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Periodically textured planar waveguides Cowan, Allan R. 2000

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PERIODICALLY T E X T U R E D P L A N A R WAVEGUIDES By Allan R. Cowan B. Sc. McGi l l University, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA September 2000 © Allan R. Cowan, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B .C. , Canada V6T 1Z1 Date: Abstract A heuristic formalism is developed for efficiently determining the specular reflectivity spectrum of planar waveguides textured with an arbitrary number of two dimensionally ( 2 D ) periodic gratings. The formalism is based on a Green's function approach wherein the electric fields are assumed to vary little over the thickness of each textured region of the waveguide. Its accuracy, when the thickness of each textured region is much smaller than the wavelength of relevant radiation, is verified by comparison with a much less efficient, "exact" finite difference solution of Maxwell's equations. In addition to its numerical efficiency, the formalism provides an intuitive explanation of Fano-like fea-tures evident in the specular reflectivity spectrum when the incident radiation is phase matched to excite leaky electromagnetic modes "attached" to the waveguide. By as-sociating various Fourier components of the scattered field with bare slab modes, the dispersion, polarization properties, and lifetimes of these Fano-like features are explained in terms of photonic eigenmodes that reveal the renormalization of the slab modes due to interaction with the 2 D grating. The model has been used effectively to directly simulate experimental reflectivity data obtained by others. ii Table of Contents A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f F i g u r e s v A c k n o w l e d g e m e n t s v i i 1 I n t r o d u c t i o n 1 1.1 Motivation 1 1.2 Textured Waveguides 6 2 T h e o r y 13 2.1 Basic Formalism 13 2.2 The Self-Consistent Fields in the Grating 18 2.3 Fields Above the Structure 25 2.4 Numerical Formulation 27 2.5 Multiple Gratings 32 iii 3 Discussion and Examples 36 3.1 Resonant Modes 36 3.1.1 Square lattice 36 3.1.2 Honeycomb grating 48 3.1.3 Summary 51 3.2 Bound Modes 52 4 Accuracy of the Model 57 4.1 Conservation of Energy 57 4.2 Accuracy of Approximations 60 4.3 Thin Grating Approximation 64 5 Applications 74 5.1 Resonant Filter 74 5.2 Textured Waveguide Properties 78 6 Conclusions 84 Bibliography 87 A Derivation of the Green's Fiinction 91 B The Reciprocal Lattice and Fourier Components 95 iv List of Figures 1.1 Schematic dispersion diagram for an untextured slab waveguide 6 1.2 Schematic dispersion diagram for a ID textured slab waveguide 8 2.1 Schematic of a 2D textured planar waveguide 14 3.1 Reflectivity spectra from a waveguide textured with a square grating. . . 38 3.2 Fourier field components included in the calculations for a square grating. 39 3.3 Reflectivity spectra near the 2nd order TE-like gap of a square grating. . 43 3.4 First order Fourier amplitudes when detuning in the X symmetry direction. 46 3.5 Fourier components of the honeycomb grating 49 3.6 Reflectivity spectra near the 2nd order TE-like gap from a honeycomb grating 50 3.7 Dispersion diagram in the M symmetry direction for a honeycomb grating. 52 3.8 Reflectivity spectrum near the Brillouin zone boundary ((3/Pg = 0.485s) for a square grating 54 3.9 Bandstructure of bound modes near the zone edge (X point) of a square grating 55 v 4.1 Reflectivity and transmission demonstrating conservation of energy . . . 59 4.2 Testing convergence of a finite number of Fourier components 61 4.3 Reflectivity spectra comparing Green's function and "exact" calculations. 62 4.4 Comparison between Green's function and experimental data 63 4.5 Comparison between Green's function and "exact" method as a function of grating thickness 65 4.6 Reflectivity from an untextured waveguide 67 4.7 Reflectivity spectra at the upper edge of the 1st order gap 69 4.8 Reflectivity spectra demonstrating the difference between adding and sub-tracting material 70 5.1 Polarization insensitive notch filter 76 5.2 Linewidth of notch filter as a function of grating location 77 5.3 Reflectivity spectra demonstrating effect of filling fraction and grating profile. 81 5.4 Effect of grating profile on radiated fields 82 vi Acknowledgements I would like to thank my supervisor, Dr. Jeff Young, for his guidance and support in carrying out this work. His enthusiasm, and knowledge of the subject, were both motivating, and instrumental to the research. I would also like to mention the members of Dr. Young's lab including, most notably, Dr. Paul Paddon and Vighen Pacradouni, with whom I had many helpful discussions regarding the work in this thesis. vii Chapter 1 Introduction 1.1 Motivation Photonic crystals are structures in which a periodic dielectric constant is used to modify and control the optical properties of the host material. Currently there are two main classes of host material: bulk dielectrics and planar dielectric waveguides. In bulk photonic crystals, propagating electromagnetic ( E M ) radiation with a wave-length on the order of the pitch of the periodicity experiences Bragg scattering. It is therefore convenient to describe the excitations in terms of photonic Bloch states, in analogy to the electronic states in normal crystals. In the vicinity of the Brillouin zone edges of the relevant reciprocal lattice the superposition of forward propagating and backscattered radiation results in standing wave-like eigenstates with zero group velocity right at the zone edges. As a result, gaps are created in the photonic dispersion within which classical electromagnetic radiation cannot propagate along the direction of the 1 Chapter 1. Introduction 2 periodicity. Such gaps are referred to as photonic band gaps (PBG)[1]. Classically, inci-dent E M radiation, at a frequency within the gap, will decay exponentially with distance into the crystal. A bulk dielectric, periodically textured in one dimension (ID) is often referred to as a Bragg reflector, or Bragg mirror, as it is possible for such structures to perfectly reflect incident radiation. These structures are currently used in a variety of applications, including vertical cavity surface emitting lasers (VCSEL)[2], and filters for wavelength division multiplexing (WDM) systems[3]. Although it may seem straightforward to extend these ideas to 2D and 3D, there are in fact many nontrivial effects that occur in these higher dimensional structures. In 3D periodically textured dielectrics it is possible to realize a full photonic band gap (expo-nentially decaying radiation within a band of frequencies, regardless of the polarization or propagation direction). The implications of a full P B G are numerous. If a point defect is introduced into the otherwise infinite periodic lattice it is possible to trap light; any light injected into this defect cannot propagate away due to the full band gap of the sur-rounding, defect free, photonic lattice[l, 4]. Likewise, line defects can be used to localize light. It has been suggested[l, 4] that propagating line-defect modes, lying within the band gap, can be redirected around micro-bends with zero radiation loss, stimulating the idea of an all-optical photonic circuit. These effects are also possible in 2D P B G material if the radiation propagates in the plane perpendicular to the texturing. Even in the absence of a defect, a full photonic band gap results in a variety of novel phenomena. Quantum mechanically, if an electronic resonance (eg. a dopant atom Chapter 1. Introduction 3 or quantum well) is introduced, the full P B G causes strong localization of photons[5], and the corresponding inhibited spontaneous emission of the excited atom[6], or exciton. Furthermore, the photon localization results in the strong interaction of radiation and matter forming dressed photon-atom bound states called polaritons[7, 8]. These bound states, being isolated from the environment by the P B G , form a coherent quantum me-chanical two level system. This characteristic makes this system an excellent candidate for a quantum logic gate[8], and thus they may form the building blocks of quantum computing. These intriguing and potentially very useful properties of bulk 3D P B G material have stimulated much research activity, both theoretical[4, 9, 10, 11] and experimental[12, 13, 14]. Despite this considerable effort, a good-quality bulk 3D photonic crystal useful in the optical or near infrared part of the spectrum is still not available. A n alternative to 2D and 3D photonic crystals in bulk dielectric hosts is the textured planar waveguide, or photonic crystal waveguide. Photonic crystal waveguides are easier to fabricate than bulk P B G material. They are fabricated by etching a 2D periodic array of holes through the core and/or cladding layers of a planar slab waveguide. The idea is that the 2D scattering potential should renormalize the slab modes establishing a photon bandstructure for electromagnetic modes "attached" to the porous waveguide[15]. An additional advantage of this system is that it is compatible with a whole class of existing waveguide-based optoelectronic components such as distributed feed back (DFB) lasers and multiple quantum well modulators. Chapter 1. Introduction 4 The semiconductor waveguides used for developing planar photonic crystals consist of a high index guiding core surrounded by lower index cladding layers. They are most often fabricated in GaAs/AlGaAs[16, 17] or InGaAsP/InP[18] heterostructures, where the GaAs or InGaAsP layers comprise the guiding cores. The cladding AlGaAs layers can be oxidized to obtain a higher index contrast[19], enabling strong confinement of the radiation. Alternatively, the cladding layers may be etched away making a free-standing core[20]. The 2D grating is fabricated by etching holes in the waveguide. The holes can either penetrate through the guiding core or be restricted to a layer in the vicinity of the core. Modern lithographic techniques make it possible to define 2D gratings with a periodicity on the order 100-500nm. With such a periodicity it is possible to control the propagation of infrared frequencies in semiconductor hosts that typically have a refractive index of approximately 3.5. As these are the frequencies used in the fields of quantum electronics, and optoelectronics, P B G waveguides may ultimately have a large impact o n technology. The major structural difference between bulk P B G material and P B G waveguides is the lack of translational invariance perpendicular to the texturing. This lack of transla-tional invariance allows total internal reflection of radiation at the layer interfaces result-ing in guided modes bound to the waveguide core: hence the name waveguide. The effect of the grating on these modes is similar to that of propagating radiation in bulk crys-tals: modified dispersion and band gaps. However, the lack of translational invariance introduces coupling to propagating modes outside of the waveguide. At any frequency Chapter 1. Introduction 5 there exist modes that propagate directly through the waveguide. Thus, even though the bound modes characteristic of the waveguide photonic crystal may exhibit gaps, they are truly just pseudo gaps[21] since other non resonant states exist in the gaps. The practical implication of having a pseudo gap rather than a true gap depends upon the property one is interested in exploiting. It has been established that strong gap-related phenomena can be observed in P B G waveguides. For example Painter et al.