizations shown in Fig. 4.2. The area integral of these distributions is a measure of the coupling, and is only non-zero for the dipole-like mode in (a). Chapter 4. Experiment 57 1 output polarizer 90° sample input polarizer Figure 4.4: Cross-polarized detection schematic for resonant scattering from a mi-crocavity mode polarized at 0°. The output polarizer is crossed with respect to the input polarizer to filter out the non-resonantly scattered light. The mode is optimally excited at a 45° angle to its polarization, although any incident polarization except parallel or orthogonal to the mode will give a non-zero resonant feature in the scat-tered spectrum. Note that the polarization of the mode cannot be determined using this method. input polarizer to filter out the scattered light from the laser excitation, which would otherwise swamp the resonant signal. One consequence of the cross-polarized detection is that the mode polarization cannot be determined from the scattered spectra - to wit, in Fig. 4.4, resonant features at 0° cannot be distinguished from those at 90°. Another consequence occurs in certain multi-mode experiments, for which the cross-polarized detection technique dictates the relative phase between the different spectral features, as described in Chapter 5. Chapter 4. Experiment 58 \Z> 100x V Fibre taper wg (a) (b) Figure 4.5: Schematics of a microcavity (pc) excited via (a) evanescent coupling with a fibre taper to an on-chip waveguide, and (b) free-space resonant scattering. Free-space versus evanescent coupling Most of the experimental literature on photonic crystal microcavities utilizes evanes-cent coupling from ID waveguide channels to access the resonant microcavity modes. These waveguides are directly incorporated onto the chip in the form of ridge waveg-uides or line defects in planar photonic crystals. Laser light is coupled into the on-chip waveguide either by butt-coupling a fibre to the end facet of the waveguide [31], or via a fibre taper brought close to the top surface [73]. The latter method, depicted in Fig. 4.5(a), is premised on evanescent coupling between two matched waveguide modes [73]. The two approaches to PPC microcavity excitation, evanescent coupling from on-chip waveguides and free-space resonant scattering, are compared schematically in Figure 4.5. In the first approach, fibre tapers offer very efficient transfer of radiation on and off the chip. The fact that the microcavity and waveguide are essentially OD and ID defects in the same photonic crystal allows lithographic control over the coupling parameters. This has the natural advantage of being fully integrable with other chip-based devices. It also has the advantage, if designed properly with single mode ID defect waveguides, of having a single channel continuum being the dominant leakage (Q-determining) channel that couples to the defect state. In other words, once radiation is coupled into the ID waveguide channel, excitation/probe radiation can Chapter 4.' Experiment 59 be very strongly coupled to the defect state. The total efficiency of the fibre-to-cavity coupling has been reported to be 44% for a microcavity that is end-coupled to an on-chip waveguide [74]. This high in-coupling efficiency cannot be matched by free-space resonant scat-tering, which has a calculated efficiency of power transfer from a C W laser to a microcavity of approximately 2% [75]. This is because the stand-alone microcavities studied here have no single continuum channel through which they couple to the out-side world. In terms of out-coupling, this also necessarily means that the collected radiation is not funnelled into a single ID output channel, but radiates into multiple channels. Despite the lower efficiency, there are several advantages to the free-space resonant scattering approach which facilitate the work presented in this thesis. In the first instance, it is a natural interface to inject short, broad bandwidth optical pulses into a microcavity. It is difficult to transport a transform-limited optical pulse along a ID waveguide, which is typically designed for discrete frequencies, without suffering nonlinear power loss and temporal and spatial distortion of the pulse [76]. One of the major themes of this thesis is coherent, multi-mode excitation of photonic crystal microcavities, and the most natural experimental approach for this regime is free-space resonant scattering. The free-space resonant scattering approach also facilitates the nonlinear experi-ments presented here. The complexities of second-order nonlinear spectroscopy would be considerably heightened if independent waveguide channels had to be designed for both fundamental and second-order frequencies. Instead, the second-order radiation is detected by collecting the out-of-plane scattered light with a high numerical aper-ture lens. With the necessity of a free-space collection path, it is convenient to also use free-space incident coupling. Finally, the pump-probe experiments discussed in Chapter 7 rely on control of the relative timing between two laser pulses to a precision better than 1 picosecond, which is readily achievable with free-space short-pulse laser Chapter 4. Experiment 60 beams, and facilitates the use of a pump beam that is above the bandgap of silicon. This ultrafast spectroscopy based on linear absorption would not be possible if all coupling were done via waveguides [45, 44]. 4.1.1 Alignment To precisely align the incident beam with respect to the microcavity, the sample is imaged in reflection to a CCD camera positioned as indicated in Fig. 4.1. The mag-nification of this imaging system is about 500 x, so a given 10 pm wide hexagonal photonic crystal fills about half the CCD camera chip. When the sample is posi-tioned at the focus of the 100x objective, the incident beam appears as a tightly focussed spot, and the hexagonal boundary of the patterned photonic crystal region is not visible, as there is nothing illuminating it. Wider illumination of the pattern without moving the sample away from the focal plane (and thus blurring the image) is accomplished by adding a second lens, with / ~ 15 cm, before the beamsplitter, as indicated in Fig. 4.1. This two lens system on the incident beam path has a dif-ferent focal length than the reflection beam path. When the sample is positioned at the focus of the reflection beam path (i.e. the focus of the lOOx), the incident beam spot is no longer focussed, and its larger spot size illuminates the full photonic crystal pattern. This can then be centred with respect to a point drawn on the CCD monitor. With the imaging lens removed, the 100x objective can then be adjusted to minimize the excitation beam spot-size at the same point on the camera monitor. This technique ensures the incident laser is focussed to the centre of the photonic crystal pattern. Ideally, to optimize coupling into the spectrometer, a collimated beam would be directed to the parabolic mirror focussed on the spectrometer input (not shown). When the beam is aligned in the fashion described, it is not actually collimated, because it is focussed at the camera imaging plane. However, the camera position is so far from the sample (50 cm) that the beam divergence is minimal, and this Chapter 4. Experiment 61 procedure is sufficient for optimizing the resonant excitation of the microcavity. Although the beamsplitter is nominally unpolarizing, the cross-polarized tech-nique was only successful for 0° or 90° polarizations of the incident beam, indicating that some depolarization may have been introduced at other incident polarizations. 4.1.2 Samples A l l of the optical data presented in this thesis are obtained from either InP or Si photonic crystals. The InP samples, fabricated by Robin Williams' group at the Institute for Microstructural Sciences at the National Research Council in Ottawa, are free-standing membranes mounted on glass [77], allowing transmission experiments to be performed. The membrane is composed of 223 nm of undoped InP separated by ~ lum from a glass substrate. In the middle of the InP membrane, there is a wetting layer of InAs that is 1.03 monolayers thick, which hosts the self-assembled InAs quantum dots [52]. The silicon photonic crystals, which were fabricated by the author, are etched in free-standing membranes separated from a thick, unpolished silicon substrate by 1.2 fim of Si02- As described in Chapter 2, the sacrificial Si02 layer is removed below the textured regions of the membrane. These samples are not suitable for transmission experiments because of diffuse scattering from the rough back surface, and so they are studied in a reflection geometry. For the pump-probe experiment, these samples are grease-mounted to a thermo-electric cooler to provide temperature stability. Two main sample holders are used. For reflection experiments, the samples are grease or clip-mounted to a 1 inch aluminum disk placed in a mirror mount holder, which enables a fine adjustment for the sample tilt and rotation. For the transmission geometry experiments on the glass-mounted InP samples, the edges of the ~ 1 cm 2 glass slides are taped to a 1 inch disc with a centrally hollow core through which optical radiation can pass. As sketched in Fig. 4.6, the disc is pinched in a lens holder with 5 mm of travel in the x/y directions, which is invaluable for the coarse Chapter 4. Experiment 62 nanopositioner Figure 4.6: Schematic of the transmission sample mount which is fixed to the nanopo-sitioner stage. The coarse x/y alignment screws have 5 mm of travel. The sample can be rotated to control its orientation with respect to the incident beam polarization. positioning of the sample. The sample holder is supported by a base fixed to a Melles Griot nanopositioner stage, which allows sub-micron control of the sample position. The nanopositioner is a 3-axis stage which provides ~ 100 nm spatial control of the samples, and greatly facilitates the precise alignment demands of the experiment. The mechanical stage adjustment knobs are usually controlled manually, giving 1 um resolution, and the piezo controllers are used to fine tune the alignment. The drawback of manual adjustment is that only the piezos have a digital display monitor. If needed, a computer interface can be used to fully control and monitor both the mechanical and piezo x/y/z coordinates. 4.1.3 Laser sources The second-harmonic C W data are obtained with an Agilent 81682A laser diode, tunable from 1460-1580nm, with a wavelength accuracy of O.Olnm and a resolution of 0.1 pm at 1550 nm. The laser has a maximum output power of 6.3 mW (8 dBm) which, after losses in the fibre, mirrors, beamsplitter, and 100 x microscope objective, corresponds to a power of 0.3mW focussed onto the microcavity. The C W resonant scattering spectra in Section 4.2 are acquired with a New Focus TLB-6600 tunable laser diode that is swept at 100 nm/s with a wavelength accuracy of 0.015 nm. Chapter 4. Experiment 63 The pulsed laser is a Spectra Physics Opal femtosecond synchronously pumped optical parametric oscillator, abbreviated as "OPO" in this thesis. It is tunable from 1420-1585 nm, with a maximum output power of 190 mW at 1550nm, and a signal pulse length of about 130 fs. It is pumped by a Spectra Physics Tsunami Ti:sapphire laser which outputs 100 fs pulses at 810 nm with an average power of 2W. Typical spectra are acquired at about 1 mW average power, although high signal-to-noise ratios are maintained at excitation powers below 100/iW. No nonlinear shifts or power dependencies have been observed over this power range. A C W HeNe laser (632.8 nm) with output power of 3 mW was used as the exci-tation source for the photoluminescence experiments. 4.2 Single-mode resonant scattering Resonant scattering spectra from an InP three-missing-hole microcavity with a single localized mode in the energy range of interest are shown in Figure 4.7. When the short-pulse OPO is used as the laser source, and the scattered light is focussed into the Bomem Fourier transform spectrometer, the spectrum in Fig. 4.7(a) is obtained. Physically, when the ~ 130 fs pulse impinges on the microcavity, it drives an "in-stantaneous" polarization in the material while it passes through. Most of this ~ 130 fs polarization pulse radiates directly out of the sample, giving rise to the broad spectrum in the figure. However, frequency components of the polarization which are resonant with the microcavity mode radiate fields within the structure that are trapped for many optical periods, only radiating away a small fraction of their total trapped energy per optical cycle. This trapped field, and the associated polarization in the material, give rise to the sharp feature in the spectrum. This process is re-peated at the laser pulse repetition rate of 80 MHz. The mode shown here has a Q of 3800, which corresponds to a lifetime r = Q/UJ = 3.1ps, much shorter than the 12.5 ns interval between pulses. Therefore, the mode has long decayed from the cavity Chapter 4. Experiment 64 before the next pulse arrives. In Fig. 4.7(b), the resonant scattering spectrum is acquired by sweeping the fre-quency of a tunable C W laser across the resonant region, and detecting the scattered light with a photodiode. Because the C W laser is monochromatic, no spectrometer is used. Although the Lorentzian mode feature is observed in both spectra, there is a profound difference in the physical mechanisms of the two experiments. In the pulsed scenario, a fraction of the initial pulse energy given by the ratio of the mode linewidth to the pulse linewidth is scattered into the microcavity. Because the timescale of the pulse is so much shorter than the lifetime of the microcavity mode, there is no local-field enhancement of the driving field, as evidenced by the 3-to-l contrast of the mode amplitude compared to the amplitude of the non-resonant scattered laser spectrum. In the CW scenario, the continuous pumping of the microcavity yields a local-field enhancement which experimentally results in the very high visibility feature observed. It is important to note that the mode spectra are physically obtained in very different fashions for the CW versus the OPO excitation source. The line-width of the C W laser is ~ 50 MHz, which is much smaller than the mode width, and so the resolution of the spectrum is provided by the laser step-size of 0.04 c m - 1 (recall that there is no spectrometer involved). For very high Q modes, care must be taken that thermal loading from the local-field enhancement doesn't distort the spectrum during the sweep and lead to errors in the width (and therefore Q). In the pulsed experiment, the spectrum is determined by Fourier transforming a time-domain in-terferogram, and the resolution is determined by the path length of the scanning arm inside the spectrometer. High resolution spectra are more time-consuming than similarly resolved C W laser sweeps due to the long path-length difference required in the spectrometer. For the results presented in this thesis, the most important difference between the short-pulse and C W experiments is revealed in the case of a cavity with more Chapter 4. Experiment 65 l i -n e -Wavenumber (cm ) (a) 351 1 1 — 30-->25 3 3-20 Wavenumber (cm 1) (b) Figure 4.7: (a) Short-pulse and (b) C W resonant scattering spectra from the funda-mental mode of the same L3-cavity. The thick background line in the C W spectrum is not noise, but residual Fabry Perot fringes, and they can be fit out of the spectrum. Fringes with a lower modulation depth are also visible in the short-pulse spectrum. Chapter 4. Experiment 66 than one spectrally distinct mode within the pulsed laser bandwidth. As described in Chapter 5, the modes can be coherently excited by a short-pulse laser, whereas a single C W laser is limited to exciting each mode separately. 