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Nonlinear optics in high refractive index contrast photonic crystal microcavities Cowan, Allan Ralph 2004

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Nonlinear Optics in High Refractive Index Contrast Photonic Crystal Micro cavities by Allan Ralph Cowan B.Sc, McGill University, 1998 M.Sc, The University of British Columbia, 2000 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Department of Physics and Astronomy)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1, 2004 © Allan Ralph Cowan, 2004  ii  Abstract  Abstract This thesis describes theoretical and experimental research on the nonlinear response of high refractive index contrast (HRIC) optical microcavities. An intuitive, numerically efficient model of second harmonic reflection from two dimensional (2D), planar photonic crystals made of sub-wavelength thick, non-centrosymmetric semiconductors is developed. It predicts that appropriate 2D texture can result in orders of magnitude enhancement of the reflected second order signal when harmonic plane waves are used to excite leaky photonic crystal eigenmodes. Local field enhancement in the textured slab, and other physical processes responsible for these enhancements are explained. A different formalism is developed to treat the Kerr-related bistable response of a 3D microcavity coupled to a single mode waveguide. This model predicts that optical bistability should be observed using only milliwatts of power to excite a cavity fabricated in Alo.1sGao.s2 As, having a quality factor of Q = 4000 and a mode volume of 0.05 /j,m . Two-photon absorption is 3  shown to only slightly hinder the performance in Alo.isGao.82As.  By includ-  ing nonresonant downstream reflections in the model, novel hysteresis loops are predicted, and their stability is analyzed. A coupled waveguide-microcavity structure is fabricated by selectively  Abstract  iii  cladding a silicon ridge-Bragg grating waveguide with photoresist. Threedimensionally localized optical modes are realized with Q values ranging from 200 to 1200, at ~ 1.5 pm. Using 100 fs pulses, the transmission spectra of these structures is studied as a function of input power. The output power saturates when the cavity mode and pulse centre frequencies are resonant, and the output exhibits super linear growth when they are appropriately detuned. These results are explained in terms of the filtering action of the microcavity on the nonlinear spectral distortion of the input pulse as it propagates through the waveguide. PbSe nanocrystals are deposited on a microcavity in order to enhance its nonlinear response. Asymmetrical spectral broadening and saturation of the transmitted power is observed with ~ 10 000 photons in the cavity. A time-dependent third order model of the microcavity's impulse response is developed to describe this behaviour. This work contributes to a quantitative understanding of how HRIC dielectric texture can be engineered to obtain useful nonlinear optical responses at moderate power levels.  iv  Contents  Contents Abstract  ii  Contents  iv , . . . . vii  List of Tables  viii  List of Figures Acknowledgements 1  xi  Introduction  1  1.1  Overview  1  1.2  Background  4  1.2.1  Controlling the flow of light  4  1.2.2  Nonlinear optics  18  1.2.3  Nonlinear optics in microcavities  23  1.3 2  . . . . . . . .  Thesis outline  24  Second Harmonic Generation in Planar Photonic Crystals  26  2.1  26  Introduction 2.1.1  Conceptual overview of the specular SH conversion process  28  Contents 2.2  Theory  32  2.3  Results and discussions  36  2.3.1  Mode-matched second harmonic generation  36  2.3.2  Effect of the cavity Q of the incoming mode  45  2.3.3  Influences other than local field enhancement  50  2.3.4  Comparison with experiment  54  2.4  Chapter summary  58  3 Waveguide and Nonlinear Resonator Coupling  61  3.1  Introduction  61  3.2  Theory  65  3.3  3.4 4  v  3.2.1  Linear coupling regime  65  3.2.2  Non-resonant background  78  3.2.3  Coupled waveguide and cavity in nonlinear regime . . . 81  Results and discussions  86  3.3.1  Kerr effect and bistability  86  3.3.2  Non-resonant background effect  92  3.3.3  New stability analysis  95  Chapter summary  102  Nonlinear response of coupled waveguide-cavity  105  4.1  Introduction  105  4.2  The coupled waveguide-cavity structures  107  4.2.1  Silicon ridge waveguides  107  4.2.2  Designing the cavity  110  4.3  Experimental results  '  114  Contents  4.4 5  vi  4.3.1  Optical set-up  114  4.3.2  Linear transmission  115  4.3.3  Nonlinear waveguide plus filter  122  4.3.4  Impulse response of nonlinear cavity  128  Chapter summary  143  Conclusions  147  5.1  Summary  148  5.2  Conclusions  151  5.3  Directions for future work  153  Bibliography  155  A Orthogonality of basis states  167  List of Tables  vii  List of Tables 2.1  SH material parameters  54  3.1  Material parameters used in simulations  87  viii  List of Figures  List of Figures 1.1  Optical fibre dispersion diagram  6  1.2  Slab waveguide dispersion diagram  7  1.3  Silicon-on-insulator waveguide structure  8  1.4  I D PC dispersion diagram^ . . .-.  10  1.5  Planar photonic crystal  11  1.6  PPC dispersion diagram  13  1.7  Microcavity. mode field profile .:. . . . . . . .  2.1  Geometry of SH calculation  27  2.2  Bandstructure for SH calculation  30  2.3  End-coupling geometry  37  2.4  Overlapping fundamental and SH bandstructure  39  2.5  Contour plot of radiated SH  40  2.6  Surface plot of radiated p-polarized SH  41  2.7  Surface plot of radiated s-polarized SH  42  2.8  Linear reflectivity at fundamental mode  46  2.9  Grating coupling  47  . . . . . . ;. . . 16  2.10 SH strength versus Q of mode  49  2.11 Fano-like feature at leaky-to-leaky condition  53  ix  List of Figures  2.12 P-polarized SH from fabricated structure  55  2.13 Measured p-polarized SH  57  3.1  Waveguide-resonator coupling geometry  62  3.2  Reflectivity of waveguide-resonator system  88  3.3  Nonlinear reflectivity of waveguide-resonator system  90  3.4  Hystersis loop for waveguide-resonator system  91  3.5  Nonlinear fano-like reflection spectra for waveguide-resonator . 92  3.6  Hystersis loop for fano-like reflection spectra  93  3.7  Effect of TPA on hysteresis loop for fano-like reflection . . . .  95  3.8  Graphical solution for lorentzian reflectivity  97  3.9  Graphical solution for Fano-like reflectivity I  99  3.10 Graphical solution for Fano-like reflectivity I I  101  4.1  Schematic of waveguides and SEM image of Bragg grating  . . 108  4.2  Schematic and optical microscope image of cavity  113  4.3  Schematic of optical set-up  116  4.4  Linear transmission spectra of air clad guide  118  4.5  Linear transmission spectra of photoresist clad guide  119  4.6  Photoresist and air clad linear transmission  120  4.7  Cavity mode linear transmission  121  4.8  Guide fl-3 power dependent spectra  123  4.9  Guide fl-3 total transmitted power  124  4.10 Power transmitted through each cavity mode  125  4.11 Uniform guide power dependent transmission spectra  126  4.12 Total power transmitted through a uniform guide  127  List of Figures  x  4.13 Nonlinear power dependence of uniform guide spectra  128  4.14 Linear transmission through waveguide fl-4  129  4.15 Output vs input power for fl-4 before q-dots are applied . . . 130 4.16 Output spectra of fl-4 before q-dots are applied. .  131  4.17 Transmission of Q-Dot doped cavity  132  4.18 Power dependent transmission of q-dot doped cavity ... . . •:_ . 133 4.19 Power dependent spectra of q-dot doped cavity mode . . . . . 134 4.20 Power transmitted through cavity mode of q-dot sample. . . . 135 4.21 Power transmission curves for q-dot sample  136  4.22 Off-resonance transmission spectra  137  4.23 Time evolution of photon population  141  4.24 Time evolution of cavity mode frequency  142  4.25 Transmission spectra obtained from simple mode  144  Acknowledgements  xi  Acknowledgements The research environment in Dr. Jeff Young's lab has always been a rewarding and stimulating place to work. For this I would like to thank all the people that have come and gone during my time in the lab. Most importantly, I would like to thank my supervisor Dr. Jeff Young. His wisdom and practical experience provided guidance without which this work would not have been possible. It has been a pleasure to learn from such a distinguished scientist. I would also like to thank Dr. Georg Rieger, Murray McCutcheon, Mohamad Banaee, and Dr. Andras Pattantyus-Abraham for the many fruitful discussions. In particular, Dr. Rieger's suggestions in the optics lab, and Dr. Pattantyus-Abraham's guidance in the clean room, proved particularly helpful for the success of this work. I would like to also acknowledge Dr. Javed Iqbal for many enlightening discussions during the development of the analytic theory presented in chapter 3. Both Weiyang Jiang and Jessica Mondia deserve special mention as their combined efforts provided the excellent experimental results shown in chapter 2.  Chapter  1.  Introduction  1  Chapter 1 Introduction 1.1  Overview  Optics as an industry and a technology has expanded over the past few decades in much the same way as the microelectronics industry.  Mirrors  and lenses are now integrated in highly sophisticated systems that can be found in doctor's offices, industrial plants, home electronics - just about everywhere.^ Completely new components, such as fibre optic cable, have revolutionized the way society functions. This revolution was based on an elegant but simple physical principle (continuous total internal reflection of light), combined with a materials and process engineering solution for massproducing very high quality glass heterostructures. While fibre optic cable is capable of transmitting light with very low loss over long distances, it can only do so in straight lines. Once the light signals reach their destination, rather bulky optical elements are currently used to separate different channels (carrier frequencies) and convert the information from light intensity to electric current so that computers can perform switching and routing functions. Once the incoming information is processed, another set of bulky components is required to regenerate optical signals and launch them into outgoing fibres. Another revolution in optical technology  Chapter  1.  Introduction  2  could be realized by replacing all of these bulk optical components, and perhaps much of the computing power, with relatively small integrated optical circuits. The fundamental obstacle to producing high-density optical circuits is the tendency that light has to leak out of confining geometries when forced to abruptly change directions. Although very effective, the confinement of light in optical fibres is quite fragile: even slight bends in the fibre will cause the light to scatter into the cladding material. This fragility is a result of the low refractive index contrast between the fibre core material and cladding material that confines the light. While increasing the index contrast of guided wave structures allows much tighter bending before losses become prohibitive, high index contrast does not, by itself, allow light to be perfectly confined in arbitrarily complicated geometries. Only with the invention of a new class of artificial materials, known as photonic crystals, has it become possible to (theoretically) achieve complete localization of electromagnetic (EM) radiation. These photonic crystals make use of periodic patterns with high refractive index contrast to produce a frequency band that is void of any E M excitations - the photonic analogue of the electronic band gap in semiconductors. Appropriately designed spatial defects embedded within periodic photonic crystals can be used to localize light because it is rigorously forbidden from propagating away from the defect into the uniform photonic crystal cladding material. Line defects act like miniature waveguides, while point defects act like resonant microcavitiesfl, 2]. By integrating different defect waveguides with one or more microcavities, it has been shown that many of the filtering and routing functions required of optical communication sys-  Chapter  1.  Introduction  3  terns can in principle be implemented in miniature photonic crystal "chips" [3, 4, 5, 6]. This unprecedented control over the flow of light could also have a profound impact on fundamental physics. Electromagnetic energy in photonic crystal microcavities can be confined for thousands or millions of optical periods to volumes less than a cubic wavelength". Under such conditions it should be possible to probe the "strong coupling" regime of quantum electrodynamics. This refers to the onset of a new quantum mechanical state in which the cavity photon state becomes coherently coupled with a resonant electronic transition, in an embedded atom or quantum dot, at an energy scale  ujRabi,  that is larger than the dephasing rates of the dipole and the  cavity [7]. High index contrast photonic crystal microcavities could therefore provide a medium in which to practically implement some of the photonic based quantum information processing schemes that have been so elegantly studied using cold atom technology in conjunction with macroscopic optical cavities [8, 9]. Photonic crystals also provide a medium in which to explore and potentially exploit nonlinear optical phenomena at relatively modest power levels. The relatively small confinement geometry means that low-cost, umamplified laser sources could generate electric field strengths sufficient for lightby-light switching applications, for instance. Many proposals for nonlinear optics-based all-optical devices already exist in the literature[10, 11], and the incorporation of such devices in photonic chips would truly revolutionize the signal processing industry. It is therefore clear that the linear and nonlinear optical properties of  Chapter  1.  Introduction  4  high-index-contrast photonic structures can potentially have a major impact on both science and technology. This thesis describes both theoretical and experimental studies of some nonlinear responses of strongly confining microcavities designed in high index contrast periodic structures. It will be shown that compared to bulk material, the efficiencies of nonlinear processes can be drastically enhanced, leading to power thresholds for all-optical switching that are comparable to the power levels carried by current optical fibres. This work provides a pathway towards implementing nonlinear optics based all-optical components on optical integrated chips.  1.2  Background  This section presents technical background material intended to help nonexperts follow the presentation of original research contributions in subsequent chapters. One subsection deals with the essential physics behind dielectric structures defined with high refractive index contrast (HRIC), and the other deals with the basic concepts of nonlinear optics.  1.2.1  Controlling the flow of light  Optical design fundamentally involves managing the interfaces between regions with different dielectric constants. This subsection focuses on dielectric geometries designed to control the propagation of radiation that is confined to wavelength-scale waveguides and microcavities.  Chapter  1.  Introduction  5  Confinement of light by total internal reflection The dispersion diagram for all stationary solutions of the Maxwell equations in a vacuum containing one straight optical fibre is schematically shown in Fig. 1.1. The dashed lines, known as light lines, correspond to the dispersion of plane waves in bulk forms of the core and cladding materials. The refractive index of the core in a typical silica-based optical fibre is only A n ~ 0.01 larger than the surrounding cladding layer, so the two light lines are very close together. The solid line, which corresponds to the important (and lossless) mode bound completely within the fibre, only exists in the phase space between the two light lines. This mode exists because light in the higher index core experiences total internal reflection (TIR) from the core-cladding interface when it propagates along the fibre with sufficient momentum. Each point in the shaded region of Fig. 1.1 corresponds to a delocalized, "radiative mode" solution of the Maxwell equations that extends indefinitely in the vacuum surrounding the cladding layer [12, 13]. Any vacuum plane wave with a wavelength not too much smaller than the dimensions of the guided wave structure will, when striking it at an arbitrary angle of incidence, experience mild refraction as it passes through the core and cladding. The field intensity of these mildly modified plane waves is not substantially different inside or outside of the confinement region: these are essentially the radiation modes. In ideal, translationally invariant waveguides, these modes do not couple to the guided modes of the structure. In practice, coupling of the guided and radiative modes is a source of loss in imperfect or bent waveguides,.and controlling this loss is the key to achieving a viable optical chip technology.  Chapter  1.  Introduction  6  Wavevector Figure 1.1: Schematic dispersion diagram (energy vs. on-axis wavevector) showing solutions of the Maxwell equations in a vacuum containing one straight optical fibre. Due to the limited phase space available for bound modes in silica fibre, only a small perturbation (momentum shift due to bends or imperfections in the fibre) will couple confined light to modes of the continuum that carry energy away from the fibre. Consider instead a high refractive index (n > 3) semiconductor slab with an air or a low index glass cladding. The corresponding dispersion in the single bound mode regime is schematically shown in Fig. 1.2. In this geometry bound modes can be assigned distinct polarization labels; either transverse electric (TE), or transverse magnetic (TM). Transverse electric (magnetic) modes have electric (magnetic) fields oriented completely in the plane of the slab, perpendicular to the modal propagation vector. Only the lowest T E mode is shown in Fig. 1.2. Due to the large index  Chapter  1.  Radiation modes C  7  Introduction  / A3  Bound mode  LU  Wavevector Figure 1.2: Schematic dispersion diagram (energy vs. in-plane wavevector) showing solutions of the M a x w e l l equations for a semiconductor slab surrounded w i t h a low index cladding material. contrast (An > 2), the phase space for guided modes is much larger, therefore a larger momentum transfer (A/3) is required to couple the guided mode to the continuum. Sophisticated silicon processing techniques developed by the microelectronics industry make it possible to fabricate semiconductor ridges that have smooth surfaces and dimensions on the order of 100's of nanometers (see schematic diagram i n F i g . 1.3) [14]. In these high index contrast "fibres", the confined mode has a characteristic lateral extent less than the vacuum wavelength of light. T h e radiation loss of guided light transmitted around tight bends (lfxm radius of curvature) i n these ridge waveguides has been shown to be less than  0.1GLB[15].  Experiments discussed i n chapter  4  of  the thesis w i l l involve ridge waveguides like these, made from silicon. If the ridge waveguide described above is bent continuously back on itself  Chapter  1.  Introduction  8  Silicon  Silica Figure 1.3: Schematic diagram showing a silicon ridge waveguide and a microring cavity on a silica substrate. (see Fig. 1.3) it becomes an annular ring cavity that supports only a discrete set of quasi bound modes that are localized to the annulus[16, 17]. For cavities with perimeters much longer than the wavelength of relevant radiation, the eigen-frequencies of the nearly bound cavity modes are those on the straight waveguide's dispersion curve at wavevectors corresponding to wavelengths that fit an integer number of times along the perimeter. However, even in the large radius limit, these are only quasi bound modes because there is always some coupling to radiation modes that will eventually lead to the dissipation of electromagnetic energy stored in any one of the localized modes. The larger the normalized radius, the longer is the lifetime (or higher the quality factor (Q)), of the quasi bound mode, but also the larger is the mode volume. In a quantum electronics context, one typically seeks modes with high Q values and small mode volumes in order that a single photon in that mode has a well defined frequency, and generates as large an electric  Chapter  1.  Introduction  9  field as possible. Complete confinement of radiation is needed in order to bend light around arbitrarily tight bends, and thus to achieve full 3D localization of photons in small volumes. The complete 3D confinement of radiation can be achieved in materials that have been referred to as photonic crystals. H R I C periodic texture: the photonic crystal A photonic crystal is a dielectric material within which the index of refraction is spatially periodic[l]. The propagation of light having wavelengths commensurate with the pitch of the dielectric texture is influenced dramatically by the modulation. Consider the I D periodic structure shown in the inset to. Fig, 1.4. When an integer multiple of half wavelengths fit into one period of the structure, coherent Bragg scattering back-diffracts' the radiation, strongly modifying its group velocity in the material. Right at the Bragg condition the group velocity is zero (the eigenmodes are standing waves), and it increases to higher and lower wavelengths, as illustrated by the schematic dispersion diagram shown in Fig. 1.4. The standing waves right at the Bragg condition, (/3 /2), can be either in or out of phase with the texture, so there is 9  a discontinuity in the dispersion at this point in reciprocal space. The lower (higher) energy mode is predominantly in the high (low) index material. The energy difference between these two modes, and thus the width of the "optical bandgap" where no propagating modes are allowed, is determined largely by the index contrast of the texture; the larger the index contrast, the wider the gap. If a dielectric medium is periodic in all three dimensions, then with proper  Chapter  1.  Introduction  10  Wavevector Figure 1.4: Schematic dispersion diagram (energy vs. wavevector normal to the slabs) of the ID periodic stack shown in the inset. design and sufficiently large index contrast, the gaps for every direction of propagation can overlap one another [1, 2]. This means that there is a range of frequencies over which no E M excitations are allowed to propagate through the material in any direction. In analogy to the electronic band gap of semiconductors, a range of frequencies void of E M excitations is referred to as a full photonic band gap. A structure exhibiting a full photonic band gap was first realized at microwave frequencies in a dielectric crystal with a pitch on the order of centimeters[18]. Three dimensional crystals with pitches on the order of hundreds of nanometers resulting in mid-infrared band gaps have subsequently been fabricated [19], but the processes are still extremely challenging and imperfect. A mid-infrared band gap is desirable as it would correspond to  Chapter  1. Introduction  11  Figure 1.5: Schematic diagram of a planar photonic crystal. wavelengths used in communication networks, and would be compatible with convenient light sources. While some groups still work towards fabricating high-quality three dimensinal (3D) photonic crystals, there has been an overwhelming shift towards 2D photonic crystals made in planar waveguide structures. A planar photonic crystal (PPC), schematically shown in Fig. 1.5, is obtained by etching holes into a semiconductor slab waveguide. The fabrication of planar photonic crystals (PPCs) is much easier than full 3D crystals because they are based on advanced thin film growth and etching techniques developed in the microelectronics industry. Furthermore, their planar geometry is ideal for direct integration with current electro-optic devices like semiconductor lasers and quantum well heterostructure modulators. Therefore these structures are both easier to fabricate, and are more practical from an applications point of view.  Chapter  1.  Introduction  12  The planar photonic crystal If a thin semiconductor slab is textured with a 2D square lattice, in-plane Bragg scattering renormalizes the properties of the untextured slab modes. The resulting spectrum of E M excitations is schematically illustrated in the dispersion diagram of Fig. 1.6, where all bands have been folded using the reduced zone scheme. Due to the 2D periodicity, the Brillouin zone (irreducible unit cell in momentum space) of this structure is two dimensional (see inset to Fig. 1.6), thus the complete dispersion diagram includes all unique lines of symmetry in the reduced zone. Scattering from the 2D texture induces gaps at the boundaries and the centre of the reduced zone, and there is a corresponding modification of dispersion (group velocity) away from these high symmetry points. All photonic modes in this planar geometry are 2D Bloch states. Each mode is labeled by a well defined in-plane wavevector in the first Brillouin zone, P, and has an electric field distribution that is two dimensionally periodic in the plane. It is therefore convenient to express each photonic mode as a finite sum of Fourier components, each component being labeled by a well defined in-plane momentum /3 + Gi, where G; is a reciprocal lattice vector of the 2D lattice. There are essentially three distinct types of photonic modes in this geometry, characterized by "their out-of-plane behaviour [20, 21, 22], Purely bound modes, shown as solid lines below the cladding light line (dashed lines), are completely localized to the slab. They can not be excited by plane waves incident from the top or bottom half spaces. Resonantly bound (leaky) modes, depicted by solid lines above the cladding light line, are substantially lo-  Chapter  1.  Introduction  13  Figure 1.6: Schematic dispersion diagram for a 2D periodic semiconductor slab surrounded with a low index cladding material. Dashed line is the cladding light line, and shaded region denotes the continuum of radiation modes. Solid lines below the light line are the purely bound modes, while those above are the resonantly bound modes. Inset depicts the high symmetry directions in the Brillouin zone. calized to the slab, but do contain components that radiate away from the slab. There is also a continuum of radiation modes (shaded region), much like those in the plain slab geometry, that are infinite in extent, and are not preferentially enhanced within the slab layer. The dominant Fourier components of leaky mode Bloch states correspond to evanescent fields confined to the slab. What makes these modes distinct is that they also contain one or more radiative Fourier components that are typically small in comparison to the dominant evanescent components. Leaky modes are essentially remnants of bound modes that diffract out of the plane as they propagate within it. Leaky modes can be conveniently excited by light incident onto the surface of the textured slab from the upper half space.  Chapter  1.  Introduction  14  The texture effectively acts as a grating coupler that scatters incident plane waves into the evanescent components of the mode. For sufficiently weak coupling (long leaky mode lifetime), the local field in the slab can be much stronger than the field associated with the CW (continuous wave) excitation wave. The nonlinear response involving leaky photonic eigenstates will be the subject of chapter 2. The reduced symmetry of textured slabs means that the eigenmodes can not be assigned simple T E or T M polarization labels. However, along high symmetry directions in momentum space the component (s) of the Bloch states within the first Brillouin zone have a well defined polarization with respect to the surface of the slab. These modes can therefore be labeled s(electric field parallel to the surface) or p- (electric field perpendicular to s direction and propagation direction) polarized [20, 22]. For textured slabs that retain a horizontal symmetry plane, the modes can be further labeled as even or odd with respect to this plane [23]. W i t h proper texture design, a range of frequencies void of any in-plane purely bound modes of a given polarization can be achieved, as indicated by the shaded region labeled "Band gap" in Fig. 1.6. Since the radiation modes exist at all frequencies this gap does not represent a complete photonic band gap. Nonetheless, in a uniformly periodic structure there is no coupling between radiation modes and purely bound modes. Thus the band gap can be considered complete for all purely in-plane modes of a given polarization. Therefore, many of the intriguing properties of rigorously complete band gaps in 3D crystals can be achieved for the in-plane modes in planar structures. For this reason, along with the relative ease in fabrication and compatibility  Chapter  1.  