[18] have demonstrated an optically pumped defect-mode laser in a 2D textured III-V semiconductor waveguide by introducing a point defect into the lattice. Therefore, P B G waveguides may be just as appealing as bulk P B G material for realizing these properties. In addition to the gap physics (e.g. defect modes, polaritons), the properties of the renormalized eigenmodes in the waveguide geometry may also result in novel devices for optoelectronic applications. The main motivation for the work presented in this thesis is to determine the basic dispersion, polarization, and lifetime properties of the bound modes of infinite planar P B G waveguides. Such research will form the basis for ultimately determining to what, extent the alluring properties of 3D photonic crystals might be preserved in the planar 2D geometry, and to what extend the properties of the bound modes themselves may be used for device applications. Chapter 1. Introduction 6 Bound Modes F i g u r e 1.1: Dispers ion diagram for an untextured slab waveguide. (3 is the in-plane wavevector, and u = UJ/C the frequency (c is the speed of light in the vacuum). T h e dashed (dash-dot) line represents the light line i n the core (cladding) material . Between the two light lines there is a discrete set of purely bound modes, and above the cladding light line is a cont inuum of radiat ion modes. Below the core light line there are no propagating E M excitations i n the structure. 1.2 Textured Waveguides T h e g u i d e d mode exc i t a t ions of an u n t e x t u r e d p l ana r waveguide are defined by the i r d i s p e r s i o n 1 , as shown i n F i g . 1.1. A d i spe r s ion d i a g r a m (plot of f requency u> = u/c vs in -p lane wavevector ) g r a p h i c a l l y depic ts the s o lu t i on to M a x w e l l ' s equat ions for a g iven geometry . T h e dashed l ine i n F i g . 1.1 is the l ight l ine i n the core m a t e r i a l . T h i s l ight l ine represents the p h o t o n d i spers ion for p ropaga t i ng r a d i a t i o n i n a b u l k m a t e r i a l w i t h a i n d e x of re f rac t ion equa l to tha t i n the core; i t ' s s lope is equa l to the speed of l igh t i n the m a t e r i a l . T h e l igh t l ine i n the c l a d d i n g m a t e r i a l is shown w i t h the dash-do t l ine . x The details of planar waveguide theory can be found in reference[22] Chapter 1. Introduction 7 In Fig. 1.1 three different regimes are evident. Below the light line of the core material is a forbidden region where there are no propagating E M excitations. Above the light line of air is a continuum of radiation modes. These excitations are not localized to the guiding core. They essentially correspond to plane waves propagating in the air that undergo multiple reflections in the waveguide layers, reflections that give rise to conventional Fabry-Perot oscillations. These modes are referred to as Fabry-Perot or continuum modes. Between the light lines there is a discrete set of guided modes bound to the guiding core. These modes are either transverse electric (TE) or transverse magnetic (TM) polarized. T E (TM) polarized modes have their electric (magnetic) field parallel to the layer interfaces. At low frequency the modes follow the light line of the cladding material and at high frequency the light line of the guiding core. In between there is a smooth transition from one to the other. A weak periodic texture introduces a layer of scattering sites that diffract the slab modes, in general leading to a renormalization of the excitation spectrum. However, for a weak texture most slab modes (continuum and bound) experience little renormalization. The continuum modes, being essentially 3D plane waves, never truly Bragg scatter from a 2D potential, they just diffract nonresonantly. The bound modes are localized to the thin guiding core; they are evancesent in the surrounding layers. These 2D excitations experience 2D Bragg scattering in an analogous fashion to 3D Bragg scattering in 3 D crystals. As in bulk P B G material, at wavelengths near the Bragg condition there is coherent Bragg backscattering of the bound modes, resulting in E M standing waves, and Chapter 1. Introduction 8 Figure 1.2: Dispersion diagram for a ID textured slab waveguide. The dashed (dash-dot) line represents the light line in the core (cladding) material. The dash-dot-dot curve is the dispersion in the extended zone scheme, the solid and dotted curves are in the reduced zone scheme. The dotted and solid curves represent the true bound modes and the resonant (leaky) modes respectively. The Bragg condition (j3 = m(3g/2) is depicted with the vertical dotted lines. pseudo gaps in the photon dispersion. The renormalization of the bound mode dispersion for a ID grating is shown in Fig. 1.2. Note that, as with electronic crystals and bulk photonic crystals, the periodic nature of the grating results in Bloch states for the renormalized E M eigenmodes. The Bloch states, being periodic with a period equal to that of the grating, enables the entire reciprocal space to be spanned by the 1st Brillouin zone. We can therefore express the dispersion in the reduced zone (solid and dotted curves in Fig. 1.2). In ID there are two standing waves that result right at each gap, one has its maxima Chapter 1. Introduction 9 predominately in the air slots and the other in the material. The energy difference between these two standing wave solutions of Maxwell's equations dictates the size of the gap. The larger the refractive index contrast of the grating (the "stronger" the texture) the larger the gaps. It is evident (compare to Fig. 1.1) that only near the zone edges (/? = mPg/2, with m an integer) is the dispersion altered by the grating; the group velocity of the modes go to zero and gaps open up. Again, the stronger the grating the further the influence of the grating extends away from the zone boundaries. Note that these are pseudo gaps as even though there are no bound modes excitation (solid/dotted curve) in the gap frequency ranges there are in fact continuum modes (grey region). Furthermore, the bound modes are split into two distinct kinds; purely bound modes (dotted curve in Fig. 1.2) and resonantly bound modes (solid curve in Fig. 1.2). Purely bound modes exist below the light line of the cladding material and thus are purely evanescent in all cladding material. The resonantly bound modes exist above the light line of the cladding material. Although they are still bound to the waveguide core one (or more) Fourier component(s) of the mode is phased matched to radiate into the cladding material. In ID textured waveguides, with the propagation wavevector of the mode parallel to the reciprocal lattice vectors of the ID grating, the solution of Maxwell's equations reduce to separate, scalar equations for the T E and T M polarized cases. Therefore the eigenmodes of the structure can be expressed as linear combinations of either the transverse electric (TE) or the transverse, magnetic (TM) polarized modes bound to the untextured waveguide. The bound and quasi-bound modes in ID textured waveguides, for Chapter 1. Introduction 10 example D F B lasers, have been studied extensively with coupled mode theory (CMT)[23, 24]. C M T solves the separate, scalar equations, for the T E and T M modes, by considering only the lowest order modes of the waveguide, one propagating in the forward and one in the backwards directions. However, if the modes have some momentum perpendicular to the grating wavevec-tors in ID, or if the texturing is in 2D, as considered in this thesis, the true eigenmodes of the textured slab can never be strictly expressed as superpositions of just T E , or just T M polarized slab modes: the eigenvalue problem is inherently vector in nature, and scalar coupled mode theory cannot be used to properly describe the electromag-netic excitations [25, 26]. A 2D vector coupled mode formalism has been developed in this lab[27]. This formalism, with the use of a resonant pole approximation, reduces a general integral equation solution to an eigenvalue problem, enabling the eigenmodes to be intuitively expressed as a superposition of the T E and T M polarized excitations of the untextured waveguide. Although the solution provides a transparent understanding of the physical processes involved in the nontrivial vector coupling which occurs in 2D textured waveguides, the solution is, typical of coupled mode theories, not completely self-consistent and limited to weak texture. The approach of this thesis will improve on these limitations. The 2D vector coupled mode formalism previously developed in this lab[27], and the models described in references [25, 26, 28] directly calculate the dispersion of the Chapter 1. Introduction 11 renormalized photonic eigenstates attached to textured waveguides. However, recent ex-perimental work[17, 20, 29, 30] has shown that specular reflectivity spectra from strongly textured waveguides exhibit characteristic Fano-like features when the frequency and in-plane momentum of the incident radiation matches the dispersion of quasi-bound (reso-nantly bound) electromagnetic excitations attached to the waveguide. Angular-resolved reflection spectroscopy therefore provides a very powerful means of determining the influ-ence of the 2D texture on the dispersion of the slab modes characteristic of the untextured planar waveguide. This effect has stimulated the development of various formalisms that describe the specular reflectivity from ID and 2D textured waveguides[15, 31, 32, 33]. This thesis will develop a new, simple approach to calculating the specular reflectivity from textured waveguides. The solution is entirely self-consistent and uses only two ap-proximations. The formalism is both accurate and numerically efficient for both weak and strong gratings, so long as they are thin compared to the wavelength. However, as we will see, by breaking a thick grating structure into a number of thinner gratings, an extension of the single-thin-grating formalism, it can accurately handle strongly textured thick waveguides as well. One particular merit of this formalism is the spectacular level of agreement between its predictions and the experimental spectra. Furthermore, the formalism makes use of an intuitive Green's function approach that provides a transpar-ent physical interpretation of the results leading to a unified understanding of resonant diffraction and photonic bandstructure in this scattering geometry. The thesis is organized as follows. In Chapter 2 the formal solution of Maxwell's Chapter 1. Introduction 12 equations is developed and all required Green's functions derived. As well, the solution for a single grating is generalized to the case when the waveguide is textured with many planar gratings. Chapter 3 presents calculated reflectivity plots, 1st order diffraction plots, and dispersion curves for sample structures, and the interpretation of the plots in terms of guided eigenmodes is given. We discuss the accuracy of the model and investigate some of the subtleties of the numerical calculation in Chapter 4. Chapter 5 discusses some of the many application of the model. Finally Chapter 6 gives some conclusions and thoughts for the future. Chapter 2 Theory In this chapter the formal solution of Maxwell's equations for a slab waveguide textured with a 2D periodic grating is presented. Section 2.1 sets up the equations and describes the strategy of the Green's function based solution, outlining the steps that follow. In section 2.2 we solve for the fields at the grating, and give the formal derivation of the Green's function. Section 2.3 describes the steps needed to find the fields above the structure, thus casting the solution as a reflectivity calculation. In section 2.4 the solution is formulated as a computationally efficient matrix inversion problem, completing the solution for a single thin grating. Finally, in section 2.5 the formalism is generalized to the case when there is more than one planar grating within the structure. 