4.3 Photoluminescence The InP samples contain a dilute layer of InAs quantum dots which are grown in a thin InAs wetting layer at a density of about 1-2 um~2 [77]. As motivated in the introduction, there is much interest in performing integrated cavity QED exper-iments [19, 20, 21] and developing efficient single photon sources [48, 78] by coupling semiconductor quantum dots (which behave like artificial atoms) to P P C microcavi-ties. In a semiconductor microcavity, the localized modes can enhance the photolu-minescence (PL) from bound electron-hole pairs, called excitons. Cavity-enhanced spontaneous emission was first predicted by Purcell in 1946 [79]. Physically, the lo-calized microcavity mode increases the local density of electromagnetic modes, which translates into enhanced fluctuations of the vacuum and drives the spontaneous emis-sion process of the excitons. In the weak-coupling limit, which usually means that the emitter-cavity interaction time is limited by the lifetime of the cavity mode, then an atom-like emitter that is well-coupled spatially and spectrally to a cavity mode will have an SE rate enhanced by 4 ^ 2 ( A / n ) 3 Q / y , known as the Purcell factor [80]. When excited by a laser with an energy greater than the quantum dot optical transition energy, excitons are created within the quantum dots which can then ra-diatively decay. In the photonic bandgap, the density of states is zero in the absence of localized defects, and so the exciton spontaneous emission can be completely inhib-ited [62, 63]. In a defect microcavity, however, if there are suitably coupled quantum dots, the Purcell-enhanced PL spectra will reveal every mode of the microcavity, even non-dipole-like modes which are relatively inaccessible to resonant scattering. Chapter 4. Experiment 67 4000 6000 8000 10000 12000 Wavenumber (cm 1 ) Figure 4.8: PL spectrum from an InP L3-microcavity. The inhomogenously broad-ened quantum dot emission spectrum extends from ~ 5000 — 8000 c m - 1 . In the photonic bandgap, ~ 6500 — 8300 c m - 1 , the emission is suppressed unless it is spa-tially and spectrally matched to a localized mode. Five such modes are visible in the bandgap (see inset). The PL from the InAs wetting layer appears at higher energy. Also, the resonant features in the spectra are polarized according to the mode prop-erties, revealing polarization information not available in resonant scattering (see Section 4.1). Although it is not a major focus of this thesis, some photoluminescence (PL) data are presented (all in the weak-coupling regime). The PL spectroscopy is performed exactly in the resonant scattering geometry, with a HeNe laser (emitting at 632.8 nm) as the excitation source and a 40x microscope objective to focus the incident light. Compared to the 100 x objective, the 40 x objective provides a larger excitation and collection spotsize, and was found to yield greater PL intensity. The half-wave plate and polarizer are not needed on the incident beam path because the quantum dots are insensitive to the excitation polarization. Chapter 4. Experiment 68 A typical PL spectrum is shown in Figure 4.8. Due to the broad size dispersion of the quantum dots, the exciton emission spectrum'is inhomogeneously broadened from ~ 5000 — 8000 c m - 1 . This low intensity, featureless spectrum is visible up to the lower edge of the photonic bandgap, inside which the emission is suppressed except where there is coupling to a localized mode. The five sharp peaks in the bandgap, which are highlighted in the inset, are indicative of the resonant enhancement of quantum dot PL by the localized modes. At higher energy, there is a large peak due to the PL from the contiguous, 2D wetting layer in which the quantum dots are nucleated [52]. 4.4 Second-order techniques The second-order nonlinear properties of InP-based PPC microcavities can be studied using a technique very similar to that described in the previous section. A laser beam at the fundamental frequency is used to excite one or more modes, as in the linear experiments. The scattered radiation from the microcavity is imaged through a spectrometer tuned to a spectral window around twice the mode frequency in order to detect evidence of a second-order nonlinear polarization being generated in the microcavity when the modes are excited. This technique is unique to the work presented in this thesis, and to the au-thor's knowledge, there is no comparable experiment in the literature which addresses harmonic generation from a microcavity that confines light in three dimensions. A schematic of the optical set-up is shown in Figure 4.9. The transmission geometry is very useful for the second-order experiments, because it facilitates the use of two different microscope objectives to account for chromatic dispersion between the fun-damental beam (u) and the harmonic light (~ 2w). A triple grating spectrometer and cooled charge-coupled device (CCD) array detector are used to measure spec-tra of the second-order radiation. During the course of a second-order experiment, the reflected resonant scattering signal is monitored at the fundamental frequency to Chapter 4. Experiment 69 image Probe -B-H W2 Pol. plate Pol. 2(0 \ 100x sample 40 x Pol Grating Spectrometer CCD detector camera FTIR Figure 4.9: Second-order spectroscopy set-up. The experiment is performed in trans-mission, allowing independent excitation and collection lens focal lengths. The second-order radiation can be spectrally-resolved with a grating spectrometer, or imaged directly onto the CCD array. optimize and maintain the coupling to the desired mode. For the C W experiments, the second-order radiation is directly imaged on the CCD array. This combines excellent sensitivity at the near infrared wavelengths of the harmonic radiation (~ 700-800 nm) with 2D spatial resolution of the radiated light. Narrow band filters (10 nm full-width at half-maximum) centred at the second-order frequency of interest are required to filter out the fundamental beam and the residual second-harmonic generation (SHG) from the OPO. The CCD detector is placed directly behind the collection lens, which is positioned at its focal length behind the sample plane. In this configuration, a thin collection lens would precisely image the Fourier transform (i.e. the far-fields) of the near-field distribution to a point a distance / behind the lens. Although the 40x microscope objective (a compound lens) used in collection is not a thin lens, it was found empirically to serve as a Fourier lens in this manner. For practical reasons, the image plane of the CCD array is more than a distance / away from the collection lens, and so the precise Fourier transform, which is equivalent to the far-field pattern of the radiation, is not imaged. However, the shape changes little as the collection lens is moved away from / and the image moved further away. The spatial pattern of the imaged radiation can be related to Chapter 4. Experiment 70 the spatial distribution of second-order polarization generated in the microcavity, as discussed in Chapter 6. The most efficient way to align the signal through the spectrometer is by back-aligning with a reference HeNe laser. Because the harmonic radiation is too weak to be visible, it is difficult to properly align the optic axis without this approach. To back-align, the spectrometer is tuned to about 634.0 nm (there is an offset of about 1.2 nm), and the flip mirror on the output port of the spectrometer is adjusted to block the detector and pass the laser. Once the red HeNe beam is visible emerging from the input lens of the spectrometer, it can then be directed via turning mirrors to strike the sample collection lens on axis. The next stage of alignment exploits the fact that the residual second-harmonic generated light from the OPO (hereafter called the "reference signal") is collinear with the fundamental OPO signal beam, and is very close to the frequency of interest radiated from the sample. Because it is much stronger than the second-order mi-crocavity radiation, the residual SHG is a very useful reference signal for alignment purposes, but care must be taken not to saturate and potentially damage the CCD detector. As a" precaution, the detector is initially set to an integration time of 0.05 s, and continuous shutter mode is used while the optical alignment is optimized. The 100 x lens is positioned to focus the fundamental ~ 1550 nm light on the microcav-ity. When the collection lens is well-aligned with the reference signal, a peak of 4000 counts/0.05 s can be measured through the spectrometer with the laser signal wave-length set to 1570 nm and a power of ~ 1 mW at the sample. It should be noted that while the reference signal is invaluable for lateral alignment of the collection lens onto the optic axis, the collection lens focal position which optimizes the reference signal strength on the detector is not the correct one for the experiment. There is considerable chromatic dispersion between this reference wavelength and the ~ 1550 nm fundamental beam when passing through the 100x focussing lens, which means that the collection lens (aligned on the reference signal) is not actually imaging the Chapter 4. Experiment 71 sample plane. After the alignment process, the reference signal is blocked to avoid obscuring the second-order microcavity radiation. This is achieved by placing a piece of GaAs which is polished on the front and back sides in the excitation beam path near the laser source. The 872 nm bandgap of the GaAs acts as an extremely effective filter which passes only the fundamental radiation, and the polished surfaces ensure the beam profile is not distorted. Ideally, the GaAs filter should be oriented at Brewster's angle, but this slightly deflects the beam and changes the alignment of the focussed beam on the microcavity, and so the filter is oriented normal to the laser beam. Note that if resonant scattering spectra are taken simultaneously with the second-order spectra, strong Fabry Perot fringes due to reflections inside the GaAs filter modulate the scattered linear spectrum and obscure the resonant features. The CCD detector operates at -120°C and is cooled by a liquid nitrogen reservoir. If the detector is initially at room temperature, after filling with liquid nitrogen it should be maintained at -80°C for 2h to guard against any residual water vapour condensing on the CCD window. Subsequently, the internal temperature can be set to -120°C, resulting in minimum noise level operation. The nitrogen dewar requires filling once daily to maintain the low temperature. 4.5 Pump-probe spectroscopy Pump-probe spectroscopy is a common ultrafast technique used to study dynamics in semiconductors [81]. The electrons of a semiconductor are excited into the con-duction band with a high energy pump laser, and then some material property such as reflectivity or absorption is resonantly probed as a function of the delay time after the pump beam. The same principles are used here in a series of experiments to investigate switch-ing and perturbations of a high-Q microcavity mode. The microcavity is pumped Chapter 4. Experiment 72 Pump F T I R Delay line Probe Pol A/2 Pol. plate BS1 B S 2 N 1 0 0 x sample Figure 4.10: Pump-probe spectroscopy set-up. The beam from the 810 nm Ti-sapphire pump laser for the OPO is brought onto the probe beam path via beam-splitter B S l . The relative timing of the pump and probe pulses can be controlled to better than 1 picosecond with the optical delay line on the pump beam path. with an above-bandgap laser, which excites free carriers and changes the complex refractive index h = n + ik through the Drude effect. Changes in n and k change the mode frequency and lifetime (Q), respectively. There are a number of reports in the literature on similar experiments, and these are discussed in the context of the present work in the introduction to Chapter 7. The pump-probe experimental setup is shown in Fig. 4.10. The pump and probe beams come from different exit ports of the OPO laser enclosure. The probe (or signal) beam is the usual tunable, pulsed OPO beam. The pump beam is the residual Ti-sapphire 810 nm laser which pumps the parametric down-conversion process inside the OPO. The relative timing of the pump and probe pulses is controlled by a retro-reflector mounted on a translation stage, which acts as an optical delay line for the pump beam, and allows sub-picosecond control of the relative time between the two pulses. The signal and pump beams are recombined at a beamsplitter (BSl), after which they propagate collinearly. To align the beams, their path lengths were determined by measurements of the beam paths internal to the laser. The difference was then compensated by aligning the beam paths on the optical table. The exact point of zero delay was found by scanning the delay line and using a BBO nonlinear crystal placed after B S l to generate sum-Chapter 4. Experiment 73 frequency light indicative of temporal and spatial overlap of the two beams. The 100 x focussing objective is positioned as usual to focus the probe beam on the microcavity and thus optimize the resonant scattering signal. Due to chromatic dispersion, the collinear 810 nm pump beam is not focussed on the microcavity, and its spot-size is estimated to be 5-10 pm. The pump beam intensity emitted from the OPO is about 300 mW, and after traversing the delay line optics and two beamsplitters, about 15-20 mW is delivered to the sample. With an estimated spot-size radius of 10 pm, this corresponds to a peak intensity of ~0.6 GW/cm 2 . The collinear propagation of the pump and probe beams is constrained due to chromatic dispersion in the 100 x lens. Future experiments may require a greater pump intensity than can be achieved in the current design, in which case a com-pensating lens would be required to focus the pump beam at the sample plane. An alternative approach would be a non-collinear geometry, with the pump beam incident from the back-side of a thin sample. One advantage of a defocussed pump excitation is that it is relatively insensitive to spatial translation as the delay line is scanned. This would likely not be true if the spot were tightly focussed. In that case, it would be preferable to put the delay line on the signal beam path so that any spatial drift as a function of delay would only affect the resonantly scattered intensity and not the perturbation strength. This could be compensated by optimizing the resonant scattering signal at each delay setting. The translation stage is operated manually with a control wheel marked with numbers. There are 25 numbers per 1 mm of stage travel,7 which corresponds to a delay path length of 2 mm, or a delay time of 6.7 ps. Therefore, each number corresponds to a delay of 267 fs. The experimental procedure consists of acquiring resonant scattering spectra for different delays of the pump pulse relative to the probe pulse. Each spectrum consists of a 10 scan average acquired at a resolution of 0.2 c m - 1 , which takes about 10 7Confusingly, there are 20 numbers per revolution of the control wheel Chapter 4. Experiment 74 minutes. The mode frequency is highly sensitive to temperature, and so the sample is mounted on a thermo-electric cooler block to stabilize the temperature. 4.6 Summary This chapter expounded the experimental techniques used for the work reported in this thesis. A basic optical set-up, in which a free-space laser beam (pulsed or CW) is focussed by a 100x microscope objective to excite a microcavity from normal inci-dence, forms the backbone of each of the methods of linear resonant scattering, pho-toluminescence, second-order nonlinear spectroscopy, and pump-probe spectroscopy. The method of cross-polarized detection has an important bearing on the scattered spectra. Only modes with a component orthogonal to the driving polarization can be detected, and their polarization cannot be determined unambiguously. Although it provides weaker input and output channels to the microcavity than evanescent coupling from a ID waveguide, free-space resonant scattering is naturally suited to the use of short-pulse laser sources. It can also easily be configured for a C W laser source, and a comparison was made of the physics of a single-mode microcavity that is excited by a short-pulse versus a C W beam. The two methods differ in whether the local field is enhanced, and the means by which the spectral resolution is determined. Moreover, as described in the next chapter, only the short-pulse excitation can coherently excite multiple modes. The scattering set-ups for both the second-order nonlinear spectroscopy and the pump-probe experiments were described, demonstrating the flexibility of the free-space resonant scattering approach for a diverse set of experiments. 