15  Introduction  with existing technology, it is planar photonic crystals that will most likely form the backbone of future optical integrated chips. Rigorous localization of light Complete confinement of radiation in high index contrast photonic crystals can be achieved by introducing spatial defects into the otherwise uniform periodic lattice. A local perturbation of unit cells repeated along a symmetry direction in the crystal can result in a I D waveguide[24, 25, 26]. Depending on its design, it can support one or more I D localized modes, each with its own characteristic dispersion within the bandgap of the surrounding crystal. The virtue of such waveguides is that, in principle, a connected series of them oriented in different directions can route light through complex paths without loss, since there are no modes in the surrounding crystal to carry energy away from the bends. Appropriate localized modification of just a few unit cells can result in completely confined electromagnetic modes[l, 27]. These are the photonic analogues of bound impurity states in semiconductors. In 3D photonic crystal hosts, the lifetimes of these defect states are limited only by the finite size of the crystal. Their mode volumes can be less than a ( A / 2 ) , where 3  0  A is the vacuum wavelength of radiation at the defect's energy. Because 2D 0  planar photonic crystals support at best pseudo-band gaps, it is not possible to produce infinite-lifetime localized defect states in textured slabs. However, with appropriate designs, it is possible to achieve very small volume 3D microcavities in 2D planar photonic crystals with Q values in excess of 100,000(28]. Figure 1.7 shows the spatial distribution of the localized elec-  Chapter  1.  16  Introduction  415nm  b 300nm  Figure 1.7: Calculated spatial electric field profile for a cavity mode of a PPC based microcavity. Figures show in-plane (a), and out-of-plane (b), magnitude squared of the electric field. Inset shows the real space lattice, dark regions being air holes and light regions being semiconductor. White box in lower figure illustrates extent of semiconductor slab.  Chapter  1.  17  Introduction  trie field in such a planar photonic crystal microcavity . The inset shows a 1  top view of the PPC surface, the diameter of the central air hole has been increased to define the microcavity. The in-plane cross section of part (a) shows that the field is localized to the high index material surrounding the central hole; the centre of the enlarged hole is at the centre of the figure. Part (b) shows a cross section perpendicular to (a), with the white box depicting the extent of the high index slab. While some radiation away from the slab can be seen, predominately the field is strongly confined to the slab. The main advantage of HRIC photonic crystal based microcavities over the microring cavities discussed above, is that the mode volumes can be much smaller,for cavities with Q < 100,000[29].  .. .  It is this long lived, tight confinement of radiation that may have profound impacts on fundamental physics, from cavity QED to quantum computing. From an applications point of view, the fine control over the flow of light achievable in high index contrast periodic structures is very exciting. There are several proposed designs for add-drop filters and wavelength division multiplexing components based on I D waveguides coupled to point defect cavities [5, 6, 30, 31, 32]. These components could all be defined on a microscopic optical chip integrated with laser sources, electro-optic modulators and receivers. Such a chip would drastically reduce the cost of current telecommunication networks. In chapter 3 the nonlinear response of such a microcavity coupled to a I D waveguide channel is considered. 1  Calculation courtesy of Murray W . McCutcheon  Chapter  1.2.2  1.  Introduction  18  Nonlinear optics  For optical chip technology to be viable, it should do more than just control the flow of light. Switches, logic gates, amplifiers etc. are required to fully process information. There are two fundamentally different ways to incorporate these features in optical circuits. One approach is to miniaturize the same basic architecture that is currently implemented using bulk optical components. This might, for instance, involve connecting on-chip optical monitors to on-chip logic that can control on-chip electro-optic transduscers to effect switching or signal leveling functions. A different approach is to use light rather than electronics to condition the optical signals. This necessarily relies on the nonlinear optical response of materials in the photonic circuit because there is no interaction of light with itself in a purely linear medium. It is this latter approach that is of particular interest in the context of HRIC optical chips. Chapters 3 and 4 deal with theoretical and experimental aspects of this problem respectively. Below we give a brief overview of nonlinear optics; thorough introductions can be found in Refs.[33, 34]. A simplified but intuitive picture of a dielectric material describes the medium as a collection of dipoles. Electric fields polarize the dipoles and they in turn generate electric fields. The Maxwell equations describe the fields generated by the dipoles, but they are awkward to work with on a dipole by dipole basis. The macroscopic Maxwell equations result from spatially averaging the fundamental Maxwell equations over length scales larger than atomic but less than a wavelength of radiation at the relevant frequency. These macroscopic Maxwell equations relate the spatially averaged electric and magnetic field distributions to the polarization and magnetization den-  Chapter  1.  19  Introduction  sities in the medium. In non magnetic materials relevant to this thesis, the equation that relates the (harmonic) electric field, polarization density, "  :  P(f,t)  = E(r)e~ , zwt  to the  is,  — P(f)e~ , lwt  V x V x E{f)  E(f,t)  - uj E{r)  = 4nu P(f)  2  2  (1.1)  where Co = u/c, c is the speed of light in vacuum and u) the angular frequency of the electric field. The magnetic field can be obtained from the electric field through the Maxwell equations. Equation (1.1) is written to emphasize the fact that the polarization density acts as a source term that generates electric fields. This is even more evident when the solution to this wave equation is expressed as, E(f)  = E {f) h  + 4TOD J 2  df  G (f, f')P(f)  (1.2)  where Eh(f) is a homogeneous solution associated with externally driven fields in vacuum, and G {f, f ' ) is the Green's function that propagates the field due to an individual dipole in vacuum. To obtain a self consistent equation for the electric field it is necessary to specify a "constitutive relationship" between the electric field and the polarization density. This can be generally written as, (1.3)  P = (E)E X  where the material susceptibility, x, is allowed to depend on the electric field (the tensor nature of the relationship has been suppressed in Eqn. (1.3)). All of linear optics is based on a field independent x (x - * Xi) that; is related to the material's dielectric constant as, e = 1 + ATT I X  (1.4)  Chapter  1.  20  Introduction  the real part of e determines the index of refraction of the material, while the imaginary part describes absorption of light by the material. A host of nonlinear effects result when the electric field is strong enough to perturb the host material sufficiently far from its equilibrium state that the dipoles no longer respond linearly to the local electric field. In this case it is convenient to express the total polarization as, P = lE  (1.5)  + Pnl =Pl + Pnl  X  —*  — *  where Pi is the leading order, linear polarization, and P i is the polarization n  due to the material's nonlinear response. If the Maxwell equations are written as, V x V x E(f)  - u e(f)E(r)  = 4nu P (f)  2  2  nl  (1.6)  the nonlinear polarization, acts as a source term for the homogeneous Maxwell equations that describe field propagation in the linear medium. Note that e can be an arbitrary function of r, so if the solution of the linear problem in a complex geometry has somehow been found, then the solution of the nonlinear problem can be written as,  E(f)  where G '(f,f')  = E \f) h  +  4nu j df' 2  G '(f,f")P (r') nl  (1.7)  is the Green's function that propagates the field produced  by a dipole in the dielectric medium,  and Eh'(r)  is a homogeneous solution  of the linear problem. P i is often written in terms of a nonlinear (field-dependent) susceptibility n  as, P  nt  = xm(E)E  =  EE  {2) X  + WEEE X  + ...  (1.8)  Chapter  1.  21  Introduction  and Xni is expanded as a Taylor series in powers of the electric field with the anharmonic response of the dipoles dictating the strength of the higher order susceptibility tensors. A detailed derivation of the susceptibility tensors can be found in any text book of nonlinear optics, as in Ref. [33]. There are three nonlinear processes dealt with in this thesis. Second harmonic generation keeps track of the field radiated at twice the driving frequency of the anharmonic dipoles. The second harmonic polarization is related to the fundamental field strength by, P ( f , 2 u / ) = \r,-2u;u,u)E{r,u)E(r,u) (2)  {2  X  ' ~  (1.9)  where the u> dependence has been explicitly shown. This Fourier component of the total"polarization {P(f, t)) is taken as the driving term in Eqn. (1.7) when calculating the Fourier component of the total electric field  (E(f,t))  that oscillates at 2u. When the total amount of power converted to second harmonic radiation is small compared to the power in the fundamental, one solves the linear Maxwell equations at the fundamental frequency to find the primary field, and then the linear Maxwell equations are separately solved at the second harmonic frequency using the source term defined by Eqn. (1.9). This procedure is carried through for a 2D planar photonic crystal structure in Chapter 2. It is shown there that the strength of the second harmonic fields can be dramatically enhanced when either the fundamental or second harmonic fields are resonant with one of the leaky modes of the crystal. The other two nonlinear processes considered in this thesis have to do with the in-phase and quadrature components of the third order polarization at the fundamental frequency. The relevant Fourier component of the nonlinear  22 Chapter  1.  Introduction  polarization is given by P  ( 3 )  ( r > ) = 3x  Hr,-u]u,-u,u)E(r,u)E\r,u)E(r,uj)  (3  where the real and imaginary parts of x^  a r e  (1.10)  responsible respectively for  the Kerr effect and two-photon absorption (TPA). The Kerr effect describes an intensity-dependent refractive index. This physics, and the TPA physics, can both be easily understood by grouping the terms in Eqn. (1.10) and taking them over to the left hand side of Eqn. (1.6), V x V x E(r,u) - <2> (e(f» 2  12nx \r,-u]u,-u,u)\E\ )E(r,u) (3  2  (1.11) Written this way it is clear that the refractive index of the medium (real part of e) is modified by an amount proportional to the magnitude of the electric field and the real part of x^- Likewise, the imaginary part of x^ corresponds to an additional source of absorption by the material. The solution of Eqn. (1.11) is more difficult than the second harmonic problem because the nonlinear polarization is at the fundamental frequency, and has to be treated on the same footing as the linear polarization. An iterative approach is usually used, where the linear solution is found in the first step, and the intensity-dependent nonlinear polarization for the second step is approximated using the linear solution for the fields. This process is repeated until the solution converges. Under certain conditions, one of which is described in chapter 3, it is possible to avoid this iteration procedure.  = 0  23 Chapter  1.2.3  1.  Introduction  Nonlinear optics in microcavities  This thesis studies how HRIC texture can be used to substantially enhance the nonlinear optical response of a material over what i t would be in the absence of the texture. In the end, most of the enhancement that is achieved is fundamentally due to microcavity effects. This background chapter therefore concludes with a general discussion of nonlinear cavity optics. The net effect of nonlinear interactions on a light field depends not only on the value of the nonlinear susceptibility and the intensity of the incident field, but also on the interaction distance (or time) over which the effects can accumulate. An optical switch based on the Kerr effect can be built by splitting a beam into two equal intensity replicas, and sending them through two distinct paths before recombining them with interferometric precision. If one of the paths contains a Kerr active medium, this represents a Mach Zehnder optical switch[35, 36]. At low input power the two arms of the interferometer are balanced so that total destructive interference occurs at the output, yielding no transmission. At elevated intensities the effective refractive index in the nonlinear arm will be different than the control arm, so as the intensity is increased the transmission will gradually increase and eventually reach unity when the accumulated differential phase shift is n. At some prescribed operating power the transmission of the nonlinear Mach Zehnder is unity, while at lower powers it is negligible: this is perhaps the simplest example of an all-optical switch. One of the reasons these switches are not used much in commercial applications is that either high optical powers or very long lengths of nonlinear material are required. It was recognized long ago that this situation could be  Chapter  1.  Introduction  24  improved considerably by placing the nonlinear medium inside a high finesse (or. high Q) Fabry Perot cavity. The interferometer and nonlinear medium now share a single path, and the mirrors cause light at the cavity resonance frequencies to make several passes through the nonlinear material. A t the expense of only working at distinct frequencies, the use of the microcavity greatly reduces the size of the switch. There are additional benefits to the microcavity approach, such as the fact that it works as a bistable switch, and these will be discussed in chapter 3. The cavity's apparent enhancement of the nonlinear effect for a given amount of material can be understood in a different, but equivalent way. As with any cavity, the quality factor is defined as the ratio of the energy stored on resonance to the power flowing through, all normalized by the mode's angular frequency.  The multiple reflections at resonance allow energy to  build up within the cavity. On resonances, the local field intensity inside the cavity is enhanced by a factor of Q times the incident intensity (under CW conditions). This local field enhancement corresponds directly to an enhancement of the nonlinear polarization within the cavity.  1.3  Thesis outline  This thesis will study the nonlinear response of HRIC photonic crystal based microcavities.  The goal is to determine how the efficiencies of nonlinear  optical processes are modified by the superior control over the flow of light that is attained in HRIC periodic structures. In particular, it will be shown that nonlinear efficiencies can be significantly enhanced by the confinement  Chapter  1.  Introduction  25  of light to small volumes for a relatively long period of time. As this strong confinement of light is only achievable in HRIC periodic structures, these materials have the potential of being the host medium for future all-optical devices. Second harmonic generation involving the resonantly bound modes of planar photonic crystals will be studied in chapter 2. It will be shown that the nonlinear response is microcavity-like, and that the efficiency of the conversion process can be greatly enhanced in comparison to untextured slabs. In chapter 3 the geometry of a microcavity coupled to a single mode waveguide is considered. This geometry is of particular importance for integrated optical circuitry. A new and simple-model for describing coupling in this geometry is developed. Among other advantages, the most attractive feature of the model is that the nonlinear response of the cavity can be included in an intuitive and analytic fashion. It is found that for a moderately high-Q cavity it is possible to observe a nonlinear response in the cavity with input powers that are achievable with current telecommunication light sources. In chapter 4, a coupled waveguide-cavity system is designed and fabricated. Using 100 fs pulses the nonlinear properties of this system are experimentally studied. When nonlinearities in the waveguide dominate over those of the cavity, measurements reveal response functions that are particularly suited for all-optical logic applications. The nonlinear response of the cavity when excited with ultrashort pulses is studied by doping the cavity with highly nonlinear quantum dots. Chapter 5 provides conclusions and a discussion of future work.  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 26  Chapter 2 Second Harmonic Generation in Planar Photonic Crystals 2.1  Introduction  This chapter describes a formalism developed by the author to calculate the efficiency of specularly reflected second harmonic (SH) generation (SHG) from two dimensional (2D) high index contrast membrane-like photonic crystals. Figure 2.1 schematically illustrates the geometry of relevant experiments. A harmonic, CW laser beam at frequency u is incident on the textured slab from the top half space. The formalism described below evaluates all of the scattered fields at u and 2u>. Of particular interest here is the specularly reflected power at 2u when the photonic crystal pitch is small enough that all, or at least most of the other components of the SH field are evanescent in the top half space. Such a structure thus functions as a mirror that doubles the frequency of the incident radiation upon reflection. Such a mirror could find application as part of a photo-detector embedded within a 3D electronic or optoelectronic device. Optical information encoded in a below-band gap beam (e.g. a clock) could be delivered without absorption to this embedded nonlinear mirror, which would direct the above-band gap SH  Chapter  2.  Second Harmonic  Generation  in Planar Photonic  Crystals  27  Figure 2.1: Schematic diagram of geometry considered i n S H calculation. Plane waves w i t h frequency u are incident from the upper half space, and S H at 2u radiates away from surface as plane waves. light to an embedded photo-detector. T h e point is that the second harmonic conversion efficiency of an untextured nonlinear t h i n film is typically very low in this specular geometry. T h e work described i n this chapter, various parts of which have been published elsewhere[37, 38], thoroughly treats the manner i n which 2 D texture can be used to engineer the efficiency and bandwidth of specular S H generation for different polarizations and incident angles. There is previously published theoretical work that deals w i t h the nonlinear response of weakly corrugated (low index contrast i n I D ) t h i n films [39, 40, 41, 42, 43]. T h e new model developed i n this chapter is valid for a more general class of dielectrics that contain high index contrast 2D texture. In addition to generalizing the domain of application, the current formalism  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 28 provides an intuitive interpretation of the underlying physics, and places it clearly in the context of the photonic crystal terminology. This chapter also contains a comparison of the experimentally measured SH reflected from a 2D planar photonic crystal (PPC) designed using the model described below. The HRIC GaAs-based photonic crystal was fabricated at UBC by Weiyang Jiang and the SH experiments were done by Jessica Mondia at the University of Toronto. The overall agreement of experimental and theoretical results is excellent. The results of the model described in this chapter have also stimulated experimental work by other research groups [44, 45, 46, 47, 48]. Their results are also in good agreement with the model described below. The principal result of this chapter is that the SH specular reflectivity of thin III-V semiconductor films can be enhanced orders of magnitude by appropriately texturing them with a periodic, 2D HRIC pattern.  2.1.1  Conceptual overview of the specular S H conversion process  The 2D photonic crystal used in this chapter to illustrate the enhancement of specularly reflected SH was chosen primarily for pedagogic reasons. In particular, its linear bandstructure is relatively simple in regions relevant to the second harmonic conversion process. The sample made by Weiyang Jiang, and studied by Jessica Mondia, has a more complicated bandstructure that would be less useful for introducing the basic concepts. A portion of the calculated photonic bandstructure for the planar photonic crystal shown in Fig. 2.1 is presented in Fig. 2.2. The crystal consists of  Chapter  2.  Second Harmonic  Generation  in Planar Photonic  Crystals  29  a 130nm thick free-standing slab of GaAs (n ~ 3.5) textured with a square array of air holes, 202nra in radius, and with a spacing of A = SOOnm. The magnitude of the smallest primitive reciprocal lattice vector is therefore f3 = 2n/A. The dotted "light line" represents the dispersion of light in vacg  uum. The shaded region corresponds to the continuum of radiation modes. The solid and dashed curves represent the dispersion of the bound and leaky photonic modes of the planar photonic crystal. This bandstructure was calculated using the author's intuitive Green's function based solution of the Maxwell equations in this geometry, as described in detail in the following references[20, 21, 49]. It should be noted that for clarity, Fig. 2.2 only shows the bands relevant to the nonlinear conversion process'considered below. In general the 2D bandstructure contains more bands over the frequency range shown in the figure. If one of the crystal's eigenmodes is somehow excited at a fundamental frequency u> and in-plane wavevector 0 , a nonlinear polarization will be O  induced in the GaAs at frequency 2u proportional to its second order susceptibility, x^(—2u;u,u>). This second order polarization will have numerous (but discrete) in-plane Fourier components at 2(5 + G{ where Gi is one of the 0  2D reciprocal lattice vectors of the crystal. Each of these second harmonic polarization sheets act coherently as source terms in Eqn. (1.6), with the net result that one of the photonic crystal eigenstates at (2u; 2(3 ) will be 0  excited with some efficiency (if 2(5 falls outside the first Brillouin zone, then Q  it must be zone-folded back into the first zone to identify the SH mode that is excited). The formalism described in this chapter can be used to calculate this efficiency precisely in the limit where the overall conversion efficiency is  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 30  Figure 2.2: Linear photonic bandstructure along the X direction of the 1st Brillouin zone for the free-standing photonic crystal waveguide described in the text. For clarity only bands involved in the SHG calculation are shown. Solid curves denote s-polarized modes and dotted curves p-polarized modes. The shading represents the continuum of radiation modes. The two shaded boxes show the areas of phase space involved in the SH calculation. low enough not to deplete the pump field at u>. Although the SH formalism developed below can be used to treat the (—2co; LO,  LO)  coupling between any combination of fundamental and second  harmonic modes (e.g. bound-to-bound, bound-to-leaky, radiation-to-leaky etc.), this thesis does not consider any processes involving purely bound modes. Thus in the following sections, both the fundamental and second harmonic modes of interest are always located above the cladding light line;  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 31 the shaded boxes in Fig. 2.2 represent the region of bandstructure over which the fundamental and phase-matched SH fields are resonant with leakymode bands in the following calculation. This means that the fundamental field inside the slab can be excited directly from the top half space with a harmonic plane wave, and that the SH mode excited by the fundamental will also radiate a field into the upper half space, in the specular direction (straight lines through the origin in dispersion curves like Fig. 2.2 correspond to a fixed angle of incidence and reflection). The results discussed in this thesis are also restricted to the excitation of modes along high-symmetry directions of the photonic crystal. This is again pedagogically driven, since all of the modes along these directions have welldefined polarization labels (the radiative components of the Bloch states are either p- or s- polarized with respect to the slab), which facilitates an intuitive discussion of the polarization selection rules for SHG in these structures. The nonlinear formalism developed in this chapter is a natural extension of a Green's function based solution of the linear Maxwell equations in this 2D photonic crystal geometry previously published by the author[20, 21]. While the discussion here is limited to second harmonic generation, other members of Dr. Young's lab have used variants of this basic formalism to study the optical Kerr effect [50], Raman scattering, and difference frequency mixing in HRIC planar photonic crystals. Section 2.2 describes the self-consistent Green's function based solution to the Maxwell equations for the second harmonic field radiated from a textured slab when excited with plane waves incident onto the upper surface of the slab. In section 2.3.1 a sample calculation is presented that quantitatively  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 32 illustrates the effect of the periodic texture. Section 2.3.2 elaborates on how the conversion efficiency depends on the Q values of the relevant photonic modes, while section 2.3.3 focuses on the relative importance of out-coupling the nonlinear polarization via radiation and leaky modes. A comparison to experiments carried out by collaborators is presented in section 2.3.4, and section 2.4 concludes the chapter.  2.2  Theory  The calculation of the radiated SH from arbitrary 2D textured planar photonic crystals (PPCs) proceeds as follows. Taking the periodic polarization in the textured region as an inhomogeneous driving term, the Maxwell equations, in the waveguide geometry, are solved using a Green's function technique. Consider a homogeneous driving electric field, E (u}, hom  f3i , z), with nc  frequency, ui — u/c, and in-plane wavevector, /3j . This driving field, which nc  is assumed to be excited by some external light source, could be any one of the radiation modes of the vacuum. When this driving field is incident onto the periodic slab, the self-consistent solution for a single Fourier component of the field in the grating {—L/2 < z < L/2) is given by _  -, '  'r/  E(u;P z) =,.E (u,p z)+ u  hom  u  L 2  J-L/2  ••<->•-.'  dz'9 (P u,z,z') u  j  where $ = /3  inc  — Gi is the in-plane wavevector of the Fourier component, Gi  are the reciprocal lattice vectors, and L is the thickness of the grating. The periodic susceptibility has been expanded as a Fourier series with coefficients  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 33 Xij (z) (i.e. Xij (z) = J dp X (p, z)exp[-i(Gi  - Gj) • p[).  9  0 u,z,z') u  is the Green's function that propagates the electric fields radiated by the polarization source at z' to the point z, within the slab (—L/2 < z < L/2). An accurate, computationally efficient, solution to this infinite set of integral equations is obtained by converting them to a finite set of scalar equations through the following approximations. First, the textured region is assumed to be sufficiently thin (L «  27r/(<Dn ), n being the average refrac9  g  tive index of the textured layer) so that the electric field can be considered as constant over its extent; E(z) —» E(z ) 0  where z denotes the centre of the Q  grating. The grating is assumed to be symmetric (i.e. vertical side walls), so that Xij (z) -*Xij is independent of z over the extent of the grating. Finally, the in-plane field structure is approximated with a finite number, N, of Fourier components. Projecting onto the orthogonal {si,(3i,z}  (see Fig.  2.1) directions the infinite set of equations is reduced to a finite set of 3N scalar algebraic equations. Unit vectors §i and $ are labeled with subscript i as they are unique for each Fourier component (i = 1..N).  