2.1 Basic Formalism We first seek the solution for the reflectivity of harmonic plane waves from a dielectric structure consisting of a single planar 2D periodic grating located anywhere within an 13 Chapter 2. Theory 14 Figure 2.1: Schematic of a planar waveguide textured with a square 2D planar grating. The grating has thickness tg, and pitch A. The dielectric constant of the cylinders is eg, and they are embedded in a layer of the material that has a dielectric constant ec. z0 is the center of the grating, and z = 0 is the center of the cavity, of thickness L, that contains the grating. arbitrary series of homogeneous layers, as schematically illustrated in Fig. 2.1. The only restriction is that the grating must reside entirely within a single layer of the structure (i.e., —L/2 + tg/2 < zQ < L/2 — tg/2). However, the generalization of the formalism to multiple grating in section 2.5 will remove this restriction. The refractive indices and thicknesses of the uniform dielectric layers are typically chosen to support bound slab mode excitations in the absence of the grating. The solution must satisfy the inhomoge-neous Maxwell equations: V - D ( r ) = 0 V x B(r) + —D{r) = 0 c V - B ( r ) = 0 Chapter 2. Theory 15 V x E(f) B(f) = 0 (2.1) c We are concerned with the linear response of nonmagnetic materials so we have as-sumed all fields oscillate as e~luJt, and have put p = 1, and thus H(f) = B{r). In equation (2.1), and throughout the remainder of the thesis, the u> dependence of all the fields has been suppressed in order to simplify notation; it is understood that E(f) = E{u\ r). The displacement field is given by D(f) = E(r) + 4irPtot(r), (2.2) and the total linear polarization is Ptot(r) = X(r)E(f). (2.3) As with the fields, the frequency dependence of v, and thus the frequency dependence of the dielectric constant has been suppressed. However, it should be noted that the solution is not restricted in this manner, dispersion in the form of a complex, frequency dependent dielectric constant may be included in our numerical solution. In order to cast Maxwell's equation in the form where the grating is considered as a polarization source we break up the total linear susceptibility, m t ° two components, x(^ = Xs(z) + Ax9(p,z) (2.4) where Xs(z) describes the linear response of the untextured slab waveguide, and Axg{p, z) represents the deviation from Xs(z) in the textured region, z0 — tg/2 < z < z0 + tg/2, that Chapter 2. Theory 16 we designate with a subscript "g". It is subtle but important to note that the grating is represented as a modification of, rather than an addition to, one of the homogeneous layers that make up the layered slab. The deviation in the susceptibility may be either positive of negative depending on the dielectric constant of the untextured layer and of the material within the holes of the grating. The inhomogeneous Maxwell equations can now be written in the form: V • (es(z)E(r)) = -4vrV • APg(r) V x B{r) +iu).es{z)E(f) = -4iriojAPg(r) V - B ( f ) = 0 V x E(f) - iujB(r) = 0 (2.5) where Co = u/c, es(z) = 1 + 4-KXs(z), and APg(r) = Axg{r)E(f). The total solution consists of a homogeneous part (when APg{f) = 0), plus the particular solution when the deviation of the polarization in the grating region, APg(r), is considered as a source of fields radiating outward from the grating. Because the grating is in general surrounded by dielectric multi-layers, the fields originating from APg(r) will undergo multiple reflections before eventually escaping into the upper half space. The equations are cast in the form (2.5) to facilitate a Green's function solution that isolates the difficult 2D textured polarization source, APg, from the non-trivial, but much simpler ID texture, es. As shown below, the Green's function which solves equations (2.5) for a non-uniform sheet of polarization localized at a plane (i.e., APg oc 8(z — z')) is both Chapter 2. Theory 17 quite simple and intuitive. It basically involves multiply-reflected plane waves where the multiple reflections in the ID layered structure can be treated analytically using Fabry-Perot-like formulae involving simple Fresnel reflection and transmission coefficients. The general solution of equations (2.5), for a single grating that is much thinner than the wavelength of interest, proceeds in two steps. First, the self-consistent polarization in the grating region, APg, is obtained using a Green's function that propagates the fields generated by APg back to the grating itself. Knowing APg, the fields anywhere can then be obtained using a general Green's function that propagates the exact polarization, APg, to any layer in the structure. With slight modification our formalism can also deal with the case when the grating is not necessarily much thinner than the relevant wavelength of radiation. This will be discussed in detail in section 2.5, where the formalism is generalized to treat waveguides textured with more that one grating, again maintaining self-consistency. We consider plane-wave excitation with a well-defined in-plane wavevector, (3. The incident field is of the form Einc{0;z) = Eoe-iv"'lsei^ (2.6) in the upper half plane, where w0 — \Ju)2 — ft2, and p — xx + yy. We assume the plane wave has unit amplitude (\E0\ = 1), but it can have any polarization. Due to the well defined in-plane wavevector, and because the 2D periodic texture can be expanded in a Fourier series characterized by a set of 2D reciprocal lattice vectors, {Gm}, it is natural. Chapter 2. Theory 18 to work with the Fourier components of the electric field at wavevectors 0 + {Gm}, a s defined by E(p, z) = £ E0 + Gm; z)ei{^+6m>p. The solution first involves finding the Fourier field amplitudes, EC(P + Gm; z), within the "cavity" that is defined as the uniform layer containing the grating. In the next section we solve for these amplitudes, and present the formal derivation of the cone-sponding Green's function, 9C {P',z,z'). The subscript c denotes the cavity. The second step is to propagate each component of the field outwards to the upper half space (UHS) above the structure. This step of the calculation, based largely on the Green's function 9 UHS (P', zi z')i is presented in the subsequent section. 2.2 The Self-Consistent Fields in the Grating In an infinite, homogeneous material with a dielectric constant equal to that in the cavity layer, i.e., es(z) = ec, Maxwell's equations would take the form ecV-Ed(p,z) - - 4 T T V - A P 9 ( p , z ) V x Bd(p, z) + iuecEd(p, z) = -4mujAPg(p, z) X7-Bd(p,z) = 0 V x Ed(p,z)-iuBd(p,z) = 0 (2.7) assuming there is an externally controlled polarization source described by APg(p, z). The subscript d denotes the "direct fields", generated by this sheet of polarization, which are Chapter 2. Theory 19 a combination of local fields and propagating fields that radiate energy away from the polarization source. There are no interfaces to reflect these outward propagating fields, hence the label "direct fields". By taking the curl of the fourth equation above, and using the second equation to eliminate Bd one can derive the wave equation for the system as: V x V x Ed{r) - u2ecEd{r) = ^u2APg{r) (2.8) Making use of a vector identity and the first of equations (2.7), to eliminate V • E, the wave equation becomes: ( V 2 + u2ec)Ed{r) = -47r(d>2ec + V ( V - ))APg(f)/ec (2.9) As discussed in section 2.1 we solve for the Fourier field amplitudes in the grating. Thus Fourier transforming in the plane, the wave equation becomes 82 ~ 8 8 - -* ( - P 2 + u2ec + ^ ) E D ( P , z) = -4TT(-W + u2ec + —( —• ))APg(p; z)/ec (2.10) where (3 is the in-plane wavevector. The symbol E is used throughout the thesis to denote both the Fourier amplitude, and the real space electric field vector, as the context should always make it clear which is being referred to. When the right hand side of equation (2.10) is considered as an inhomogeneous driving term, the Green's function solution for the direct fields is -* -* -* -* r 0 0 <-* -* -* ~* ED(P; z) = E%°m((3; z) + dz' 9d (/?; z, z') • APg(f3; z') (2.11) J — OO Chapter 2. Theory 20 where 9d 0',z,z') is the Green's function solution of equation (2.10), and Edom is the corresponding homogeneous solution. The derivation of 9d 0;z,z') proceeds as follows[34]. Taking the Fourier transform of equation (2.9) one has {-k2 + u2ec)Ed{k) = -A7r(-kk + u2ec)APg(k)/ec (2.12) where k is the 3D wavevector of the Fourier amplitude, Ed{k), in momentum space. The Green's function solution to (2.12) is 9 d = + ( 2 - 1 3 ) where 1 is the identity tensor. The infinitesimal factor iS is included, in the standard fashion, to define the pole. It will be set equal to zero at the end of the calculation. The planar geometry of the waveguide naturally defines a z axis along which there is no translational symmetry. Thus it is convenient to write the 3D wavevector as k = /3 + kzz. With only a few lines of algebra, and using 1 = ss + J3J3 + zz, one can write the Green's function (2.13) as[35] o r/f-u * *\ - 2 ™ 2 {HPHP) + Pc+0)Pc+0) _ s0)s0) + Pc-0)Pc-0)\ _ 4TT y<iW + KzZ) k z _ w M _ i S k z + Wc(p) + i S J 6 c (2.14) and then inverse Fourier transform, by integration in the complex kz plane, to obtain 9d0;z,z') = ^ [ ^ - ^ ) e i w « W ( . - * ' ) ( S ( j g ) S ( ^ + ^ ^ e + ( ^ ) Chapter 2. Theory 21 + 8{z' - z)e^^'-^{s(P)S0)+pc.0)p^0))] -—S(z-z')zz (2.15) where 8{z — z') is the Dirac delta function, and 8(z — z') is the Heaviside function. The s and p unit vectors