75 Chapter 5 Multi-mode microcavity characterization using linear spectroscopies 5.1 Introduction Multi-mode microcavities are of interest for both classical and quantum optical in-formation processing schemes. One reason for exploring beyond single mode cavities is to facilitate conditional operations, in which an operation (or gate) is applied to one mode contingent on the occupancy of another mode. Classically, this multi-mode interaction could be mediated by the optical Kerr nonlinearity or the Faraday ef-fect [24], or through free-carrier dispersion. In fact, the latter effect has already been exploited for all-optical switching in two mode cavities, where one mode is resonant with the signal beam and the other mode with the control pulse [45, 44]. In light of the recent demonstrations of optical confinement for more than 1 nanosecond in high-Q PhC microcavities [4], a multi-mode microcavity could serve the function of storing multiple optical signals in an integrated optical circuit. In such a device, the intrinsic second-order nonlinearity of the host medium could be used to generate radiation indicative of the occupation of those modes, as will be shown in Chapter 6. There are also a number of quantum information processing schemes that are premised on multiple photon modes [51, 23], and these have the potential to be im-plemented in high-Q, small-V photonic crystal microcavities. Given these exciting Chapter 5. Multi-mode microcavity characterization 76 prospects, there is surprisingly little work reported on the multi-mode properties of P P C microcavities. Although most of the literature focusses on optimizing the Q of single modes in PPC microcavities, nearly all of these microcavities are, in fact, multi-mode, and therefore potentially useful for multi-mode optical- information processing. The multi-mode properties of two types of PPC microcavities, single-missing hole (S) and three-missing-hole (L3) defects, are studied in this chapter. In the first section, the mode structures of each microcavity type are discussed by drawing on results from FDTD simulations, photoluminescence data, and resonant scattering data. This makes use of the complimentary aspects of the experimental techniques, and allows a comparison between the numerical simulations and data. Two types of modes will be discussed: truly localized (bound) modes, of which there are a small number in the photonic bandgap, and quasi-localized modes, of which a plethora can exist near the upper edge of the dielectric band, depending on the nature of the defect potential. The second part of the chapter describes the coherent excitation of two modes of a single microcavity using short-pulse excitation with a bandwidth that spans the two modes. Coherent multimode excitation is a powerful aspect of the short-pulse reso-nant scattering approach, and markedly different from cw-excitation. It allows two modes to be excited simultaneously, and their relative amplitudes to be controlled both by the central frequency of the incident laser pulse and by its polarization, assuming the modes have different polarizations. The spectral coherence of the res-onant scattering spectral features, which are not simply the sum of two independent Lorentzians, is revealed by a phase shift induced in the cross-polarized measurement. The experiment raises interesting questions for future cavity QED research regarding how single photon sources might be used to initialize the state of a single photon in a multi-mode cavity [82]. Chapter 5. Multi-mode microcavity characterization 77 CO O Q C0 0 CO Figure 5.1: Schematic photonic density of states (DOS) in a defect microcavity, showing the air band (AB) and dielectric band (DB), two bound modes (B), and two quasi-localized modes (QL). 5.2 Mode profiles Understanding the mode structure of all localized and quasi-localized modes associ-ated with a given type of photonic crystal defect state is essential to properly charac-terize the experimental method and to optimize the design of the samples. It permits the identification of spectral regions where a single mode, or more than one mode, can be excited within the bandwidth of the excitation beam. It is also a crucial step for future cavity QED experiments, which will require well-characterized resonant cavities for integration with quantum dots. When a defect is introduced into an otherwise uniform photonic crystal lattice, it acts like a localized potential [83] which mixes the air or dielectric band Bloch states, depending on whether there is a local increase or decrease of the dielectric constant. If the potential is strong enough, modes can be split off from the continuum to form bound modes in the gap. A schematic resonant scattering spectrum is shown in Fig. 5.1, showing two bound modes in the photonic bandgap. In a uniform PPC, there is an increase in the density of states at both the air and dielectric band edges, and this can give rise to quasi-localized (QL) continuum modes in the presence of a defect potential [83, 84]. Quasi-localized modes are centred at the defect site, but Chapter 5. Multi-mode microcavity characterization 78 because they reside in the in-plane continuum of the surrounding uniform photonic crystal, they have propagating components in the plane, and are not purely bound modes. 5.2.1 Localized modes In general, the larger the microcavity, the more localized modes exist, but even for the simplest conceivable cavity, formed by removing a single hole (5-cavity), the microcavity is not single-mode. This can be intuitively understood with a symmetry analysis, as outlined in reference [11]. The defect destroys the translational symmetry of the photonic lattice and mixes the band-edge Bloch modes. When a single hole is removed to create an 5-cavity in a 2D hexagonal lattice, localized modes are "pulled" down from the air band, since mode frequencies tend to decrease with an increase in the dielectric constant. Specifically, the relevant air band basis modes that are mixed to form the localized modes are those at the M-point, which defines the upper edge of the bandgap. In a deeper defect potential, a more extended set of basis states would be required to fully describe the possible localized mode excitations. For the 5-cavity, the band-edge basis states are sufficient. The M-point air-band basis modes consist of three plane waves oriented along the three unique M-point axes (there are 6 M-points which define 3 unique axes through zone centre). The symmetry operations of the point group Cev of the defect can be applied to these three plane-wave basis states to give their representation in this symmetry group. This representation can then be decomposed into the irreducible representa-tions of the C6„ group to give a set of three symmetry basis functions for the localized air-band donor modes, as described in ref. [11]. A similar analysis can be used to define the acceptor modes formed by local decreases of the dielectric constant. One limitation of this simple analysis is that it doesn't predict where the modes with the corresponding symmetry are located spectrally, and for the 5-cavity, only 2 of the donor modes are actually in the photonic bandgap. These modes are the degenerate Chapter 5. Multi-mode microcavity characterization 79 pair shown in Fig. 5.2. Microcavities can be designed for single-mode operation if the desired mode is spectrally isolated from the rest. Tweaking the symmetry of a cavity is a com-mon strategy to split degeneracies or to bring modes of desired symmetry into the bandgap [13]. In larger microcavities, such as the L3-cavity, the symmetry is usually lower, but higher-order modes are brought into the bandgap, and so there are generally more modes with more complicated spatial intensity distributions. D 0 a <> o O o o J y> (a) U x (b) (c) Figure 5.2: FDTD simulation results from the 5-cavity, showing (a) the refractive index mesh, and (b,c) the degenerate (x', y')-dipole pair of modes. The normalized electric field intensity \EX<(LU)\2 + |.Ey(a;)|2 is plotted. The simulated structure has index n = 3.16, thickness d = 223.5 nm, pitch a — 500 nm, and hole radius r = 150 nm. Figure 5.3: FDTD simulation of a different 5-cavity (with smaller air holes) in which the holes are elliptical, yielding the mode profiles shown in (b) and (c). The dashed arrows show the approximate directions of the mode polarizations. The energies of the modes are split by 40 c m - 1 . The simulation parameters are identical to Fig. 5.2, except the elliptical holes have semi-axes r\ = 115 nm, r 2 = 135 nm. Chapter 5. Multi-mode microcavity characterization 80 5-cavity The two localized modes of a perfect 5-cavity are dipole-like, degenerate, and have the electric field intensity distributions shown in Fig. 5.2. Their relatively small Q-factors (200 - 500) can be attributed to two effects. Firstly, although the dominant Fourier components mixed by the defect perturbation are at the Brillouin zone boundary, and therefore below the vacuum light line, a separate Fourier coefficient near zone centre can be coupled to via scattering from the photonic lattice. The strength of this component is determined by symmetry. The symmetry of the dipole-like modes is such that there are large Fourier components with low in-plane momentum causing significant out-of-plane loss [14]. A second factor is the very small volume of these modes, for which V ~ 0.35(A/n)3, about half the volume of the L3-mode calculated in Section 3.4. The small real-space distribution leads to a more extended Fourier-space distribution of the mode about the zone boundary points, and thus a greater overlap with the vacuum light cone. Despite the modest Q-values, the simple mode spectra makes these cavities an ideal test platform for analyzing multi-mode excitation (Section 5.3). The degeneracy of the two modes can be broken by perturbing the photonic crystal lattice surrounding the defect. To render the microcavity single-mode for the purpose of making a laser, Painter et al. enlarge two holes to push the x-dipole mode frequency out of the bandgap [33]. Another way to break the degeneracy is to make the air holes elliptical rather than circular. A simulation of this geometry is shown in Fig. 5.3, where the holes are stretched by 20 nm along an axis 45° to the oZ-axis of the cavity [85]. The new mode intensity profiles clearly reflect the change of symmetry in the cavity. A vector indicates the polarization axis of each mode. L3-cavity The three-missing-hole cavity design is ideally suited to resonant scattering exper-iments in both the linear and nonlinear regimes. Most of the localized modes are Chapter 5. Multi-mode microcavity characterization 81 dipole-like,8 and therefore capable of being efficiently excited by a normal inci-dence laser beam. In some other cavity designs [86, 87], the localized modes have whispering-gallery profiles which are more difficult to excite from the top half-space. The L3-cavity is also the simplest reported design for generating high-Q modes, and can readily be excited into either a one-mode state or a multimode state, depending on the tuning of the excitation spectrum. In Chapter 3, the L3-cavity was discussed in the context of using F D T D simula-tions to optimize the Q of a single mode. However, there are several, other localized modes in this cavity which are revealed in resonant scattering experiments, and an accurate identification of these modes is needed in order to discriminate the high-Q mode from the other features observed. In addition, the other modes are interesting in their own right. They have unique spectral, spatial, and polarization properties, some of which will be important for the nonlinear work in this thesis. Figure 5.4 shows a spectrum of all the truly localized modes from a three-missing-hole microcavity like the one shown in Fig. 5.5. The L3-cavity is fabricated in an InP slab containing a dilute layer of InAs quantum dots, and the side-holes (s) are shifted out by 60 nm to increase the Q of the fundamental mode [12]. Simulations of this sample show that the bandgap extends from 6290 to 8760 c m - 1 , and therefore all of the features shown in Fig. 5.4 correspond to localized modes within the bandgap. The spectral range of the features exceeds that which can be probed in resonant scattering, because at energies greater than 6700 c m - 1 , a profusion of sharp water absorption lines appear in the laser spectrum which obscure the narrow resonant features of the cavity. To cover the full spectrum, the experimental data, shown in the lower portion of the graph, are presented as a combination of resonant scattering and photoluminescence spectra. The black spectrum shows a resonant scattering spectrum with a sharp resonant feature (A). This is the fundamental mode of the cavity, with Q ~ 7, 200. The contrast 8 S e e F i g u r e 3.5 a n d t h e d i s c u s s i o n o f S e c t i o n 4 . 1 . Chapter^Mm.mode microcavity characterizati 82 6400 6600 6800 7000 7200 7400 Wavenumber (cm - 1) Figure 5.4: Experimental (lower) and simulated (upper, green) spectra showing all localized modes of a three-missing-hole microcavity. The bandgap extends from 6290 -8760 c m - 1 . The experimental spectra are stitched together from a resonant scattering spectrum (black), and PL spectra with a y'- (blue) and x'-oriented (red) polarization analyzer. The simulated spectrum was obtained from an F D T D simulation with a point monitor at a point of low-symmetry in the cavity. Data to the left of the vertical line are divided by 10. between the mode signature and the non-resonantly scattered laser background is quite low in this particular sample.9 The red and blue traces are P L spectra from the same microcavity. The high-Q mode A is not revealed by PL, likely due to the low signal-to-noise ratio of the data. The blue (red) spectrum was taken with a y'- (x'-) polarizer in the output, clearly showing the polarized nature of the radiation from these dipole-like modes. An FDTD simulation of the cavity generates the spectrum shown in green, which corresponds very well to the experimental data. Because this simulation spectrum is 9 I n o t h e r s a m p l e s , i t w a s f o u n d t h a t a n g l i n g the s a m p l e t o g i v e a n o f f - n o r m a l i n c i d e n c e e x c i t a t i o n h e l p e d inc rease t h e c o n t r a s t . Chapter 5. Multi-mode microcavity characterization 83 » • t ~ j <r Figure 5.5: SEM image of an InP L3-cavity nominally identical to the one studied in Fig. 5.4. The side-holes marked s are shifted out by 60 nm (the other parameters are summarized in Table 5.1). The scalebar is 1 p,m long. Table 5.1: Summary of experimental and simulated parameters for the data of Fig-ure 5.4. Al l units are in nm. The uncertainty in the SEM measurements is about ±5nm. Parameter SEM Simulation pitch (a) 514 510 radius ( r ) 180 185 shift (s) 60 60 thickness (d) 231 223 index (n) - 3.16 taken by fast Fourier transforming the time data at a single point in the cavity, the relative magnitudes of the different features are partially determined by the choice of that point. The magnitude of the peak at 6400 c m - 1 dominates the other features in the simulation spectrum, and so the dynamic range below 6600 c m - 1 is divided by 10 to show this peak on the same scale. Table 5.1 shows a comparison of the parameters used in the simulation with the corresponding values measured from an SEM image. Within the ±5nm uncertainty in the SEM measurements, the agreement with the simulation parameters is good. The parameter with the greatest uncertainty is the thickness, d, which can vary by Chapter 5. Multi-mode microcavity characterization 84 -1 -0.5 0 0.5 1 < 0 > l o o , , 0 0 c o «•=* *=• - o o o O - o Do* j. 6 6 o O 0 a Figure 5.6: Spatial intensity profiles of all localized modes of a three-missing-hole microcavity. Features A, B, and E are y'-polarized, and C, D i , and D2 are rr'-polarized. Their spectra are shown in Fig. 5.4. 5% across a wafer and is difficult to measure locally. The spatial intensity profiles of the microcavity modes can be elucidated from the FDTD simulation by spectral filtering using a DFT monitor (see Chapter 3 for more details), and they are shown in Figure 5.6. Because peak D in Fig. 5.4 actually consists of two unresolved mode peaks (and peak C is also not fully resolved), a few simulations employing different symmetric boundary conditions were required to extract the pure mode profiles shown in Fig. 5.6. Modes Di and D 2 are actually degenerate in the unshifted L3-cavity, but are slightly split by the 60 nm side-hole shift in this sample, creating one broad peak in the scattering spectrum. It is interesting to note the similarity in the field intensity profiles of mode A and the x'-dipole mode of the 5-cavity shown in Fig. 5.2(b). The fact that these modes Chapter 5. Multi-mode microcavity characterization 85 have the same symmetry indicates that they are actually the same localized resonant excitation, in spite of the different defect potentials. In the 5-cavity, the x'-dipole mode is near the middle of the bandgap. When the two nearest-neighbour holes in the x'-direction are removed to transform the 5-cavity into an L3-cavity, the average dielectric constant of the mode increases substantially, pulling the mode down in frequency [2] from the middle of the bandgap to ~ 50-100 c m - 1 above the dielectric band edge. 5.2.2 Quasi-localized modes Resonant scattering spectra which probe energies more than 200 c m - 1 below mode A, near the band edge and further into the dielectric band continuum, show a forest of densely packed features that are spatially localized to the defect region. This is illustrated in Figure 5.7 by a series of broadband spectra from a silicon microcavity. Each spectrum has been normalized by its non-resonantly scattered laser background, and five such spectra are stitched together to span a large energy range. The band edge is determined from simulations which match the position of mode A, and is uncertain to within 50 c m - 1 . Mode A is clearly isolated in the lower part of the bandgap, about 50 c m - 1 above the dielectric band edge, which is marked by the dashed line. The numerous sharp features in the dielectric band are quasi-localized (QL) modes. Their Q factors are measured to be relatively high, typically ~ 2,000 - 20,000. In 5-cavities, FDTD simulations reveal the existence of QL features, but they have not been observed experimentally. Because there are many QL modes (due to the large density of states near the band-edge), it is difficult in general to ascribe a given spectral feature to a given mode. However, many of the L3-cavities in InP show two doublets of y'-polarized QL modes about 200 c m - 1 below the fundamental mode A. These are revealed clearly in Fig. 5.8, which shows a C W spectrum taken from the InP microcavity studied in Chapter 5. Multi-mode microcavity characterization 86 10 8 =! 