If required,  thick gratings can be modeled by splitting the textured region into I thinner regions, each of which satisfy the constant field approximation. When this is done the set of scalar algebraic equations to solve is of size 3NI. The simple matrix algebra required for the solution of the 3NI set of equations, and the derivation of the Green's function 9 (A,u>, z, z'), are described in detail in references [20, 21]. These Fourier field components are then used as the source fields in calculating the Fourier components of the 2nd order polarization in the textured  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 34 region: P( \2uj ,z)  = £  nl  n  (-2w,w,w) :  xl  771,/  E(u>, p  inc  - {G - Gi), z)E(u, p m  inc  T h e in-plane wavevectors of p( \(3 )  - G z)  are given by /?„ = 2/3  nl  n  (2.2)  h  inc  — G. n  Note  -(2)  that X  nm  is a 3rd rank tensor whose components are labeled by the symmetry  axes of the electronic lattice. T o obtain the proper vector components of P i n  one has to rotate the fundamental field into Cartesian coordinates defining the electronic lattice, or the susceptibility tensor into the P P C natural coordinates of {SJ, 0i, z}. T h e vector components of P i w i l l therefore depend on n  the angle between the P P C symmetry axes and the electronic lattice axes. In the following calculation the angle between the [001] axis of the electronic lattice and the F — X (see inset to F i g . 1.6) direction of the P P C lattice is assumed to be 20°. See section 2.3.3 for further discussion of this point. Using b o t h this nonlinear polarization, as well as the linear polarization at (2u), p ) n  as inhomogeneous driving terms, the Fourier components of the S H  field in the grating can be self-consistently calculated using a similar Green's function approach. Therefore, w i t h i n the undepleted p u m p approximation, the S H Fourier components i n the grating satisfy: E(2U,  •E  P , Z) = J-L/2 n  xl  /  ( - 2 w , 2u)E(2u,  dz' 9  (P , 2U, Z, z') n  p , z') + /*<«<)(JM), L m  z')}  (2.3)  m  In this equation, the nonlinear polarization term, p( \ nl  is acting as the ho-  mogeneous driving field for fields at 2a>, i n direct analogy to E  h o m  i n the  linear equation (2.1). T h e difference w i t h this driving term is that its source  Chapter  2.  Second Harmonic  Generation  in Planar Photonic  Crystals  35  is internal to the slab rather than being excited by external sources. Secondly, rather than a single specular driving component  (Pine),  there are N'  (number of Fourier components used to approximate the field oscillating at 2cu) coherent driving terms each with a unique wavevector p . n  By using N' Fourier components for the in-plane field structure, and /' thin textured and/or nonlinear regions over which the field is effectively constant, equation (2.3) is reduced to a system of 3N'l' algebraic equations and solved with an analogous matrix method as was used for equation (2.1). The Fourier components of the total SH field in the textured regions are then transferred to the top surface of the structure by E (2u,P ,z )= UHS  n  YI /  t  i = 1  &  ..,, Aj-tj/2  (Pn, 2w, z z')[^  X  u  nm  E(2u>, p , zl) m  m  (2.4)  + pM(2u>J zi)] ni  where z denotes the surface of the structure, z and t respectively denote t  l  l  Q  g  the centre and thickness of textured region i (i.e.  Yli=i..v t  l g  = L),  and 9uHS,i  is the Green's function which propagates the polarization from inside region i to the surface; it is also derived in Refs.[20, 21]. Subscript UHS refers to the upper half space, and serves to distinguish this electric field and Green's function from those internal to the slab. As before this equation is solved by making some simple approximations in order to convert it to a finite set of algebraic equations. The specular component, EUHS(2U,  2{3 , z ), of this inc  t  solution is the SH field propagating away from the slab. The field E(2u!,P ,z ) in Eqn. (2.4) is the total electric field in the i  m  0  textured region and is self-consistently given by (2.3). Physically this field  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 36 corresponds to light that, once radiated from the internal polarization source p( \ nl  experiences further 2D Bragg scattering by the texture, as well as lat-  eral confinement to the slab by T I R . The first term on the right hand side of Eqn. (2.4) describes how this field scatters from the texture to couple to a radiation mode that carriers the field away from the slab. The second term physically corresponds to light that, once radiated from the internal polarization source, directly couples to a radiation mode that carries light away from the structure, and does not first undergo T I R and/or Bragg scattering. It is the relative strength of these two contributions to the radiated S H field that is the topic of section 2.3.3.  2.3  Results and discussions  This section presents the predictions of the model and discusses the physical processes that determine the overall S H G conversion efficiency in planar photonic crystals. Subsection 2.3.1 presents model results and describes the origins of the enhancement. Subsection 2.3.2 presents a calculation verifying the microcavity nature of the nonlinear process. A discussion of the mode overlap and out-coupling processes is found in subsection 2.3.3. A comparison between the model and experimental measurements is shown is subsection 2.3.4.  2.3.1  Mode-matched second harmonic generation  The formalism described above can be used to calculate the second harmonic fields generated in this P P C geometry under a variety of physically distinct  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 37  Bound mode Figure 2.3: Schematic illustration of end-coupling a light source in order to excite purely bound modes that exist below the cladding light line. situations. If (to, (5) of the homogeneous driving field is below the cladding light line but above the core light line, it physically corresponds to a localized (in the vertical direction) excitation within the plane of the PPC. If it is resonant with one of the purely bound bands, it can effectively be thought of as representing a bound mode that might have been launched, for instance, by end-coupling a focussed laser field onto the cleaved edge of the 2D PPC as schematically illustrated in Fig. 2.3. In the undepleted pump approximation, the SH fields generated by this bound fundamental field could in general be i) local to the slab and insignificant, ii) local to the slab but coincident with another bound mode of the PPC, in which case there would be a phase-matched conversion of photons from the fundamental propagating mode to the SH propagating mode, or iii) both local and nonlocal, when coincident with a leaky mode of the PPC. In this latter case, the conversion of fundamental photons would be to fields that both propagate in the plane, and simultaneously radiate coherently into the top and bottom half-spaces. The case of SHG due to a bound fundamental mode is not considered further in this thesis. Third order nonlinear processes involving bound modes are considered in chapters 3 and 4 using a different formalism.  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 38 Model predictions for the SH conversion efficiency when CW plane waves are used to excite the PPC structure of Fig. 2.1 above the cladding light line are now presented. The calculation considers s-polarized incident radiation oriented along the F — X axis of the photonic crystal. The regions of the bandstructure over which the fundamental incident field is varied, and the corresponding range of the SH polarization generated by these fields, are shown by the two shaded regions in Fig. 2.2. These shaded regions are plotted together in Fig. 2.4. The solid curve is the low frequency s-polarized mode from Fig. 2.2, while the dashed and dotted curves are respectively the high frequency p and s-polarized modes plotted at half their frequency and in-plane  wavevector.  As s-polarized incident radiation oriented along the X-axis of the photonic crystal is assumed, the solid curve in Fig. 2.4 represents the photonic band with which the fundamental field can be resonant. Likewise, the dashed and dotted curves represent higher lying photonic bands which the SH can resonate with when the fundamental has the frequency and in-plane wavevector given by the axes values. For simplicity, the index of refraction of GaAs was taken to be 3.5 throughout the calculation. If the actual refractive indices at both OJ and 2u were used, the location of the photonic bands and crossing points would shift slightly, but the physical processes leading to enhancements that are discussed below would be unchanged. Figure 2.5 plots the intensity of the second harmonic field radiated into the UHS when the incident fundamental field is s-polarized, and the output second harmonic field is p-polarized (s-p scattering). The calculation covers the same area of the 1st Brillouin zone shown in Fig. 2.4. Figure 2.5 is a plot  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 39  0.45 0.448 0.446 0.444 ^0.442  3 0.44 0.438 0.436 0.434 0.145 0.15 0.155 0.16 0 165 0.17 0.175 0.18  Figure 2.4: A n enlarged and overlapped depiction of the two rectangular regions in Fig. 2.2. The solid curve is the low frequency s-polarized band in Fig. 2.2. The dashed and dotted curves are respectively the high frequency p and s-polarized bands plotted at half their frequency and in-plane momentum. Circles denote points where fully mode-matched nonlinear conversion can occur. of log[ \EUHS{2U, P pec, z )\  2  S  t  ] as a function of the fundamental frequency, u),  and fundamental in-plane wavevector, (5 (f3 = 0 as the incident field is along X  y  the photonic crystal's X-axis). The subscript "spec" denotes the component of the field which is phase-matched to radiate into the UHS in the specular direction. The incident field intensity is taken to be 1 statvolt/cm, corresponds to an intensity of 475 x 10 W/m . 4  2  which  For this calculation the  130nm grating was split into two 65nm thick gratings (/' = 2 in Eqn. (2.4)).  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 40  0.434 0.145  0.15  0.155  0.16  0.165  0.17  0.175  0.18  w  9  Figure 2.5: P-polarized radiated SH from the free-standing photonic crystal waveguide described in the text. Contours show enhancement in the radiated SH when the fundamental frequency and in-plane wavevector intersect one of the bands in Fig. 2.4. Each contour corresponds to 3.92dB. These two layers produced well-converged results; converged in the sense that calculations with 3 layers gave the same results up to a fraction of a percent. The fundamental and SH fields were approximated with 81 and 25 Fourier components respectively. Table 2.1 on page 55 summarizes the parameters used for the calculation. The contours in Fig. 2.5 illustrate the enhancement in the magnitude of the radiated SH that occurs when either the fundamental or SH fields are  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 41  x10 E —  o  6-  'c  o E  x  T3 C T3  2-  4—•  -o CO DC  0  045  0445  P/P,  co/cp  Figure 2.6: Plot of the s-p radiated SH. resonant with photonic modes. Notice that the ridges follow the solid and dashed dispersion curves of Fig. 2.4. The greatest enhancement in the SH occurs when both fundamental and SH fields are resonant with leaky modes. Figure 2.6 and 2.7 show 3D plots of the radiated SH intensity for the s-p and s-s scattering processes respectively. The vertical axis is the intensity of the field on a linear scale in units of (W/m)  2  of 475 x  10  4  (W/m) , 2  and for a x  (2)  for an incident field intensity  value of 238prn/V[51]. In Fig. 2.6 (s-p  scattering), the large enhancement of the SH at the mode-matched condition, as shown in the contour plot of Fig. 2.5, is obvious. In Fig. 2.7, there is  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 42  -3  Figure 2.7: Plot of the s-s radiated SH. again an enhancement when each of the relevant fields is separately resonant with a photonic mode. Notice that for the s-s scattering process the ridges now follow the s-polarized SH modes instead of the p-polarized modes as was the case in Figs. 2.5 and 2.6. This is due to the photonic modes having a well defined, s or p, polarization. These numerical results can be qualitatively understood as follows. If the homogeneous driving field is above the cladding light line then it corresponds to a plane wave incident onto the slab. If (UJ,(5) is resonant with a leaky mode of the PPC, it will excite that mode and generate relatively large  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 43  fields internal to the slab. If (u>,(3) is not resonant with a leaky mode, it will couple simply to one of the radiative eigenstates of the PPC with no significant buildup of internal field strength. In either case, the second order polarization generated within the slab at (2u, 2(1) can in turn excite either radiation modes or leaky modes. The SHG processes excited by a plane wave incident from the upper half space (UHS) can therefore be divided into the following four categories of mode conversion: radiation-to-radiation, radiation-to-leaky, leaky-to-radiation, and leaky-to-leaky. The radiation-to-radiation mode conversion process occurs when both the fundamental field and the internally generated SH fields correspond to non-local radiation modes. As these modes experiences no T I R and minimal Bragg scattering, the HRIC periodicity has virtually no effect on this conversion process, which is therefore very weak. I t results in a SH conversion efficiency comparable to that from a uniform slab of nonlinear material only a couple of hundred nanometers thick. Most of the "background", low level signal in Figs. 2.5, 2.6, and 2.7 corresponds to this radiation-to-radiation mode conversion process. The radiation-to-leaky mode conversion process involves a SH polarization not significantly different from the untextured case. However, since this polarization can excite one of the leaky modes at (2a>,2/3), it is possible that there can be an enhancement of the total radiated power at the second harmonic. When this occurs it is because Bragg scattering of the electric fields generated by the nonlinear polarization induce additional polarization at {2u,2(3). This enhancement process is evident by the ridges in Figs. 2.5 and 2.6 that follow the higher lying photonic bands (dashed curves in Fig.  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 44 2.4). This modest enhancement (as compared to SH radiated from an untextured slab) is found to be ~ 10 . In the s-s process of Fig. 2.7, weak ridges 2  corresponding to this enhancement process are also seen (follow dotted curves in Fig. 2.4), here the enhancement is ~ 10 but is still much smaller than 3  the process leading to the large ridge evident in the figure. The leaky-to-radiation mode conversion process involves a substantial increase in the strength of the nonlinear polarization at (2u>, 2(3) due to the large local field buildup at (u, (3) when a CW plane wave excites a leaky mode. Even without resonantly exciting a localized mode at (2a;, 2(3), this local field enhancement at the fundamental frequency will' therefore lead to an enhancement of the radiated second harmonic, directly via a radiation mode. A ridge of enhanced SH that follows the dispersion of the incoming s-polarized mode is clearly evident in the s-s process shown in Fig. 2.7. This calculation finds an enhancement factor of ~ 10 over untextured waveguides. 5  A ridge of enhancement (this time ~ 10 ) associated with the incoming mode 4  is also evident in the s-p process shown in Figs. 2.5 and 2.6, although it is masked somewhat by an extremely large peak that is discussed next. Finally, the leaky-to-leaky mode conversion process involves strong local fields at the fundamental frequency that enhance the strength of the nonlinear polarization at (2u, 2(3) and the possibility of further enhancement due to this relatively strong nonlinear polarization resonantly exciting a leaky mode, which induces yet more polarization at (2LO,2(3). This "double resonant" condition, when leaky modes are involved at both the fundamental and SH frequencies, offers the greatest potential for SH enhancement. At the resonant to resonant condition in Fig. 2.5, {to/(3g = QAAA,(3/(3 = 0.164), g  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 45 the peak intensity is 1.5 x 1 0 " (stv/cm) , 9  2  1  or 7.1 x 1(T W/m , 3  2  3.83 x 10  6  times greater than the maximum intensity radiated from an untextured version of the same waveguide. In the s-s process of Fig. 2.7 the leaky-to-leaky condition does not show any further enhancement over that of the leaky-toradiation condition. The reason for this is discussed in section 2.3.3. The enhancement factors quoted above are particular to this structure, these specific photonic modes, and the incident power used in the calculation. No attempt has been made to optimize the conversion efficiency in any way. The following subsections elaborate on several of the physical processes that contribute to resonant enhancement effects demonstrated in this section.  2.3.2  Effect of the cavity Q of the incoming mode  The linear reflectivity spectrum of this PPC in the vicinity of the incoming resonance, and for an incidence angle of 30°, is shown in Fig. 2.8. The phase of the reflectivity is also shown. The PPC is clearly acting like an optical cavity with a quality factor Q ~ 134. In contrast to more traditional cavities, the input/output coupling in this geometry is mediated by a grating, and the cavity itself is otherwise lossless. More specifically, an incident plane wave is scattered by the texture into an evanescent field localized to the plane of the membrane. If the plane wave is phase matched to a leaky eigenmode of the PPC, the evanescent field will build up in the slab (via T I R ) to a level where C a l c u l a t i o n assumes the value of x  ( 2 )  for GaAs is 238pm/V[51]. Since the incident  field is assumed to have an amplitude of 1 statvolt/cm corresponding intensity of the 2nd harmonic is 7.08 x 1 0 approximation of an undepleted pump is valid.  or 475.6 x 10 W/m , 4  -3  W/m , 2  2  and the  it is concluded that the  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 46  0.4  0.41  0.42  0.43  0.44  0.45  0.46  ro/cp  g  Figure 2.8: Linear reflectivity spectrum and phase of reflected field in vicinity of fundamental resonance. Angle of incidence is 30°.  Chapter  2.  Second Harmonic  Incident  Generation  in Planar Photonic  Crystals  47  Reflected  S 81 S  Bound  Bound  transmitted Figure 2.9: Schematic illustration of grating coupling of light into Fourier components bound to the slab. rescattering from the grating couples light out (reflects it) at the same rate that it is incident. The weaker the grating coupling, the higher the Q of the cavity, and the larger the internal field strength is at resonance. Figure 2.9 schematically illustrates this grating coupling process. As long as the exciting beam diameter is greater than the in-plane decay length of the leaky mode (which is clearly satisfied for the plane wave excitation considered here) then there is no appreciable in-plane propagation before the light scatters out of the slab via the grating. Hence the textured slab will support an electric field strength that is enhanced with respect to the incident field strength by an amount proportional to the Q of the mode. To demonstrate the large influence of this local field enhancement of the fundamental field on the overall SH conversion associated with the incoming resonance, the following artifice was adopted. To keep the internal conversion efficiency and phase matching conditions fixed, while substantially varying  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 48 the Q of the incoming resonance, model calculations for the same 130 nm thick textured slab are repeated, but they now include a remote GaAs substrate, separated from the textured slab by a vacuum layer with variable thickness d (see inset to Fig. 2.10). This approach is based on previous work (Ref.[20]) where it was reported that the linewidth of resonant coupling in 2D textured planar waveguides can be controlled over a wide range by varying the location of the grating within the slab waveguide structure, while leaving the resonant frequency almost unchanged. The linear bandstructure of this modified structure is effectively the same as shown in Fig. 2.2 so long as the vacuum layer thickness is greater than the evanescent decay length of the modes attached to the slab. However, the Q of the incoming s-polarized resonance changes substantially as the thickness of the vacuum layer is varied, as shown by the solid curve in Fig. 2.10. The dashed curve in Fig. 2.10 shows the intensity of the upward propagating component of the SH field generated by the fundamental mode excited in this geometry. The SH field is for the s-s scattering process, and the inplane wavevector is held constant at (3/f3 = 0.164x as the cladding thickness g  was adjusted. The strength of the SH clearly correlates strongly with the Q of the incoming resonance. The slight phase shift is due to the fact that some of the other factors involved in the self consistent calculation are also influenced by the vacuum cavity, but to a much smaller extent. This calculation verifies that the enhancement in the radiated SH field is directly related to the Q of the incoming leaky mode. As this is a measure of the field strength of the fundamental field in the slab, it is clear that nonlinear conversion efficiency in this HRIC periodic structure is strongly influenced  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 49  Cladding Thickness (d) [nm] Figure 2.10: The dependence of the radiated SH on the Q of the incoming photonic mode. The solid curve is the Q value of the fundamental s-polarized mode (right axis), and the dashed curve is the peak intensity of the s-s radiated SH (left axis). The inset depicts a schematic cross section of the PPC illustrating which layer thickness is being varied. by this local field enhancement. This can be easily understood from Eqns. (2.3) and (2.4), where it is clear that when all symmetry effects are held constant, the radiated SH is directly related to the strength of the nonlinear polarization. But from Eqn. (2.2), P i is proportional to the internal electric n  field intensity at the fundamental frequency, which is in turn proportional to the leaky mode Q value when on an incoming resonance.  Chapter  2.3.3  2. Second Harmonic  Generation  in Planar Photonic  Crystals  50  Influences other than local field enhancement  Local field enhancement is not the only process that contributes to the overall SH conversion efficiency. In Eqns. (2.3) and (2.4) for instance, the magnitude of the nonlinear polarization might be fixed, but its vector orientation with respect to the Green's tensor can still have a major impact on the overall radiated SH intensity. Also, from Eqn. (2.2), depending on the relative orientation of the electronic and photonic crystal axes, a given fundamental field strength inside the slab can generate very different SH polarization intensities. This subsection illustrates some of these additional influences with reference to the model results presented in the previous section. The symmetries of the photonic and electronic crystals, and their relative orientation can have a big impact on the nonlinear polarization generated under fixed excitation conditions. The local field enhancement considered in the previous section depends on the scalar dielectric constant of the GaAs, and the photonic crystal: it does not depend on the orientation of the electronic crystal. However, the magnitude of the SH polarization can actually be zero in the presence of strong local fields when the two crystals are aligned. For example, in this work the nonlinear waveguide is assumed to be made of GaAs, and the photonic crystal to have square symmetry. The GaAs crystal is of cubic symmetry and a member of the 43m point group. Therefore, when the [001] axis of the GaAs crystal is aligned with the X-axis of the square symmetric photonic lattice, it is not possible to excite p-polarized SH modes along the T — X direction of the square Brillouin zone. Likewise, for the same orientation of the two lattices, no s-polarized SH field is generated along the r — M direction. Thus along these symmetry axes, the fundamental polariza-  Chapter  2.  Second Harmonic  Generation  in Planar Photonic  Crystals  51  tion will either be preserved or rotated by 90° in the SH when the photonic and electronic crystal axes are aligned. For arbitrary orientations of the two lattices, either s or p-polarized incident radiation will in general produce an elliptically-polarized SH field. For a given magnitude and orientation of the nonlinear polarization, as determined by the excitation conditions, the photonic crystal parameters, and the relative orientations of the electronic and photonic crystal axes, the remaining factors that determine the overall conversion efficiency can be loosely classified as internal conversion and out-coupling efficiencies. The internal conversion efficiency is a measure of how effectively the nonlinear polarization density at 2u generates internal electric fields at 2u> (Eqn. (2.3)). The outcoupling efficiency has to do with how effectively the nonlinear polarization couples out of the slab. It can directly radiate (described by the second term on the right hand side of Eqn. (2.4)), or indirectly radiate by first exciting a new leaky photonic mode (described by the first term on the right hand side of Eqn. (2.4)). The direct out-coupling efficiencies (2nd term on the right hand side of Eqn. (2.4)) of the nonlinear polarization generated along the incoming resonance of Figs. 2:6 and 2.7 can be clearly seen to favour the s-polarized SH radiation mode. This is because all along the incoming resonance, where the nonlinear polarization in both graphs is identical, the radiated intensity is far greater in the s-polarized output channel away from the double resonance conditions. Further comparison of these two graphs near the double resonances reveals evidence of the two out-coupling pathways. Near the resonance condition in Fig. 2.6, it is clear that the internal conversion efn-  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 52 ciency of the nonlinear polarization to the p-polarized outgoing leaky mode is high, since the double resonant peak is much larger than the incoming resonance alone. In this case the indirect out-coupling process dominates. In contrast, the internal conversion efficiency of the nonlinear polarization to the s-polarized output leaky mode is almost nonexistent in this example. There is virtually no additional enhancement, of the incoming resonance on going through the resonant s-polarized outgoing modes (no increase in SH at mode-matched conditions in Fig. 2.7). It is the direct term that dominates in this case,.  '  ,  -  '  To illustrate what happens when the direct out-coupling efficiency is comparable to the internal coupling efficiency to the outgoing resonant mode, Fig. 2.11 shows a double resonance calculated for the fabricated sample discussed in the following section. The dominant ridge in this contour plot is the incoming resonant enhancement via direct coupling to radiation modes. The prominent Fano-like feature at the double resonance condition occurs because the phase of the directly coupled SH radiation is only slowly varying over this range of frequencies, whereas the SH radiation due to scattering of the outgoing leaky mode excited by the nonlinear polarization changes phase by 7r radians as the frequency is swept through this outgoing resonance. In this case the peak resonant coupling efficiency is comparable to the direct, nonresonant conversion efficiency, so the two interfere destructively on one side of the resonance, and constructively on the other. This completes the discussion of the processes that determine the overall conversion efficiency for radiated SH generation involving resonantly bound modes of HRIC planar photonic crystals. While all processes are not easily  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 53  Figure 2.11: Calculation of radiated s-polarized S H field from the fabricated membrane P C when excited w i t h s-polarized radiation. T h e superposition of the non-resonant and resonant out-coupling processes gives rise to a Fano-like feature at the mode-matched condition. separated i n the fully self consistent model, the above discussion presented examples that illustrate the processes of local field enhancement, internal conversion, out-coupling strength, and a v i r t u a l phase matching between photonic modes.  