are defined as s(0)=J3xz (2.16) and . c ± 0 ) = 0j^dM_ ( 2 1 7 ) with wc(P) = \ZuJ2ec ~ P2 • These unit vectors describe the polarization of propagating radiation with respect to the plane of the polarization source. For s(p)-polarized radiation the electric (magnetic) field is parallel to the plane. Equation (2.15) is the direct Green's function describing radiation propagating away from the polarization source. The 8(z — z') term in (2.15) describes waves propagating upward from the polarization source located at z', and the 8(z' — z) term describes waves propagating downward. The zz term is due to the V • APg(p,z) term in Maxwell's equations and describes the depolarization effects of the grating, or the local fields. If the polarization sheet is embedded between two multi-layer dielectrics, each of the outward propagating solutions in 9 a will undergo multiple reflections within the grating cavity. The effect of the multiple reflections can easily be taken into account by adding an infinite series of terms to 9d, each of which is proportional to Fresnel reflection Chapter 2. Theory 22 coefficients for s- and p-polarized radiation. We describe the reflectivity (as seen from the grating cavity) from the multi-layer dielectrics above and below the cavity, with effective Fresnel coefficients rup, and r^ oum respectively. The multiple reflections generate an infinite geometric series which can be summed to find the Green's function in the layer containing the grating, 9c: gc(P,Z,z') = ^ r r X WC{P) 9{z - z'ywcmz-z'] + 6{z' - z ) e - ^ c ( / 3 ) ( * - 2 ' ) + -iwc(0)(z+z'-L) , jwc(0)(z+z'+L) Sdozun iwc(0)(z-z'+2L) + r a u p r s d o w n I E _j_ e-iwc({3)(z-z'-2L) 1 + 9(z - z')e^m(z-z') + iwc(0)(z-z'+2L)-, ' Pup ' Pdown C 1 pc+0)pc+0) + 9(z' - z)e-^m^') + Pup Pdown -iwc(0)(z-z'-2L) A , PUP)PC-{P) I r P u p e + rv. e iw^z+z'+L)pc+0)pc-0) 4.TT 5(z — z')zz (2.18) where L is the thickness of the layer containing the grating, Ds,p — 1 r s , p u p r s , P d o w n e iwc(0)2L (2.19) Chapter 2. Theory 23 and r S ) P are the effective s and p Fresnel reflection coefficients from the top (up) and bottom (down) multi-layers. The details of how the terms are added to the Green's function are described, for p polarized radiation, in Appendix A . The Green's function, 9c (B; z, z'), describes the fields at z due to the fields radiated directly from the externally-driven polarization source at z', and all the fields at z due to the polarization induced in the cavity and the surrounding multi-layer materials which themselves have been polarized by the fields radiated from the source at z'. This Green's function is entirely self-consistent, and thus lifts the limiting approximation made in the 2D vector coupled mode formalism[27]. This Green's function is completely general in that it accounts for a grating located anywhere within the multi-layer structure. A l l of the boundary conditions and multiple reflections within the multi-layer slab that were implicit in the left hand side of equation (2.5), but absent in equation (2.7), have been included in the Green's function given by equation (2.18). With this Green's function the solution of equation (2.5) for the total field amplitudes in the cavity, —L/2 < z < L/2, is given by Ec(p; z) = Ehc°m{B- z) + / dz' 9C (/?; z, z') • APg(P; z'). (2.20) J-L/2 The subscript c is used to denote total fields in a restricted interval of z, the cavity. The homogeneous field, E^om(P; z), is the field within the cavity of the multi-layered structure due to an incident plane wave, in the absence of the grating. It is a solution of (2.5) Chapter 2. Theory 24 when APg = 0, and is given by: EtT&z) = e ^ ^ ( * - * ) ( ^ ( l + r a d o l B B e i w ^ ( i + 2 . ) ) 5 ( ^ s ( ^ Up + r P d o w n e i w < m L + 2 z ) P c + 0)Po-0))) • Einc0 (2.21) As with the Green's function, its derivation requires an infinite sum over the multiple reflections which occur within the cavity. The factors Dsp are given in equation (2.19), and the Fresnel reflection coefficients rsv , and rsv, are the same as in the Green's function; they describe the reflection from the multi-layer stacks above, and below the grating cavity respectively. The Fresnel transmission coefficient tSin, and tPin, respectively describe the transmission of incident s- and p-polarized radiation from the UHS to the cavity, as seen from the UHS. The unit vector p0± is analogous to pc± in (2.17) but with ec —> 1.0 and wc —> w0. It is the unit vector for p-polarized radiation in the UHS where the dielectric constant is unity. The incident field, Einc(P;z) is given by (2.6). Note, with the homogeneous field, the polarization is no longer treated as external, thus the solution is completely self-consistent. Finally, for a periodic grating, Axg{r) can be expressed as a Fourier series AXg(f) = j:XGjz)ei^. (2.22) m Substituting this into equation (2.20), and using that APg(B\ z) = S XGm(z)E{f3 — Gm] z), we can write the fields radiated by the grating as a sum over the Fourier components Chapter 2. Theory 25 of the periodic grating. Then the solution of equation (2.5) for the fields in the grating cavity is -. - _ rL/2 _ „ Ec(p; z\ = E'°m(p; z)+ dz> gc (/?; z, z') £ XsmW)Ee{B ~ Gm; z>) (2.23) J—L/2 m where E^om is given by equation (2.21) and the Green's function is defined in equation (2.18). Equations similar to equation (2.23) can be written for each Fourier component of the field, P + {Gm}, but E^om(P; z) is nonzero only for the specular component. Recall that in the absence of the grating the only nonzero component of the field is the one with an in-plane wavevector equal to that of the incident in-plane wavevector. From (2.23) one can clearly see that the effect of the grating is to couple together Fourier components of the field that differ by Gm, and that the strength of this coupling is described by X<g (2)-2.3 Fields Above the Structure Once equation (2.23) is solved for all the Fourier components, Ec(p — Gm), in the grating, the fields anywhere can be obtained simply by finding the appropriate Green's function that propagates the self-consistent grating polarization to the z value of interest. For a reflectivity calculation the relevant fields are those in the upper half space (UHS) above the structure. Thus rL/2 _ _ _ EUHS(P; z) = E%?s(8; z) + / dz' 9UHS (/?; z, z') • AP9(P; z') (2.24) J-L/2 Chapter 2. Theory 26 where the subscript UHS denotes the total fields in the restricted interval of z, the upper half space. The Green's function 9UHS 0',z,Z') describes waves which emanate from the polarization at z' and are transmitted out of the top of the structure. It is derived from gd 0;z,z') in a similar way as before: by summing the infinite series in Fresnel coefficients which arises due to the multiple reflections within the slab. It is given by 9UHS0,Z,Z') = ^ ^ ( F F L M E ^ ^ ) X ts Ds + Pc+ 0)pc- 0)rPdown E*»c(FL(I+*o (2.25) where i S i P u p are the Fresnel transmission coefficient for the layers above the grating cavity, as seen from the cavity, and zt denotes the top of the structure. The homogeneous solution of the wave equation in the UHS is given by $UHS0\ z) = ( 1 + e i w ° ^ 2 ^ 7wg (/?)) • Emc0; z) (2.26) —* —* where Einc(P; z) is given by equation (2.6), and rwg= rss0)s0) + rppo+0)po^0) (2.27) with rs and rp the Fresnel reflection coefficients as seen from above the structure. The tensor Vwg effectively represents the strength of the field radiated into the upper half space by the polarization induced in the multi-layer slab (in the absence of the grating) by the incident plane wave. Chapter 2. Theory 27 To this point no approximations have been made. The infinite system of equations, implied in (2.23), gives an exact, self-consistent solution for the fields within the layer containing the grating, with the required Green's function given by equation (2.18). Once this solution in the grating region is found, the infinite set of equations, of which equation (2.24) is one, gives an exact solution for the corresponding fields reflected into the upper half space. Although most of the calculations in this thesis concern the reflected field, we note that the formalism is completely general and for completeness we now present the transmitted The total field transmitted out the bottom of the textured waveguide is given by field. ELHS(0\Z) = E%Ts0-,Z) + •L/2 dz' *gLHs 0;z,z')-APg0;z') (2.28) -L/2 where the homogeneous field is ELFS0\Z) = ( ( / ? ) e ^ W * + * > ) • Einc0;zt) (2.29) with t w g = tss0)s0) + tppo-0)po-0), and the Green's function 9LHS 0;Z,Z'), which is similar to (2.25), is: 2-Kiuj2 (2.30) Chapter 2. Theory 28 2.4 Numerical Formulation This section shows how the infinite set of integral equations (Eqns. (2.23) and (2.24)) can be cast as a finite set of algebraic equations that can be numerically solved by inverting a matrix. To simplifying notation we rewrite equation (2.23), for z in the grating cavity, and for all Fourier field components, as rL/2 *-+ E{c)n(z) = Effi(z) + / dz' g[c)n (z,z')J2Xnm(z')E{c)m(z') (2-31) where the wavevector has been denoted by a subscript, i.e., Ec{(3 — Gm,z) — E^c\m{z). and Xnm(z) is the Fourier coefficient of the grating susceptibility that couples the rnth field component to the nth one. To simplify the solution we transform the system of integral equations (2.31) into a system of algebraic equations with the use of one simplifying assumption. If the grating thickness is much less than the wavelength of light in the material, i.e., tg <C 27r/(uy/7g~), then the field intensity will be approximately constant over the grating. We can then replace E^c\m{z') by the field at the the center of the grating, E^m(z0), where z0 denotes the center of the grating. Furthermore, for a symmetric grating, i.e., the profile of texture has vertical side walls, Xnm is independent of z and thus can be taken outside of the integral in (2.31)1. With this approximation, the solution for the field at the center 1 For the thin gratings considered here the vertical extent of the grating is effectively negligible, thus the assumption of Xnm being independent of z is appropriate. The implications of the grating profile for thick gratings will be investigated in Chapter 5. Chapter 2. Theory 29 of the grating, z0, is En{z0) = E% i,Zo)-\- In {Zp, ZQ) y IXnmEm^Zo) (2.32) m where In (zQ, z0) is the Green's function integrated over the thickness of the grating: We have dropped the subscript (c), referring to the grating cavity, as it is implied by specifying z = z0, where z0 always refers to the center of the grating. The second simplification is to consider only a finite number, N , of Fourier components of the field. This reduces the infinite set of vector equations implied in equation (2.32) to N 3-vector equations. We then reduce the N 3-vector equations to a system of 3N scalar equations by projecting equation (2.32) onto the orthogonal s, j3 + Gm, z coordinate system of the problem. The set of 3N scalar equations can be written in matrix form as: Here E is a 3N component vector, M {z0,z0) =1 (z0,z0)- X is a 3N