6 -2->> <n c 4 £ 2 0 6300 6350 6400 6450 6500 6550 6600 6650 Wavenumber (cm - 1) Figure 5.7: A stitched sequence of broadband resonant scattering spectra from a Si L3-cavity for which the laser line-shapes have been normalized out. The dielectric band edge is shown by the dashed line. The high-Q mode A appears at 6555 c m - 1 , and a plethora of quasi-localized (QL) modes are visible below this in the dielectric band. The photonic crystal has a = 470 nm, r = 180 nm, d = 198 nm, and side-hole shift s = 23nm. Fig. 4.7(b). These modes can be simulated with the FDTD code, and plots of their | £ y | 2 distributions are shown in Fig. 5.9. As expected, the modes have large areas that extend well beyond the microcavity boundary. From a cavity QED perspective, QL modes are not of great interest because they have relatively large mode volumes (as evident in Fig. 5.9). However, they have many interesting properties which make them attractive test beds for the physics of this thesis. Because of their large mode areas, QL modes are relatively easy to excite by resonant scattering. Physically, the large mode volumes imply that QL modes have relatively localized distributions in fc-space, which leads to a reduction of radiative components inside the light cone and good Q factors. The fact that their field distributions extend well beyond the cavity gives different scattering properties to their second-order radiation, as discussed in Chapter 6. Moreover, their spectral density can make for an impressive demonstration of coherent multi-mode excitation. Dielectric band Bandgap Chapter 5. Multi-mode microcavity characterization 87 50h DB Bandgap 40h A c 20[ 10 0 1 6200 L J 6300 6400 Wavenumber (cm-1) 6500 Figure 5.8: A C W resonant scattering spectrum from an InP L3-cavity, showing two pairs of quasi-localized modes in the dielectric band (DB). A close-up spectrum of mode A in this microcavity was shown in Figure 4.7(b). The sample parameters are summarized in Table 5.1. An example is shown in Fig. 5.10, which displays one of the spectra from Fig. 5.7 which has not been normalized to remove the laser line-shape. Four modes (at least), each with a Q between 6,000 and 20,000, are coherently excited in a single spectrum. Having covered the zoology of localized modes in the S- and L3-microcavities, the details of the coherent multi-mode excitation will now be explored. 5.3 Multi-mode resonant scattering For a variety of classical and quantum information processing applications, it is de-sirable to be able to store and manipulate the population of more than one mode. In specific, practical implementations, the various modes of a cavity would be popu-lated independently by various sources in the optical circuit (as would be the case in a wavelength-division multiplexed (WDM) chip). For the purposes of studying the nonlinear properties of multi-mode cavities pop-ulated by more than one mode, a broadband, sub-picosecond pulsed laser was chosen Chapter 5. Multi-mode microcavity characterization 88 Figure 5.9: (a,b) Images of the y'-electric field intensity, l-Eyl 2, of two QL modes near the dielectric band edge, which are attributed to the pair of features (doublets) in Fig. 5.8. The refractive index map of the FDTD simulation area is shown in (c), and a plot of \Eyi\2 of the localized mode A is shown in (d) for comparison. The spatial dimensions are in microns, and the intensity colours arbitrary. to simultaneously inject electromagnetic energy into two or more modes having en-ergies within the bandwidth of the excitation pulse. While the nonlinear interactions are discussed in Chapter 6, this sub-section describes the excitation process itself, which is coherent. Although the coherent nature of this excitation is not critical to the nonlinear properties described in later chapters, it may be relevant to photonic qubit initialization mechanisms in the quantum limit. This will be discussed briefly at the end of this chapter. The samples studied here are 5-defect microcavities in an InP slab containing InAs quantum dots. As discussed above, tweaking of the microcavity symmetry can split the degeneracy of the two localized modes. PL and resonant scattering data from an 5-cavity with clearly separated modes are shown in Figure 5.11 [88]. The mode polarizations cannot be resolved by resonant scattering (see Section 4.1), but the PL Chapter 5. Multi-mode microcavity characterization 89 1 cd 6300 6400 6500 Wavenumber (cm - 1) Figure 5.10: (a) Un-normalized version of one of the spectra in Fig. 5.7, showing four quasi-localized modes, each with a Q factor between 6,000 and 20,000, coherently excited by a short-pulse laser beam. data reveal that the modes are oriented at ±45° with respect to the photonic lattice. The SEM image of the microcavity, shown in Fig. 5.11(a), suggests that elliptical holes may be the source of the mode splitting. Although there is considerable disorder, on average the holes are stretched along the —45° degree direction by about 20 nm. F D T D simulations from such a cavity are shown in Figure 5.3, and the polarizations of the modes are broadly consistent with these results. The mode-splitting determined from the simulation is only about half the experimental value of 60 c m - 1 , which likely reflects the effect of the hole size and shape fluctuations, which were not reproduced in the simulated structure. 5.3.1 Coherent multi-mode excitation model To analyze the physics of the multi-mode resonant scattering, the spectra shown in Fig. 5.11 are fit with combinations of Lorentzian functions. When no polarization analyzer is used for the PL measurements, the resulting spectrum can be simply fit by the (incoherent) sum of two Lorentzian lineshapes. However, it is apparent from the shape of the resonant scattering spectrum that it cannot be modelled by a simple Chapter 5. Multi-mode microcavity characterization 90 1.0 0.5 0 45° +45° a) 1 am b) 0 45° / \ 0.5 >, 6350 6400 6450 6500 6550 1.0r c) 7\ / \ out: <— 6^50 6400 6450 6500 Wavenumber (cm-1) 6550 Figure 5.11: Spectra from an InP S'-cavity with two non-degenerate dipole modes. The SEM image of the microcavity is shown in (a), revealing an elliptical hole per-turbation to the 6-fold symmetry. In (b), the PL spectra acquired with an output polarizer show the modes are polarized at ±45°. The resonant scattering spectrum is shown in (c), with the indicated orientations of the input and output polarizers. The sample parameters are d=223 nm, a=500 nm, and elliptical holes with semi-axes (ri,r2) = (120,140) nm, and r 2 oriented along the —45° direction. Chapter 5. Multi-mode microcavity characterization h • 91 sum of Lorentzians (the intensity is too high between the peaks). This would not account for the coherence between the two resonances. Instead, the spectrum must be fit by a function which accounts for the relative phase between the two resonant features. As shown in Fig. 5.12, an excellent fit is given by the coherent sum of three functions: a hyperbolic secant, which describes the line-shape of a mode-locked laser pulse, and two Lorentzian functions with complex coefficients: I = a0sech((ui - UJ0)/T0) H — — + u — u>i + iTi u> — u>2 + iT2 (5.1) To analyze the fundamental response of the cavity, the spectrum is normalized by the laser line-shape function extracted from the fit, leaving the two signature features of the resonant modes. The resulting response function, which is a measure of the intrinsic properties of the cavity, exhibits a well-defined relative phase 62 — Q\ = IT between the two Lorentzian features. Clearly, this response function must be independent of the centre frequency of the OPO excitation spectrum. This is verified in Fig. 5.13, which shows a series of spectra acquired as the OPO is tuned through the resonant region. When each spectrum is fit to eq. (5.1), the relative phase between the two modes is determined to be IT in each case. The response can be explained in terms of a harmonic oscillator (HO) model of the oscillating electrons that comprise the radiating material polarization. The electron motion projected along the two cavity mode directions, x\ and x2, for which there will be a resonant enhancement, is first considered. From the canonical HO equations of motion, the solutions for the electron displacement are: - / ^ Fl/m Xi(uj) = — : j— = 2 F 2 2 ' m , , (5-2) where F\ and F2 are the driving forces due to the incident laser electric field, m is the Chapter 5. Multi-mode microcavity characterization 92 Figure 5.12: Two-mode resonant scattering spectrum, showing the experimental data (dots) with the coherent Lorentzian fit superimposed (blue). The three constituent functions of the fit are shown as red dashed curves. electron mass, k{ is the restoring force constant, ji is a phenomenological damping term, and u>i = y/ki/m. Since it is the phase of the system which is relevant, the magnitudes of the parameters are not discussed. Letting (xi,x2) refer to the electron displacements along the (#i,#2) = (-45°,45°) directions in Fig. 5.11(a), then a resonant excitation along the 0° (y) direction drives a polarization proportional to P(u) oc (XI(UJ) cosf^ + x2(u) cos#2)?/, which essentially corresponds to two spectrally in-phase oscillators in the 0°-direction. Because the signal is analyzed in the cross-polarization (90°), two spectrally anti-phased oscillators are measured, with polarization proportional to P(u>) oc (XI(LU) sin#i + x2(u) sin92)x, which gives rise to a 7r-phase shift between the two Lorentzian features. Relative amplitude control of the modes can be achieved by tuning the excitation centre frequency, as shown in Fig. 5.13(a), but because the pulse bandwidth exceeds the mode spacing, both modes will always be excited with this method. Complete Chapter 5. Multi-mode microcavity characterization 93 <xT 90 CM 45 0 6400 Laser line centre (cm ) I I (b) 6440 6480 Figure 5.13: (a) Spectra acquired as the OPO is tuned through the resonant spectral region, (b) Phase shift between the modes, calculated from fits to the response functions determined from (a). control of the amplitudes could be provided by adjusting the polarization of the incident beam. The spectra in Fig. 5.13(a) are all acquired with a vertically polarized incident beam, but aligning the excitation polarization to be parallel with one of the modes (and thus orthogonal to the other) could be used to selectively excite a single mode. The single resonant feature would not show up in the resonant scattering spectra because it does not have a component orthogonal to the driving field, but this configuration has been studied by detecting the second-order radiation [88], which is bound by no such constraint. These experiments demonstrate control of the initial phases and amplitudes of the modes, and as such may have implications for future non-classical experiments when few or single photons are trapped in a two mode P P C microcavity. To give a relevant example from the cavity QED literature, Rauschenbeutel et al. have demonstrated the entanglement of two orthogonally-polarized modes of a superconducting Fabry Perot cavity that share a single photon [82]. A single rubidium atom that can be tuned into resonance with either mode is used in the preparation of the entangled state. A similar experiment might be achievable using two orthogonally-polarized modes of a PPC microcavity that are coherently initialized, as demonstrated here. A Chapter 5. Multi-mode microcavity characterization 94 single photon source with appropriate statistics could have a frequency uncertainty sufficient to span the modes, and a resonant quantum dot [21] embedded in the microcavity could be used to manipulate the quantum state. 5.4 Summary The characteristics of the localized modes of the S- and L3-microcavities were dis-cussed, including both bound and quasi-localized (QL) modes familiar from semicon-ductor theory. Bound modes are localized at the defect site, forbidden to propagate by the photonic bandgap. Their number and intensity profiles depend on the strength and size of the defect potential. The QL continuum features, although not purely lo-calized, are interesting for their favourably high Q-values, multi-mode structure, and extended spatial profiles. They are prominent features in experimental and simulated resonant scattering spectra near the lower band edge of L3-cavities. Experimentally, QL modes have not been observed in S'-cavities, although FDTD simulations suggest they do exist. Most of the literature has focussed on PPC microcavity design to optimize the quality factor and mode volume of single localized modes. Motivated by new classical and quantum information schemes premised on multi-mode microcavities, the multi-mode properties have been considered here in detail. The experiments take advantage of the broadband facility of short-pulse resonant scattering for multimode excitation. An S'-cavity with two modes oriented orthogonally to each other was studied with PL and resonant scattering measurements. An analysis of the resonant scattering spectra in terms of a coherent harmonic oscillator model reveals a well-defined phase shift between the modes. Control of the relative amplitudes of the modes is achieved by tuning the excitation pulse frequency and controlling its polarization. These results may be relevant in future classical or quantum experiments which require well-defined coherence between two modes of a microcavity. Chapter 6 Second-order responses 95 6.1 Introduction This chapter describes several novel second-order nonlinear measurements in InP pho-tonic crystal microcavities excited by low average power C W and pulsed lasers. The second-order microcavity response is studied in two different regimes, the impulse response (pulsed) regime and the frequency domain (CW) regime. The second-order impulse response is probed by scattering short pulses from the OPO into the micro-cavity, and observing the subsequent radiation emanating from the cavity near twice the excitation frequency. As discussed in Chapter 4, there is no local-field enhance-ment in this case, but second-order processes involving the electromagnetic energy stored in the microcavity mode(s) generate harmonics radiating over the lifetime of the mode(s). The principal advantage of observing the nonlinear impulse response lies in the fact that the short-pulse excitation can populate more than one microcav-ity mode at a time, as described in detail in Chapter 5. This allows a study of the second-order interaction between various modes, which is central to all-optical infor-mation processing. On the other hand, with a CW laser as the excitation source, the microcavity serves as a build-up cavity to achieve high local-field enhancements which facilitate single-mode nonlinear processes driven by relatively simple, low-power diode laser sources. There have been a number of literature reports which take advantage of the local-field enhancement, or "cavity build-up", for the enhancement of second-order nonlin-ear conversion in ID microcavities defined by bulk mirrors [89, 90], and 2D photonic Chapter 6. Second-order responses 96 crystals [91, 92]. It has also been shown theoretically that when both the fundamen-tal and second-harmonic beams are mode-matched to leaky modes of the structure, there is a significant resonant enhancement of the radiated second harmonic [93]. This effect was shown to be due to the local-field enhancements associated with the incoming and outgoing resonances. Second-order frequency generation has yet to be demonstrated in microcavities which confine light in three dimensions. To the author's knowledge, the results pre-sented here are the first demonstrations of these effects from fully localized modes in both the impulse and frequency domain regimes. These results demonstrate that a variety of low-power microcavity-based second-order nonlinear processes can be achieved in PPC structures using modest optical powers. This chapter begins with a discussion of the non-resonant second-order interac-tion between a Gaussian laser pulse and the InP sample, and shows that the bulk nonlinear susceptibility governs the non-resonant laser interaction in both untextured and textured regions of the slab. Section 6.3 discusses the impulse response regime in the resonant case, when either single or multiple modes are excited, and explains why the second-order radiation from the microcavity modes is polarized. The section finishes with a demonstration of how the detection of second-order fields could be used to monitor the joint occupation state of different microcavities, which would be a useful function in an integrated optical circuit. In Section 6.4, the frequency domain response of a single-mode microcavity is described, and local-field enhancements are exploited to generate second-harmonic radiation with a low power C W laser. It should be noted that although all of the data presented in this chapter are taken from samples which contain a dilute layer of InAs quantum dots, samples with no quantum dots show the same non-resonant response as described here, and the quantum dots appear to play no substantive role in any of the processes observed. Chapter 6. Second-order responses 97 12500 12600 12700 12800 Figure 6.1: Fundamental and second-harmonic generation signals from the untex-tured InP slab. The fundamental spectrum shows Fabry Perot interference fringes typical of the cross-polarized measurement. The lower x-axis refers to the fundamen-tal spectrum, and the upper x-axis to the SHG spectrum. 