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 54 Parameter  Theory  Experiment  L  130 nm  140 nm  A  800 nm  770 nm  hole diameter  404 nm  320 nm  r - X to < 001 > angle  20°  65°  oxide thickness  none  950 nm  oxide index  N.A.  ~ 1.6  Table 2.1: Material parameters for sample consider in theoretical calculations above, and those for the sample fabricated by Weiyang Jiang.  2.3.4  Comparison with experiment  This subsection presents a comparison between the model described above and experimental measurements of specular SH generation from 2D HRIC planar photonic crystal waveguides. A planar waveguide structure very similar to the one described above was designed and fabricated in the clean room facility at UBC by a M.Sc. student, Weiyang Jiang. The main difference is that rather than being fully clad with air, the real sample is clad below with a ~ 1 pm thick aluminum oxide layer on a thick GaAs substrate[38, 52]. A l l of the parameters for both samples are tabulated in table 2.1. This sample was designed to exhibit the double resonant enhancement effect described above. Using the linear and nonlinear codes developed by the author [20, 21, 37], an appropriate crystal was found that had a low order leaky band that could be resonant with a fundamental excitation beam at a wavelength near 2/xm. This allowed the higher order band resonant with  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 55  20  25  30  35  40  INCIDENT A N G L E (deg)  Figure 2.12: Contour plot of the calculated radiated p-polarized S H from the fabricated waveguide structure. Dashed lines show the dispersion of photonic mode that the S H field can be resonant w i t h . Labels A , B , and C refer to points of mode-matched conversion. the S H polarization to also fall below the bandgap of G a A s , thus avoiding complications due to absorption. Figure 2.12 shows the model prediction for the specularly reflected S H as a function of the energy and incident angle of the fundamental excitation beam. Note that the S H signal from this sample exhibits very similar features to those described i n the preceding section. T h e experimental measurements of the specularly reflected S H from this sample were done by Jessica M o n d i a , a P h . D . student i n Professor Henry van  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 56 Driel's group at the University of Toronto. The experiments were done by focusing 150/s fundamental pulses with a bandwidth of ~ 2 1 0 c m , onto the -1  surface of the structure at a repetition rate of 250kHz. The centre wavelength of this beam was tunable around 5300 c m , and the beam was focused to a - 1  35 pm (FWHM) diameter spot, with an average power of 1.6 mW. The angle of incidence was varied from 16° < 9 < 40° in a plane defined by the sample normal and t h e P — X axis Of the PC. The SH spectra were analyzed and recorded with a monochromator and an InGaAs detector cooled with liquid nitrogen.-  '  <"•.-'•'  A contour plot of the p-polarized SH spectra obtained by varying the centre wavelength and incident angle of the fundamental is shown in Fig. 2.13. It is clear that there is very good qualitative agreement between the experiment (Fig. 2.13) and the model results described above (Fig. 2.12). The locations of the strong incoming resonant ridge, and the doubly resonant peaks agree to within a few percent. This level of error results from uncertainties in the precise material parameters used in the model, which leads to band energies that are off by a few percent despite carrying out detailed fits of the measured linear bandstructure [38]. The higher frequency bands discussed here are particularly sensitive to the precise values of these parameters, as well as to disorder effects. The main discrepancies between the model and experiment are the linewidths, and details of the SH intensity variation along the incoming and outgoing resonances. Considering that the experiment involved short pulses with a bandwidth larger than the resonant linewidths, whereas the model deals with harmonic plane waves, the overall agreement is quite remarkable.  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 57  INCIDENT ANGLE (deg)  Figure 2.13: Contour plot of the measured radiated p-polarized S H from the fabricated waveguide structure. Labels A , B , and C refer to points of mode-matched conversion. More than half of the increased linewidth can be accounted for by averaging the plane wave model results over the appropriate range of in-plane wavevectors that mimic the 35 /zra spot size. Fabrication imperfections i n the sample also contribute to the line broadening (and correspondingly lower Q values). W i t h regard to the S H intensities, the background (or fully non-resonant) S H signal consistently fell below the noise floor of the detector so it was not possible to extract absolute enhancement factors. However, peak A i n F i g . 2.13 is 1200 times stronger than the noise floor. T h i s therefore represents a lower limit to the actual enhancement  obtained under double resonant  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 58 conditions. Even if the enhancement factor could be accurately measured, it would almost certainly be less than the calculated value. This speculation is based on the relatively large measured linewidths that correspond to lower Q values, and thus lower local field enhancement factors. The fact that the relative intensities of peaks A and B are different in the model and experiment is not unexpected because the precise SH intensity is very sensitive to the Q values of the leaky modes. The net impact of imperfections and the nonplanar excitation on the effective Q values is different for different bands. The pulsed nature of the excitation also reduces the local field enhancement in a Q-dependent fashion (this is discussed further in chapter 4). In general, the experimental results described in this subsection largely substantiate the predictions of SH enhancement made using the model described earlier in this chapter.  2.4  Chapter summary  In summary, the efficiency of the specular SH excited by plane waves incident on 2D textured semiconductor membranes has been studied theoretically and experimentally. The influence of leaky photonic eigenstates attached to the porous membranes is evaluated using a Green's function technique that self-consistently includes both the linear and nonlinear polarizations in one heuristic calculation. The combined effects of quasi phase-matching and strong local field enhancements allow the SH conversion to become strong particularly when the fundamental and SH fields are resonant with leaky photonic eigenmodes of the planar photonic crystal. The strong dependence  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 59 of the SH conversion efficiency on the Q factor of the incoming resonance illustrates that a significant amount of the enhancement is essentially due to the microcavity nature of the leaky modes. The SH polarization generated by the local fields at the fundamental frequency can be coupled to the specular radiative SH mode either directly, or indirectly via excitation of a high-lying photonic crystal band. The overall SH conversion spectrum is strongly influenced by the relative strength of these two out-coupling processes. Experiments carried out by collaborators on samples fabricated at UBC support the predictions made using this nonlinear model. The basic formalism can be easily extended to higher order nonlinear processes, and to include the truly bound (infinitely long lived) modes outside of the light cone. Periodically textured thin films have been designed by others as notch filters, using both I D [53, 54] and 2D gratings[55, 56]. The results of this chapter indicate that it is possible to design nonlinear resonant reflection filters with such textured thin-films. When light incident onto the structure is sufficiently intense, the reflected signal will contain a SH component that could be detected independent of the fundamental reflection. Such nonlinear filters could be used in optical sensing applications. More generally, this chapter illustrates that the strong confinement of light to sub-wavelength dimensions that is achievable in HRIC periodic structures can result in large enhancements in the efficiency of nonlinear processes. To fully explore the impact this might have on optical circuitry, the next chapter considers the nonlinear response of an ideal, high Q, low mode volume, photonic crystal based 3D microcavity. In contrast to the free space excitation considered  Chapter 2. Second Harmonic Generation in Planar Photonic Crystals 60 above, the cavity w i l l be excited by evanescently coupling it to a single mode waveguide channel.  Chapter  3.  Waveguide and Nonlinear Resonator  Coupling  61  Chapter 3 Waveguide and Nonlinear Resonator 3.1  Coupling  Introduction  This chapter describes a thorough theoretical analysis of a linear, single mode optical waveguide coupled weakly to a nonlinear 3D microcavity. The scattering geometry, which also includes a nonresonant scattering source downstream of the microcavity, is illustrated schematically in Fig. 3.1. A n analytic expression for the nonlinear reflectivity of harmonic modes launched from the left hand side of this structure is obtained when the nonlinear response of the material is described by the real and imaginary parts of its third order susceptibility. This result is expressed in terms of several "matrix elements" that couple distinct optical eigenmodes of the waveguide and resonator via the linear, in general HRIC dielectric susceptibility that defines the waveguide and the cavity, and the nonlinear dielectric susceptibility. In many respects this represents a significant generalization of much earlier work that considered the third order nonlinear transmission of I D Fabry Perot resonators [10, 11, 57]. As indicated in chapter 1, recent advances in microfabrication technol-  Chapter  3.  Waveguide and Nonlinear Resonator  Cavity  Coupling  62  Background PC  h  Waveguide  Source of  Figure 3.1: Schematic diagram of the scattering geometry. A single-mode high Q cavity is side-coupled to a single-mode I D waveguide that contains a downstream scattering centre, which is the source of Rnr i n section 3.2.2. ogy allow these microcavity-based structures to be realized w i t h mode volumes less than a cubic wavelength, and corresponding Q values i n excess of 10 .  T h e principal objective of this study was therefore to determine  4  whether Fabry Perot-like bistable switching might be obtained i n this wavegu i d e / 3 0 microcavity geometry at optical power levels compatible w i t h planar lightwave circuitry. T h e answer is "it should be", based on the offresonant intrinsic  values of optoelectronic semiconductor materials such  as A/o.i8 j ao.82^4 t an operating wavelength near 1.5fim. <  ;  s &  T h e basic HRIC waveguide-resonator coupling problem has recently received considerable theoretical and experimental attention i n the linear response  regime.  Experiments have successfully coupled waveguides  to  resonators[6, 58, 59, 60]. M u c h of the relevant modeling work is based on finite difference time domain ( F D T D ) simulations[3, 61]. A n a l y t i c a l treat-  Chapter  3.  Waveguide and Nonlinear Resonator  Coupling  63  ments have used either coupled mode theory[62, 63], or a Hamiltonian-like formulation of the Maxwell equations wherein a first order differential equation is solved in terms of spinor-like photonic eigenmodes that included both electric and magnetic field components[30, 64, 65, 66]. A perturbative solution of this Schrodinger-like equation results in an eloquent description of the coupling in these structures. There are two reasons why a different approach was developed in the present work. First, as in chapter 2, most of nonlinear optical theory is based on solving the electric field wave equation with nonlinear polarization driving terms. Second, there is an ambiguity in the explicit expressions for optical "matrix elements" that appear in published versions of the Hamiltonian-like approach. This chapter describes a completely new formulation for describing waveguide-resonator coupling. It is based on a Green's function solution to the electric field wave equation rather than a Hamiltonian-like approach. In the linear regime the formulation results in well defined overlap matrix elements and it can therefore be directly used to simulate realistic structures. In fact, the reason for the ambiguity in the Hamiltonian approach is revealed in the present development. The analytic' model provides an intuitive description of waveguide-resonator coupling. Most importantly, the new formulation makes it relatively straightforward to analytically include nonlinear optical effects. The model describes the nonlinearity in the microcavity using a complex third order susceptibility. The real part represents a field dependent index of refraction, the optical Kerr effect. Earlier work on Kerr-active optical cavities showed that the index change induced by stored field energy near resonance  Chapter  3.  Waveguide and Nonlinear Resonator  Coupling  64  can render the transmission through the structure multi-valued for a range of incident frequencies [10]. It has been proposed that the resulting hysteresis loops (output power vs input power at a fixed frequency) can be used to control the transmission through the structure "all optically". However, it has also been known for some time that the utility of this effect is limited by linear and nonlinear absorption[67]. The linear absorption can be neglected if the operating frequency is well below the bandgap of the semiconductor used to fabricate HRIC waveguides. It is impossible to completely avoid nonlinear absorption (multiphoton absorption), and in GaAs, InP and Si, the two photon absorption (TPA) represented by the imaginary part of the third order susceptibility is expected to dominate when operating near 1.5/xm. Finally, in any realistic optical circuit, back reflections from downstream devices on the chip can not generally be ignored. Previous work by Mohamad Banaee[50], who generalized the nonlinear model described in chapter 2 to include a third order nonlinearity, showed that these nonresonant back reflections have a profound impact on the nonlinear hysteresis curves associated with leaky mode excitation in 2D PPCs. By including these downstream reflections in the current, analytic model, it is possible to more fully explore their nature, and potential utility in nonlinear optical devices. Section 3.2 presents the derivation of the nonlinear analytical model. The predictions of the model are discussed in section 3.3. The work presented in this chapter is published in Ref.[68].  Chapter 3. Waveguide and Nonlinear Resonator Coupling  3.2  65  Theory  A detailed explanation of the Green's function based solution to the electric field wave equation, and the approximations used to reduce the solution to an analytic form are most easily described in the linear optics regime. This is done in section 3.2.1. The solution is reduced to an analytic form by expanding both the electric field and the Green's function in an intuitive set of basis states. The set of basis states can be rigorously justified within the regime of weak coupling between the cavity and waveguide, as described in Appendix A. In section 3.2.2 the solution is generalized to include a downstream scattering source in addition to the resonant cavity. The addition of the nonlinear terms, as described in section 3.2.3, is quite straight forward.  3.2.1  L i n e a r c o u p l i n g regime  This subsection is organized as follows. First the equations to be solved are set up. Following that is a detailed discussion of the approximations made in defining a set of basis states used to expand both the Green's function and the electric field that solves the wave equation. After evaluating the reflected and transmitted fields there is a discussion of how this derivation differs from related linear reflectivity calculations already present in the literature. While this derivation assumes the general case where both the waveguide and the microcavity are surrounded by a photonic crystal, the solution is valid for any waveguide-resonator system that satisfies the weak coupling criterion, and whose resonator exhibits low radiation losses (high-Q).  Chapter  3.  Waveguide and Nonlinear  Resonator  66  Coupling  Waveguide-cavity system Starting from the macroscopic Maxwell equations it is straightforward to show that the electric field, V  x V  E(r,u>),  x E(f,u)  satisfies the following wave equation,  =0J D(r,uj) 2  (3.1)  = u e (r)E(f,u) 2  t  where the fields are assumed to have a harmonic time dependence, e \ wt  OJ/C,  where c is the speed of light in vacuum, and e (f) t  OJ =  is the dielectric  constant distribution that includes the background photonic crystal, the I D waveguide, and the localized cavity. The transmission and reflection can be found by solving Eqn. (3.1) for the electric field at the ends of the waveguide, x —> oo for transmission and x —> — oo for reflection, assuming that a waveguide mode was launched from one end (x = —oo). To facilitate a Green's function solution the total dielectric constant is expanded as e (r) = e (f) + 47rx (0> where e (r) is the dielectric constant 0D  t  w  w  of the photonic crystal including only the I D line defect waveguide, and X (f) 0D  =  describes the change in the dielectric constant  {e (r) — e (r))/4ir t  w  that is needed to further introduce a OD cavity. Equation (3.1) can then be written as: (V x V x -u e (r))E(r,oj) 2  w  = 4Troj x (r)E(r,oj) 2  0D  (3.2)  Note, in the above, and therefore for the rest of the derivation, it is assumed that the cavity is side-coupled to the waveguide. However, with a minor redefinition of e (f), w  the formulation can easily be modified to treat the  case of a cavity imbedded within the waveguide. Fig. 3.1 is a schematic  67 Chapter  3. Waveguide and Nonlinear Resonator  Coupling  illustration of the waveguide-resonator geometry considered in the following calculation. A simplified notation is achieved by converting to an operator formulation of Eqn. (3.2) (£-a) e )|*)-47rcu x |*) 2  2  (3-3)  0 D  U )  where C = V x V x , the operators e and x°  D a r e  e (f)8^(r w  - r') and  (r\x \f') 0D  - ?') respectively, where  = x° (r)S (r D  defined as ( f | e | f ) = w  w  (3)  5^(f — f') is the three dimensional Dirac delta function. The vector electric field is given by ( f | ^ ) = E(f,u>). Since £ is a linear self-adjoint operator over real space, the homogeneous part of (3.3) defines the orthonormal set of eigenstates of a PC containing a I D waveguide: (3.4) where u)j are eigenvalues of the eigenstates \4>i). These eigenstates can be calculated with a variety of techniques; one common approach, is numerical FDTD simulations. The completeness and orthogonality relations for these eigenstates are:  <  ($i\i \$j) w  ,  = Sij  •  ,  (3.5)  where 1 is the unit tensor, and Sij is the Kronecker delta function. The sum over i in Eqn. (3.5) is over all possible (physical and unphysical) solutions of Eqn. (3.4). The Green's function solution to Eqn. (3.3) is based on the full Green's  Chapter  3. Waveguide arid Nonlinear Resonator  68  Coupling  function, G, characteristic of a PC containing both the I D and OD defects: |*> = \$ )  +  hom  4nLd Gx \$ ) 2  0D  hom  (3.6)  where l^' ' ), the homogeneous solution, is an eigenstate of the system de10 71  fined by Eqn. (3.4). The Green's function G ( f , f ) = (r\G\f') is defined by . . - . ••• . ; , (C - u i 2  w  - ATXU )G  = {£-  2 0D X  oj e )G 2  t  =1 .  (3.7)  The solution for the electric field of the coupled waveguide-cavity PC is then reduced to finding the Green's function that satisfies Eqn. (3.7) and using it to solve Eqn. (3.6).  The Green's function A simple analytic solution of this problem, valid in the limit of weak coupling, can be obtained by expanding G in terms of a restricted set of intuitively chosen basis states. Quite generally, G can be uniquely expanded in terms of any orthonormal basis, {|</> )}, as n  G = ^2b ,m\<j>n)(<£m.\, n,m n  (3.8)  where both sums extend over all states in the basis. The restricted, intuitive set of basis states is found as follows. Assume that the cavity introduced by x 0 £ \ in the absence of the I D waveguide, supports only one OD localized mode. This mode is at a frequency Defining  td{r) — e {f) — 4 7 r x ( ^ ) 1D  t  u>i,  and is.denoted by the eigenket  \(f>i).  as a dielectric function that describes the  background PC and just the OD cavity, the homogeneous equation that the localized state satisfies is: £ $ ) = w ? e  d  $ ) .  (3.9)  Chapter  3.  Waveguide and Nonlinear  Resonator  Coupling  69  It follows that the localized state is normalized as (4>i\id\<pi) = 1. Also assume that in the absence of the OD defect, x  W  supports only one  band of I D waveguide modes, labelled ki, in the frequency range of interest. The subscript i runs from zero to infinity, representing the infinite number of distinct wave vectors of the I D guided modes, denoted by \4>ki)- These are solutions of (3.4). The localized mode eigenstate  —*  is normalized as follows:  <f>i{f)  = -i^=m  (3.io)  V *mode  where vi(f) is a unitless function that describes the shape of the localized eigenstate. It might be obtained from a FDTD calculation, for example. The effective mode volume, V de, is given by the normalization condition mo  following Eqn. (3.9): Vmode-  dre^m^?-  f  (3-11)  J all space  To be consistent with earlier definitions of the mode volume we assume that the maximum of the product ed(r)\vi(f)\  is scaled to unity.  2  For the guided mode it is natural to express the orthonormal states as k(r~) = - ^ u where  k  i  {f)^\  (3-12)  ( f ) = u*fc (r + Ax) is a unitless Bloch function periodic along the i  direction of the waveguide, L is the length of the guide, and A ff is an effective e  area of the mode. Substituting this form for ^ ( r ) into the orthogonality relation for the guided modes, and converting the integral over all space to one over a unit cell by multiplying by N = L/A, the number of unit cells,  70  Chapter 3. Waveguide and Nonlinear Resonator Coupling one finds *S =  A  \i  „^(r1fe(0|, 2  (3.13)  1 \ J unit cell  where the maximum of e^(r)|Mfc (f)| is scaled to unity. A is the length of 2  i  the unit cell along the waveguide direction. If these I D and 0D defect modes all exist well within the photonic bandgap of the host PC, then it is a good approximation to neglect all propagating, bulk PC modes in the expansion of G. Hence the infinite set of I D waveguide modes, and the single 0D localized mode comprise the intuitive set of basis states. Because | ^ ) and \4>i) are solutions of  different  wave equations, they  are not strictly orthogonal. If this is disregarded, the derivation proceeds and {10**)}. Substituting this  by expanding G, using (3.8), in terms of  expansion into the defining Eqn. (3.7), and projecting onto the state (4>i\ from the left and e \4>j) from the right, one arrives at t  Yl[ul(4>i%\4>n)  -  v ($i\e \$n)}bn, ($m\et\(£j) 2  t  m  (3-14)  = {fa\it\$j),  n,m  where here the subscripts i and j refer to any state n in the basis. In the first term, i = e if n = {ki} corresponding to a waveguide mode, and e = id if n  w  n  n = I, corresponding to the 0D localized mode. Equation (3.14) can be written in matrix form as follows: MbT=T  •  (3.15)  For the physical scattering problem of interest here, T will have an inverse, so the expansion coefficients of G are given by  Chapter 3. Waveguide and Nonlinear Resonator Coupling where det(M) is the determinant of matrix M , and A  m > n  71  is the cofactor of  element M „ . Directly from (3.14), the elements of matrix M are m i  M , m  = u ((f) \€ \(f) ) - o ) 2 ( 0 | e t | 0 n )  (3.17)  2  n  m  w  m  n  for n ^ I and M ,, m  =  (3.18)  u ($rn\£d\$i}-tf($ \eM 2  m  for n = I. However, unitarity (M ,i = M* .) requires that ki  k  (3.19)  u?(<j>kMd\<j>i) = u)l.{$i\e \4> )* = ul.($ \e \$i), w  ki  ki  w  which means that there is in fact no distinction to be made for n =  except  for the diagonal, n=m=l term, which does require the distinct expression, Eqn. (3.18). The derivation to this point has only been restricted by the assumption that the guide and cavity support a single mode each, and that these modes are deep within a band gap so that coupling to bulk and radiation modes can be neglected. Neglecting the orthogonality of the basis set will now be justified by making some approximations "characteristic of a weak coupling regime. In the weak coupling regime the localized and guided eigenstates are only weakly perturbed by each other and therefore are themselves very close to being eigenstates of the full photonic crystal described by e . Within this t  approximation it follows that the intuitive basis states approximately satisfy the following orthogonality relation {}n\£t\$m)  = $n,m-  (3.20)  Chapter  Waveguide and Nonlinear Resonator  3.  Coupling  72  Appendix A presents further justification of this approximate orthogonality relation, and gives a discussion as to what physical coupling mechanisms are retained and neglected by the approximation. Using (3.20), the matrix elements of M can be expressed as U) (4>m\ew\<f>n) ~ W <5 ,„  M "=  (3.21)  2  mtn  m  n  for all but the n=m—l term, which is instead ,  Mi,i =  '  Cof^^'-Cb . 2  .  (3.22)  Making use of the rigorous orthogonality condition for the guided modes, as well as the normalization condition for the localized state, one arrives at M,  (3.23)  = (u ~ u )5 , 2  m n  2  n  m n  for n,m ^ I, M  (3.24)  = uf - u  1  u  for n = m = I, and M ,i ki  =  Mt = -4nul($ \x \fy.  (3.25)  0D  M  ki  The definition of i , and the approximate orthogonality relation between the w  localized and guided modes, have been used in deriving (3.25) from (3.21). The overlap function xl^i  1S  defined as  x23 = <4lx0DlA = / drx (m^ 0D  • 5(0-  (3-26)  Adopting an indexing convention for the basis states where the localized mode is labelled 1, the matrix M has a dense first row and column followed by an infinite diagonal block.  Chapter 3. Waveguide and Nonlinear Resonator Coupling  73  This concludes the derivation of the Green's function. I t is valid within the weak coupling approximation, and only applies when the guide and cavity each support a single mode deep within a band gap defined by the surrounding PC. Guided mode expansion coefficients The solution to the wave equation can now be expressed as |*) = | ^ - ) +  ^47r^ [M- 1 ] n , m |^)(4|x° |* 2  D  /lom  ),  (3-27)  n,m which suggests that | ^ ) = one of {i} = {I, hi, k ,...}. 2  a»(a>)|^i), where the subscript i can be any  I f the homogeneous field is expressed as \ ^  h o m  ) =  o>h\4>k ), then the expansion coefficient of an arbitrary waveguide mode is h  given by a (u)  a5  =  ki  h  + J2^2[M'%^rn\x° \^h )ah . D  kukh  (3.28)  m  Due to the block diagonal form of M the only nonzero term in the sum over m in Eqn. (3.28) is for m = I. The coefficient [ M  (-l) *(fet(Mi,fcJ <+J  M  '  L  =  En^(-l)^(-l)  1+n+fc<  det(M)  lki 1  - 1  ] ^ is  M , det(M „ fci  n  fc  nAfci  )  det(M) (3.29)  where the factors of (—1) in the second equality are for hi > I and n < ki. As above, only the n = I term contributes to this sum. The  det{M i/i ) kit  yki  term represents the determinant of a diagonal matrix and is therefore given by the product of the diagonal elements. Thus Eqn. (3.28) becomes a (a>) = a 8 fci  h  kukh  +  det(M) (3.30)  3. Waveguide and Nonlinear  Chapter  Resonator  74  Coupling  The determinant of M can be written as det(M)  = M det{M ) hl  = M det(M ) lA  l t i  (3.31)  i+j  Ui  The sub-determinant det(Mj i/i ) t  l A  J2 {-l) M M det{M ).  -  u  £(1)'M det(M )  -  ul  jtl  jmi  is nonzero only when j = i due to the  ti  diagonal form of the waveguide eigenstate block in M. The sub-determinants in  (3.31)  are just the product of the remaining diagonals, therefore TT  J  f~2  ~2  ~2 ~2 U  det(M) = Ui^iiuf-u^i^  -u  V - / , s2~4($i\X \$i)(&\X° \(tl)i 0D  -2J4TT) w  -  t  oo\  D  2  _  ~  J-  2  (3.32)  The guided mode expansion coefficient is thus given by (47r) ^ ^(^.