x 3N component matrix, where / and X are also 3N x 3iV component matrices. / is composed of N different 3 x 3 matrices, the In from equation (2.32), along the diagonal. Each 3 x 3 block in the matrix X has the structure of the identity matrix scaled by the Fourier components of the grating, the Xn,m in equation (2.32). Equation (2.34) is the self-consistent solution for the fields in the grating generated by polarization throughout the entire structure, in response to all non-zero polarization sources. This expression can be (2.33) E {z0) =E (z0)+ M (z0, z0)- E (z0) (2.34) Chapter 2. Theory 30 manipulated to give the total field in the grating in matrix form as: E (z0) = (1 - M (z0, Zo))-1 E {z0) (2.35) Following a similar approach, the fields at the surface of the structure, zt, can be written as: E (zt) =E (zt)+ N (zt, z0) E (z0) (2.36) Here E (zt), and E (zt) are 3N x 1 column matrices of the field components, and •» •» AT (zt, Zo) =L \Zt, z0)- X, where L {zt, zQ), and X are 3N x 3N component matrices. L is composed of N different 3 x 3 matrices, L n , along the diagonal, where L n is analogous to J n in equation (2.33) with 9 (c)n replaced with 9(UHS)U- X is the same matrix as described above. Again, we have dropped the subscript (UHS) in (2.36) as it is implied with z = zt. Substituting equation (2.35) into equation (2.36) we find the solution for the field at the top of the structure, zt, in matrix form: E (zt) =E (zt)+ N (zt, z0){ 1 - M (z0, Zo))'1 E {z0) (2.37) where E (zt), given by equation (2.26), is written as a 3N x 1 component vector. Equation (2.37) gives the field above the structure produced by the polarization in the grating acting on the full multi-layer structure, and from all other sources. This equation is the general self-consistent solution for the Fourier field amplitudes above an arbitrary multi-layer slab waveguide that contains one planar 2D periodic grating: the two inherent Chapter 2. Theory 31 approximations are i) the field variation over the thickness of the grating is neglected, and ii) a finite number of Fourier components of the in-plane field structure have been retained. The field E (zt) contains all N Fourier components included in the calculation and thus equation (2.37) can be used to calculate each Fourier component of the field excited by the incident plane wave (NTH order scattering). The specular field is the zeroth order component (zeroth order scattering) of the NTH order vector E (zt), and only includes =>/iom the upward propagating component of the homogeneous field, EUHS • Thus the total specularly reflected field, ESR, from the structure is ESR(zt) = N (zt,z0)(l - M (z0,z0)) 1 EHOm (z0) + rwgexp[-iw0zt]. (2.38) specular where the subscript "specular" denotes the component of the matrix in the square brack-ets which has an in-plane momentum equal to the in-plane momentum of the incident field, the zeroth order component. The in-plane wavevectors of all the other Fourier components differ from the incident in-plane wavevector by the reciprocal lattice vectors: Bm = p i n c + Gm. Because there is no homogeneous field with an in-plane wavevector Pm, m ^ O , the mth order field is simply given by >hom Emth(Zt) X (zt, z0)(l — M (z0, z0)) E (z0) (2.39) mth In summary, the specular reflected field from a 2D textured planar waveguide, as calculated with our Green's function formalism, is given by equation (2.38), and all Chapter 2. Theory 32 higher order diffracted fields are given by equations of the form (2.39). 2.5 Multiple Gratings In this section we generalize the solution developed in the preceding sections to account for an arbitrary number of thin gratings, £, within the waveguide. As a particular applica-tion of this generalized formalism, the numerous gratings can all be adjacent, effectively allowing the treatment of thick gratings. If the waveguide contains £ gratings, or, equivalently, £ planar polarization sources, then we can write the susceptibility as Xif) = Xs(z) + Ax9l (p, z) + AXg2(p, z) + ... = Xs(z)+ £ &X9I(P,z) (2.40) i=i...e At any given value of z, only one of the Ax9< terms is nonzero. The solution of Maxwell's equations in analogy with equation (2.23) is Ec(p;z) = E*°M(P;z)+ dz[gc (/?; z, z[) £ XSm(z[) EC(P - GM, z[) J-oo m dz'2 9 c (P; z, 4) XGJZ'2)Ec(P - GM] 4) + ... (2.41) where the integral over dz\ goes from zQ. — t9i/2 < z < z0i +t9i/2. The center, and thickness of grating i are denoted by z0i, and tg. respectively. Upon making the thin grating approximation in each textured region (see equation (2.32)), equation (2.41) can Chapter 2. Theory 33 be written as 4 W = ^ o m W + £ I , ( ^ , ^ ) E t t K ) (2.42) j=i...e ™ The susceptibility Xnm, as before, is assumed not to depend on z within each grating, however it can be different for each grating depending on the index contrast and filling fraction. It is therefore labelled with the superscript gi. To handle the multiple gratings the cavity has been defined as extending from the top of the uppermost grating to the bottom of the lower most grating. With this definition we avoid having to derive any new Green's functions to describe coupling between gratings in different layers. However, in doing so there is no longer a unique ec to the problem. One can choose an arbitrary ec, and once chosen the susceptibility in each grating is unambiguously determined by Ax9Mz) = e9M*l~€c (2-43) where e9i (p, z) is the actual dielectric constant at any point within the grating. The implications of choosing different values of ec will be investigated in Chapter 4. It should be noted that we can even incorporate an untextured layer within this cavity which has a real dielectic constant different from ec by setting all Fourier components to zero except for the D C component, xoo-As all the gratings exist in a cavity c, with background dielectric constant ec all of the tensors, In (z0i,z0j), are simply the Green's function (2.18), integrated over the thickness Chapter 2. Theory 34 of grating j, and evaluated at grating i. When i — j , it is identical to the tensor defined in equation (2.33). Note that the zz, term is only nonzero when i = j. Equation (2.42) represents £ infinite sets of scalar equations, an infinite set for each grating. Taking only N Fourier components for each grating we can cast the solution in matrix form as: E=E + M • E • (2.44) The vector E is a 3N£ component vector of the Fourier field amplitudes. To clarify; components 1 to 3N are the field amplitudes at grating 1, z = z0l, components, 3N(i-1)+1 to 3Ni are the field amplitudes at grating i, z = z0i. The vector E is a 3N£ component vector of the homogeneous field amplitudes at each grating. Finally, the tensor M is a 3N£ x 3N£ component matrix with each square 3N x 3iV block describing the coupling between gratings. The diagonal blocks represent the fields at each grating as a result of itself, they are identical to the 3N x 3iV matrices in equation (2.32). Solving this equation by matrix inversion, as in section 2.4, we find the field amplitudes in grating i to be EM= £ [1 - M]y E (z0j) (2.45) where the subscript ij refers to a specific 3iV x 3iV block of the 3N£ x 3N£ matrix [ l — M ] _ 1 . Notice that when £ = 1 this equation reduces to equation (2.35). The field amplitudes at the surface of the structure are given by a sum of the fields Chapter 2. Theory 35 which emanate from each grating that get transmitted out of the top of the structure: E(zt)=E (zt)+ £ N (zt,z0i) E(z0i) (2.46) i=i...e =>• =$-hom The matrices E (zt) and E (zt) are 3 A'' x 1 column vectors of the total, and homo-geneous fields respectively. The 3N x 3N matrix jV (zt, -z0 i) is analogous to N (zt, zQ) in section 2.4, and takes different values for each i. Substituting (2.45) into (2.46), the final answer for the field amplitudes above a slab waveguide textured with £ planar periodic gratings becomes => =$.horn k ^ =>hom E(zt)=E ( * * ) + £ £ N (zt,z0i)[l — M]ij E (z0j) (2.47) i=i...ej=...e in analogy to equation (2.37) in the case of a single grating. The total field above, E (zi), is a sum of the fields radiated out of the upper surface from the polarization in each grating, and all the polarization they induce in the surrounding multilayer, including the polarization they induce in each other. Equation (2.47) is the general self-consistent solution for the Fourier field amplitudes at the surface of an arbitrary, multi-layer, planar structure textured with I planar 2 D gratings, located anywhere within the structure. The only approximations are, as in the single grating case: i) each grating is thin enough that the field variation over each may be neglected, and ii) a finite number of Fourier components of the field structure have been retained. It is important to note that approximation i) no longer limits the thickness or profile of the gratings that can be simulated, because it is now possible to split a thick grating into many thinner gratings, all with the same reciprocal lattice vectors {Gm} but Chapter 2. Theory 36 with different Fourier components Xnm- The only added cost of subdividing the gratings is in computation time. Chapter 3 Discussion and Examples In this chapter various examples are used to illustrate how the formalism developed in the previous chapter can be used to understand the nature of the E M excitations characteristic of waveguides textured with 2 D periodic gratings. 3.1 Resonant Modes 3.1.1 Square lattice We first illustrate some of the basic scattering properties of 2D-textured waveguides using calculated reflectivity spectra from a one-layer, free-standing slab waveguide of thickness ts, with a buried layer consisting of a square array of cylindrical dielectric perturbations of thickness tg « ts. The pitch, A, for a square grating is the real space distance between the dielectric scattering centers. The filling fraction is the ratio of the amount of material within the grating, z0 — tg/2 < z < z0 + tg/2, with dielectric constant eg to that with 37 Chapter 3. Discussion and Examples 38 dielectric constant e g 1 . The parameters for this structure are: A = 500nm, ts/A = 0.26, tg/A = 0.05, es = ec = 12.25, eg = 2.25, z0 = 35nm, and the filling fraction of the square symmetric grating is 25%. This structure was chosen primarily for pedagogical reasons; we will refer to it as structure 1. For a grating with square lattice symmetry the reciprocal lattice vectors are given by Gnm — nf3gx + rhBgy with Bg = 2 7 r / A being the principle lattice vector. Given ec, es, eg, A, and the filling fraction, the Fourier coefficients of the grating, Xnm, c a n be calculated easily. Appendix B outlines this calculation for an arbitrary 2D periodic lattice of circular holes. The specular reflected field,from this structure, when excited with an incident plane wave with a fixed in-plane wavevector of j3inc/'Bg = O.Olx, is shown in Fig. 3.1. Such an incident in-plane wavevector probes the 1st Brillouin zone at a point slightly detuned from zone center in the X symmetry direction. This figure plots the squared magnitude of equation (2.38) versus the normalized frequency. We have included 9 Fourier components of the field at B = Binc + nBgx + mBgy for n, m = 0, ± 1 . A sketch of these components in momentum space is shown in Fig. 3.2a for /3; n c = 0.0, corresponding to an incident plane-wave perpendicular to the surface which excites a zeroth order Fourier component at the center of the Brillouin zone (the dotted line in Fig. 3.2). Figure 3.2b is a diagram of the Fourier components when Binc = {\Binc\,Q}, corresponding to an incident plane wave 1 I n the remainder of the thesis eSi refers to the actual value of the dielectric constant in grating i outside of the holes, e 9 i refers to the actual value of the dielectric constant in grating i inside of the cylindrical holes, and ec refers to the arbitrary dielectric constant used to calculate the Green's functions and homogeneous fields. When only one grating is considered, as in this section, the subscript i will be dropped. Chapter 3. Discussion and Examples 39 1 > • l-H -t—> o 0 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Re{S/pg} Figure 3.1: Reflectivity spectrum at B/Bg = 0.01a; for structure 1. The solid curve is for s-polarized radiation and the dotted curve is for p-polarized. Each resonance is due to coupling into the radiative component of the renormalized slab modes with: (1) the lowest order TE-like modes, (2) the next higher order TE-like modes and (3) the lowest order TM-like modes. oriented along the X symmetry direction of the Brillouin zone. Figure 3.2c corresponds to detuning in the M symmetry direction: P i n c = {\pinc\/V2, \t3inc\/\/2}). The solid arrows in Fig. 3.2 depict the in-plane wavevector of each field component, and the dashed arrows depict the polarization of its transverse electric field. Sketches of the Fourier components in momentum space, like the one in Fig. 3.2, are extremely useful in describing the physics of the eigenmodes attached to textured planar waveguides. We will refer back to Fig. 3.2 frequently in the subsequent sections. Chapter 3. Discussion and Examples 40 F i g u r e 3.2: Fourier field components included i n the calculations shown in F i g . 3.1, for a square grating. Do t t ed lines trace out the first B r i l l o u i n zone of the lattice. M , and X denote the points of symmetry w i th in the B r i l l o u i n zone. Exc i t a t ion at normal incidence is depicted i n (a), and detuning in the X and M directions is depicted in (b) and (c) respectively. T h e 1st order Fourier components are labeled by A , and the 2nd order by B . T h e component at the center of the B r i l l o u i n zone is the 0th order or radiative component. T h e solid arrows are the in-plane wavevectors for each Fourier component, and the dashed arrows, the polar iza t ion of the electric field associated w i t h s-polarized Fourier components. Chapter 3. Discussion and Examples 41 Some of the features of the spectrum in Fig. 3.1 can be understood in terms of simple kinematics and resonant grating-coupling to bound modes characteristic of the slab. These features will be described first. Other, more subtle features of the true modes result from renormalization effects. These properties are discussed in the subsequent subsection. Kinematic Effects The specular reflectivity spectrum from an untextured multi-layer dielectric slab is mod-ulated by the constructive and destructive interference of multiply reflected fields with a unique in-plane wavevector, /?jnc(u)), for each driving frequency a). This gives rise to the slowly varying background in the spectrum in Fig. 3.1. In the presence of a periodic grating with Fourier components {Gm}, incident plane waves with in-plane wavevec-tors, Pinc(Cj), generate a polarization source at the grating that excites the surround-ing layers with fields oscillating at the same frequency, but with in-plane wavevectors Pinc(&) + {Gm}. For most in-plane wavevectors these locally driven fields are weak, and they have little effect on the specular reflectivity spectrum. Under these circumstances the dominant fields excited by the plane wave are essentially the same as those that would be excited in the absence of the grating. They represent the continuum, or Fabry-Perot, Bloch eigenstates of this dielectric structure. However, even for a weak perturbation, if one of the /?i„c(u)) + {Gm} is close to a wavevector at which the untextured slab supports a bound mode, the amplitude of the corresponding field can become very large. This Chapter 3. Discussion and Examples 42 resonant field influences the specular reflectivity spectrum over a range of frequencies in the vicinity of this phase-matched condition. The sharp features in Fig. 3.1 correspond to frequencies at which various Fourier components of the 2D periodic grating add to the incident wavevector, 0inc = 0.018gx, to effectively resonantly excite various T E and T M slab modes. As can be seen in Fig. 3.1, near zone center the resonant modes are grouped in sets of 4. This is a consequence of the symmetry of the 2D square reciprocal lattice, as is evident from Fig. 3.2a. Recall that of the 9 lowest order Fourier components of the field, 4 have \fi\ ~ 3g near zone center. The 4 resonances labeled (1), in Fig. 3.1, are due to coupling into 4 eigenmodes of the textured waveguide which are predominantly composed of distinct linear combinations of these 4 TE-polarized Fourier components of the field, components labeled by A in Fig. 3.2 a. The resonances labeled (2) are the eigenmodes which are composed primarily of the TE-Fourier components with \B\ ~ y/2Pg, labeled B in Fig. 3.2a. The set of peaks labeled (3) are primarily made up of the 4 TM-polarized Fourier components with \B\ ~ Bg. The T M slab modes are at a higher energy as the effective index for the T M slab modes is less than for T E slab modes. At an even higher-energy one would observe the 4 eigenmodes resulting from the T M Fourier components at \p\ ~ y/2Bg, etc. On the basis of this purely kinematic argument, the four distinct eigenmodes should be degenerate in energy at zone center, as the in-plane wavevectors of each dominate Fourier component making up these eigenmodes are all equal in magnitude there. When Chapter 3. Discussion and Examples 43 detuning in the X direction, 2 of the 4 eigenmodes should be degenerate (see Fig. 3.2b), and in the M direction there should be 2 pairs of degenerate eigenmodes (see Fig. 3.2c). To understand why there are in fact 4 distinct modes in both the X and M directions, 3 distinct modes at zone center, and why different modes are excited by either s- or p-polarized radiation, it is necessary to go beyond the simple kinematic effects of the 2D grating and consider how it modifies the basic nature of the eigenstates. Mode Renormalization Effects In order to further investigate the more subtle physics of these E M excitations we con-centrate on only one set of peaks. The reflectivity spectrum around the set of resonances labeled by (1) in Fig. 3.1 is shown on an enlarged scale in Fig. 3.3, for a variety of in-plane wavevectors extending 2% of the way to the X point (Fig. 3.3a) and M point (Fig. 3.3b) of the Brillouin zone. Note that the sharp, Fano-like features that signify coupling to waveguide modes all rise precisely to unity reflectivity. This has been previously noted by others in the context of ID and 2D gratings on waveguides[31, 33, 36], and it always occurs when only one of the components of the corresponding Bloch state is kinematically allowed to radiate into either cladding region. Note also that different resonances have quite different linewidths. These linewidths are inversely related to the lifetimes of the associated Bloch states, as limited by the strength of their radiating component. The 4 modes evident in all of these spectra are primarily T E polarized, although all have some admixture of T M polarization. Chapter 3. Discussion and Examples 44 O LU _1 UL LU > r-O 111 111 DC 0.00125 0.000625 0.396 0.398 0.400 0.402 0.404 R e { S / p J 0.01 0.005 0.0025 0.00125 0.000625 0.396 0.398 0.400 0.402 0.404 Re{S/pB} F i g u r e 3.3: Reflect ivi ty spectra for increasing \B/Bg\ away from zone center for a square grating (structure 1) when detuning i n the X (a) and M (b) symmetry directions. For clar i ty the curves have been offset vertically. The solid curve is for s-polarized incident radiat ion, and the dotted curve is for the p-polarized case. A s resonances approach zone center their w id th approaches zero or they become degenerate. The different dispersion properties and lifetimes of the eigenmodes are evident. Chapter 3. Discussion and Examples 45 At zone center the two narrow modes become infinitely long lived (true bound modes) and the two broad modes become degenerate, resulting in only three distinct modes. This is a fundamental consequence of the symmetry of the lattice: no isolated mode at the zone centre could have a finite lifetime because there can be no preferred polarization of its radiative component. Modes that radiate (have finite lifetimes) at the zone centre must be degenerate. Recall, from equation (2.31), that the polarization of the zeroth order component responsible for coupling to the radiative fields is given by P0 ocmJP N XomEm (N=4 when only considering the dominant components). Thus, with reference to Fig. 3.2a, a mode consisting of a fully symmetric superposition of the dominate field components, i.e., eigenvector oc (1,1,1,1), will result in a cancellation of the electric field of the zeroth order component. This is the case shown in Fig. 3.2a, and is demonstrated by the vector addition of the small, dashed arrows representing the electric field polarization. This is one of the infinitely longed lived modes at zone center. A n in-phase superposition of components in the x direction that are out of phase with those in the y direction, i.e., (1,1,-1,-1), also results in the an infinitely longed lived mode. The two degenerate modes can be composed of superpositions of two out-of-phase components in either the x or the y direction, i.e., (1,1,1,-1) and (1,-1,1,1). In this case the weighted, sum of the electric fields can not cancel and thus these modes are leaky at zone center (have a non-zero Oth order component). Now consider what happens when detuning away from zone center along a direction Chapter 3. Discussion and Examples 46 of symmetry, albeit a reduced symmetry compared to the zone center. The eigenmodes are still linear combinations of the Fourier components but now all of the amplitudes are no longer equal. Each mode gradually becomes dominated by less than 4, and perhaps just one Fourier component. Figure 3.4 again shows the P/Pg = O.Olx reflectivity spectra from Fig. 3.3a, but with the corresponding 1st order scattering spectra from components labelled A in Fig. 3.2b. From Fig. 3.4a it is clear that the mode which disperses to higher energy (resonance (iv) in Fig. 3.3a) becomes dominated by a single Fourier component, the component in the +x direction. Therefore, the mode's polarization and dispersion properties are then dominated by this component alone. Referring to Fig. 3.2b, significant dispersion to higher energy is illustrated by noting that the magnitude of the wavevector of th i s component increases at the maximum possible rate with detuning. As well, since th i s component is predominantly T E polarized (dashed arrow in Fig. 3.2b), the zeroth order polarization it drives must radiate s-polarized light. This can be interpreted from Fig. 3.2b by noting that the direction of the electric field of this radiative component is perpendicular to the direction of propagation, which corresponds to an s-polarized field. Likewise, the mode labeled (i) is dominated by the Fourier component in the —x direction (see Fig. 3.4a). This mode's significant dispersion to lower energy is illustrated in Fig. 3.2b by its wavevector now decreasing in magnitude at the maximal rate upon detuning. As before this mode is s-polarized. Chapter 3. Discussion and Examples 47 CD »rj 1e+4 O C/3 O > o CD CD 1e+2 1e+0 1e-2 1e-4 > o CD CD a) | A ../ ; K ! / • \ <\ I I 3 \ H H I 1.0 0.8 • l—H o CD 0 . 