6.2 Bulk response To understand the second-order response involving the microcavity modes, it is help-ful to first discuss the response of the untextured InP slab when excited at normal incidence by a tightly focussed pulsed laser beam. This seemingly simple experiment yields some surprising and non-intuitive results, and, when considered in conjunction with spectra taken from the microcavity, shows that the bulk nonlinear susceptibil-ity is the dominant mechanism for all the second-order processes observed. This is important to properly explain the data, but also paves the way for future nano-scale dielectric engineering to optimize these bulk second-order effects. The optical set-up shown in Fig. 4.9 is used to measure the spectrum of the second-harmonic radiation generated when the OPO beam is focussed onto a uniform region of the InP slab. As shown in Fig. 6.1, the spectrum mimics the shape of the incident Chapter 6. Second-order responses 98 1 Incident polarization: — x+y — y — x — x-y o 1 -x-y y x+y Output polarization x x-y Figure 6.2: Non-resonant laser SHG amplitude for different incident polarizations, x and y refer to the InP crystallographic axes. The amplitudes for each polarization are normalized to 1. pulse, but it is centred at exactly twice the frequency. This radiation arises from the second-order polarization induced in the slab in response to the 130 fs incident laser pulse. Its polarization properties can be used to study the mechanism of the second-order process, and, in particular, to determine what symmetry tensor governs the process. In Figure 6.2, the peak values of the second-order spectra are plotted as a function of the output polarization angle for four different polarizations of the incident beam. It is evident from Fig. 6.2 that the harmonic radiation produced by an incident laser beam polarized along x or y is cross-polarized with respect to the input. In contrast, when the incident laser is polarized with equal components along both crystallographic axes i.e. either x + y or x — y, the harmonic response is maximal in the parallel polarization, and has a non-zero offset in the cross-polarization. The bulk nonlinear susceptibility tensor for InP, a zincblende (43m) crystal, has only three non-vanishing components: xfX^ J k. Naively, one might therefore Chapter 6. Second-order responses 99 expect only unpolarized SHG radiation from a [001] zincblende slab which is excited at normal incidence (a normally incident plane wave would have only in-plane electric field components), and only when both Ex and Ey field components are excited in the material. However, Fig. 6.2 shows that the radiation is clearly polarized, and does not require both Ex and Ey fields simultaneously. Assuming only bulk tensor components, in order to generate second-harmonic radiation at incident polarizations along x or y, one of the two driving fields must be in the z-direction. The induced in-plane nonlinear polarization must be given by Pi(2u) = e0XijlEj(cj)Ez(uj), where = (x,y), i ^ j. This polarization radiates light that is cross-polarized (index i) with respect to the transverse driving field (index j), as observed in the data. . The existence of longitudinal (£) fields is well known for tightly focussed Gaussian beams [94, 95], for which the paraxial approximation breaks down. Near the beam waist, it can be shown that Ez oc xEx [94], so the longitudinal field is anti-symmetric about the y-z mirror plane. Because of this odd symmetry of Ez, the second-harmonic polarization which is generated is of opposite sign on either side of the beam centre, and is oriented in a direction orthogonal to the driving field component. When the incident field is parallel to x or y, the resulting second-order field radiated by these in-plane components of the polarization will thus be oriented perpendicular to the incident field. For x + y or x — y incident polarizations, the SHG amplitude in Fig. 6.2 peaks in a direction parallel to the incident polarization, with a small isotropic background that is evident when the output polarizer is crossed with respect to the input. This behaviour is consistent with the bulk x^ response in which the three non-vanishing tensor elements are equal in magnitude. The second-order polarization generated with these incident polarizations can be expressed as P^\2uj)Kxi%Ey(Lu)Ez(Lu)x+ Xy%Ex(uj)Ez(u)y+ xi%Ex(uj)Ey(co) z. (6.1) Chapter 6. Second-order responses 100 When the incident field is oriented along x + y, Ex(u) = Ey(uj) and both Py(2co) and Px(2u) polarizations are generated. The resulting second-order field radiated by these in-plane components of the polarization will thus be oriented parallel to the incident field, a response that will be made clearer below. There will also be a z-oriented dipole that radiates with no preferred polarization, which explains the non-zero background in this polarization configuration. The physics of a focussed Gaussian beam can be simulated with FDTD by mod-elling a Gaussian beam of waist diameter 2 /xm that is travelling in the z direction and focussed to the middle of a 230 nm thick InP slab. Figure 6.3 shows the simulation results for two distinct cases: in row 1, the beam is polarized parallel to one axis [x) of the InP crystal, and in row 2, it is polarized along x + y, so that it has equal components along both axes of the InP crystal. In the first case, the second-order po-larization distribution (lc) is cross-polarized with respect to the incident beam, and in the second case (2c), it is polarized parallel to the incident beam. If the polarization distributions shown in column (c) for the two cases are each modelled by two point electric dipole sources oriented like the arrows, the far-field intensity distributions (see Section 3.6) shown in column (d) are generated. The far-field images are polar plots, and should be interpreted as the 2D projection of the fields on the surface of a hemisphere of radius 1 metre centred on the microcavity. To experimentally verify this analysis, the far-field second-harmonic radiation is measured directly, by placing the CCD detector immediately behind the collection lens to image the 2D far-field radiation pattern, as described in Chapter 4. Figure 6.4 shows the imaged radiation for the two different polarization configurations that dominate the bulk response. These polarizations are denoted by the arrows - dashed refers to the polarization of the incident beam, and solid to the polarization of the second-order radiation. Fig. 6.4(a) displays the far-field radiation from the radiating (2) Py component for the case when the incident beam is polarized parallel to the a>axis of the InP crystal, and should be compared to Fig. 6.3(ld), which shows the FDTD Chapter 6. Second-order responses 101 Figure 6.3: FDTD simulations of a Gaussian beam focussed to a 2 ptm waist diameter. Row 1 shows a series of images for a Gaussian beam polarized along the x-axis of the InP crystal. Row 2 corresponds to a Gaussian beam polarized along x + y. The first three columns show spatial images of (&)Exx + Eyy, (b) Ez, and (c) PJ 2 ) x + P y ( 2 ) y. Note that P^ is polarized orthogonally to E in row 1, but parallel to E in the case of row 2. The last column shows the far-field radiation | -Ex| 2 + \Ey\2 from two dipole sources oriented as shown by the vectors in (c). The intensity scale in (d) is normalized. The amplitude scale for the first three columns, shown by the bar in the third column, is normalized for each figure. In absolute terms, \EZ\ ~ 0.05|£' x|. The spatial scales are in units of /xm. far-field intensity pattern for the same case. In the other scenario, Fig. 6.4(b) is the experimental measurement of second-order far-field radiation in the case when the incident laser has equal components along both the x- and y-axes of the InP crystal. It should be compared to the FDTD result shown in Fig. 6.3(2d). Although there is some disparity in the relative intensity of the signals for the two orientations, the two sets of images agree well in symmetry and polarization, proving that the second-order process in the untextured slab is well-described by the bulk x^ symmetry. In such thin membranes, it is important to consider the role of surface second-harmonic generation (SHG). This is a commonly used probe of elemental semiconduc-tors like silicon [96], for which the bulk x^ tensor is zero due to centrosymmetry, but has also been applied to non-centrosymmetric crystals [97]. For a zincblende surface, which has symmetry 4mm, the non-zero surface symmetry susceptibility elements Chapter 6. Second-order responses 102 Figure 6.4: Measured far-field patterns of the second-order radiation from an untex-tured region of the InP slab for two different polarization configurations. In each figure, the dashed arrow indicates the polarization of the incident beam, and the solid arrow the polarization of the detected second-order radiation. The symmetry and polarization of these results agree well with the far-field plots shown in the FDTD simulations of Fig. 6.3. Image (a) here should be compared with (Id) there, and (b) here with (2d) there. are Xzzz, Xzii, a n d Xizi, where i = x,y. If the incident beam is polarized parallel to x or y, the latter tensor element would give rise to radiation polarized parallel to the excitation field. There is little sign of this in Fig. 6.2. When direct images were taken to observe radiation from the Xxzx or Xyzy components, some signal was detected, with an intensity about 40x weaker than shown in Fig. 6.4(a) and (b). However, this is not necessarily indicative of surface effects, because the measurement would give the same result if there were a slight mismatch between the (x,y) InP axes and the particular polarization axes of light used in the experiment (which is likely given that the sample was aligned by eye). When the OPO beam is focussed onto a P P C microcavity, non-resonant second-harmonic light is detected with similar intensity and polarization properties as in Fig. 6.2. This indicates that the bulk x^ is & ^ s 0 dominant in the microcavity, at Chapter 6. Second-order responses 103 least for the nonlinear Gaussian beam interaction. And as will be shown below, the resonant second-order radiation is also consistent with a nonlinear mechanism mediated primarily by the bulk InP. A l l of these results indicate that the bulk InP nonlinear susceptibility is the dom-inant nonlinearity at play, both in textured and uniform regions of the slab. This is extremely useful information because it greatly simplifies the second-order picture. There is a very high surface-to-volume ratio in PPC microcavities (this is one of the contributing factors to the very short (400 ps) free-carrier lifetimes measured in Chap-ter 7), and if the surfaces played an important role, their widely varying symmetries would make for a very complex second-order conversion mechanism. From the per-spective of a device designer, the result is important because, in contrast to surface symmetries, the bulk symmetry is both simple and uniform throughout the microcav-ity, and this should facilitate engineering of the mode fields to optimize second-order processes. 6.3 Microcavity impulse response 6.3.1 Single-mode excitation When the OPO beam excites a single resonant mode in a L3-microcavity, the second-order spectrum (not the far-field patterns just discussed) observed at twice the fun-damental frequency looks very similar to the scattered linear spectrum, as shown in Fig. 6.5. In these spectra, which are from the same microcavity as discussed in Figure 4.3, the resonant mode corresponds to the high-Q cavity mode A. In the 2u spectrum, the broad background corresponds to the non-resonant laser SHG discussed in the previous section. The sharp feature at exactly twice the mode frequency results from second-harmonic conversion of the fields scattered into the mode while irradiated by the short pulse, and trapped there for a time determined by the mode Q. Chapter 6. Second-order responses 104 12600 12800 13000 13200 Figure 6.5: Fundamental (LU) and second-order (2a;) spectra from the high-Q mode (A) of an InP L3-microcavity. The arrows indicate the energy scale corresponding to each trace. 6.3.2 Polarization properties of second-order mode radiation It is important to analyze the polarization properties of the second-order radiation from the microcavities to understand the later results of this section. Unless otherwise explicitly noted, the InP crystal axes (x,y) are at 45° to the photonic crystal axes (x1, y'), as shown in Figure 6.6. If an output polarizer is used to analyze the polarization of the second-order ra-diation shown in Fig. 6.5, the series of spectra shown in Fig. 6.7 are obtained for different orientations of the polarizer. The incident laser is polarized along the x' + y' direction, which corresponds to the x-axis in the InP frame. A careful look at Fig. 6.7 reveals that the resonant and non-resonant processes have different polarization prop-erties; the laser SHG is polarized in the —x' + y' direction, whereas the mode SHG is polarized along y', parallel to its linear fields. Chapter 6. Second-order responses 105 500 nm Figure 6.6: Relative orientation of photonic crystal axes [x',y') and InP crystal axes (x,y) for the data in this chapter (with the exception of Section 6.4.2). The polarization of the laser second-harmonic radiation is consistent with the bulk description given above. The incident laser is polarized parallel to the x-axis, and gives second-order radiation polarized along y, as expected (Py(2u) = toXvxzEx(u)Ez(u)). To interpret the mode polarization, the FDTD method described in Section 3.5 is used. In that section, the second-order polarization distribution of mode A is calculated for the case when all of the principal field anti-nodes drive strong Pz (2) components via the Xzxy tensor component. Figure 6.8 summarizes these results, showing the microcavity, mode intensities, and second-order polarizations for this mode, and also for the z'-polarized mode C as a comparison. The Pz distributions are remarkably similar for the two modes. It is perhaps unsurprising, then, that a polarization-resolved series of second-order spectra from mode C show the same properties as mode A in Fig. 6.7 - i.e., the second-harmonic radiation is polarized in the y'-direction. The far-field radiation from the multipolar Pz distributions of Fig. 6.8 can be simulated using F D T D according to the method of Section 3.6. For mode A, the far-field intensity patterns were presented in Fig. 3.8, and are shown again here in Fig. 6.9, rescaled to show only the central 40° cone of radiation that is collected in the experiment. The radiation is almost purely polarized along the y' direction, as Chapter 6. Second-order responses 106 6550 Figure 6.7: Second-order spectra from the fundamental y'-polarized mode of a L3-microcavity as a function of output polarization angle. Figure 6.8: Simulated field intensity profiles and associated Pz distributions for the y'-polarized mode A (b-c) and the x'-polarized mode C (d-e). An image of the microcavity is shown in (a) for perspective. Chapter 6. Second-order responses 107 in the experiment. If the air holes did not significantly modify the radiation from these z-oriented dipoles, one would expect the harmonic radiation collected along the 2-axis to be weak and largely unpolarized. But because the z-dipole distribution radiates predominantly in-plane, scattering from the lattice of air holes surrounding the microcavity plays an important role in directing light out of the plane, and in determining the polarization properties of the far-field radiation, as discussed in the context of Fig. 3.8. The simulations also reveal that absorption of the above-bandgap second-order radiation is an important determinant of the far-field pattern. This is not surprising, given that the absorption depth of InP at 12,900 c m - 1 (775 nm) is 280 nm [72]. Therefore, the majority of the second-order radiation that is directed down the long-axis of the microcavity is absorbed before it can be scattered out of the plane. This helps to explain the spatial distribution of the light in Fig. 6.9. As will be shown in Section 6.4 describing the frequency domain studies, these far-field simulations agree well with experimental measurements using a C W excitation laser (see Fig. 6.14). Figure 6.9: Simulated far-field intensities (a) \EX>\2 and (b) | £ y | 2 radiated by the second-order polarization of mode A (Fig. 6.8). These images are apertured versions (NA=0.65) of those shown in Fig. 3.8(3c). 