|x 2  a  k i ( t i )  =  d h S k i , ^  H  2  (3.33)  |^)(^|x  O D  |4j^  . n - o n i l w l i - n n u ^  1  Since the sum over i in  0 P  (3.33)  •  does not include the localized state I, it  can be written as J2i —* Y, - This sum, evaluated by converting the sum to ki  an integral, yields  a (u) ki  = a8 , h  ki kh  +  i  u  ~  u  —unWLt.joD  .JOD  $~g [X+ ,iXi,+ ku  ku  •  -r  V  O D  .JOD I X-k ,lXi- \ u  (  3  -  3  4  )  kw  as the final expression for the expansion coefficient. Localized mode amplitude Although not needed to calculate the reflection and transmission of the guided mode in the linear response regime, the amplitude of the localized  Chapter  3. Waveguide and Nonlinear Resonator  75  Coupling  mode is important when deriving the nonlinear response in the cavity. The amplitude coefficient of the localized mode is, from (3.27), ai(Q) =  £4™ [M-V<4|x°Vfc h K2  (3-35)  m  The only nonzero term in the sum is again m = I. The required element of M-  1  is [  M  h  ~ \M(M)  ~  det(M)  '  ( 3  '  3 6 )  where det(M) is given in Eqn. (3.32). Evaluating the sums in the denominator as in the previous section, the expression for the localized mode expansion coefficient becomes:  4ir*2xjg a fc  ~2  ,~,2  i47r £ L r^OD 2  3  h  ~,0D  ,  ^OD  0D  V  1 '  Reflected and transmitted fields The transmitted field is found by evaluating the fields at x = + 0 0 . As only the waveguide modes carry energy, far away from the localized defect the fields at x = + 0 0 are found by summing over only the ki states in the basis: ,  (x -> oo|*> = $ >  fc4  (u0<* -  °°l4>'-'  ' -•  •  -(3.38)  The sum overfcjis similar to that which appeared in the denominator above; however, now there is an exponential factor, e , coming from the eigenstate lhiX  (f\4>ki)  •  Assuming that both the Bloch function of the eigenstate and the  overlap integrals vary slowly with the in-plane momentum ki, the integral reduces to L  1  r .,  or  dkiul.l1  e  ikiX  w  [2me{x)e - Res(k ) ik  e  lkiX  ; : ki -h — ie x  ul  ; -] ki + k + ze w  + 2iri6(-x)e- " Res(-k )} ik  x  ul  (3.39)  Chapter  3.  Waveguide and Nonlinear  Resonator  76  Coupling  where 6 is the step function, and Res(k ) is the residue of the integral evalUJ  uated at k . Upon carrying out the integrals one finds w  (x -  oo|*> = [1+  "'  **jj  s  i 4  "  , f c  }a {x -> c x # J , f c  h  .  (3.40)  where the in-plane momentum of the homogeneous field has been denoted as a forward propagating guided mode. The field at x —* —oo is  "•  (x^-oo\*)  inc  ,~.2  r  ki  J47r iZ. L r^OD 2  3  ^  —>• - o o | 0 J +  = a {x  -> -oo\(j) )  ki  ~  •  + (rl# e/>  = (r\$ )  = ^2a (uj)(x  ~ 12  •  f c  h  0D  ,  0D  0D  K •  1"  l A + f c ^ A i . + f c ^ + A-Jfe^.iXj.-fcJ V  V  V  )  Invoking a mirror symmetry of the 0D defect structure along the waveguide axis, overlap integrals involving +k  and — k  u  other. That is,  are set equal to each  w  (0_fcJxOD|(^) = (0+fcJx° |0j)- The transmission and reflecD  tion coefficients are then _  (x -> oo\V)  a ^ x -»  _  z  X ,iXi,  g  oo|</>+cJf  wj -  w  ^  —  w  x^Xj,™  and p/. ,\ _  lT  /  \  \ ~» -OQ|Wre/) g  a ^ x -> - o o | 0 _ J fc  i87r o> L 2  _  ii,  ^  3  0 £ > OP  *i»>,tX<,«> ^—Xw,iXi,  w  where the subscript w simply denotes the waveguide mode at frequency Co. Before generalizing this solution to include downstream reflections and a third order nonlinear response, the present derivation in the linear response regime is compared with those previously published by others [64].  Chapter  3.  Waveguide and Nonlinear Resonator  77  Coupling  Equations (3.43) and (3.42) are respectively the reflection and transmission of a guided mode that is weakly coupled to an otherwise lossless resonant cavity. The lineshape is Lorentzian, R(u>) = ,  2  ! f  -  r  ,  (3.44)  with a linewidth of T=  l X w  '  1 1  .  (3.45)  Note that the Q of the resonance is given by Q — tDj/T. This result presented here is a direct solution of the wave equation, and has involved approximations that are physically justified if the guide and cavity are weakly coupled. If the denominator is appropriately factorized, Eqn.  (3.44) has exactly the same form as in reference [64], with the ex-  x?S, >  ception that the overlap function,  * well-defined in terms of overlap s  integrals involving eigenmodes and the dielectric perturbation that defines the OD cavity. In the Hamiltonian formulation of this scattering problem, the corresponding coupling term is given in terms of the difference between a dielectric function, e (r), "associated with the unperturbed Hamiltonian", 0  and the total dielectric function, e (r). However, the "unperturbed" dielect  tric function, e (f), is ill-defined, since there is no unique dielectric function 0  that at once has the waveguide and localized modes as exact solutions. The author's derivation actually sheds some light on the resolution of this ambiguity. If (3.17) and (3.18), as well as the orthogonality condition of Eqn. (3.20), are used in the remainder of the development, without invoking the unitarity condition, (3.19), then the final solution does not conserve flux. Within the weak coupling approximation then, it is necessary to adopt (3.19)  Chapter 3. Waveguide and Nonlinear Resonator Coupling  78  to conserve flux. Comparing this unitary result with that derived in reference [64], it becomes clear that their e (r) can be taken as either e {f) or Cd{f) 0  w  (either of the two basis photonic crystals assumed in the current derivation) in order to obtain a physically well-defined coupling matrix element.  3.2.2  Non-resonant background  In this section the above solution is modified to treat the more general situation when there is some downstream perturbation of the I D waveguide that introduces a frequency-dependent (linear) background reflectivity, TZ , that nr  is assumed to be known a priori  This background reflectivity is incorpo-  rated in a manner consistent with the Green's function formulation of the scattering problem: Fan[69] has previously used a transfer matrix approach to include the effects of downstream reflection on linear resonator-waveguide coupling. The full expression for the field in the waveguide plus cavity system, before taking the asymptotic limit to x = — oo that yields the reflectivity, and before any of the sums over ki are carried out, is given by l*>  = J2 k \$k )  + ai\$i) = a \$ )  a  i  i  h  kh  + +  1  ^ - -  2  - E  f  c  i  ^ \ $  l  ) ( $  l  \ x °  D  \ $ k  h  ) a  h  ]  (3.46)  ( 4 7 r ) ^ ^ M &  which can be written in operator notation as (3.47)  Chapter  The operator  3.  Waveguide and Nonlinear Resonator  V,  {$i\V\$ki)  with matrix element  79  Coupling  = 47nD^.(<^|x |</O (note: 0£)  Cbk = OJ), is defined to conveniently group factors associated with the driving h  term in a Green's function solution. The operator G  0  is defined as G  Q  =  Sfcj ^-i^-i" • In the regime where contributions from non-guided modes of the I D waveguide PC can be neglected, the sum over ki can be taken to be the sum over all eigenmodes of the waveguide PC. Therefore the operator G  0  is just the bare Green's function of the I D waveguide PC. From Eqn. (3.47) it is evident that the full electric field depends on a bare Green's function G , and a corresponding homogeneous field \<pk ), both h  0  characteristic of the exact structure of interest, minus the local mode whose resonant coupling is being sought. A generalization of the previous result can therefore be obtained by finding the homogeneous solution and corresponding bare Green's function characteristic of the I D waveguide, including the downstream perturbation, but excluding the perturbation responsible for the local mode of interest. These will be substituted in place of the Green's function and homogeneous field presently used in Eqn. (3.47). This approach is valid as long as the non-resonant source does not alter the operator x° D  Therefore, the non-resonant source must be external to the cavity. The new Green's function and homogeneous field are found as follows. Assume that the source of the non-resonant background can be described by some susceptibility, x.nr- The wave equation can be written as (£ - U 6 )\V ) 2  w  nr  = 47TcD nr|Cr). 2  X  (3.48)  The subscript nr serves to make it explicit that this is the electric field of a waveguide PC with a non-resonant source, not the full electric field as in  Chapter  3.  Waveguide and Nonlinear  Resonator  80  Coupling  Eqn. (3.47). The Green's function solution is | t f ) = (1 + 4iru G nr)\$h).  (3.49)  2  nr  nrX  Defining N = 4ncd G Xnr,  Eqn.. (3,.49) becomes  2  nr  \$ )  (3.50)  = (l + N)\$ ).  nr  h  Furthermore, the Green's function for the" full waveguide plus non-resonant perturbation is related to the Green's function of the I D waveguide by Dyson's equation, = (1 + N)G .  G  nr  (3.51)  0  The non-resonant background can then be included in the above solution by simply substituting \<j> ) - • (1 + 'N)\$ ) kh  and G -»• (1 + N)G  kh  0  0  in (3.47).  The field in the coupled waveguide-cavity system, with some non-resonant background present, is then = (1 + N)a \$ ) h  + [(1 + N)G V\fy($i\V(l  kh  a  x  \$i){$i\V(l + N)\$ )a ] kh  h  + N)\$ )a kh  •  h  +  (3-52)  uf - u  2  - {(j)i\V{l +N)G V\(pi) 0  •  At a spatial location, f , upstream from the non-resonant scattering source - that is x < x , nr  where x  nr  is the spatial coordinate at which the non-  resonant scattering source begins - the operator TV acts on an eigenstate of the waveguide in the following way: {f(x < x )\N\$ ) nr  where lZ (u ,x) nr  ki  ki  = K (Cd ,x)$- (r), nr  ki  ki  (3.53)  is a complex scalar function of fcj, and position x. This  is nothing more than the definition of a reflection coefficient for the guided  3. Waveguide and Nonlinear  Chapter  Resonator  81  Coupling  mode reflecting from the non-resonant source. The present context is only concerned with the value of the reflection coefficient at x = x , the location Q  of the resonant cavity; 'R {Cjk ,x nr  i  RnrfakJ.  = x) = Q  W i t h this definition of the operator N, the evaluation of the sums, and of the fields at x = —oo, follows the approach already presented. The result, for the reflectivity of a guided mode coupled via x° to a resonant cavity, D  and scattered by a downstream perturbation with reflectivity i?„ (u)), is r  „,  *M  .  „  ...  -^  M  +  Zl  _ _^ _ ^ ?  1 + Rnr{U))  X ,lXl,w  1  +  K r (  ,  w  .  W J  ,  i X ?  n•  o r / 1  .  (3.«)  which has the simple form of a renormalized Lorentzian lineshape coherently added to a non-resonant background. Owing to the interference of the downstream and resonant contributions, the lineshape of the resonance in the reflection spectra is generally Fano-like. Finally, from Eqn. (3.52), the amplitude of the localized mode is found to be  "  M  "  <Sr-o»--^xSxl!2(i + Ar(o))-  ( 3  '  This amplitude will be important for the nonlinear discussion of the next section.  3.2.3  C o u p l e d waveguide a n d c a v i t y i n n o n l i n e a r regime  In this section the above solution is extended to include a third order nonlinear response of the host material.  Chapter  3. Waveguide  and Nonlinear  Resonator  .Coupling  82  A general nonlinear polarization, \P ), is included by modifying (3.3) NL  as: (C - Cu e )\$) = 4iru 2  \$) + Anu \P ).  2 0D  w  X  2  NL  (3.56)  The third-order degenerate nonlinear polarization is given in real space by <—>(3)  —* \T T  P (f,u) = 3X  —*  — »  —*  (r,-u\u,-u),u))E(f,u))E {r,u))E(r,uj),  NL  m  (3.57)  and by grouping terms, it can be seen that the net physical effect of this nonlinearity is to introduce an intensity dependent susceptibility, <-»AfL  X  <-+(3)  (f) = 3 X  (3.58)  (r) : E(f)E*(f)..  ~(3)  Note that X  is in general complex. The real part leads to an intensity  dependent refractive index, while the imaginary part quantifies the amount of two-photon absorption. In the weak coupling limit it is quite reasonable to assume that the only mode that will have enough intensity to induce a substantial nonlinear susceptibility will be the localized mode, when excited near resonance. Thus to a good approximation the intensity dependent susceptibility can be written as X  NL  (r) =  3X  ( 3 )  (TO:5(f)0r(O|ai(w)|  = a x ^ C f O r W T O  1  a  ^ .  (3-59)  'mode ^NL  From (3.59), X  «->0D  if) will, like X  ( f ) , be localized in the vicinity of the  defect mode, but it will not have exactly the same shape. I n this formulation, it's dynamic behaviour is determined completely by the localized mode amplitude, a/(a>). Equation (3.56) can thus be written, (£ - uH )\$) = 4KCO {X 2  w  OD  +  (\ai(Cu)\ )}\$).  NL X  2  (3.60)  Chapter  Waveguide and Nonlinear Resonator  3.  Treating |a/(u;)| as a parameter in x i 2  NL  a s  o  n  e  83  Coupling  might in an iterative  solution to the nonlinear Eqn. (3.60), one can formally replace x°  with  D  X°  D  + X  NL  m  a  of the linear development presented above. Because the  u  new expression largely preserves the local nature of x°  D  > t  n e  discussion of  which matrix elements can be neglected due to the weak coupling approximation carries over, and further approximations can be made due to the relative size of x°  D  a n  d X NL  I* particular, when considering different ma1  trix elements, {(f> \x \4 m), if n n d m are both I D guided modes, then the NL  a  >  n  resulting overlap function describes a third order polarization generated from the evanescent tail of the guided mode. This is certainly negligible given the approximation that only the field in the cavity is strong enough to generate a significant nonlinear polarization. If one of n or m is a localized and the other a guided mode than the overlap function represents a nonlinear modification of the coupling to the localized mode, and an associated modification of the resonant linewidth. While this is relevant, the present analysis will only retain the largest effect of the nonlinearity, the direct renormalization of the resonant frequency of the bound mode through matrix elements" of  x  NL  that involve the localized state twice. W i t h this assumption, the only element of M that is altered from those of the purely linear derivation is the Mij element, which becomes M;^ = ojf — to — 4.7ru Xi!i , and this modification of M is the only change that is 2  2  L  encountered in the linear analysis. Hence, the nonlinear reflectivity is given by Til \ D l~\ , R((J) = Rnr{u) + ^ 0  iuT(l + Rnr(u)) , ,^i --TVI 2  ~o  2  2  , P  (~W>  ^.61)  where the nonlinear overlap function, x$ , has been written as C K | Q | . The L  2  Chapter  3.  Waveguide and Nonlinear Resonator  Coupling  84  coefficient a is defined as 3 a =  f  "(3)  / drv?(r)- X  (r)u,(r)i5?(r)^(r),  (3.62)  •mode ^  which serves to separate the renormalized material response from the dynamical variable'Q = ai(u)/ y/V d , mo  e  associated with the localized mode am-  plitude. Thus the'modified M y element results in an extra factor in the denominator of the reflectivity that renormalizes the localized mode resonant frequency by an amount proportional to the intensity of the electric field in the cavity. Recall that formally, the local mode amplitude was assumed to be a parameter in the original equations of motion in order to obtain (3.61). To find the self-consistent value(s) of ai(Co) that satisfy the full set of equations at a given frequency and incident field strength, one must solve n  a i [ U )  4 ^ X ^ ( 1  =  uf - Co  2  + Rnr(C0))  -* * & a ^ - «x^f,S(l  + ^(^))'  which is obtained by incorporating the modified M y matrix element into the earlier presented derivation of ai(u). Taking the amplitude squared of Eqn. (3.63) results in a cubic equation whose roots are the values of ai(u) that self-consistently solve the third order nonlinear equation, Eqn. (3.63). These solutions are used in Eqn. (3.61) to find the reflection spectra in the presence of the nonlinearity. Finally, it is useful to express the amplitude of the homogeneous driving field, ah, in terms of the average power of the incident waveguide mode. The total electromagnetic energy of the incident waveguide mode within one unit cell is w  =  1.  f  o7T J unit cell  d ^ c o i ^ O I ' + l^t)! ] 2  3. Waveguide and Nonlinear Resonator  Chapter  =  ±~f  85  Coupling  (3.64)  dre {r)\E {f)\\ w  h  Z7T J unit cell  where \Eh(r)\  2  = \ah(j>k if)\  2 1 S  h  the time averaged electric field of the mode.  The time for the energy to move from one unit cell to the next is A/(v c), and 9  therefore the power carried by the incident waveguide mode is P =  v cW/A, 9  or  A Z7T  J unit cell A ffL e  = ^l^l2Thus,  an  =  Furthermore,  ^2nLP/vgC.  = ^v r/L87r oj , 2  2  g  (3-65) from  Eqn.  (3.45),  so Eqn. (3.63) for ai(u) can be re-expressed in terms  of simple physical parameters as + Rnriu))  U^TFjcil adu) =  . ,...  2  .  (3.66)  Equations (3.61) and (3.66) represent the final result of the derivation . 1  For a given incident power, the local mode amplitude in the cavity is obtained by solving (3.66), and the corresponding local field strength is Q = ai(u)/y/V demo  Equation (3.61) then yields the reflectivity of a waveguide  mode, in a I D waveguide PC, that interacts with a non-resonant scatterer and a localized, nonlinear cavity that supports a single bound mode in the frequency range of interest. 1  D u e to confusion about conventions for specifying the nonlinear susceptibility, some  of the numeric factors i n the above equations differ from those i n Ref. [68]. The following figures are the same as i n Ref. [68], however all quoted powers are reduced by a factor of 4.7T. A n Erratum is i n progress.  Chapter  3.  Waveguide and Nonlinear  Resonator  Coupling  86  Losses due to the nonlinear process of two photon absorption are included ~(3)  in the formalism through a complex X  (r). If required, linear material  losses could be included by assuming a complex linear susceptibility. In any realistic structure there would be some radiation losses that would cause a finite resonant linewidth even in the limit of vanishingly small coupling between the waveguide and the cavity. This could be included in the formalism by including in the basis another set of modes with a continuous dispersion (in addition to the I D waveguide modes treated above), and allowing them to couple to the localized mode. The net result would be an additional contribution to the linewidth, r —> T + T in the denominator of Eqn. (3.61), 0  with no corresponding change to the coupling (the T in the numerator of (3.61) would remain unchanged). This too is consistent with the linear result obtained using the Hamiltonian approach [64]. Finally, the general formalism presented above can be used to treat any order of nonlinear polarization. In practice, the approximations needed to render a simple analytical result, when possible, will depend on the nonlinearity considered.  3.3 3.3.1  Results a n d discussions K e r r effect a n d b i s t a b i l i t y  For the purpose of illustrating the nonlinear reflectivity properties of realistic PC waveguide structures, the set of material parameters summarized in Table 3.1 is adopted. The third order susceptibility corresponds to a  Alo.iaGao.mAs  host at a wavelength of 1.55 pm. The following neglects the order unity  • Chapter  3. Waveguide and Nonlinear  Parameter  Value - 1 9  5.5 x 10~  Vmode  1.94 x 10  m  20  3  Hz  14  none  4000  Q  87  Coupling  Units  4.08 x l O  Ana  Resonator  Table 3.1: Material parameters used in simulations. <-(3)  renormalization to the bulk value of X  due to the nonuniform localized  state. That is, over the extent of the localized mode i t is approximated that a ~ 3x^  1 v  d  J rfr|^(f)|  4  «  where e  is the average dielectric  avg  constant of the cavity region. A typical photonic crystal based cavity could have an air to material filling fraction of roughly 30%. This leads to e  avg  8.11 for an Alo^Gao^As  =  index of 3.34. The author chooses to focus on  18% AlGaAs since it has been shown to have the greatest ratio of nonlinear refraction (n ) to two photon absorption ((3) at 1.55 /im and is therefore 2  of particular interest for optical switching applications [70, 71]. The value used for the real part of x^ (7.1 x 1 0  _ 1 9  (m/V)  2  ) can befound from the  calculations and data presented in reference [70]. When the background, non-resonant reflectivity is ignored, the linear and nonlinear reflectivity in this scattering geometry are essentially identical to the nonlinear transmission that has been studied extensively by others in the context of nonlinear I D Fabry-Perot cavities [10, 11]. Of significance here are the absolute powers required to observe bistable behaviour in this PC geometry where the localized mode volume can be less than a cubic wavelength. The discussion below first illustrates that Fabry-Perot-like bistable  Chapter  3.  Waveguide  and Nonlinear Resonator  Coupling  88  Energy [ c m " ] 1  Figure 3.2: Reflection spectra for an incident mode power of 0.2/JW (dashed), and 1.23 mW (solid). T h e plot at 0.2 uW exactly coincides w i t h the purely linear calculation, the plot of E q n . (3.44). behaviour can be observed at power levels as low as 3.2 mW described i n Table 3.1.  i n the structure  The nontrivial influence of including downstream  reflections w i l l be considered next. Figure 3.2 shows reflectivity spectra i n the absence of any non-resonant background reflection for incident waveguide mode powers of 0.2 pW  (dashed  curve) and 1.23mW (solid curve). T h e dashed curve is the linear result that occurs when the peak field excited within the localized defect causes a negligible shift of the bound mode's resonant frequency. A s the incident power is increased, the nonlinear term renormalizes the cavity mode resonance by an  Chapter  3.  Waveguide and Nonlinear Resonator  Coupling  89  amount proportional to the renormalized susceptibility in the cavity region. As the energy approaches the resonance from below, the field strength in the cavity increases, which causes a nonlinear increase in the effective refractive index in the cavity region, because the third order susceptibility of AlGaAs is positive at 1.55pm. The increase in the refractive index decreases the resonant mode frequency, pulling it towards the incident frequency which in turn further enhances the coupling to the cavity. This positive feedback increases the slope of the rising edge of the reflectivity spectrum as compared to the linear result. As the frequency extends beyond the renormalized resonant mode frequency the field amplitude in the cavity decreases and the mode shifts back towards its linear frequency. This negative feedback keeps the resonant frequency close to the incident guided mode frequency, resulting in a (relatively) shallow slope on the falling edge of the resonance. The most interesting consequence of the Kerr-induced resonant frequency shift is the onset of bistability at higher powers. Figure 3.3 plots reflection spectra for incident powers up to 10.5raW. In the current example the reflectivity becomes multi-valued when the,power is increased above ~ 3.2 mW. This low threshold for bistability is a result of a large local field confined to a volume that is less than a cubic wavelength. The curve of circles in Fig. 3.4 is a plot of the reflected power as a function of incident power at a fixed energy on the low energy side of the resonance. As the incident power is increased, the reflected power gradually increases along the bottom branch of the curve until it reaches about 12.3 mW.  At this point, the reflected power jumps to around 5.5 mW due  to the instability of the interior branch of the curve. Decreasing the incident  Chapter  3. Waveguide and Nonlinear Resonator  90  Coupling  6460 Energy [cm" ' 1  Figure 3.3: Reflection spectra for incident mode powers of 0.18mW(dotted), 3.5mW(dashed), 6.97 mW(solid), and 10.47 rnW(dash-dot). power from above 12.3 mW,  the reflected power follows the upper branch  of the curve, dropping to minimal reflected power at about 4.4 mW.  The  dramatic variation from low to high reflected power, which corresponds to a switching from near zero to unity reflectivity, would be ideal for nonlinear switching applications. However, this simulation does not include the imaginary part of x ^ \ which accounts for two photon absorption. When 3  this is included, the corresponding hysteresis loops become smaller since the absorption reduces the peak reflectivity. Using a bulk two-photon absorption (TPA) coefficient for Alo.iaGaowAs  of 0.34 cm/GW  [71], the hysteresis loop  91  Chapter 3. Waveguide and Nonlinear Resonator Coupling  Energy = 6447.8cm"  „ 8S88«»»* » 8  !88  £ 5  jL | 4 o CL  ~§ 3r o <u «=  .  CC  ^  10 Power [mW]  15  20  Figure 3.4: Hysteresis loop for Lorentzian resonance at an energy of 6447.8 c m (circles). The stars and diamonds show the effect of two-photon absorption(TPA) when the TPA coefficient is assumed to be 0.34 cm/GW and 1.46 cm/GW respectively. The arrows indicate the bistable loop. - 1  width is reduced by approximately 1.1 mW (the curve of stars in Fig. 3.4). A theoretical prediction suggests that the TPA coefficient for Alo.1sGao.82As should be 1.46 cm/GW  [70]. When this is used the resulting hysteresis loop,  is given by the diamond curve in Fig. 3.4. The calculation shows that as long as the TPA coefficient is not much larger than the latter value, then TPA does not quench bistability in  Alo.i8Gao.82As  at this wavelength. However,  TPA can significantly alter the hysteresis loop if it is greater than the former,  Chapter  3.  0> 6440  Waveguide and Nonlinear  ' 6445  Resonator  ' 6450  ' 6455  Coupling  92  ' 6460  Energy [ c m " ] 1  Figure 3.5: Reflection spectra, in the presence of a downstream nonresonant scattering source R , for an incident mode power of 0.18 m W (dotted), 3.5 mW(dashed), 6.67 mW(solid), and 10.47 raW(dash-dot). nr  experimentally observed value.  3.3.2  Non-resonant b a c k g r o u n d effect  Now consider the impact of including a non-resonant downstream scattering source, with a reflectivity, Rn — 0.6e / . Figure 3.5 shows the reflectivity -l7r  2  r  spectra in this case, for the same set of incident powers as in Fig. 3.3. As in the Lorentzian  the power is increased the change in the refractive  index of the material shifts the resonant frequency to lower energy. However,  Chapter  7i  Waveguide and Nonlinear  3.  i  i  i  Resonator Coupling  93  i  Power [mW] Figure 3.6: Bistability of Fano reflectivity lineshape at 6447.8 c m . Arrows indicate the bistable loop the system follows as the incident power is increased from zero and then decreased again. Circles(o), crosses(x), and asterisks(a) depict the three distinct solutions to the cubic equation found from Eqn. (3.66). The 'o', 'x', and 'a' labels introduced in this caption are for relating each solution to Fig. 3.10. - 1  the way in which the shifted resonance coherently adds to the stable nonresonant background results in drastically different lineshapes than in the Lorentzian case. When bistability occurs it is possible for loops to appear in the spectra, and these loops result in drastically different hysteresis loops. The reflectivity, at 6447.8 c m , as a function of incident power is shown - 1  in Fig. 3.6. This bistable loop is very different than for the Lorentzian lineshape. In this example the threshold power for the bistable loop has  Chapter 3. Waveguide and Nonlinear Resonator Coupling  94  decreased significantly. The 'on' switching occurs at 5.9 mW while the 'off' switching occurs at only 2.8 mW. As is evident in Fig. 3.5, for high incident powers the nonlinear reflectivity can be close to unity over a broad range of frequencies. This translates into distinct output power characteristics in the hysteresis loops. In this example, at high incident power, the reflected power becomes almost linear with the incident power (Fig. 3.6). Figure 3.7 illustrates the effect of TPA in this particular example of a nonresonant reflecting source. In this example the nonlinear absorption process significantly quenches the unity reflectivity portion of the reflection spectra. The entire upper branch of the hysteresis loop is therefore significantly reduced in maximum power due to the absorption process. It is therefore evident that it is important to include such non-resonant sources in any model of the nonlinear performance of coupled waveguides and cavities. On the other hand, non-resonant sources could be designed into the structure in order to engineer desired reflection spectra and hysteresis loops. The reflected field producing the Fano-lineshape is a result of interference between the sharp resonant field and the non-resonant background field that is slowly varying in both amplitude and phase. The nature of the Fanolike hysteresis curves depends strongly on the amplitude and phase of the downstream reflectivity, hence there is a rich diversity of behaviours that can be generated. The next section presents a stability analysis of the solutions involving nonzero downstream reflections; the analysis verifies the hysteresis loop that the system will follow.  