6 ^ 0.4 CD 1e+4 O O H-l 0.396 0.398 0.400 0.402 0.404 Re{S/p 2} 1e+2 1e+0 1e-2 1e-4 0 3 /. "">A ^^^^ , — 1.0 h 0.6 > -4—» o CD CD 0.4 0.396 0.398 0.400 0.402 0.404 Re{S/R} F i g u r e 3.4: Reflect ivi ty spectra (linear scale) at B/Bg = O.Olx from F i g . 3.3a (curve 0) and the first order Fourier components (log scale) in the +x direction (curve 1), — x direct ion (curve 2), +y d i rect ion (curve 3), and — y direction (curve 4). The two components i n the y direct ion are exactly equal due to symmetry. S-polarized field components are shown i n a) and p-polarized components i n b). Chapter 3. Discussion and Examples 48 The two less dispersive modes, labeled (ii) and (iii), quickly become linear super-positions of Fourier field components oriented almost perpendicular to the direction of propagation (see Fig. 3.4a and b). We must then consider the linear superpositions of these vector components to qualitatively describe the physics of these modes. In Fig. 3.2b it is evident that the magnitude of the wavevectors of each of these components change very little upon detuning, therefore these modes experience relatively little dispersion. The polarization of the radiative component they drive is found by considering the vector addition of the electric fields (see Fig. 3.2b). A symmetric (anti-symmetric) superposi-tion yields an electric field perpendicular (parallel) to the direction of propagation, which yields a s- (p-) polarized radiative component. The M direction is the other direction of high symmetry in the square lattice. Figure 3.3b shows the reflectivity spectra, and Fig. 3.2c, the relevant Fourier wavevectors when detuning in this direction. Away from the zone-centre along this direction all of the modes gradually become dominated by a linear superposition of 2 symmetric or antisymmetric pairs of slab modes oriented at ~ 45 degrees from the propagation direction. Modes which disperse to greater (lower) energy are dominated by the Fourier components in the +x and +y (~x and —y) directions. The symmetric and anti-symmetric combinations of these components yield one s- and one p-polarized mode for each pair. Away from these high symmetry lines in the Brillouin zone the eigenmodes are no longer purely symmetric and anti-symmetric superpositions of the untextured modes. The corresponding eignemodes are in general elliptically polarized. Chapter 3. Discussion and Examples 49 3.1.2 Honeycomb grating As a second example consider a grating with even greater symmetry, based on a hon-eycomb lattice. We consider the same structure as above but with the square grat-ing replaced with a honeycomb grating with a real space hole spacing, a, of 333.3nm. This results in a lattice pitch of A = y/3a = 577.7nm and thus ts/A = 0.225, and tg/A = 0.043. We refer to this as structure 2 (note: the value of A was chosen so that the distance between the principal Bragg lines of the honeycomb and square grat-ings is the same). The primitive reciprocal lattice vectors for this lattice are given by G = 27r /a{l /3 ,1 /V3}, {47r/3a,0}. Note that the calculation of the Fourier components of the grating, although similar to the square lattice, now involves the Fourier transform over a unit cell containing two circles. Included in the reflectivity calculation are 13 Fourier components of the field as depicted in Fig. 3.5. Figure 3.5a is at zone center, and 3.5b is detuning in the K direction of the Brillouin zone (dotted line). In contrast to the square lattice, there are now 6 lowest order Fourier components that define the grating, and there are therefore 6 resonances zone-folded to the origin of the Brillouin zone. These resonances are evident in the reflectivity spectra shown in Fig. 3.6a, which were calculated in the vicinity of the lowest order TE-like eigenmodes (the same as in Fig. 3.3a) when the exciting radiation is slightly off normal incidence along the M direction. From Figs. 3.6a and 3.3a it is clear that qualitatively similar behaviour is observed for both the honeycomb and square gratings. At the zone centre there are infinitely long Chapter 3. Discussion and Examples 50 Figure 3.5: Fourier field components included in the calculations for a honeycomb grating. Dotted lines trace out the first Brillouin zone of the lattice. M , and K denote the points of symmetry within the Brillouin zone. Excitation at normal incidence is depicted in (a), and detuning in the K direction is depicted in (b). The solid arrows are the in-plane wavevectors for each Fourier component. lived (truly bound) modes. As they evolve away from the zone centre they develop finite lifetimes. Modes which have finite "lifetimes at the zone centre become degenerate in pairs of s- and p-polarizations. Away from the zone centre, bands composed primarily of Fourier components with wavevectors parallel to the propagation direction are labeled by the s-polarized nature of their radiative component. Modes consisting primarily of Fourier components with wavevectors not parallel to the direction of propagation form pairs of bands with similar weak dispersion; one band has an s- and one has a p- polarized radiative component. In the square grating there was only one such pair of bands, for the honeycomb grating there are two pairs. The dispersion and polarization properties can be explained by considering the vectoral coupling of the dominate Fourier components as done in Fig. 3.2 for the square grating. Chapter 3. Discussion and Examples 51 &3 • i—< > 2 O CD <D 1 0 • i—i > -y . . . . | 11- 3 •7-— r-M a) 0.04 0.02 p/p, 0.01 0.005 0.38 0.39 0.40 0.41 0.42 R e { S / R } 3 O ^ « <D 1 0 < ^ . B I I 1 ! I.L. Li If _ s '•I r-K —-SL 1 " t b) 0.04 0.02 0.01 0.005 p/p g 0.38 0.39 0.40 0.41 0.42 Re{S/R} F i g u r e 3.6: Reflectivity spectra for increasing \0/0g\ for a honeycomb grat ing (structure 2) when detuning i n the M (a) and K (b) symmetry directions. F O r clari ty the curves of be offset vertically. T h e higher degree of symmetry causes 6 resonances i n contrast to the 4 found for the square grating. T h e solid curve is for s-polarized excitation, and the dotted curve is for p-polarized exci tat ion. Resonances exhibit quali tat ively the same behaviour as w i t h a square grating. Chapter 3. Discussion and Examples 52 For the honeycomb lattice, the K direction is the other high symmetry direction. Fig-ure 3.6b shows the reflectivity spectra in this direction. The eigenmodes labeled by A and B become primarily symmetric and antisymmetric superpositions of the two slab modes propagating almost perpendicular to the propagation direction (ie: superpositions of the components labeled A l and A2 in Fig. 3.5b). They therefore exhibit similar, weak dis-persion with one labeled s and one labeled p. The other 4 modes (2 s- and 2 p-polarizecl) experience intermediate dispersion, but again, because none contain components exactly-parallel to the propagation direction, they occur in s-p polarized pairs. 3.1.3 Summary To summarize the physics of resonant modes attached to planar textured waveguides as investigated with our reflectivity calculation, we note that each reflectivity spectrum probes the dispersion of the resonant eigenmodes along a vertical line in ui — Q space. Thus a plot of the value oiuijQQ at each resonance, as a function of ft, yields the dispersion diagram. The dispersion diagram for the honeycomb grating, found by plotting uj/0g for each resonance in Fig. 3.6a, is illustrated in Fig. 3.7. The slope of the dispersion curve for each eigenmode is the group velocity of that mode. Near the zone center the group velocity of the eigenmodes is modified due to the renormalization effects of the index contrast. The extent to which the modified group velocity extends away from the zone center increases with increasing scattering strength (index contrast). Right at the zone Chapter 3. Discussion and Examples 53 0.415 0.385 -I • • • 1 0.00 0.01 0.02 0.03 0.04 P/P8 F i g u r e 3.7: Dispers ion diagram for a honeycomb grating in the M direct ion w i t h i n the first B r i l l o u i n zone. The solid lines are s-polarized eigenmodes and dotted lines are p-polarized. The symbols represent the positions of corresponding Fano resonances i n F i g . 3.6a. center the group velocity is zero, and photonic standing waves result from the highly symmetric coupling of the bare slab modes. 3.2 Bound Modes In the last section we demonstrated that the specular reflectivity spectrum of propagating plane waves provides a convenient probe of the photonic dispersion of the leaky modes Chapter 3. Discussion and Examples 54 (resonant modes) attached to the porous waveguide. By considering the reflectivity of evanescent waves, it is possible to probe the true bound eigenmodes that exist below the light line. In this section we will use our model to probe these bound modes and discuss some of the nontrivial coupling which can occur in these 2D-textured waveguides. Experimentally, an incident evanescent wave can be achieved using a prism. In our formalism we can model an incident evanescent wave by simply setting pinc large enough so that w0 = \Ju>2 — p?nc is imaginary. This corresponds to probing the band structure below the light line where there is no continuum of modes but rather just discrete bound modes, as shown in Fig. 1.2 (Chapter 1). Instead of appearing as Fano resonances, bounded by unity reflectivity, the bound modes are revealed as poles in the specular reflectivity. Referring to Fig. 3.2b, if pinc = 0.5Pgx (detuning to zone edge) then three of the 9 components considered in the previous section have relatively large in-plane wavevectors so we only include six components (p = {± /? g /2 ,0} , and {±Pg/2, ±Pg}) in the present calculation. Note that at the zone edge there are no Fourier components of the Bloch states that are phase matched to radiate into the substrate or the vacuum. This is another way of understanding why the corresponding eigenstates are pure bound modes. As an example, we consider the same structure as in Section 3.1.1 (structure 1). Figure —* _ 3.8 shows the reflectivity spectrum at Pinc/Pg = 0.485£. There are two resonances excited by s-polarized evanescent waves near u>/Pg ^ 0.25. These are eigenmodes essentially composed of T E slab modes near P = {±Pg/2,0}. The second set of peaks near ui/Pg ~ Chapter 3. Discussion and Examples 55 Id g 1e+4 O 1e-4 0.20 0.25 0.30 0.35 0.40 R e { S / R } •J ^ j - s-pol p-pol 1 ' 0.45 F i g u r e 3.8: Reflect ivi ty spectrum near the B r i l l o u i n zone boundary (P//3g = 0.485x) for the square grat ing (structure 1). T h e true bound modes appear as poles i n the reflectivity spectrum. O f the 4 p-polarized modes, the 2 at lower CJ are T E - l i k e modes and the upper 2 are the lowest order T M - l i k e modes. 