6.3.3 Monitoring of multi-mode excitation When more than one mode is simultaneously excited by the broadband laser pulse, the nonlinear scattering spectrum is richer. Two different examples of multi-mode Chapter 6. Second-order responses 108 second-order spectra from a L3-cavity are presented in Figure 6.10. In both, there are two features visible in the linear spectrum, and three features in the second-order spectrum. The additional feature is due to sum-frequency mixing between the two excited modes of the microcavity. To examine the two cases in turn, the x'-polarized modes of the three-missing-hole cavity are the subject of Fig. 6.10(a). As shown in Fig. 5.4, these modes are separated by less than 60 c m - 1 , and so they can all be simultaneously excited with a single laser pulse. The linear scattering spectrum shows resonant features at 6700 c m - 1 and 6780 c m - 1 . The laser, polarized along the x'-axis with a full-width at half-maximum of ~ 90 c m - 1 , was centred at 6610 c m - 1 , below the mode energies, in order to emphasize these separate features. The lower feature is attributed to mode C, and the upper feature to mode D i (see Fig. 5.6 for reference). Note that mode D 2 has two in-plane reflection symmetry planes, and so is not excited in resonant scattering. When the laser is tuned to 6735 c m - 1 to more evenly excite the resonant modes, and with the output polarizer aligned in the y' direction, the nonlinear scattered spectrum at 2u> is obtained. The lowest and highest energy peaks are at exactly twice the frequencies of the microcavity modes evident in the linear spectra, and the central sharp feature, at 2 x 6740 c m - 1 , is at precisely their sum frequency. The extra peak at the sum frequency therefore corresponds to the second-order nonlinear interaction of fields resonant in the two modes. In Fig. 6.10(b), two quasi-localized modes are excited. The modes are quite different from those in (a): they have much larger mode volumes, their Q factors are about two times larger, and they are not purely localized modes in the photonic bandgap. However, similar physics in terms of the nonlinear mode interactions is observed. Again, there is an extra peak in the nonlinear scattered spectrum indicative of the two mode excitation condition. This demonstrates the generality of this result for different microcavity excitation conditions, and suggests an application. In an integrated optical circuit, detection of this second-order radiation using narrow band Chapter 6. Second-order responses 109 13000 13200 13400 13600 M o d e 2 12500 6600 6700 6800 Wavenumber (cm-1) (a) 12600 12700 12800 6400 Wavenumber (cm (b) Figure 6.10: Second-order impulse response of two-mode L3-microcavities. In (a), the x'-polarized modes are excited. The laser is tuned to 6610 c m - 1 for the linear spectrum (lower x-scale), and 6735 c m - 1 for the nonlinear spectrum (upper x-scale). In (b), two quasi-localized modes are excited by the laser tuned exactly between the modes. The nonlinear data show an extra feature due to sum-frequency generation (SFG) between the modes. The apparent shift between the two spectra in (a) may reflect a slight miscalibration of the grating spectrometer used to acquire the harmonic spectra. Chapter 6. Second-order responses 110 filters would provide a weak, non-destructive means of monitoring the joint occupation state of the microcavity, providing a logical AND gate operation. Although this functionality is demonstrated here in the case where a single short pulse is used to simultaneously populate the two modes, the sum-frequency signature would also occur if the modes were populated by independent sources. 6.3.4 Monitoring of multi-cavity occupation In this section, a more sophisticated use of nonlinear mode mixing is executed that demonstrates how optical logic gates could be applied to monitor the populations of modes in different microcavities. The experiment involves conditionally generating a signal dependent both on the occupation of one microcavity mode (or many) and the overlapping presence of a transient (non-resonant) signal in the cavity. C O i , C 0 2 , C 0 3 0 ) 2 , CO3 T Figure 6.11: Schematic optical circuit for the joint state nonlinear monitor. (1) Cavity 1 drops channel u\ from a multi-mode waveguide. (2) ui\ is coupled into cavity 2 (non-resonantly). Cavity 2 supports two resonant modes, ui^ and LU5, and when u>\ arrives, (3) new frequencies are conditionally generated at ui + uA or u\ + u5, depending on the population of cavity 2. This process is illustrated in the schematic optical circuit of Fig. 6.11. The figure shows two microcavities, 1 and 2, with 1 acting as a single mode channel drop fil-ter [35], and 2 acting to store photons at frequencies LU4 and co5. Signal ui is dropped from a multi-mode channel through the microcavity filter and into the middle waveg-Chapter 6. Second-order responses 111 uide. When signal u>i reaches cavity 2, it can interact with modes u>4 and u>^ via the second-order nonlinearity. New signals are generated at u>i + o;4 or uii + u)$ that are conditional on the combined occupation state of cavities 1 and 2. The details of the various coupling schemes are left unexplored here, but the nonlinear physics is demonstrated in the following experiment. Figure 6.12 shows a series of four second-order spectra from the microcavity sup-porting the quasi-localized modes discussed in Fig. 6.10(b). The spectra are acquired as the microcavity is being simultaneously excited by short, resonant pulses (as in Fig. 6.10(b)), and longer, picosecond (ps) pulses tuned far off resonance with the microcavity modes. This two-colour source is readily available from the unfiltered "signal" beam output of the OPO when it is tuned near the degeneracy point (where both signal and idler frequencies are close to half the pump frequency). An example of this unfiltered spectrum when the signal is tuned to 6335 c m - 1 is shown in Fig-ure 6.13(a). The short OPO signal pulses are accompanied by relatively long (a few ps) OPO idler pulses. The idler pulses in this spectrum are dominated by a single spectral feature at Ui = 6090 c m - 1 , but there are several smaller intensity peaks at lower frequency which are also revealed in. the second-order nonlinear processes discussed below. The centre frequency of these ps pulses (u;;) converges with the signal beam frequency (u>s) at half the pump frequency (u>p/2), which is known as the degeneracy point, as the OPO is tuned. The resonant scattering spectrum from the two-mode microcavity studied in Fig. 6.12 is shown in Fig. 6.13(b). The resonant enhancement of the mode features is evident in comparison to the non-resonantly scattered idler signal. The off-resonant ps pulse from the idler plays the role of ui\ in the schematic of Fig. 6.11, while the two modes play the parts of co^ and u>5. The second-order spectra in Fig. 6.12 show three principal groups of features that are marked by lines A, B, and C to guide the eye. Feature A and the broad background in group C (the fit of which is plotted separately as a dashed blue line), are due to non-resonant second-harmonic generation of the ps and fs features, respectively, in Chapter 6. Second-order responses 112 3 c A I B t \ A •1 S 4 \ (xi°>. X i i i 1 i i S3 \ — V \ (X10) \ i i i i AJ/V A S 2 V \ \ (x10) i \ 1 I 1 1 A i« r ^ J ~ \ (1 i i l rt 1 / *\ S 1 i, 6000 6100 6200 6300 - 1 ' 6400 Wavenumber (cm ) Figure 6.12: Spectra given by the interaction of a non-resonant, narrow linewidth idler pulse; a broad, resonant laser pulse; and a two-mode P P C microcavity. The four second-order spectra (red) are plotted at half the measured energy. The signal pulse is tuned to higher energy for each spectrum S1-S4, and so the idler tunes to lower energy. The amplitude of the low energy region has been multiplied by 10 for clarity. The solid blue curve shows the laser spectrum scattered from a non-textured region of the sample, as in Fig. 6.13(a). The dashed blue curves schematically show the signal pulse spectrum tuning to higher energies. Chapter 6. Second-order responses 113 (a) 3 •f 0.5 c CD 0 Idler, Co, (ps pulse) Degeneracy frequency • | Signal, cos (fs pulse) / \ ,,A-.-AA,,^ .y 1 . J . V (b) 1 6000 6200 6400 Frequency (cm-1) 3 ~ 0.5 cn c 0 2 modes ( 0 J • -Idler, (Oj I I A j u 1 , 6000 6200 6400 Frequency (cm 1) Figure 6.13: (a) Spectrum of the OPO laser beam, showing the femtosecond (fs) pulse at 6335 c m - 1 and the prominent picosecond (ps) pulse at the frequency of the idler. A number of smaller features are also visible near the main idler peak, (b) Resonant scattering spectrum from the microcavity studied in Fig. 6.12 (the tuning is slightly different from (a)). the excitation spectra. The three sharp features in group C that do not shift, and the two sharp features in group B that shift at half the rate of the excitation beam(s), are specific to the microcavity modes. They all reflect second-order processes that involve the fields "trapped" in at least one of the modes. The three (fixed) peaks in group C corre-spond to the mode SHG (2a>4, 2^5) and SFG (CJ 4 + UJ5) features, as in Figure 6.10. The features in group B are then easily understood to result from the second-order radiation of the two field distributions trapped in the microcavity modes respectively interacting with the ps pulses that irradiate the cavity during the ring-down to pro-duce peaks at u\ + and tui + u$. To understand the difference in the shift rate Chapter 6. Second-order responses 114 between features A and B, consider a ps pulse at toA interacting with a microcavity mode at wc- When the ps pulse is tuned from uA to UJA — Aw, the second-order feature A shifts from 2uA to 2(w^ — Aw), which is a shift of —2Aw, whereas feature B shifts from UJA + u>c to u>A — Aw + u>c, a shift of just —Aw. The processes illustrated here demonstrate that the fields stored in microcavity modes can be nohlinearly mixed with non-resonant signals to produce sum-frequency radiation. If the non-resonant fields were generated from light previously stored in a different microcavity, as in the schematic of Fig. 6.11, this principle could be used to conditionally generate information at new frequencies which depend on the joint occupation state of two different microcavities. 6.4 Frequency domain microcavity response 6.4.1 High-Q localized mode In this section, the use of a PPC microcavity as a build-up cavity to facilitate low-power harmonic generation is demonstrated. Figure 6.14 shows the first demonstra-tion, to the author's knowledge, of cavity-enhanced second-harmonic radiation from a wavelength-scale microcavity that confines light in three dimensions excited by less than 1 milliwatt (300 pW) of C W laser power. The direct 2D imaging technique is used, with a narrow bandpass filter centred at 780 nm placed between the detector and the collection lens to pass only the SHG signal. A 2 hour integration yields the pattern shown in Fig. 6.14. This image is remarkably similar to the FDTD simulation of Fig. 6.9, and experimentally verifies the multipole model of the nonlinear polar-ization within the microcavity, which is premised on the bulk \ ^ symmetry. The faint pattern inside the two main lobes is thought to result from diffraction from the aperture (NA=0.65) of the collection lens, which truncates the radiation at a solid angle of 40°. When the diode laser is tuned off resonance, the harmonic signal becomes ex-Chapter 6. Second-order responses 115 x10 4 Wavelength (nm) Figure 6.14: (a) CCD image of the measured second-order radiation pattern collected from the high Q mode A of the L3-microcavity over a 2h integration. The concentric rings show the solid collection angle up to 40°, the limit of the NA=0.65 collection objective. The C W linear resonant scattering spectrum for this mode is shown in (b), along with the orientation of the microcavity. tremely weak. By comparing the intensity on and off resonance, the second-order enhancement factor is estimated to be about 1000 x. This is in stark contrast to the pulsed case, where the difference between on and off resonance is small in the inte-grated second-order signal, because it only changes whether or not there is a small narrow peak in the spectrum. This 1000 x enhancement is necessarily less than the value of ~ Q2 one would expect if the Q of the microcavity were controlled com-pletely by its coupling to a single continuum decay channel, as might be the case if a single channel ID waveguide were used to couple to it. The enhancement factor is consistent with FDTD simulations that indicate only 2% of the normally-incident excitation beam is coupled into the cavity (the rest being scattered into radiation modes) [75]. In this experimental geometry, the on-resonance conversion efficiency, which is determined by the ratio of the collected harmonic radiation power to the average excitation power, is ~ 10 - 1 3 . This reflects the relatively weak coupling into the mode, and the fact that the second-harmonic frequency in this demonstration is actually above the bandgap of the host InP, which therefore absorbs it strongly. Optimization Chapter 6. Second-order responses 116 of several key factors could greatly enhance this efficiency. A waveguide-coupled excitation geometry would increase the incident power coupling to the microcavity by about a factor of 20 [73], which translates to a factor of 400 in the SHG power. The photonic crystal and excitation source could be designed such that the SHG was below the InP bandgap to prevent linear absorption. And more elaborate engineering of the microcavity could optimize the collection of the harmonic radiation. In Figure 3.8, the far-field SHG distributions from a slab with and without material absorption were shown. The difference in the total integrated signals of these distributions is a factor of 20. (Over the 40° numerical aperture of the collection lens, this is reduced to a factor of 13x, because of the difference in the radiation patterns.) A design optimized for these coupling and absorption effects could lead to an efficiency of ~ 10~9. The ultimate design would be a doubly-resonant microcavity, for which both the fundamental and second-harmonic fields are locally enhanced [93]. While designing the second-harmonic signal to be below the bandgap would in-crease the overall efficiency, there might be advantages to having it above the bandgap. For example, a photodetector circuit could be built around the microcavity to detect the free-carriers excited by the second-harmonic. 6.4.2 Quasi-localized mode Thusfar, C W results have been shown from just the high-Q mode of the L3-cavity. This was motivated by the desire to get quantitative confirmation of the dominant second-order polarization pattern by comparing the measured to the simulated far-field patterns. This comparison was easiest for the high-Q mode due to its relatively simple shape, and relatively large Q-value and small mode volume. Similar resonant harmonic generation can be observed from other modes of InP L3-microcavities. Here, for comparison purposes, the far-field SHG image from one of the QL modes with Q = 1300 is shown in Fig. 6.15. The mode was excited on resonance with the 300 yuW C W excitation laser, and the SHG radiation was collected Chapter 6. Second-order responses 117 Figure 6.15: CCD image of the second-order radiation collected from a y'-polarized quasi-localized mode pumped on resonance at 1576.3 nm and integrated for 15 min-utes. over a 15 minute integration window. The relative orientations of the photonic and InP crystal axes in this microcavity are different than for all the previous data, as shown in Fig. 6.16. There is only a 15° rotation between the two coordinate systems. The signal is weaker, as expected due to the larger mode volume, but this may also reflect a weaker coupling into the mode. It is also largely unpolarized; and has more intensity at relatively small momenta, which is qualitatively consistent with the more extended real-space distribution of the mode. Although not easily quantifiable, the pattern likely reflects both the relatively delocalized intensity distribution of the mode (Fig. 5.9), and the influence of the different relative orientation of the two coordinate systems. Two important conclusions can be drawn from this result. Firstly, it demonstrates that the local-field enhancement in a P P C microcavity can be used to enhance SHG frequency-conversion in different types of microcavity modes. Secondly, the nonlinear far-field radiation pattern can be controlled by engineering both the properties of the microcavity mode, and the orientation of the microcavity with respect to the crystal lattice of the host semiconductor. To summarize this section, local-field enhancement in PPC microcavities could be exploited for low-power frequency-conversion devices in integrated optical circuits. To optimize the design of a particular nonlinear device, the results presented here demon-strate that there are many degrees of freedom important in the process, including the Chapter 6. Second-order responses 118 Q factor of the microcavity mode, its spatial intensity distribution, and its orienta-tion with respect to the axes of the semiconductor host. In addition to these intrinsic factors, extrinsic factors such as the in- and out-coupling of the microcavity radiation are important determinants of the overall conversion efficiency. ••WW >• •< 500 nm Figure 6.16: Relative orientation of photonic crystal axes (x', y') and InP crystal axes (x, y) for the data of Fig. 6.15. 