95  Chapter 3. Waveguide and Nonlinear Resonator Coupling  1  1  —1  1  1 o°  Energy = 6447.8cm"  1  t——-i  5 £  o°  _  5  o  o  o **  o  5 " o  0  a.  o  o  0  "° 3 £  * o  o  o  *  o  * *  * *  # * *  ** **  *  * o  '  :  0 0 0 'ooooooo  <l>e»®e®®®® 3  4  Power [ m W ]  0  5  Figure 3.7: Hysteresis loops showing effect of two-photon absorption for Fano resonances. The curve of circles is the result in the absence of TPA while the diamonds and asterisks are with a TPA coefficient of 1.46 cm/GW and 0.34 cm/GW respectively. Energy is 6447.8 c m - . 1  3.3.3  New stability analysis  The bistable response of I D Fabry-Perot cavities is often discussed in terms of a graphical solution that clearly reveals the three allowed solutions in the multivalued reflectivity regime, as well as the stability of these solutions. This analysis, described in references [10] and [11], is shown below to be a limiting case of a more general stability analysis that applies to the Fano-lineshape case. Defining a control parameter, /3  =  Po +  fi' \ i\ i a  2  2  the expression for the  96  Chapter 3. Waveguide and Nonlinear Resonator Coupling reflectivity in,the Lorentzian case becomes  i-T^r.  (3-67)  p — IUJL  and therefore (3 = OJ — OJ is the detuning from resonance in the linear limit, 2  2  0  and (3'  a  1S 2  factor representing the Kerr-effect. Plotting \R\ as a function of 2  (3 — (3 , one obtains the solid curve in Fig. 3.8. Since (3 — (3 is proportional Q  0  to \ai\ , the x-axis can be taken to be \a,[\ in arbitrary units. This curve 2  2  illustrates that, for some initial detuning from resonance, as \ai\ is increased 2  the system is pulled into resonance, as described above. Using expression (3.63) for ai to eliminate the resonant frequency dependence from the reflectivity, the following independent relationship between R and the incident power is found: R =R  iT[\ + Rnr) ai n  r  +  i  L  )  _  :  Z  -  r  (3-68)  -  In the Lorentzian limit, Rn = 0, this can be used to obtain the following r  power-dependent relationship between (3 and R:  \ \ = 7nA>  -  R 2  w  h  e  r  e  (3 = z^h\"^K\2 2  mode  v  (3 69)  each value of \a \ , Eqn. 2  F  =  o  r  h  1  (3.69) defines a linear relationship between |i?| and j3 — /3 , where the slope 2  0  depends on the incident power. The lines for \a \ corresponding to 4.38, 2  h  6.97, and 12.2 mW are plotted as dashed lines in Fig. 3.8. The intersection of these lines with the curve are the allowed solutions to the problem. The stability of the solution can be found from the following considerations [10]. For the passive optical system considered here, the rate of change  Chapter  Waveguide a n d Nonlinear Resonator  3.  97  Coupling  1 0.9-  / B  0.8-  A /  0.7 0.6  1  1  0.5 0.4 0.3 0.2 0.1 0  4  10  6  W  n  [A.U.]  x10  4  Figure 3.8: Graphical solution for the Lorentzian lineshape. Solid curve is independent of power and at a constant energy of 6447.8 c m (Eqn. 3.67) and the dashed curves are independent of frequency and at a constant incident power (Eqn. 3.68) of 4.38 mW (A), 6.97 mW(B), and 12.2 mW(C). - 1  of the control parameter (3 is proportional to the difference between its driving function and its steady state value. Therefore, (3 satisfies the following dynamical equation:  T ^  =/ ?  2  M  2  +  (3.70)  where r is the cavity response time. Perturbing (3 from its steady state value by j3 = J3 + S(3(t), one arrives at the following equation for 5(3{t):  .*« > t  dt  +  (i-  A W  »«W  )5(3(t) = 0, d(3  (3.71)  Chapter  3.  Waveguide and Nonlinear  which has solutions S(3(t) = exp(-^t) ets of (3.71). For  <  Jjj^pj  Resonator  Coupling  98  where 7 is the expression i n the brack-  it is easy to see that 7 is less than zero.  Therefore the solutions to 8(5{t) grow exponentially and are thus unstable. > j^p^ the solutions are stable. T h i s analysis indicates that the  For  d  negative sloped branch i n the hysteresis loop of F i g . 3.4 is unstable. W h e n there is a nonzero downstream reflection, E q n . (3.67) easily generalizes to it! = Rnr + -—,!J"^° j  (3.72)  r  but E q n . (3.69) does not generalize. T h i s is because the non-resonant contribution introduces a phase .shift between the reflected field and the field i n the cavity. T h i s can be seen from E q n . (3.68), from which it is clear that \R\  is not directly proportional to |aj| i n the Fano case.  2  2  Therefore, the  graphical solution cannot be expressed i n a 2-dimensional plot of | i ? | versus 2  |a(| , because such a plot lacks any information about the phase of the two 2  field  components. The  graphical solution i n this  more  general  situation  requires  a  4-dimensional plot of the real and imaginary parts of R as a function of real and imaginary ai. T h e author has verified that the three m u t u a l intersections of the four surfaces (real and imaginary R at constant frequency and power, respectively independent of power and frequency) indeed yield a graphical solution of the nonlinear reflectivity problem. However, the stability arguments for the Fano case would have to be generalized from a comparison of slopes i n the graphical solution to the comparison of 2-dimensional gradients. Instead of proceeding i n this fashion the following introduces a simpler stability argument that is essentially the same as i n the Lorentzian limit.  Chapter 3. Waveguide and Nonlinear Resonator Coupling  99  Figure 3.9: Graphical solution for the Fano-lineshape. Solid curve is independent of power and at a constant energy of 6447.8 c m and the dashed curves are independent of frequency and at a constant incident power (Eqn. 3.73) of 2 . 8 m W ( A ) , 5.28 mW'B), and 5.9mW(C). - 1  If one uses \R — Rnr\  2  instead of |i?| in Eqn. (3.68), then there is a 2  power-dependent proportionality to j3 — (3 , namely 0  ... - .- > • '*'~^ Sh^p  :  ja  I"  (  3  -  7  47r<Z)(V " 0  where 0™ = P'I\T{I+RZ)\ • ^ power-independent relationship for the function \R— Rnr\  2  6447.8 c m  is obtained directly from Eqn. (3.72)., Plotting the latter curve at - 1  and the former at incident powers of 2.8, 5.28, and 5.9 mW, one  arrives at the graph shown in Fig. 3.9. In contrast to the Lorentzian case, this  3  )  100  Chapter 3. Waveguide and Nonlinear Resonator Coupling  diagram does not represent a full, graphical solution for the reflectivity since one cannot'extract the reflectivity from a knowledge of \R — Rnr\ - However, 2  this is a graphical solution to \R—Rn \ . 2  T  Upon solving the full cubic equation  for an incident power of 5.28 mW-and plotting \R — Rn \  2  r  rather than |i?| , 2  the curve shown in Fig. 3.10 is obtained. The solutions shown in this figure correspond exactly to the crossing points of Fig. 3.9. The solid vertical line is at an energy of 6447.8 cm" and illustrates this equivalence. Therefore each 1  of the three solutions found graphically in Fig. 3.9 can be directly associated with one of the three distinct solutions to the cubic equation derived from Eqn. (3.66). The three distinct values of \R - Rn \  arising from the three  2  r  distinct analytic solutions to the cubic equation are labeled 'o', 'x', and 'a' in Fig. 3.10. The 'o', 'x', and 'a' solutions correspond to the circle(o), cross(x) and asterisk(a) solutions for the reflected power that results from the same three solutions to the cubic equation. There is therefore a clear link between the graphical solutions in Fig. 3.9 and the numerical hysteresis loop in Fig. 3.6. This is possible since Rn is single-valued. A stability argument of the T  three graphical solutions in Fig. 3.9 can then be used to investigate the stability of the three branches in the hysteresis loop. Assuming the same feedback relaxation equation given in Eqn. (3.70), the equation for 5@(t) in the Fano-case becomes +  ( 1  "® > \ h  A R  d/f ' ^ n r  which again has the solutions 5(5{t) = exp(-^t). the region for which ^pO^f < ^ ^ j f  a r e  2 )  ( t ) =  °'  (3  -  74)  It follows that solutions in  unstable while solutions when  the opposite is true are stable. Since these unstable solutions correspond to the internal branch of the Fano-derived hysteresis loop in Fig. 3.6 it is  Chapter 3. Waveguide and Nonlinear Resonator Coupling  101  Figure 3.10: Plot of \R — R \ as a function of energy for an incident power of 5.28 mW. The arrows point to the three distinct sections of the curve that originate from the three distinct solutions to the cubic equation derived from (3.66). The dashed is labeled 'o' corresponding to the solutions depicted by circles in Fig. 3.6. The solid and dash-dot are respectively labeled 'x' and 'a', and correspond to the crosses and asterisks in Fig. 3.6. The solid vertical line is a slice at 6447.8 cm" and illustrates that these numerical solutions are the same as the graphical ones found from curve B in Fig. 3.9. 2  nr  1  Chapter  3.  Waveguide arid Nonlinear Resonator  Coupling  102  concluded that this internal branch is unstable and thus the loop follows the path depicted by the arrows in Fig. 3.6. The approach presented here greatly simplifies earlier stability arguments for hysteresis loops associated with Fano-resonances, approaches that relied on absorption within the nonlinear material [72] or phenomenological parameters [73]. Finally, one of the most striking features of these Fano-derived hysteresis loops is the fact that different branches of the curve can cross each other. These crossing points in plots of output power versus input power do not correspond to degenerate solutions. This is because each solution still has a unique phase with respect to the incident field. It is not enough that the amplitude of the electric field (proportional to power) for each solution is the same, but their phases must also be equal to render the solutions degenerate. These crossing points therefore represent no critical switching point for the system. In fact, the stability analysis above indicates that one of the two solutions is unstable and therefore there is only one allowed solution at these crossing points. Nevertheless, as is evident from the example presented here, bistable loops resulting from Fano-resonances can have significantly different properties than the usual Lorentzian-derived loops.  3.4  Chapter summary  This chapter derived a simple analytic solution for the reflection of a guided mode that interacts with a Kerr-active nonlinear resonant cavity and a downstream non-resonant scattering source. A second order wave equation for the electric field is solved using an intuitive expansion of the associated Green's  Chapter  3.  Waveguide and Nonlinear  Resonator  Coupling  103  function and the field, rather than solving the equivalent first order equations for both the electric and magnetic fields, as has been reported by others [30, 64, 65]. All of the relevant linear and nonlinear coupling mechanisms are clearly and explicitly associated with well-defined overlap integrals involving electric field Bloch states and dielectric perturbations. The approximations required to obtain this simple analytic solution are made clear. The simple form of the solution avoids the need for iterative solutions. Instead, an independent cubic equation for the localized mode amplitude is solved first, and the result is used to obtain the reflectivity for a given incident power. For moderately high-Q (Q ~ 4000) resonant cavities with mode volumes on the order of 0.05/^ra , which should soon be attainable using various PC 3  fabrication technologies, the model predicts Kerr-related bistable behaviour at incident power levels of ~ 3.2mW in Alo.isGao.82 As. Although two photon absorption reduces the maximum range of the hysteresis loops, the reduction is estimated to be only a few percent. This low threshold for nonlinear optics is in agreement with a perturbative approach developed by others [74]. The presence of non-resonant downstream scattering sources in the waveguide results in Fano-like resonant features in the reflection spectra. In the nonlinear regime the coherent superposition of the stable background and power dependent resonant contribution result in topologically distinct hysteresis loops (in contrast to the. more common Lorentzian situation). Conventional stability arguments are generalized in order to determine which branches of these novel hysteresis loops are stable. From these calculations it is concluded, that'photonic crystals made from Alo.1sGao.82As  offer the potential for realizing bistable optical functionality  Chapter 3. Waveguide and Nonlinear Resonator Coupling  104  at power levels on the order of 3.2 mW, without significant impairment due to two photon absorption.. It is also clear that non-resonant, downstream reflections can significantly modify the nature of the bistable reflectivity. This fact may be used to obtain more flexibility in designing nonlinear devices, but regardless, it shows that these reflections should not be overlooked in analyzing the nonlinear behaviour of waveguides that interact with resonant localized cavities. The geometry, and particular nonlinearity studied in this chapter are particularly relevant for future photonic chips containing all-optical devices. This work has shown that it is possible to observe nonlinear response with extremely modest input powers, indicating that such all-optical devices can be expected to work when driven with low power (low cost) semiconductor laser sources. With further improvements in fabrication techniques, and sufficient market demand, communication and computing networks may one day include some version of these all-optical devices. The next chapter concerns the fabrication and experimental probing of a simple waveguide-resonator device.  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  105  Chapter 4 Nonlinear response of coupled waveguide-cavity  structures  excited w i t h ultrashort pulses 4.1  Introduction  This chapter describes experimental work done by the author pertinent to the waveguide/nonlinear microcavity structures considered theoretically in the preceding chapter. Commercially-fabricated HRIC ridge waveguides in the top 200 nm thick silicon layer of a silicon-on-insulator (SOI) wafer were obtained through an industrial collaboration with Galian Photonics Inc. Some of these waveguides contain I D HRIC periodic texture designed to introduce optical stop bands at vacuum wavelengths near 1.5 pm. Optical lithography processes were developed by the author in order to fabricate 3D microcavities in the centre of these I D grating structures, as described below in subsection 4.2.2. The maximum Q value of these cavities is Q ~ 1200. The only available source of moderately powerful and widely tunable radiation in this wavelength range was an optical parametric oscillator that produces ~ 100 fs long pulses at a repetition rate of 80 MHz. The maximum  Chapter  4. Nonlinear response of coupled  waveguide-cavity  106  average power of this source is ~ 100 mW, corresponding to peak powers in each pulse of ~ 12.5 kW.  This provided an almost ideal probe of the  nonlinear response of these structures, but its short pulse nature necessitated additional modeling, since all of the theory developed in chapter 3 was for harmonic excitation that is more relevant to optical processing chips. Section 4.3.4 describes the nonlinear impulse response model that was developed to compare with experimental results obtained using this ultrashort pulse source. Independent work by Georg Rieger and Kuljit Virk on untextured versions of silicon ridge waveguides revealed substantial spectral broadening and two photon absorption at average powers as low as a few milliwatts when the ridge width is less than ~ 2/xm[75]. This posed unforeseen challenges and opportunities. On one hand, nonlinear transmission functionality was demonstrated in a truly integrated device by utilizing the linear filtering properties of the microcavity in conjunction with the nonlinear propagation in the incoming waveguide (the opposite situation to that modeled in chapter 3). On the other hand, the situation considered in chapter 3 had to be realized by using a wider and softer waveguide as an input channel, and by introducing PbSe nanocrystals in the microcavity to enhance its nonlinear response compared to that of intrinsic silicon. Section 4.3.3 presents results in the regime where nonlinear processes in the waveguide dominate over those in the cavity. The linearly filtered nonlinear transmission response curves demonstrated in this section are attractive for applications in all-optical logic devices. Various parts of these measurements have been published elsewhere [76].  Chapter  4. Nonlinear  response of coupled  waveguide-cavity  107  Semiconductor nanocrystals with fundamental exciton resonances close to the 1.5 \im operating wavelength were introduced into the microcavity to increase the nonlinear response to a point where it could be studied at input powers low enough that nonlinear processes in the incoming waveguides could be neglected. For the 80 MHz train of pulses, the length of time between each pulse is longer than the lifetime of the cavity (~ l p s ) , while the length of each pulse is much shorter than the lifetime. In this regime the cavity effectively receives an impulsive excitation from each pulse and the detector time-integrates the spectral evolution of radiation that leaks out of the cavity, into the exit waveguide. This experiment is repeated (signal averaged) every 12.5 nsec. The results are presented in section 4.3.4.  4.2  The coupled waveguide-cavity structures  4.2.1  Silicon ridge waveguides  The SOI wafers consist of a Si substrate, a 1 pm thick SiOi layer, and a 200 nm Si capping layer. The waveguides were defined by etching the top silicon layer away through a mask that protected an image of the waveguide patterns.  The resulting waveguides are therefore clad above and on the  sides with air, and with SiOi on the bottom. A 1 mm by 2 cm sample containing over 200 different waveguides was cleaved from the full SOI wafer. All waveguides extend across the entire 1 mm sample as schematically (figure is not to scale) illustrated in Fig. 4.1. The end facets of the sample are polished to facilitate coupling light into and out of the waveguides. The input facet to each guide presents a 200 nm high by 3 pm wide rectangular  Chapter  4.  Nonlinear response of coupled waveguide-cavity  ...  108  Silicon  Figure 4.1: Top is a 3D schematic (not to scale) illustration of the waveguide sample, showing input multi-mode guides, tapers, and narrow single mode uniform and Bragg grating guides. Bottom is an SEM image of the waveguide referred to as fl-3 in the text, before being coated with photoresist.  109  Chapter 4. Nonlinear response of coupled waveguide-cavity  silicon ridge to the polarized excitation beam. This relatively wide (multimode) input guide extends for 390 pm in the particular waveguides used to fabricate the microcavities described below. It is followed by a 100 pm long adiabatic taper that gradually (monotonically) reduces in width from 3 pm down to 580nm, this sample will be referred to as fl-3. A second microcavity (discussed in section 4.3.4) is fabricated in a guide with a 650nm wide narrow section, it is referred to as fl-4. For both waveguides the narrow (singlemode) waveguide section is 10pm long, and it is followed by an identical taper that couples the light back into another 3 pm wide guide that extends 390pm to the output facet of the waveguide. Simulations done with commercially available software (Photon Design Inc.: FIMMWAVE and FIMMPROP-3D) verify that the tapers couple the lowest order T E mode of the multimode guide into the single guided mode of the narrow guides with 99.9% efficiency. Some waveguides discussed below have longer single mode sections. In these guides the multi-mode input and output guides are correspondingly reduced in length keeping the total guide length 1 mm long, with the single mode section at the centre. These guides will therefore be described by quoting only the length and width of their single mode section. The narrow section of waveguide is not homogeneous like the rest of the waveguide; it consists of a I D periodic set of 200 nm wide by 200 nm high silicon "islands" on a 400 nm pitch. Figure 4.1 shows a schematic illustration of these gratings as well as an SEM image of the Bragg grating section . The 1  periodic array of silicon islands represent a HRIC I D Bragg reflector with a 1  The SEM image was acquired by the author at the Simon Fraser University nanoimag-  ing facility.  Chapter  4. Nonlinear  response of coupled  waveguide-cavity  110  bandwidth of ~ 300 nm and the low energy edge of the stop band is near 1.5 fxm[77]. Galian Photonics Inc. determined that the dimensions of the structures agreed with their nominal values to within ± 1 nm[77]. Waveguides that contain Bragg gratings, like the ones described above, have received considerable attention in the context of nonlinear "gap soliton" switching. A t low powers where linear propagation occurs, an incident pulse at a centre frequency within the band gap of the Bragg grating is reflected. However, by increasing the pulse intensity enough to modify the refractive index of the material, the band edge can be nonlinearly shifted enough to allow some transmission of the pulse. This has been theoretically predicted[78] and experimentally observed.with ns long pulses in silicon slab waveguides [79], and in fibre Bragg gratings [80]. The nonlinear work described in this chapter does not rely on gap solitons. The Bragg gratings are used instead to form the mirrors of a microcavity embedded in the waveguide.  4.2.2  Designing t h e c a v i t y  The Bragg gratings written into the Si waveguides are uniform over their length, so additional processing is required to introduce a "defect" at their centre, that can act as a 3D microcavity. When the waveguide is clad with air and the teeth of the grating are filled with air the lower energy edge of the band gap for the waveguide shown in the SEM image of Fig. 4.1 (waveguide fl-3) is found to be at 6900 c m . If the cladding material is - 1  changed to a material with a higher index of refraction this band edge is expected to shift to lower energy. The higher index cladding increases the effective index of the mode, leading to a decrease in its energy, thus red  Chapter  4. Nonlinear response of coupled  waveguide-cavity  111  shifting the centre frequency of the band gap. Furthermore, the width of the band gap is decreased because the index contrast is reduced when the teeth of the grating are filled with the higher index material. The net effect of coating the I D Bragg grating section of the waveguide with a higher index cladding is to shift the low energy band edge down slightly in energy, but the high energy edge remains well within the gap of the uncoated waveguide grating. There is therefore an energy window within which light can propagate in an air clad grating, but it is reflected by the dielectric clad grating. A microcavity can thus be formed by leaving the central region of a section of Bragg grating uncoated, and surrounding it on either side with coated regions. The low energy band edge of the central part of the grating (region A in Fig. 4.2) therefore lies above the bandedge of the surrounding grating regions (regions B in Fig. 4.2), so light can be trapped in the central region. Photoresist is a transparent polymer that can be uniformly applied to the waveguide structure and selectively removed. The index of refraction of the carbon based photoresist polymer used in this work (Clarient: AZ P4110) is ~ 1.6. The sample was first cleaned by rinsing it with acetone, then methanol, and finally deionized water before drying it with a stream of dry nitrogen. The relatively narrow (1 mm) waveguide sample was too small to mount in the spinner apparatus usually used to apply photoresist to wafers. It was therefore mounted with double-sided tape onto a larger, scrap wafer. This wafer (with sample attached) was then mounted in""the spinner (Headway Research Inc.: EC-101D) using vacuum suction. The waveguide sample was entirely covered (~ 2 drops) with the photoresist solution. The sample was spun at 5000 rpm for 45 sec, yielding a ~ 1 fim thick photoresist  Chapter  4. Nonlinear response of coupled  waveguide-cavity  112  layer. The sample was then baked at 100°C for 10 minutes. The cavity was formed by removing a 4 um wide region of photoresist in the centre of the grating, as shown schematically in Fig. 4.2. This was done with UV lithography techniques. A 4 //m wide by greater than 2 cm long UV transparent rectangular region was located on a 3 inch UV lithography mask. Pieces of scrap Si wafer were taped (using black electrical tape) to the mask to cover up all other UV transparent regions. The mask and waveguide sample (removed from larger wafer) were mounted in a Canon mask aligner (model: PLA-501F), the 4 pm wide region was aligned perpendicular to the waveguides and accurately positioned across the centre of the Bragg grating region. The sample was illuminated (through the mask) with UV light (A ~ 400nm) for 25 sec exposing the 4 \xm wide region. The sample was then developed in a 1:6 ratio of deionized water:photoresist developer (Clarient Inc.: AZ 400K) for 65 sec, at room temperature, and with light agitation (manual swirling of beaker containing solution while holding immersed sample with tweezers) throughout. Development was abruptly stopped after 65 sec by immersing the sample in deionized water for 5 sec and then drying the sample with a dry nitrogen flow. An optical microscope image of the surface of the structure, as well as a schematic illustration of a vertical cross section is shown in Fig. 4.2. Optical microscope images were obtained with a Carl Zeiss Inc. Axiotech 100 microscope outfitted with a Diagnostic Instruments Inc. Spot digital camera. A 4 pm wide portion at the centre of the grating is clad with air while the 3/im sections at either end of the grating are clad with photoresist. Over the extent of the 3 /xm barriers the photoresist thickness varies from zero (at the  Chapter  4.  Nonlinear response of coupled  waveguide-cavity  113  Figure 4.2: Top is a cross sectional schematic of the waveguide-cavity structure. B o t t o m shows a top view optical microscope image of several waveguides after the cavity was formed. T h e sample referred to i n the text as f l - 3 is the one on the right.  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  114  inner boundary) to roughly 400 nm, as estimated using a profilometer. The flat surface of the lpm thick photoresist layer was observed ~ 10 pm from the centre of the grating. This apodization in the sidewall of the photoresist layer is evidenced by the fringes in the optical microscope image of Fig. 4.2.  4.3 4.3.1  E x p e r i m e n t a l results Optical set-up  All transmission measurements reported here were done with the optical setup shown schematically in Fig. 4.3. Pulses of 100 fs duration and a repetition rate of 80 MHz are generated in a optical parametric oscillator (OPO). The OPO was a femtosecond synchronously pumped OPO (Spectra-Physics Inc. Opal) pumped with a Mode-locked Ti:sapphire laser (Spectra-Physics Inc. Tsunami) that in turn is pumped with a diode-pumped, cw visible laser (Spectra-Physics Inc. Millennia Xs). The pulses are directed through a quarter wave-plate, a polarizer and then a 40 X objective lens that focuses the light onto the polished 3 pm by 200 nm cross section of the input waveguide. The wave-plate and polarizer are used to control the incident power. In order to obtain the fine control necessary for optimum coupling efficiency, the lens is mounted on a Melles Griot nanopositioning stage that provided 100 nm control over the lens position. The Melles Griot nanopositioning equipment included a NanoMax-TS 3 axis flexure stage (17MAX301), driving motors (17DRV001), Nanostep (17MST001) and Piezoelectric (17MPZ001) control modules, that were all run with Lab View software (17LVD002). The waveguide sample is mounted on a Melles Griot translation stage (17AMU001/D)  Chapter  4. Nonlinear response of coupled  waveguide-cavity  115  that provides 2D movement within the focus plane of the lens, as well as tilt and rotation degrees of freedom. In order to distinguish the guided light from that which travels through the substrate, the waveguide sample is tilted by 7° as shown in the inset to Fig. 4.3. A 20X objective, followed by a telescopic column with up to 12X addition magnification and a dual visible/infrared CCD camera (Electro-physics Inc: 7290A) is mounted above the sample to align the input beam with the desired waveguide (see inset to Fig. 4.3). The output facet of the waveguide is imaged to an intermediate plane with a 15 mm focal length lens. At this plane an aperture is set to block everything except for the light emerging from the waveguide. A n infrared CCD camera (Electro-physics Inc: 7290A) set in this image plane is used both as an aid for alignment and to position the aperture. In addition to the CCD camera, a photodiode is used to measure the signal strength at this image plane. The photodiode was calibrated with a power meter to allow absolute measurements of the transmitted power. The signal detected in the image plane is optimized on the CCD by adjusting the input-coupling lens with the nanopositioning system. Transmission spectra are obtained by directing the apertured signal into a Fourier Transform Spectrometer (BOMEM DA8 FT spectrometer) fitted with an InGaAs detector. The spot is focused onto the spectrometer's detector using only reflected optics in order to eliminate chromatic aberrations.  4.3.2  L i n e a r transmission  A typical low power (1 mW average power incident onto waveguide input facet) transmission measurement through one of the waveguides is shown in  Chapter  4. Nonlinear response of coupled  waveguide-cavity  Waveplate  116  Power meter  OPO source polarizer 40x lens sample Waveguide stage ' Collecting lens Aperture —  Figure 4.3: Schematic layout of optical set-up used for the experimental transmission measurements. The inset (surrounded by a dashed border) shows a side view of the sample with the incident beam to the left  Chapter  4.  Nonlinear response of coupled waveguide-cavity  ...  117  Figure 4.4: Low power (ImW) transmission spectra of an air clad waveguide containing a 650nm wide by 20pm long Bragg grating region. Blue, red, and black indicate measured spectra for different OPO centre frequencies.  Fig. 4.4. The Bragg grating portion of this waveguide was 650 nm wide and 20 pm long. For this measurement the waveguide is clad with air. This spectrum is obtained by normalizing the raw waveguide transmission spectrum with the incident OPO pulse spectrum obtained by directing the beam to the spectrometer immediately before entering the waveguide. Separate measurements for different centre wavelengths of the OPO beam are stitched together in order to cover a bandwidth larger than that of the OPO ( ~ 150cm ). -1  The low energy edge of the photonic band gap is seen at ~ 6800cm . Above -1  this energy (and extending well beyond the tuning range of the OPO source) the incident light experiences complete Bragg reflection from the grating. The modulation at frequencies below the band gap result from interference of light that is weakly localized to the 20 pm long grating region due to reflections at the Bragg grating/uniform waveguide boundary.  Chapter  4. Nonlinear response of coupled  6100  6200  waveguide-cavity  6300 6400 6500 Wavenumber [cm" ]  118  6600  1  Figure 4.5: L o w power (1 mW) transmission spectra of the same waveguide shown in F i g . 4.4 after it has been coated w i t h photoresist. Blue, red, and black indicate measured spectra for different O P O centre frequencies.  Figure 4.5 shows the measured transmission spectra after the ~ 1 pm thick photoresist layer was uniformly applied to the surface. T h e band edge shifted to lower energy by more than 400 c m  - 1  . It is w i t h i n this 400 c m  - 1  wide energy window that localized cavity modes can be supported once the cavity is defined. Figure 4.6 shows a comparison of air clad and photoresist clad transmission measurements obtained from a second waveguide.  The  Bragg grating section of this waveguide is 510 nm wide and 50 pm long. T h e band edges have been shifted up i n energy due to the lower effective index of the narrower guide. Narrower Fabry Perot resonances below the band gaps result from the longer Bragg grating section. T h e photoresist cladding causes a ~ 500 c m  - 1  shift i n the band edge towards lower energy.  A microcavity was successfully fabricated i n a guide w i t h a 580 nm wide,  Chapter  4. Nonlinear response of coupled  119  waveguide-cavity  Figure 4.6: Transmission spectra showing a ~ 500 c m red shift in the band edge between an air clad waveguide (red curve) and a photoresist clad waveguide (green curve). The waveguide has a 510nm wide and 50wm long Bragg grating section. Input power is 1 mW. - 1  10 um long Bragg grating section. An optical microscope image of this coupled waveguide-cavity structure is shown in Fig. 4.2. This structure will be referred to as fl-3 in the following discussion. The linear transmission spectrum is shown in Fig. 4.7. The relatively sharp edge around 6450 c m  - 1  is the  band edge of the photoresist clad barrier regions. Between the 6450cm and -1  6900 c m  - 1  (measured band edge when clad with air) is the window within  which light can be trapped in the air clad cavity region. Two resonances are observed within this range, one at 6771 c m  - 1  and the other at 6592 c m . - 1  These resonances correspond to light that is transmitted through the structure by tunneling through microcavity states localized to the cavity. Their spectral width is a measure of the cavity mode quality factor, or lifetime.  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  120  Figure 4.7: Normalized linear transmission spectra through the waveguidecavity labeled fl-3. The feature at 6450 c m is the band edge of the photoresist clad reflecting regions of the cavity. The resonances at 6592 cm' (Ml) and 6771 c m (M2) are the two localized modes of the cavity. The average incident power was 1 mW. - 1  1  - 1  The Q of the former mode is 250 while for the latter it is 180. The input coupling efficiency of the focusing lens set-up was determined by taking measurements on a waveguide that does not contain a cavity or Bragg grating. The propagation losses through uniform waveguides like this elsewhere on the chip are on the order of 1 dB/mm[4,  77]. Without any  additional losses due to the Bragg grating and cavity one can estimate the input coupling efficiency from the transmitted power. The waveguide used for this measurement has 275 /zm long multi-mode sections (3 pm wide) at either end, and 100 /xm long tapers that couple the multi-mode waveguides to a 250 \im long, 510 nm wide single mode waveguide section. For these measurements the waveguide is coated with photoresist. The input and out-  Chapter  4. Nonlinear response of coupled  waveguide-cavity  121  put facets of the waveguide are therefore essentially identical to the ones in which cavities were defined. When 2mW of average power was incident onto the input facet of the guide, the photodiode, positioned in the image plane of the output facet, measured 1 pW of transmitted power. The efficiency of the collection optics was measured to be roughly 40% [81]. Fresnel reflection at the output facet introduces a ~ 70% transmission factor. The power inside the guide is therefore estimated to be ~ 3.6 pW, resulting in an input coupling efficiency of ~ 0.18%. This largely reflects the relatively poor mode match between the free space OPO beam and the fundamental mode of the input waveguide (only the fundamental mode in the wide guide couples well to the single mode region).  4.3.3  Nonlinear waveguide plus filter  This section summarizes how the transmission through waveguide-cavity structure fl-3 depends on the power of the incident laser pulses. The input power is determined by deflecting the incident laser beam after the waveplate and polarizer into a power meter and adjusting the waveplate to obtain the desired input power. Figure 4.8 shows the raw (not normalized to laser spectrum) transmitted spectra for incident powers of 2 — 30 mW when the incident OPO centre frequency was 6771 c m  - 1  (recall that the spectral full  width at half maximum of the incident pulse is ~ 150 c m ) . - 1  The satu-  ration of the total transmitted power, as measured with the photodiode, is plotted in Fig. 4.9. The spectra of Fig. 4.8 have been scaled so that the area under each curve is proportional to the corresponding total transmitted power. Both the shape and energy of the two localized mode resonances are  Chapter  4. Nonlinear response of coupled waveguide-cavity  1  1  1  1  1  1  122  ...  II  Figure 4.8: Transmitted spectra through waveguide-cavity fl-3 for incident average powers ranging from 2 — 30 mW', and an OPO centre frequency of 6771 cm' . Labels M l and M2 are for reference to Fig. 4.10. 1  independent of the input power, however the relative strength of the two is power dependent, as shown in Fig. 4.10. The power through the high energy mode (labeled M2) at first increases with increasing power and then at high powers it strongly saturates. The low energy mode (labeled M l ) , which is detuned ( ~ 179 cm' ) 1  well into the lower energy tail of the exciting OPO  spectrum, grows superlinearly with input power for lower powers. At higher powers it becomes almost linear, with some saturation. This response can be explained by noting that in the absence of any significant nonlinear effects occurring within the microcavity itself, it acts essentially as a notch filter. Any nonlinear processes that modify the spectrum of the pulses propagating through the waveguide before encountering the microcavity will be filtered by the modes of the cavity. Experimental  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  123  Incident power [mW] Figure 4.9: Total measured transmitted power vs input power. Saturation is caused by TPA. Waveguide has a 580 nm wide by 10 \xm long Bragg grating section (structure fl-3), and the centre OPO frequency is 6771 cm' . 1  measurements carried out by Dr. Georg Rieger on similar (no grating or cavity) Si ridge waveguides showed spectral broadening and saturation of the total transmitted power. When those results were compared to a nonlinear pulse propagation model developed by Kuljit Virk, the saturation in the output power was attributed primarily to two photon absorption (TPA) and much of the pulse broadening and distortion was consistent with selfphase-modulation associated with the real part of the silicon's third order susceptibility[75] . Spectral broadening of the pulse results in power being 2  redistributed towards the wings of the spectrum where there is very little 2  I n addition to these instantaneous processes, some of the saturation and the asymme-  try i n the spectra could only be explained by including the optical influence of free carriers generated by the T P A .  Chapter  100  -  0)  4. Nonlinear response of coupled waveguide-cavity  1  1  1  —1  Q.  <D  CS  40  1  rvn^—  o "D  1  *  -  M2  124  ...  -  E  u>  c  CD I—  1  10  15  i  i  i  20  25  30  35  Incident power [mW] Figure 4.10: Power transmitted through cavity modes of waveguide fl-3. Red curve corresponds to power transmitted through a 100 c m bandwidth centred about 6592cm , the low energy mode ( M l ) . Blue curve is the power transmitted through a 62 c m bandwidth centred about 6771 c m , the higher energy mode (M2). OPO centre frequency is 6771 c m . - 1  -1  - 1  - 1  - 1  power in the original spectrum. This effect increases with increasing input power, so one expects the power at the centre frequency to increase sublinearly, and the power off in the wings to increase superlinearly with increasing input power, due to this pulse broadening effect alone. When the influence of TPA is included, which introduces a saturation effect across the whole spectrum, the superlinear response off centre frequency is softened, and the saturation at the centre frequency is enhanced. This interpretation is completely consistent with the nonlinear transmission behaviour reported in Fig. 4.10, where mode M l is detuned 179 c m  - 1  into the spectral tail of the pulse,  and M2 is at the centre frequency. Measurements carried out by the author on a waveguide similar to the  Chapter  4. Nonlinear response of coupled  6200  6300  6400  6500  125  waveguide-cavity  6600  6700  6800  Wavenumber [cm ] Figure 4.11: Transmitted spectra through a 1 mm long 2.5 pm wide multimode waveguide. Incident powers range from 0.5 to 20 mW. The broadening and the deep dip at the centre frequency are caused by self-phase-modulation and two photon absorption. Strong modulation is due to inteference between different guided modes. Black curve shows spectrum of exciting OPO pulse. one consider here, but lacking a microcavity or Bragg grating, indicate that spectral broadening and saturation are indeed occurring in these structures. Figure 4.11 shows spectra for different powers launched into a 2.5 pm wide multimode waveguide that extends for 1 mm with no taper, Bragg grating, or cavity along its length. Figure 4.12 shows the total output power vs input power for the same guide; saturation is observed. The power transmitted through this uniform waveguide at an energy detuned 179 c m  - 1  from the centre frequency of the input spectrum (denoted  by solid vertical line in Fig. 4.11), is plotted as a function of the input power by the red curve in Fig. 4.13. Like the power transmitted through mode M l in Fig. 4.10 this curve shows superlinear growth for low powers, and then it  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  126  Input power [mW] Figure 4.12: T o t a l measured transmitted power vs input power for 1 m m long 2.5 u m wide multi-mode guide. saturates at higher powers. T h e blue curve i n F i g . 4.13 is the power transmitted on resonance, it is similar to mode M 2 i n F i g . 4.10. Thus, if notch filters were placed after this particular waveguide, one would expect to see very similar power transmission curves as actually observed i n sample f l - 3 (Fig. 4.10 M l and M 2 ) . T h i s analysis strongly suggests that the nonlinear transmission observed i n sample f l - 3 is indeed due to the filtering action of the two microcavity modes, M l and M 2 , on the nonlinear spectral modifications occurring i n the input section of the waveguide, before the pulses encounter the microcavity. Some care must be taken here though, as the waveguide used to obtain the nonlinear spectra i n F i g . 4.11 is not identical to the waveguide preceding the cavity i n sample f l - 3 (there i n no taper, it is longer, and slightly narrower). T h e fact that the shape and the centre frequency of the resonances do  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  127  Input power [mW] Figure 4.13: Red curve is power transmitted at an energy red shifted from the centre frequency by 179 cm' . Blue curve is power transmitted on resonance. Curves correspond to the 1 mm long 2.5 pm wide uniform guide spectra shown in Fig. 4.11. Inset shows red curve plotted by itself. There is a significant superlinear growth at low power. 1  not change with power strongly suggests that there are negligible nonlinear effects occurring within the cavity itself. Based on the theory of chapter 3, any significant Kerr activity due to local field enhancement in the cavity should directly alter the centre frequency and shape of the transmission resonances. The dominance of the waveguide nonlinearity in these experiments therefore seems at odds with the theory of chapter 3. The resolution of this paradox has mostly to do with the fact that these experiments were done with ultrashort pulses, shorter than the cavity lifetime, whereas the theory in chapter 3 was based on harmonic (CW) excitation. The peak local field trapped inside the microcavity is not enhanced in proportion to the cavity Q  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  128  factor in this impulse excitation limit. At best, the energy injected into the cavity when the bandwidth of the excitation source exceeds the linewidth of the mode is that fraction of the total incident pulse energy in proportion to the relative linewidths of the resonance and the excitation pulse. In addition, by the time the incident pulse reaches the microcavity, its intensity is already substantially reduced from what it is at the entrance to the waveguide due to TPA and scattering at the waveguide/Bragg grating interface. Thus, in these experiments, the field intensity is substantially greater in the incoming waveguide than it is inside the cavity.  4.3.4  I m p u l s e response o f nonlinear c a v i t y  Semiconductor nanocrystals were introduced to the cavity in order to enhance its nonlinear response with respect to that of the waveguide. Small crystals of narrow bandgap semiconductors like PbSe, with radii on the order of just a few nanometers, exhibit excitonic absorption thresholds that are shifted by quantum confinement effects well up towards the visible end of the spectrum[82, 83]. PbSe nanocrystals having a diameter of ~ 5nm are essentially transparent at wavelengths above ~ 1.5«m, so slightly smaller nanocrystals should represent a blue-shifted, resonant two level system when placed in the microcavities described above. The dipole transition moments of these nanocrystals are known to be large, and they scale as r [84, 85] (r is 3  the radius of the nanocrystal), therefore it is expected that the nonlinear response of the microcavity should be considerably enhanced over its intrinsic behaviour due to the silicon alone. Evident technologies sell PbSe "IR-Evidots" with a nominal diameter of  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  129  Figure 4.14: Linear transmission spectra of f l - 4 cavity before quantum dots are introduced. Q of the mode is ~ 325. 5.5 n m that have a nominal absorption peak at 1550 nm, w i t h a spectral full w i d t h at half m a x i m u m of 100 n m [83]. T h i s considerable w i d t h is due to the inhomogeneous distribution of particle sizes i n the solution. T h e author designed and fabricated a new microcavity w i t h a localized mode close to 1550 n m . T o shift the microcavity mode to this wavelength range, a different single m o d e / B r a g g grating section w i t h a w i d t h of 650 n m , 70 n m wider t h a n before, was used. T h e cavity is fabricated following the same steps as outlined above. T h e linear transmission spectra through this new photoresist-covered structure, as presented i n F i g . 4.14, shows a localized mode at 6475 c m  - 1  (~  1.544um, Q ~ 325). T h e light localized i n this cavity was therefore expected to be very close to resonant w i t h the exciton transition i n the quantum dots. T h i s waveguide-cavity structure w i l l be referred to as f l - 4 . T h e nonlinear response of this new cavity was studied before introducing the quantum dots. W i t h the incident O P O spectrum tuned to 6475 c m  - 1  ,  Chapter  4.  Nonlinear response of coupled  0  130  waveguide-cavity  10 20 30 Incident power [mW]  40  Figure 4.15: Output power vs input power for fl-4 before q-dots are applied. Red curve of circles is total transmitted power, while blue curve of squares is power transmitted through a 40 c m bandwidth centred about 6475 cm' . OPO centre frequency is 6475 c m . - 1  1  - 1  and for incident average powers ranging from 2 to 40mW, there was very little saturation of either the total transmitted power (red curve in Fig. 4.15), or the power transmitted through the microcavity mode (blue curve in Fig. 4.15). The corresponding spectra are shown in Fig. 4.16. This response is ideal for the quantum dot experiment, where the intention is to mimic the conditions of chapter 3 (a linear waveguide coupled to a nonlinear microcavity) as closely as possible. The PbSe nanocrystals are suspended in a hexane solution with a concentration of 2.5 mg/mL.  Each nanocrystal is coated with hydrocarbon  (aliphatic) ligands, and there are excess ligands in the hexane solution which are continually interchanged with those chemically attached to the nanocrystal surface. These ligands improve the lifetime of the nanocrystals by pre-  Chapter  4. Nonlinear response of coupled waveguide-cavity  6300  6400  6500  6600  ...  131  6700  _1  Wavenumber [cm ] Figure 4.16: Output spectra for waveguide-cavity fl-4 for input powers ranging from 2 to 40mW; before the q-dots are applied. OPO centre frequency is 6475 c m . The spectra have been vertically offset for clarity. - 1  venting them from clumping together while in solution[86]. Using a pipette, 10 pL of the solution was dropped onto the surface of the waveguide sample. Within a few seconds the hexanes would evaporate leaving a thin film composed of ligands and embedded nanocrystals (see below). While mounted in the optical transmission set-up the sample was continuously kept under a dry nitrogen flow to limit photo-oxidation of the PbSe crystals. The sample was stored in a dark vacuum chamber when not mounted for measurements. A linear transmission spectrum taken after the solution was applied is shown in Fig. 4.17. The cavity mode has shifted from 6475 c m  - 1  to 6442 c m  - 1  and  the linewidth was reduced to 5.4 c m ; Q = 1200. Atomic force microscope - 1  (AFM) images indicate that the surface of the structure was completely covered in a film; there was no evidence of texture in the grating region. X-ray  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  132  Figure 4.17: Low power (lmW) linear transmission spectrum for waveguidecavity structure fl-4 after quantum dots have been added. photoelectron spectroscopy (XPS) measurements on the sample show that in addition to Pb and Se there is a substantial amount of carbon in this covering layer, presumably associated with the ligands in the original solution. The shift and narrowing of the microcavity mode is thus attributed to the influence of a thin, uniform dielectric layer that coats the entire sample. Its influence is only significant in the region of the microcavity though, since it is separated from the core of the waveguide by the thickness of the photoresist everywhere else. This is evidenced by the fact that the bandedge of the barrier regions was unchanged after being coated (compare band edge feature just below 6 4 0 0 c m in Figs. 4.14 and 4.17). The dramatic increase in mode -1  Q is mostly caused by a reduction and smoothing of the index contrast in the microcavity itself, and at the boundary between it and the tunnel barriers on either side.  Chapter  6320  4. Nonlinear response of coupled waveguide-cavity  6360  6400 6440 Wavenumber [cm ]  ...  133  6480  1  Figure 4.18: Output spectra of q-dot doped microcavity (sample fl-4) for input powers of 0.5 — 20mW. OPO centre frequency is 6442cm . The spectra have been vertically offset for clarity. -1  Spectra for incident powers ranging from 0.5 to 2QmW, and a laser centre wavelength corresponding to 6442cm , are shown in Fig. 4.18 . The incident -1  laser spectrum is ~ 150 c m  - 1  wide, so transmission occurs through the cavity  mode and at frequencies outside of the band gap. Figure 4.19 highlights the transmission through the cavity mode alone. There is a very clear broadening to higher energies and strong saturation at the nominal cavity frequency evident in these spectra. The low energy edge remains sharp independent of the incident power. This nonlinear distortion of the transmission spectra was observed with only a few milliwatts of average power incident onto the input facet of the wavguide. This is estimated to correspond to exciting the cavity with ~ 10 000 photons, after accounting for the input coupling efficiency and the ratio of exciting pulse to cavity mode linewidths. Figure 4.20 illustrates the saturation behaviour of the total power transmitted through this spectral region. The curve of circles in Fig. 4.21 shows there is also saturation of the  Chapter  6420  4. Nonlinear  response of coupled waveguide-cavity  l  1  1  6430  6440  6450  ...  134  r  6460  6470  Wavenumber [cm ] -1  Figure 4.19: Same spectra as in Fig. 4.18 but showing only the spectral region around the cavity mode. The spectra have been vertically offset for clarity. signal transmitted outside of the band gap, but it is less than that observed on resonance. The dashed curve is the sum of the transmission above the band gap and through the mode. The curve of squares is the total transmitted power measured with the photodiode. The spectra in Fig. 4.22 illustrate that there is virtually no nonlinear response at the cavity resonance when the excitation pulse is blue shifted 78cm~ from the low power cavity mode frequency. This, combined with the l  fact that the structure exhibited virtually no nonlinear response at these input power levels before the quantum dot solution was added, proves that the nonlinear response reported in Fig. 4.19 is due to light in the microcavity region that interacts with the quantum dot suspension. A number of nonlinear experiments conducted over a period of one week from the fabrication of this sample yielded similar behaviour, but measurements taken two weeks later showed no nonlinear response. This is also evidence that the nonlinearity is  Chapter  4. Nonlinear response of coupled waveguide-cavity  0.20 0.15  ...  135  r  1  -  O CL  •o  -  -t—•  CO  c  03  0.00  -  / 1  5 10 Incident power [mW]  0  15  20  Figure 4.20: Power transmitted through a 2 0 c r a b a n d w i d t h extending from 6435 cm' to 6455 c m , the b a n d w i d t h of the cavity mode. Sample is f l - 4 and O P O centre frequency is 6442 cm' . - 1  1  - 1  1  due specifically to the quantum dots, as they are expected to deteriorate on this timescale due to photo-oxidation. T h e nonlinear response of the cavity mode when excited w i t h a t r a i n of ultrashort pulses is clearly significantly different t h a n what one would expect based on the model developed i n chapter 3. T h e nonlinear response of cavities under pulsed excitation has been studied when the spectral w i d t h of the pulse is narrow w i t h respect to the cavity resonance; that is, when the cavity lifetime is shorter than the pulse length [87, 88]. T h e current study concerns an entirely different regime. T h e 100 fs pulses used here are shorter t h a n the lifetime of the cavity, while the length of time between pulses is longer. In this regime the cavity is injected w i t h an impulse of electromagnetic energy, and the resulting nonlinear modification of the cavity resonance varies i n time as the energy leaks out. Moreover, as the pulse is shorter than the cavity  Chapter  4. Nonlinear response of coupled waveguide-cavity  5  0  ...  136  10 15 Incident power [mW]  Figure 4.21: Total power transmitted through structure fl-4 is shown in solid curve with squares. The power transmitted between 6330 — 6 4 1 5 c m , below the band gap is shown in the solid curve with circles. The sum of the latter curve and the curve shown in Fig. 4.20 is the dashed curve. OPO centre frequency is 6442 cm' . -1  1  lifetime, constructive inteference of the light leading to a steady state local field enhancement is rapidly terminated, leading to a significantly reduced internal field strength as compared to the CW excitation regime. In this regime it is relevant to kept track of how the energy (or number of photons) in the cavity evolves in time. As the incident pulse is spectrally much wider than the cavity mode, a population of photons proportional to the bandwidth of the cavity resonance is left in the cavity after the short pulse scatters. In the linear limit this population will decay with a time constant equal to the cavity lifetime. In the nonlinear regime additional losses due to nonlinear processes will also effect the dynamics of the internal field strength. A simple rate equation model was developed to qualitatively describe the time dependent response of the cavity when excited with an ultrashort pulse.  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  137  6420 6430 6440 6450 6460 6470 6480 Wavenumber [cm ] Figure 4.22: Transmission spectra through cavity mode (structure fl-4) when O P O is tuned 78 c m above the low power cavity mode energy. Input power for spectra range from 0.5 to 32 mW. T h e spectra have been vertically offset for clarity. - 1  It assumes a t h i r d order nonlinear response for the quantum dots, hence only T P A and a Kerr-related shift i n the effective refractive index of the cavity is included. Despite its simplicity, the model captures many of the features evident i n the spectra of F i g . 