6.5 Summary This chapter described a variety of low-power, second-order processes in L3-microcavities that demonstrate promise for integrated optical logic and frequency-conversion. A detailed study of the nonlinear response of the untextured InP slab revealed that the patterns and polarization properties of the radiated second-order signals can be completely explained using just the bulk second-order susceptibility of the host slab. Similarly, both the non-resonant and resonant second-order processes in the microcavities are consistent with this simple picture. Two different regimes, namely impulse (pulsed) response and frequency domain (CW), were highlighted in this work. The relevant application depends on the regime being probed. With a short-pulse excitation beam to probe the impulse response, the timescale is too short for local-field enhancement effects. However, because the broad bandwidth excitation can be used to populate more than one mode of a microcav-ity, rich second-order scattering spectra revealing sum-frequency generation between Chapter 6. Second-order responses 119 different microcavity modes are observed. By detecting the various single-mode or inter-mode nonlinear mixing signals, the occupation state of a microcavity could be weakly monitored without significantly depleting its energy. When the trapped fields in a microcavity are mixed with a non-resonant field, sum-frequency radiation is gen-erated due to the nonlinear mixing of the non-resonant and resonant fields. If the non-resonant signal were produced by light in a different microcavity, the joint oc-cupation state of two different microcavities could be used to conditionally generate information at new frequencies. In the frequency domain experiment, the local field is enhanced by the reso-nant driving of the microcavity with a C W laser source. A 1000-fold enhancement of the SHG from a 300 juW C W incident beam was observed when a microcavity was pumped on resonance, demonstrating low-power nonlinear frequency-conversion. Similar effects were observed from quasi-localized modes, suggesting the generality of this process. Chapter 7 Ultrafast mode switching 120 7.1 Introduction The initial motivation for the work in this chapter was a straightforward generaliza-tion of previous work on optical switching to the case of wavelength-scale microcavities excited by sub-picosecond, free-space laser-pulses. In the process of achieving this goal, an unexpected behaviour in the resonant scattering spectra was observed that re-vealed more subtle mechanisms at work, which have not previously been addressed in the free-carrier-mediated optical switching context. These results were analyzed and interpreted in terms of a novel mechanism for frequency conversion. More profoundly and speculatively, they suggest a possible new method for generating non-classical light using perturbed microcavities. To address the initial motivation first, the control of light with light is one of the ultimate goals of integrated optical chips. Microcavity-based optical switching would allow the implementation of sophisticated algorithms for optical signal processing, much the same way that transistors allow the electrical signal gating that is the basis of modern computing. All-optical switching on an integrated chip is premised on a signal beam and a separate control beam which, by changing the refractive index of the structure by some means, alters the transmission or reflection of the signal beam. There is a wide variety of possible mechanisms for altering the index, such as the instantaneous optical Kerr effect, the injection of free carriers, and thermal changes, and they are distinguished'by having different required power levels and timescales. The first microcavity-based "switching" experiments were in structures where Chapter 7. Ultrafast mode switching 121 the confinement was only one- (ID) or two-dimensional (2D), and consisted of the nonlinear optical tuning of band edges in ID Si/SiC>2 Bragg stacks [39], and 2D bulk photonic crystals in silicon [41, 42] and AlGaAs [40]. The band edges are shifted by pump-induced changes in the refractive index resulting from both Kerr and free-carrier effects [41]. Relatively steep band edges can lead to large changes (greater than 250%) in the differential reflectivity, AR/R. For all-optical switching, microcavities that confine light in three dimensions have advantages in terms of size and power over ID or 2D confined systems. In ID Fabry Perot or Bragg stack cavities, the Q-factor can be very high, but the operating power is too high to be commercially viable due to the large mode volumes. PPC microcavi-ties represent the ultimate limit in terms of size and scalability for optical integration. In the ideal case of coupling to a single-mode channel, the switching energies can be less than 1 pj [44, 37]. High-Q modes are clearly desirable, because the mode shift required to achieve a given switching contrast is reduced, and small-y modes reduce the power requirements of the control beam. Carrier-induced switching based on two-photon absorption (TPA) has been demonstrated using on-chip waveguide coupling to silicon-based PPC microcavi-ties [44] and integrated microring resonators [45]. These experiments take advantage of the power savings of a resonant control beam, but must rely on relatively weak TPA of the control photons to inject free carriers. Here, the all-optical tuning of a single-mode PPC microcavity is achieved by injecting a transient population of free carriers directly through the linear absorption of a non-resonant pulse with energy above the silicon bandgap. A mode with Q = 35,000 is shifted 3.0 c m - 1 , or 16 times its unperturbed line-width, by an 810 nm pump pulse with intensity 0.6 GW/cm 2 . The mode line-width is also broadened by a factor of 3.5 due to free-carrier absorption. Chapter 7. Ultrafast mode switching 122 7.2 Carrier-induced optical switching The pump-probe experiment uses an 810 nm pump pulse brought in at a variable time delay to dynamically shift the frequency of a high-tQ mode in a silicon L3-microcavity. The pump serves to inject free carriers into the cavity, which have two dominant effects on the resonant mode: (1) the index of refraction of the silicon is changed through the Drude effect, and (2) the mode is broadened due to free-carrier absorption. Three relevant .regimes of interest can be identified according to the relative delay between the pump and the probe beams. Defining to as the excitation time of the microcavity mode via the probe pulse, tp as the arrival time of the pump pulse, and r m as the lifetime of the mode, these regimes are: tp < to (I), to < tp < to + rm (II), and tp » t 0 + rm (III). Focussing first on regimes I and III, Figure 7.1 shows a series of normalized mode spectra for different values of the pump-probe delay, defined as r = to — tp. In the raw spectra, such as shown in Fig. 7.2, the background is not flat, but consists of the broad non-resonantly scattered laser line-shape. In addition, there are Fabry Perot interference fringes which obscure the line-shape of the mode (the fringes are more visible here than in the spectrum shown in Fig. 4.7(a), because the resolution is higher and the mode visibility is lower with respect to the non-resonant background). To produce the spectra in Fig. 7.1, both the laser line-shape and the fringes are normalized out by dividing each of the raw spectra by a reference spectrum.10 The combination of this normalization procedure and a slight Fano asymmetry of the mode line-shape introduces a small level shift between the high and low energy backgrounds of the spectra (this is more visible in some of the spectra in Fig. 7.6 discussed in the next section). 1 0 The reference spectrum must be taken on the cavity at the same value of the OPO energy, because the Fabry Perot fringe pattern is sensitive to both the optical frequency and the spatial position of the focussed beam. Therefore, the reference spectrum was taken by temperature-tuning the mode to a different spectral window, and this procedure successfully produced the clean spectra of Fig. 7.1. Chapter 7. Ultrafast mode switching 123 -200 6308 6309 6310 6311 6312 6313 Frequency (cm - 1) Figure 7.1: Resonant scattering spectra from a microcavity for a wide range of delay times between the pump and the probe beams. The spectra are offset so that their non-resonant backgrounds intercept the vertical axis at the corresponding pump-probe delay, T = t0 — tp. The red spectrum in the lower plot shows the bare mode spectrum with the pump off. The bare mode and the two spectra at negative delays are not fully resolved, and so appear broader than the numbers quoted in the text. The photonic crystal microcavity parameters are a = 474 nm, r = 175 nm, and s = 0. These spectra are relatively easy to interpret, and demonstrate the intuitive result for canonical all-optical switching behaviour. When the pump beam is blocked, the bare cavity mode spectrum shown in red is obtained, with Q0 = 35,000. With the pump beam unblocked, the mode is blue-shifted for both positive and negative values of the delay. This indicates that there is a steady-state background population of free carriers, iV b g , maintained by the 80 MHz train of pump pulses, as sketched in Fig. 7.3. At negative delays, where this background population is sampled, there is a 1.2 c m - 1 shift and 40% reduction in the Q-factor (Q(-200 ps) = 20,000) with respect to the Chapter 7. Ultrafast mode switching 124 1r 0.8-Figure 7.2: Example of a raw pump-probe spectrum. The data of Fig. 7.1 are ob-tained by normalizing the raw spectrum by a similar spectrum for which the mode is temperature-tuned to a different frequency. 0 -1-1 1 1 --12.5 0 12.5 t(ns) Figure 7.3: Schematic of free-carrier density as a function of time. The 80 MHz laser injects a new carrier population every 12.5 ns, and there is an equilibrium background level. Chapter 7. Ultrafast mode switching 125 2 E o I 1 $ C • 0 . 5 r 0 L - 2 0 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 Probe delay (ps) Figure 7.4: Mode energy as a function of delay time between the pump and the probe beams. The circled data are discussed in the next section. bare cavity mode. At 0 ps, there is a transient blue-shift of the mode to u2 = 6310.9 c m - 1 , and subsequently the mode red-shifts as a function of increasing delay. The transient frequency shift, UJ2 — u>\, is 1.8 c m - 1 , and the total shift with respect to the bare mode, u2 — too, is 3.0 c m - 1 , or 16 line-widths (To) of the bare cavity mode. The mode position as a function of positive delay, which is shown in Fig. 7.4 for a more complete set of spectra than are shown in Fig. 7.1, essentially corresponds to a map of the free-carrier relaxation rate. Even during the relatively rapid free-carrier decay immediately following injection (with a ~ 400 ps decay constant), the carrier density is still approximately constant during the lifetime of the cavity mode (T c(0 ps) ~ 8 ps at the peak density). Both the 400 ps initial carrier decay constant, and the fact that the decay is not a single exponential function, are consistent with other experiments on silicon microresonators [74, 45]. This free-carrier initial lifetime is several orders of magnitude shorter than the microsecond lifetimes in bulk sili-con [76, 98], and is attributed to surface recombination in the high surface-to-volume ratio cavity. Chapter 7. Ultrafast mode switching 126 The free carriers are injected by a 20 mW average power pump beam, which has a peak power of 0.6 GW/cm 2 , and contains about 109 photons per pulse (200 pJ). About 3% of the photons are absorbed (as;(775 nm) = 1.6 x 105 m - 1 ) , creating 3x 107 free carriers in the slab. The spot-size of the pump beam is not accurately known, because the 100 x focussing lens is optimized to focus the resonant probe beam. The spot radius is estimated to be 5-10 p,m, which gives a range of free-carrier densities of (4.8-19) x l O 1 7 c m - 3 . The best measure of this density can be determined from the magnitude of the shift in the microcavity mode frequency. The free-carrier dispersion changes the refractive index according to the following equation, which is an empirical model that is in good agreement with Drude theory [99, 100]: A n = -[8.8 x 10~22 • AiV e -r-8.5 x 10" 1 8 • (AJV h) 0- 8], (7.1) where Ne and are the electron and hole densities, respectively, and are assumed to be identical here. From perturbation theory, A n = — nAco/to. The refractive index of silicon at 1585 nm is n = 3.47, and the maximum shift from the bare mode frequency is Aa; = 3.0 c m - 1 , and so the index change is A n = —1.7 x 10 - 3 . From eq. (7.1), this predicts a maximum free-carrier density N = 5.2x 10 1 7 c m - 3 , in agreement with the above estimate. A similar calculation based on the 1.2 c m - 1 equilibrium shift (probed at negative delays) gives a background carrier density of N\>g = 1.6 x 10 1 7 c m - 3 . 7.3 Ultrafast perturbation of a harmonic oscillator The results shown above speak to the controlled switching of a microcavity mode on a timescale that, in principle, might be expected to be limited by the ~ 130 fs pulse-length of the pump beam. This time is essentially instantaneous compared to Chapter 7. Ultrafast mode switching 127 the time-scales of either the mode (17 ps) or the free carriers (400 ps). However, an inspection of Fig. 7.4 reveals that for delays between —100 ps and 0 ps (data circled), the switching is not, in fact, instantaneous, but has an extended ramp up to 0 delay. By further exploring this negative delay regime, some unexpected behaviour is revealed that can be understood in terms of dynamic frequency conversion. At negative delays, the pump pulse arrives after the probe pulse (0 ps) excites the mode. Because the 1/e lifetime of the mode in the presence of the equilibrium free-carrier concentration (iVbg) is 17 ps (as calculated below), there are large reso-nant material polarizations within the microcavity at probe delays of —17 ps, and measurable fields out to at least —60 ps, as depicted schematically in Fig. 7.5. There-fore, it is not a dead cavity being perturbed at negative delay, with no perceptible effects, but rather a cavity very much alive with trapped fields that, when perturbed mid-life, give rise to complex spectra. Figure 7.6(a) shows a series of spectra corresponding to dynamic perturbations during the ring-down of the microcavity mode. In this regime, the probe beam excites the mode, with frequency oj\ and width Ti (the equilibrium values in the presence of -/Vbg), and as the energy in the microcavity decays, the pump pulse arrives and changes the cavity properties by injecting free carriers. Notomi has shown how such a dynamically perturbed microcavity could be used as a device for the frequency con-version of light [101]. The effect is analogous to turning the tuning peg of a vibrating guitar string to change the note to a higher frequency. At a probe delay of -60 ps, only 3% of the initial energy of the mode at uii remains in the microcavity when the perturbation occurs, and so most of the energy collected by the spectrometer is due to the emission from the unperturbed mode. As the delay approaches 0, an in-creasing fraction of the collected energy is due to radiation from the perturbed mode, revealed by the amplitude oscillations towards higher frequency and a broadening of the initial line-shape. At -14 ps, the spectral weight is broadly distributed over the range between u>i and LO2) yielding an almost dispersive line-shape. Only at -7 ps is Chapter 7. Ultrafast mode switching 128 Probe delay (ps) Time (ps) Figure 7.5: Schematic of the amplitude of a harmonic oscillator initialized at 0 ps (by the probe) and perturbed at r = 20 ps (by the pump), illustrating the change in both the resonant frequency and line-width after the perturbation. For comparison with Figure 7.6, the delay time of the probe beam is indicated on the upper time axis, showing that the perturbation occurs at a delay of -20 ps. The real time of the experiment appears on the lower axis. the clear signature of the perturbed cavity mode at u)2 revealed. This behaviour is largely reproduced using a phenomenological model to describe the rapid shift of the resonant frequency and lifetime (Q-factor) of a damped harmonic oscillator, and hence ring-down characteristics of the cavity, as induced by the free-carrier modification to the refractive index. The simple model is summarized by the following equation for the resonant material polarization in the cavity: P{t) = e-iuJlt'Tl\9{t) - 8(t-tp)).