4.19. A s is evident from the spectra of F i g . 4.18, the broadband 100 fs pulse injects two 'kinds' of photons into the region were the quantum dots reside. One group of photons populate the cavity mode at 6442 c m  - 1  . T h e other  group, at energies just below the band gap of the barrier regions of the Bragg grating, are essentially transiting through the cavity region. T w o dimensional F D T D simulations of a similar structure show that these near bandedge photons tend to dwell briefly i n the central cavity region even though the barriers are nominally transparent at those energies. A t any given time the  Chapter  4.  Nonlinear response of coupled  waveguide-cavity  138  nonlinear response of the quantum dot-doped cavity is determined by both of these populations. The number of photons in the cavity mode at any given time is denoted N , while the number of transient photons is N . The rate c  t  equations describing the evolution of these two population are c —at dN  c  0* at  1 », N T  =  1  c  =  -L r  N t  J  TTPA  C  -J_  *r r -( -) N + Ie V  N t  c  2  c  +  -W.  (4.1)  he  TTPA  t  The first term on the right hand side of each of these equations describes the linear rate at which the photons are lost from the quantum dot/cavity region due to scattering into radiation modes, and the coupled waveguide. From experiments on identical microcavities embedded in the middle of longer I D Bragg gratings, the cavity Q of this structure is known to be dominated by radiation losses. The lifetime of photons in the cavity, mode is r = 1.0 ps, as determined from the experimental linewidth . The effective c  lifetime of the transient photon population is estimated to be r ~ 190 fs. t  This comes from the linewidth of the low energy peak in the transmission spectra of Fig. 4.18. The second term in each equation describes photons lost due to two photon absorption processes in the quantum dots. This causes saturation of the transmitted intensity in the model. The TPA loss rate is proportional to the total number of photons in the cavity, N = N + N , and to the TPA c  absorption coefficient (3: 1/T  TPA  =  )  2  §  ^ -  t  Here n is the effective index of  the cavity and of the Bragg grating waveguide, and V  m  is the cavity mode  volume. Although the effective indices and mode volumes for the two photon populations are different, the qualitative conclusions drawn from this model  Chapter 4. Nonlinear response of coupled waveguide-cavity ...  139  are unchanged if a common set of values is adopted. The derivation of 1/TTPA uses the fact that N = U/hu, where v is the frequency of the photons, h is Planck's constant, and U — J e\E\ dV is the energy in the electromagnetic 2  field. The third term in Eqns. 4.1 describes the injection of photons into the cavity region while the excitation pulse, assumed to have a duration of 17 = 100 fs, is incident. I and I are the number of photons per unit time injected c  t  into each population. The area under the cavity mode and transient 'mode' resonances in a low power transmission measurement indicate that the pulse injects roughly twice as many photons into the transient population as into the cavity mode. For the following calculation it is assumed that I = 2.2 *I . t  C  These equations are numerically solved using Matlab. The solution for I = 1.0 * 10 , b = TTPA/N 12  c  =  0.08 * 1 0 " , r = 1.0 ps, and r = 190 fs 12  c  t  is shown in Fig. 4.23. There is a rapid growth of the population for the first ~ 130 fs, corresponding to photons injected by the incident pulse, then the population decays through the TPA losses and linear leakage to radiation modes, and into the waveguides. The value of I used in these simulations was c  chosen so that ~ 10,000 photons are injected into the cavity mode by each pulse. The TPA loss rate was treated as a variable parameter. Parameters b and T (intoduced below) were varied in order to have the model qualitatively agree with the experiment. The solution of equations 4.1 can be used to determine the Kerr related index change by  where n is the unperturbed refractive index, and ri2 is the Kerr coefficient. T 0  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  140  0.12  Figure 4.23: Time dependent evolution of the two populations of photons. Solid line is for photons in the cavity mode, lifetime of r = 1.0 ps, while dashed line is for the transient photon population, lifetime r = 190 fs. An incident flux of I = 1.0 * 10 photons per sec, and a TPA time constant of TTPA = 0.08 * 1 0 * N are assumed. c  12  t  c  - 1 2  is the well known nonlinear figure of merit defined as n = /3X/T, where A is 2  the free space wavelength of the light. T — 5 is assumed for this calculation. In the limit that the cavity population changes much slower than an optical period, the instantaneous cavity mode frequency is given, to a good approximation, by, W where u = 6442 c m 0  - 1  = T T # ^ )  ( 4  '  3 )  is the unperturbed cavity mode frequency. Combining  Chapter  4.  Nonlinear  response of coupled waveguide-cavity  ...  141  6458  2  2.5  Time [sec]  5  3 x10  12  Figure 4.24: T i m e dependence of shifted cavity mode frequency. Simulation parameters are as described i n caption to F i g . 4.23, and the nonlinear figure of merit is assumed to be T = 5. this w i t h the independent solution of Eqns. 4.1 for the cavity mode populations, the time-dependence of the shifted cavity frequency is as shown i n F i g . 4.24. To compare this model w i t h the experimental results it is necessary to extract a transmission spectrum from this model. T h i s is done by combining the instantaneous population w i t h the instantaneous cavity frequency and numerically differentiating to get an estimate of the number of photons emitted into the out-going waveguide per unit time at a given cavity frequency. In detail, the total (dynamic) range of the cavity mode frequency is  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  142  first divided into many equal size frequency bins. The time dependent mode frequency and cavity population curves are fit to polynomial equations that are inverted to extract the duration of each bin, A t , the mean frequency value of each bin, 6j , and the mean number of photons in the cavity mode avg  for each bin, N™ . The energy that leaks from the cavity for each bin is thus 9  given by f(ut) ~  (4.4) ar  c  where a is a constant factor greater than unity. It accounts for the fraction of light that leaks into the waveguide mode, rather than that which is lost due to scattering. Recall that Q, and hence cavity lifetime, is limited by scattering losses in these structures. As this factor will only affect the amplitude and not the spectral shape of the transmitted field it will be assumed to. be unity in the following calculation. To take in to account cavity lifetime broadening, the results of the above procedure are convolved with a Lorentzian corresponding to the low power resonant linewidth;  where T = 1/2-KT C is the spectral width of the mode. C  The final plot of the transmitted power spectrum is shown in Fig. 4.25. As the incident energy in each pulse is increased, the high energy edge of the spectra shifts further to the blue, but there is always strong emission at the unperturbed frequency. This behaviour simulates the experimental spectra extremely well, and verifies that the measured nonlinear effect is as is expected for ultrafast pulse excitation of a nonlinear microcavity.  Chapter  4. Nonlinear  6430  response of coupled  i  1  6440  6450  waveguide-cavity  1  r  6470  6480  1  6460  143  6490  Wavenumber [cm" ]  Figure 4.25: C a v i t y output spectrum vs frequency as calculated w i t h the rate equation model. Parameters are: T = 5, TTPA = 0 . 0 8 * 1 0 * . / V , r = 1.0 ps, and r = 190 fs. Spectra are for an incident flux of I = 0.05, 0.25, 0.5, 1, 1.5, 2.0 * 1 0 photons/sec. -12  c  t  12  c  T h e m a i n feature of the data that the simple model does not capture is the apparent clamping of the m a x i m u m frequency shift of the mode. T h i s discrepancy persists even when additional nonlinear absorption is included to take account of the free carriers generated by T P A . It is likely that the nonlinear response is not well approximated by a t h i r d order perturbative expansion at high intensities.  Chapter  4.4  4. Nonlinear response of coupled waveguide-cavity  ...  144  Chapter summary  This chapter described the results of fabricating and measuring the nonlinear response of a coupled waveguide-cavity when excited with ultrashort pulses. On SOI single mode Bragg waveguide structures a. microcavity was defined by selectively coating the ends of the Bragg grating with photoresist. The band edge difference for the photoresist coated barriers and the air clad cavity defined a spectral window in which light could be confined to the air clad region. In the regime where waveguide nonlinearities dominate over those of the cavity, superlinear growth was observed in transmission through cavity modes detuned from the central exciting wavelength of the laser pulse. This was attributed to spectral broadening of the guided pulse upstream from the cavity, followed by spectral filtering by the cavity. Superlinear behaviour was observed for incident average powers up to 15 mW (corresponding to ~ 27 /JW in the guide), beyond which saturation due to TPA quenched the effect. This low power threshold for nonlinearities is attributed to the small modal area of these sub-wavelength sized Si "pipes". The nonlinear response of the cavity was studied using short pulse excitation of a microcavity doped with PbSe nanocrystals ~ 5 nm in diameter. In contrast to the case of CW excitation considered in chapter 3, where the local field strength in the cavity is enhanced by a factor of Q, the enhancement is reduced with ultrashort pulse excitation, and the nonlinear response is inherently time dependent. Transmission spectra of quantum-dot-doped microcavities showed a broadening towards higher energy and the on-set of a sharp lower energy edge with only milliwatts of average incident power. This corresponds to exciting the cavity with ~ 10000 photons. Saturation of  Chapter  4. Nonlinear response of coupled waveguide-cavity  ...  145  the total transmitted power was also observed. A simple rate equation based model that includes the third order nonlinear response of the quantum dots, and the temporal evolution of the energy in the cavity qualitatively captured many features of the data. The results presented here have illustrated a new geometry for designing all-optical elements; a nonlinear waveguide followed by a narrow band spectral filter. In the waveguide the propagating pulse experiences distortion due to self-phase-modulation and saturation due to TPA. Desired nonlinear transmission response curves can be designed by varying the centre frequency and bandwidth of the filter. When centred on resonance with the guided mode, strong saturation is observed in transmission. This type of response could be used in optical limiting devices. When the filter is tuned off resonance the transmission curve showed superlinear behaviour at very low input powers. If two waveguide channels were combined at the input to this waveguide-filter structure, a slightly improved version of this superlinear response could be used to achieve all-optical A N D gate functionality. This appears to be the first suggestion of using a nonlinear waveguide followed by a filter to design all-optical devices. This geometry is essentially the reverse of most optical device proposals where the nonlinearity is assumed to be in the cavity, the waveguides simply acting as "wires" that carry the light signals between each processing element. While the short pulse excitation source did not allow for direct comparison with the theoretical model developed in chapter 3, the nonlinear ^response of a quantum-dot-doped cavity .was observed. '• This .work has succeeded in probing quantum dots coupled to the E M excitation of a microcavity through  Chapter  4. Nonlinear response of coupled  waveguide-cavity  146  a single mode waveguide. This geometry is entirely on-chip, with the active part of the structure being less than 20 pm long. It is this architecture that has been proposed for future optics based quantum computing devices. The work presented here comprises some of the first experimental work in this direction.  Chapter  5.  Conclusions  147  Chapter 5 Conclusions The last few decades have produced dramatic growth in the field of optics. While some discoveries and inventions have not progressed beyond academic literature, others have revolutionized the day to day activities of mankind. Photonic crystals, or more generally wavelength-scale optical elements that take advantage of high refractive index contrast (HRIC) to achieve unique optical functionalities, are a relatively new class of optical materials that could possibly form the basis of an "optical chip" industry.  Proponents  of optical chips envisage the integration of numerous discrete information processing functions, currently implemented using bulk optical elements and computers, on a compact platform that employs waveguides to route optical signals. In addition to passively routing light signals, a truly revolutionary optical chip technology would have to perform some level of on-board signal processing. A particularly elegant way to realize at least some of this functionality would be to use light beams to control themselves, or other light beams directly, without converting optical information into the electrical domain. At a minimum level, this requires the host medium to respond nonlinearly to the optical signal strength it experiences. To be practical, the efficiency of the nonlinear processes have to be good enough that moderate optical pow-  Chapter  5.  Conclusions  148  ers can be used to control the operations. It has been recognized for some time that the intrinsic properties of HRIC optical materials make them an attractive medium in which to demonstrate particularly efficient nonlinear optical interactions. This thesis has explored some of the nonlinear optical properties of HRIC semiconductor waveguide structures.  Emphasis was placed on gaining a  quantitative understanding of how high refractive index contrast can be utilized as a design parameter to exact efficient nonlinear responses. Two distinct nonlinear processes were considered as much for their pedagogic utility as for their potential use in optical chips. These were second harmonic generation from planar 2D photonic crystal membranes, and the third order Kerr effect (and associated two photon absorption) in HRIC waveguides coupled to 3D optical microcavities. The following section summarizes the results.  5.1  Summary-  Chapter 2 developed a simple, yet rigorous solution for the SH field radiated from a planar, 2D GaAs photonic crystal when excited by an harmonic field incident from the upper half space. It was shown that the nonlinear conversion process could be enhanced over that of an untextured slab of GaAs by several orders of magnitude when one or both of the fundamental and second harmonic fields has the appropriate frequency and in-plane momentum to excite leaky modes of the photonic crystal. Much of this enhancement was shown to be due to the local field enhancement that occurs within the slab when leaky modes (remnants of bound slab modes that happen to diffract  Chapter  5.  Conclusions  149  out of the plane as they propagate within it) are excited. Additional processes that influenced the overall conversion efficiency were also identified, and illustrated using model calculations. These have to do with the internal conversion of the fundamental field to the second order polarization, and of the second order polarization to second harmonic electric fields. Chapter 2 also presented a comparison between the model predictions and experimental measurements carried out by collaborators on a sample designed and fabricated at UBC by Weiyang Jiang. The successful comparison substantiated the principal model predictions, and demonstrated peak enhancement factors of at least 1200 times. This was only a lower bound, limited by the noise floor in the experiments. Chapter 3 concerned the nonlinear response of a Kerr-active microcavity coupled to a single mode waveguide. This geometry is particularly relevant for future optical chip designs, as it is well known that Kerr-active cavities exhibit bistable behaviour that can be used for all-optical switching, amplification, and limiting functions. A n extensive Green's function formulation of the nonlinear scattering problem yielded a simple analytic model that accurately describes weak evanescent coupling between the waveguide and the photonic crystal microcavity in the case of harmonic fields launched from one end of the waveguide. The model predicts that bistable behaviour can be achieved with only milli-watts of CW power in the incident waveguide, for a cavity fabricated in 18% AlGaAs, having a Q of 4000, and a mode volume of 0.05 « m . The model includes downstream reflections as well as nonlinear 3  absorption, and can thus be used to model realistic structures. I t was found that two photon absorption reduces the useful power range over which the  Chapter  5.  Conclusions  150  system can be used as a bistable optical switch, but waveguides made from an 18% AlGaAs alloy should work at an operating wavelength of 1.5 p.m. Fano-like response functions resulting from downstream reflections dramatically alter the hysteretic loops leading to qualitatively different switching behaviour.  A generalized stability analysis was developed to predict the  stable branches of hysteresis loops associated with these nonlinear Fano responses. The threshold optical powers required to achieve bistable switching can be reduced by factors of two by appropriate choice of the nonresonant downstream reflectivity. Chapter 4 described the fabrication and optical characterization of a coupled waveguide-cavity system much like the one studied in chapter 3. A single mode silicon ridge waveguide that contained a HRIC I D Bragg grating section was used for these experiments, and photolithography was used to define a ~ 4 pm long cavity in the middle of a 10 pm section of this Bragg grating. The Q values of different microcavity modes made in this manner ranged from ~ 200 to ~ 1200. Two distinct nonlinear functions were demonstrated with this basic sample geometry. The dominant nonlinearity exhibited in the pure silicon structures turned out to be attributable to spectral broadening and two-photon absorption in the incoming waveguide. In this regime the microcavity acts as a tiny in-line notch filter at the end of the nonlinear waveguide. If placed at the centre of the excitation spectrum (~ 100 fs duration pulses from an optical parametric oscillator were used to study these samples), the transmission through the notch filter exhibited strong saturation behaviour suitable for optical limiting applications. When placed far off in the wings of the  Chapter  5.  151  Conclusions  excitation spectrum, the nonlinear broadening of the incident pulse results in a superlinear increase in the transmission through the notch filter. Such a transmission curve can be used to achieve all-optical AND gate logic, if the input to the nonlinear waveguide is designed to accept two independent input signals from two distinct, linear waveguides. To enhance cavity related nonlinearities to a level where they dominate the nonlinear response of the waveguide, the cavity was doped with PbSe nanocrystals having fundamental excitonic transitions slightly blue shifted from the operating wavelength of 1.5 pm.  Experimental measure-  ments showed asymmetric broadening to higher energies of the transmission spectra through the microcavity resonance. Good agreement with experimental spectra was obtained using a simple model that mimicked the third order impulsive response of a microcavity to a pulsed excitation much shorter than the cavity lifetime. This substantial nonlinear response was observed when only ~ 10,000 photons were injected into the microcavity.  5.2  Conclusions  High refractive index contrast periodic texture can greatly enhance-the efficiency of nonlinear optical processes in semiconductor waveguides. The enhancements are predominantly due to the large local field strengths that can be achieved at moderate optical powers when light is confined to small mode volumes for relatively long periods of time. Theoretical models of the nonlinear response of these HRIC periodic structures must treat the nonlinearity and the photonic eigenmodes in a self-consistent fashion if they are to  Chapter  5.  Conclusions  152  be accurate for realistic structures. A "second harmonic mirror" can be realized using very thin textured membranes of noncentrosymmetric semiconductors such as GaAs. The model developed in this work can be used to design the texture so that the second harmonic conversion is optimized for particular incident wavelengths, angles, and polarizations. Under conditions where both the fundamental and second harmonic fields are resonant with leaky eigenmodes of the planar photonic crystal, the second harmonic conversion can be enhanced by several orders of magnitude over what it would be without the texture. Still, the absolute conversion efficiency is modest, so applications of such structures will likely be in niche areas where it is crucial to convert the frequency of radiation in a thin layer, but where the absolute conversion efficiency is not critical. The concept of integrating nonlinear high refractive index waveguides with small 3D microcavity filters to achieve a variety of nonlinear optical transmission characteristics is, to the author's knowledge, new. The proof-of-principle experimental demonstrations of saturated absorption and threshold-like, superlinear transmission in chapter 4 (and Ref. [75]) show that such devices can be compact (much shorter than 1 mm in length ), and that the nonlinear responses can be achieved using pulses with peak optical powers of only ~ 3 mW.  The overall losses associated with the particular  structures used in this work are high, but different structures and materials that utilize the basic concept, could be optimized. The model results described in chapter 3 suggest that realistic 3D photonic crystal microcavities should exhibit bistable reflection when excited with only milli-watts of quasi-cw optical power in a single mode waveguide.  Chapter  5.  153  Conclusions  All-optical devices exploiting such cavities can therefore operate when driven with low-cost, low power semiconductor laser light sources. This work further showed how non-resonant reflective elements placed downstream from the cavity might be used to engineer a wide variety of nonlinear response functions not typically considered in the optical bistability literature. These findings suggest that practical all-optical integrated chips may indeed be attainable in photonic crystal hosts. This conclusion assumes that the nonlinear response of the microcavity is due only to the intrinsic, non-resonant Kerr coefficient of the host semiconductor. The experimental work in chapter 4 suggests that much lower threshold optical switching should be realizable by doping the microcavity with nanocrystals that resonantly enhance its nonlinear response.  v  "  '  Further optimization of this latter structure, to increase the Q value and reduce its mode volume, could bring the threshold for nonlinear response down to the point where only a single photon has to be injected into the cavity.  Such structures should provide new opportunities in cavity QED  and quantum information processing, as was discussed in the introduction. The waveguide-cavity coupling that was studied in this thesis may provide a controlled way to access and manipulate the quantum states of a strongly coupled cavity-quantum dot system.  5.3  Directions for future work  The waveguide-cavity structure studied in this work should be improved to reduce cavity scattering losses to the point where the Q value is controlled by  Chapter  5.  154  Conclusions  coupling to the single mode waveguide section. This will require using 2D, not just I D HRIC photonic crystal confinement, and it will also require a means to adiabatically couple (apodize) the waveguide and' microcavity.  Strate-  gies for locally placing high-quality semiconductor nanocrystals at high-field points in the microcavity also need to be developed. Once such samples are available, both CW and pulsed experiments should be performed and compared to theoretical models.  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Central to this justification is a discussion as to what coupling mechanism are included and which are excluded by the approximation. First, if n and m correspond to guided mode eigenstates then (3.20) can be written as (fki^wlfkj)  + (4lx° |4,) 4 7 r  D  =  S i n c e t h e  first  t e r m  o n t n e  left-hand side of this expression is the rigorous orthogonality of the guided mode eigenstates, it follows that (c/>fc |x |</>k ) = 0. The overlap integral 0D  i  ($ki\x \$kj) 0D  i  describes the direct renormalization of a single guided eigen-  state (if n = m = ki), or the direct coupling between two guided eigenstates (if n = ki and m = kj), by the presence of the cavity. I n the weak coupling regime this will be negligible and it is therefore valid to neglect it: A n analogous argument involving the localized state results in ($i\x  \<f>i) =  1D  0, which  is also valid within the weak coupling limit. If either n o r m corresponds to a guided mode eigenstate and the other to the localized mode, then there are two equally acceptable ways to expand equation (3.20): ( t & j e j ^ ) + 47r(0 |x |<^) OD  fei  ^($ki\x \$i) 1D  —  = 0 and {}kMd\$i)  +  0- Multiplying each by the appropriate factor of u and  subtracting, one obtains  2  ojl.{$ \x \<J>i) OD  ki  -<^i{^ki\x \^i) 1D  = °> where the uni-  Appendix  A. Orthogonality  of basis states  168  tarity condition has been used to eliminate the factors containing e. Therefore, the assumption of tik\($ki\XoD\<i>i) ~  (4>kMt\<f>i)  tii($ki\x \<f>i) 1D  =  = 0 in a physical system requires that  0- While this constraint isn't quite as  simple as the ones above, it is still very useful. Consider the unitarity condition of Eqn. (3.19). Using the definition of both i and id and some simple algebra it is possible to derive the following w  expression:  <4NA =  ul(k\x \$i) 0D  —*  Due to the regularity of the function that on resonance  col.($ki\x° \$i)  1D  (A.i)  — *  it follows from the above  (4>ki\it\<l>i),  |x l<&) 1£>  ~  D  -a?<&l* lA>-  =  0. This on-resonant  result, which is independent of the value of the function (0fcjet|<&), is simply a consequence of unitarity. The weak coupling regime is only concerned with the response of the system near resonance, since the resonance is relatively narrow in frequency. The smooth continuous nature of the function ($kMt\$i)  verifies that  Cbl.{$ \x° \^i) D  ki  is sufficiently small  ~ ^i(^ki\x \^i) 1D  near resonance. This simply expresses that near resonance the response of the structure is dominated by guided-localized mode coupling while away from resonance this coupling mechanism becomes of the order of the weak mode renormalization process, and thus the latter cannot neglect with respect to the former. It can therefore be concluded that the orthogonality condition given in (3.20) is a valid approximation within the weak coupling limit. Intuitively it corresponds to neglecting the direct renormalization of the guided and localized mode with respect to the dominate localized-guided mode coupling process. '  '  ' .  ;  :  Appendix  A. Orthogonality  169  of basis states  Furthermore, for a particular waveguide-resonator structure the deviation of the functions ( 4 l x ° | 4 > , D  ($i\x \4i), 1D  and^ (4| 2  i  0 D X  |^)-^ (4lx 2  1 I ?  l^>  from zero represent a measure of the validity of the solution for that structure. If the deviation from zero of these function is much smaller than the value of the dominant coupling mechanism quantified by (0z|x | /O> OD  (  t n e n  one can conclude that the system lies within the weak coupling limit. Note, the first two functions, being directly dependent on the spatial separation of the guide and cavity, represent a spatial constraint, while the final expression corresponds to a constraint in frequency detuning  

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