+ e-^-^e-^W-^-^eit-tp), (7.2) where 9(t) is the Heaviside step function. The first term describes the evolution be-Chapter 7. Ultrafast mode switching 129 6308 6309 6310 6311 6312 6313 Frequency (cm-1) 6308 6309 6310 6311 6312 6313 Frequency (cm-1) (b) Figure 7.6: (a) Experimental spectra of a high-Q mode dynamically perturbed by a population of free carriers, (b) Simulations of a perturbed harmonic oscillator in which the perturbation of UJ and T occurs linearly over a 500 fs time width. The dashed red line in both plots indicates the e _ 1 lifetime of 17 ps of the mode (at u>i) before the dynamic perturbation. Chapter 7. Ultrafast mode switching 130 tween time t = 0 and t = tp of the polarization associated with the fields trapped in the mode. The second term represents the evolution after the pump pulse per-turbation at time tp, with new frequency w2 = OJ\ + Auie~(-t~tp^Tc and line-width r 2 = Ti + Are~(*~ i p)/T c, where r c = 400 ps is the free-carrier lifetime. The scaling factor, g - ^ i t p - r i ^ j s necessary to match the amplitude of the oscillator between the two regimes at t = tp. The Heaviside functions in the equation represent an instan-taneous perturbation, which is a reasonable assumption in the classical limit, since the 130 fs pump pulse is more than two orders of magnitude shorter than the mode lifetime. If u>2 and T 2 did not depend on time, the Fourier transform of eq. (7.2) could be calculated analytically to yield: ' P(u) = J P(t)t ,iu>t dt 1 , i{w-U)l+iTl)T (1 _ e « ( " - " i + « r i ) T ) + e ( ? _ 3 ) OJ — U)\ + U 1 U> — U>2 + li- 2 where the Heaviside functions are replaced by the equivalent relation Bit) = lim — / did, (7.4) allowing simple evaluations of first the time integral and then the Co integral. Because u>2 and T 2 are functions of the time-dependent free-carrier density, this analytic result does not obtain. Instead, time-domain data were numerically gener-ated according to eq. (7.2) and then Fourier transformed, and the parameters were varied to provide the best match between the simulated (Fig. 7.6(b)) and experimen-tal spectra. Qualitatively, this simple model captures all the main features of the data. At 0 delay, the frequency shift determined from the best fit is ALO = 1.8 c m - 1 , and the damping parameter due to free-carrier absorption is T 2 = 2.0Fi, which gives Q(0 ps) = 10,000. In the previous section, the maximum free-carrier density in the Chapter 7. Ultrafast mode switching 131 cavity at 0 delay was calculated to be N = 5.2 x 10 1 7 cm 3 . Using this value, the free-carrier absorption can be calculated according to [99] A a = 1.45 x l f T 1 7 • A N . (7.5) The extinction coefficient is then calculated from the relation k = XAa/iir, yielding k = 9.5 x 10~5. The effect on the mode Q-factor can be estimated using the finite-difference time-domain (FDTD) method, assuming n and k do not change over the mode lifetime. This predicts that the bare cavity Q-factor of 35,000 is spoiled by a factor of 3.7, which is in good agreement with the experimental ratio of 3.5. The various parameter values are summarized in Table 7.1. For this purely classical model, the dependence of the simulation results on the finite time of the perturbation was investigated. The spectra are actually best fit when a 500 fs linear transition for both u> and T is used in place of the step function. This reflects both the pulse broadening (pump and probe) beyond 130 fs that occurs during propagation through the 100x microscope objective, and the convolution of the probe pulse shape with the integral of the pump pulse shape (the carrier density is roughly proportional to this integral pulse-shape). In comparison to the step function, the overall effect on the spectra of this finite width transition is small. However, in a quantum mechanical analysis of an instantaneous perturbation, the frequency conversion is no longer adiabatic, as discussed below. This analysis shows that the spectra from a dynamically perturbed microcavity mode can be robustly modelled as a damped oscillator which undergoes a rapid change in frequency and lifetime. Together with the experimental results, this work shows that the transition of the emission spectrum from the original to the perturbed cavity frequency is much richer than a simple redistribution of spectral weight between two well-defined Lorentzian peaks. Chapter 7. Ultrafast mode switching 132 Table 7.1: Summary of fit parameters for the simulation spectra of Fig. 7.6(b). Parameter Value 6307.9 cm" 1 LOi 6309.1 cm" 1 6310.9 cm" 1 Qo= ^o/ro 35,000 Qx = Q(-200ps) = wi /T i 20,000 Qi = Q(Ops) = u2/T2 10,000 400 ps 7.4 Implications for squeezed state generation These observations prompted a literature investigation of instantaneous shifts of the harmonic oscillator. There has been considerable theoretical interest in this phe-nomenon, and its relevance to the generation of squeezed states has been established in references [102, 103, 104]. It has been demonstrated experimentally in optical lattices in which the lattice potential is abruptly altered [105, 106], and proposed as a method to generate entanglement in nanoelectromechanical devices [107]. A microcavity-based scheme for the generation of non-classical states of the optical field quadratures would be very interesting from the perspective of creating inte-grated photonic quantum chips. Optical quantum information schemes are premised on non-classical light sources which produce single photons with high probability [48], or entangled photon pairs [108, 109]. It has been shown that squeezed vacuum states are entangled, and therefore that they can enable the teleportation of continuous ob-servables, such as the two quadratures of the electromagnetic field [49, 110]. There-fore, this experiment suggests the intriguing possibility of using perturbed microcav-ity modes to generate non-classical states for use in various quantum information schemes. These considerations are briefly explored in this section. Although specu-lative and likely challenging to implement at optical frequencies, the concept may be applicable in other electromagnetic cavity geometries operating at lower frequencies Chapter 7. Ultrafast mode switching 133 (eg. mid IR, THz, or even RF). Because the coherent state representation for the Hilbert space of a quantum oscillator is overcomplete, an oscillator of frequency a; can be represented by the equally overcomplete set of coherent states of an oscillator at frequency to'. These states are squeezed states of the first oscillator at to. Graham shows that when external changes to an oscillator frequency are made suddenly, squeezed states can be generated, but an adiabatic change has no effect on the variances of the the oscillator quadratures [103]. These two limiting cases are distinguished by the relation of the perturbation rate, 7 , to the oscillator frequency, u. The change is considered to be non-adiabatic if 2ITJ/LU 3 > 1. In the context of the pump-probe experiment, the oscillator period is ~ 5 fs, whereas the free-carrier perturbation occurs on the time-scale of the pump pulse, which is ~ 130 fs. Thus, the experiment described in this chapter is in the adiabatic regime. Even if the perturbation were to occur more rapidly, there would still be the issue of the degree of squeezing that can be obtained for a given frequency shift, from u to 8OJ. Preliminary calculations [111] suggest that the frequency shift would need to be on the order of 10% for observable squeezing in the microcavity experiment. Assuming that the non-adiabatic regime could be accessed in lower frequency cav-ities, but with less than the 10% frequency shift, there may be other ways to achieve useful squeezing levels. Averbukh et al. have shown that the periodic modulation of the frequency of an oscillator can lead to an exponentially enhanced squeezing [112]. With this approach, it may be possible to drive up the amount of squeezing by repeat-ably perturbing a microcavity mode with a sequence of pump pulses. Alternatively, Notomi et al. recently proposed a scheme for generating remarkably large wavelength shifts of up to 20%, with no significant impact on the mode Q-factor, by using a double-layer structure consisting of a stacked pair of P P C microcavities with a con-trollable separation distance [113]. Both of these ideas would be very challenging to implement at optical frequencies, but might be applicable with ultra-short laser Chapter 7. Ultrafast mode switching 134 pulses (< 10 fs) and mid-IR microcavity modes (which would have longer periods). 7.5 Summary The linear absorption of a 200 pJ non-resonant pump pulse injects free carriers with a density sufficient to shift a high-Q microcavity mode by 3.0 c m - 1 , or 16 line-widths of the bare microcavity mode. In the TPA-based switching experiments, Almeida et al. report 25 pJ switching pulse energies in a silicon micro-ring resonator [45], and Tanabe et al. report a few 100 fJ energy requirements in a PPC microcavity similar to the one studied here. The 200 pJ requirement in this experiment is inflated by the chromatic aberration in the 100 x lens, and would be on the order of 1 pJ if the pump beam could be focussed to the mode area. When the same experiment is done at negative time delays, corresponding to a very rapid perturbation of a mode that is already excited, the coherent oscillations observed in the spectra can be well-modelled by a'harmonic oscillator undergoing an abrupt shift in its resonant frequency. The phenomenon of perturbed oscillators has been shown to be relevant to frequency conversion, and might potentially be applicable to the generation of squeezed states. If the effect observed in this work could be achieved more rapidly in relation to the mode period, and with a larger magnitude (perhaps by a periodic perturbation), this approach might hold promise as a microcavity-based source for non-classical radiation. Chapter 8 Conclusions 135 This thesis focusses on the nonlinear and transient properties of multi-mode InP and Si planar photonic crystal microcavities. The coherent excitation of two orthogonally-polarized microcavity modes was demonstrated using normally incident laser pulses with a bandwidth greater than the mode separation. The mode coherence is mani-fested in a well-defined relative phase observed in the cross-polarized, scattered radi-ation. The relative amplitudes of the modes are controlled by a combination of the centre frequency and polarization of the incident pulse. This amplitude and phase initialization of two microcavity modes is expected to be relevant to integrated cavity QED experiments involving few or single photons in multi-mode cavities. When multiple modes of microcavities in InP membranes are populated using the short-pulse excitation technique, they interact nonlinearly via the second-order (x^) susceptibility of the host InP. Spectra of the radiation from the second-order polarizations driven by- the resonant modes exhibit second-harmonic features from each of the populated modes, and sum-frequency peaks from second-order mixing of the different modes. In addition, when a non-resonant (NR) pulse impinges on a microcavity in which one or more localized modes have been excited, sum-frequency radiation is observed from the respective nonlinear interactions of each populated mode with the NR pulse. For a given microcavity, the frequency of this conditionally-generated sum-frequency light can be adjusted by tuning the NR pulse frequency. The polarization properties of the radiated second-order light reveal that the bulk second-• order susceptibility is the dominant nonlinearity at work in these experiments. While the short-pulse excitation technique offers a convenient means of studying Chapter 8. Conclusions 136 multi-mode interactions, it does not take advantage of the local-field enhancement available in these ultrasmall, 3D-localized microcavities. Using a C W laser with less than 1 mW of average power, the far-field second-harmonic radiation from the reso-nantly excited fundamental mode of an L3-defect state was measured directly. The almost undetectable harmonic signal obtained from the same cavity when excited by the C W laser tuned off-resonance was at least 1000 times weaker than the on-resonant case. This result offers dramatic evidence of the internal field enhancement achieved even when using free-space excitation of a modest Q (= 3,800) microcavity mode. The measured 2D far-field patterns of the radiation agree well with those simulated using FDTD when it is assumed that the bulk InP second-order susceptibility governs the generation of the nonlinear polarization. The polarization and pattern of the har-monic radiation was found to depend strongly on material absorption and scattering from the air holes surrounding the microcavity. In a separate set of experiments done on Si-based microcavities, two optical pulses, one resonant with a Q = 35, 000 microcavity mode, and the other with an energy above the bandgap of the silicon, were both focussed on the microcavity, with their relative arrival time controlled precisely to better than 1 picosecond. These experi-ments were designed to demonstrate how the above-bandgap pulse could be used to inject free carriers into the cavity, and thus change its natural frequency, demonstrat-ing the potential for low-power, ultrafast all-optical switching. Above-bandgap pump pulses with an energy of 200 pJ were shown to shift the mode by 16 line-widths, while broadening its width by a factor of 3.5. An unexpected result from these experiments was discovered when observing the time-integrated spectra generated when the probe pulse (which injected energy into the microcavity mode) preceded the above-bandgap pump pulse. In this regime, the mode is perturbed while it is ringing down. The spectra show coherent oscillations which are well-simulated by a simple model of an instantaneously perturbed harmonic oscillator, and are relevant to optical frequency conversion. Chapter 8. Conclusions 137 8.1 Implications The knowledge gained from experiments and simulations done in this thesis work is relevant to the design of classical and quantum optical information processing chips. While it is true that the field of classical "all-optical" information processing is a bit like nuclear fusion (i.e. it's always just around the corner), the progress made over only a short period in the 1990s, when investment was high, suggests that its realization is more a matter of markets and economics than physics and engineering. On the other hand, there is no Intel established in the quantum information field, and so the prospect of cavity QED-based quantum processing is very worthy of active pursuit. In the classical context, the second-harmonic work reported here demonstrates how PPC microcavities could be used to perform low-power frequency shifting and conditional optical logic, both of which are important functions in an integrated optical circuit. The nonlinear mixing of two microcavity modes acts like an optical AND gate. The nonlinear interaction between a NR pulse and the fields confined in a microcavity conditionally generates information at new frequencies, which are signatures of the joint occupation state of two different microcavities. The second-order results suggest an investigation of the related second-order non-linear process of difference-frequency generation (DFG). Two modes within the band-width of the OPO pulse would generate DFG radiation at THz frequencies. The THz region of the spectrum is relatively difficult to access and yet has many physical and biomedical applications [114], and so engineerable sources based on PPC mi-crocavities would be highly desirable. A limiting factor for this direction is likely the relatively weak second-order nonlinearity of InP. A possible solution might be to fabricate PPCs by focussed-ion beam milling in highly nonlinear materials such as lithium niobate. The pump-probe work illustrates the frequency shift of a "signal" mode by a non-resonant "control" pulse, and, together with published work by others using Chapter 8. Conclusions 138 CW probes involving free-carrier generation, shows clearly that microcavities can be used as low-power all-optical switches. The dynamic perturbation of a microcavity in which electromagnetic energy is stored demonstrates a novel mechanism for frequency conversion, in analogy to the tuning of a vibrating guitar string to change the note. The nonlinear experiments in this thesis also have implications for quantum in-formation processing, in particular with regard to possible non-classical, localized sources of light. Parametric down-conversion (PDC) is the reciprocal nonlinear pro-cess to sum-frequency generation, and has been used to generate entangled photon pairs [115]. The second-order results reported here suggest that a two-mode PPC mi-crocavity pumped at the sum frequency might be a source of entangled photon pairs generated via PDC. Moreover, the pump-probe work identified an entirely different way in which non-classical light might also be generated using an entirely different effect that does not rely on the host material's lack of inversion symmetry. While the concept of "instantaneously" perturbing the frequency of a populated microcav-ity mode in order to squeeze the resulting radiation is intriguing, it is unlikely to be practical at near infrared frequencies. The basic idea may be worth considering if it could be implemented at mid-infrared frequencies, where it would be easier to perturb the cavity on a time-scale faster than the oscillation period, but some sort of periodic build-up technique would still likely be needed to render it useful. 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