0.43 consists of 6 resonances. Two of these are the lowest order TM-like eigenmodes, consisting of T M slab modes near 3 = {±/3 g /2 ,0} , similar in nature to the s-polarized doublet at lower energy. The other 4 eigenmodes are composed primarily of the TE-like —* slab modes at 3 = {±3g/2, ± /3 g }. Two are excited by s-polarized, and two by p-polarized evanescent fields. In Fig. 3.9 we show the dispersion of these modes as calculated with the reflectivity model. As with the resonant modes at zone center, the coupling of Chapter 3. Discussion and Examples 56 0.45 r ^ 0.44 ;h oo CO. ^ 0 . 4 3 0.42 -0.47 0.48 0.49 0.50 P/P8 Figure 3.9: Bandstructure of bound modes near the zone edge (X point) obtained by plotting the location of the poles found in the reflectivity calculation. The solid lines are s-polarized while the dotted lines are p-polarized. The anti-crossing is due to the coupling of T E and TM-like modes. the untextured modes result in renormalized eigenmodes with distinct polarization, and dispersion properties. As at zone center, at zone edge the group velocities of the modes go to zero, resulting in standing waves, and band gaps. Note that the polarization of this zeroth order component (the component within the first Brillouin zone) is still a good quantum number below the light line. Chapter 3. Discussion and Examples 57 This portion of the bandstructure illustrates some of the subtle, but significant con-sequences of this particular scattering geometry; i.e., no translational invariance per-pendicular to a 2 D periodic texture. Two of the 4 TE-like bands consist primarily of anti-symmetric superpositions of T E slab modes with wavevectors oriented away from the direction of propagation, { p g / 2 , ± P g } and {—Pg/2,±Pg}. The combined effect of these principal components of the eigenmode driving the corresponding first order Fourier com-ponents of the grating susceptibility is to excite a polarization oscillation oriented along the X direction, with a wavevector parallel to the X direction (p-polarized), near the zone-boundary. If the waveguide dispersion is such that the T E slab modes at {±Pg/2, ± P g } are nearly degenerate in energy with the T M slab modes at {±Pg/2,0}, then this oscil-lating polarization sheet excited by the anti-symmetric superposition of T E slab modes can resonantly excite the T M mode oriented along X . This somewhat unconventional coupling mechanism leads to additional anti-crossings away from the zone boundaries, as evident in the dispersion curves shown in Fig. 3.9. The true photonic eigenmodes on either side of the gap associated with this anti-crossing consist primarily of two T E slab modes at {Pg/2, ± P g } , and one T M slab mode at {pg/2,0}. The position of these anti-crossings within the first Brillouin zone can be engineered easily by appropriate choice of in-plane lattice constants and symmetries, and the vertical slab properties. The anti-crossings can occur above or below the light line, and the associated eigenmodes are interesting superpositions of TE-like and TM-like slab modes. Such properties might form the basis of waveguide-based polarization mode converters. Chapter 4 Accuracy of the Model In Chapter 3 we demonstrated how our solution can be used to investigate the properties of the E M excitations of a textured planar waveguide. As with any theoretical solution and numerical model, there is the question of whether the solution obtained is correct. Therefore, we devote this Chapter to a discussion of the accuracy of the model, and will investigate some of the subtleties of the calculation. 4.1 Conservation of Energy First we test the solution to see if it is physically sensible; does it satisfy conservation of energy? In the planar diffraction grating geometry we consider, the energy flux carried perpendicular to the interfaces by the E M fields must be conserved. A l l calculations in this thesis were done in the regime where only one of the Fourier components of the polarization is allowed to radiated. Therefore all fluxes perpendicular to the surfaces are only carried by the specular components of the fields. 58 Chapter 4. Accuracy of the Model 59 The reflected field is given by equation (2.38), and for the purposes of this section we call this field ER. The transmitted field is given by an analogous equation to (2.38); it is derived from equation (2.28) with the same steps as carried out in section 2.4 for the reflected fields. We refer to this as ET- The flux up and down, normalized to the power of the incident field, are respectively given by *ER\2y/e^cos 9inc | 2 I E{nc |2\J~£a~ COS Qinc T2 = l - E r f y / c ^ C O S 9b = ^ 2 y/&Hb ~ B2 ^ ^ \Einc\2y/ea~cos8inc \/u2ea- fi2 where eA>6 are the dielectric constants of the semi-infinite half space above and below the structure respectively, 9inc is the angle of incidence of the incident field, and 9b the angle the transmitted wavevector makes with the surface normal. When only the one Fourier component is allowed to radiate conservation of energy is expressed as R2 + T2 = 1. Figure 4.1 shows, at Pinc/Pg = 0.2x, a plot of R2 and T2 versus Co/Bg. The structure consists of a 130nm GaAs core (es = ec = 12.25) cladded below with a lOOOnm oxide layer (eox = 2.25) and a semi-infinite GaAs substrate, and above with air. The core contains a square symmetry, 25nm thick oxide grating (eg = 2.25, A = 500nm. z0 = 0) with a filling fraction of 25%. Up to the normalized frequency u/Bg — 0.533, for this structure, the sum of the square of the specular reflectivity and transmission are equal to one, and energy is conserved. However, above u)/Bg = 0.533 this is not so. This is because at Bind fig = 0-2 the value of \Joj2eox — (fiinc — figx)2 becomes real, and thus the fiinc — Bgx Fourier component is able to radiate into the oxide cladding. Chapter 4. Accuracy of the Model 60 1.2 0.0 - — ^ , 1 0.45 0.50 0.55 0.60 0.65 Re{c5/pg} Figure 4.1: Specular reflectivity (solid curve) and transmission (dotted curve) illustrating the conservation of energy, R? + T2 = 1 (dashed curve). See text for the geometry of the structure. At u> = O.5330g (dotted line) one of the first order Fourier components radiates into the substrate and energy is no longer conserved by just the zeroth order component. There are then two radiating Fourier components. Some of the power is radiated by this second component rather than the first, and thus energy is not conserved by the zeroth order component alone. Note that the Fano resonances no longer reach unity when more than one component radiates, as has been noted by others in the context of resonant scattering. [31, 33, 36] Chapter 4. Accuracy of the Model 61 4.2 Accuracy of Approximations In this section we consider the accuracy of the two inherent approximations of our for-malism: and finite number of Fourier components and a constant field over the thickness of the gratings. First consider the less severe approximation; a finite number of Fourier components used to describe the total field distribution. Higher order Fourier components have no effect on the fundamental physics of the problem, they just add to the accuracy of the numerical result obtained. Additionally, as the number of Fourier components included increases so does the computation time. Therefore, in the interest of having a compu-tationally efficient model we would like to minimize the number of Fourier components included. The requirement is that we include enough components so that the solution converges to an acceptable level. Figure 4.2 shows the reflectivity calculation for the free standing structure of chapter 3 (structure 1) when 25 Fourier components (solid curve), and 9 Fourier components (dashed curve) are included. The two solutions agree very well. We have tested convergence on a number of semiconductor air/oxide structures with gratings 0-150 nm in thickness; in general we conclude that for convergence it is sufficient to include up to the 2nd order Fourier components. Since we know our calculations including 9 Fourier components are themselves con-verged, it is then possible to compare these calculations with other reflectivity measure-ments. Below we test our approximation of a thin grating by comparing our calculation Chapter 4. Accuracy of the Model 62 0.4 i . . • 1 0.396 0.398 0.400 0.402 0.404 Re{c5/pJ Figure 4.2: S-polarized specular reflectivity, at 0/0g = O.Olx, from the free standing structure of Chapter 3 (structure 1) as calculated with 25 (solid curve) and 9 (dashed curve) Fourier components. The shift between the two calculations is 0.015%, which demonstrates sufficient convergence of the 9 Fourier component calculation. with other methods; both theoretical and experimental. We demonstrate the accuracy of this Green's function-based reflectivity calculation, when applied to a thin slab, by comparing it with an exact integration of Maxwell's equations[15, 20]. Figure 4.3 compares the specular reflectivity from the free standing 2D textured slab described in Chapter 3 (structure 1), as calculated using the approach described in this thesis (upper plot), and using the "exact" model (lower plot). The computation time required for the exact solution was approximately 2000 times longer than for the Green's function approach. In Fig. 4.3 it is clear that the spectra are in excellent agreement, both with respect to the locations and widths of the resonances. Chapter 4. Accuracy of the Model 63 F i g u r e 4.3: Reflect ivi ty spectra, at 0/0g = 0.005x, for the free standing 2D textured slab described i n Chapter 3 (structure 1) as calculated w i t h an exact integration of Maxwel l ' s equa-tions (lower plot) and w i t h the approach described i n this thesis (upper plot) . The resonance widths and shapes are i n excellent agreement. T h e gap wid th agrees to w i t h i n 7% while the center of the gap agrees to wi th in 0.6%. The inset show the resonances over a wider energy range. Chapter 4. Accuracy of the Model 64 Figure 4.4: A plot of the p-polarized mode at the 2nd order TE-like gap as calculated with the present method and as experimentally measured in a specular reflection geometry [17]. See text for the structures parmaters. The center of the gap agrees to within 0.6%, and the gap width to within 7%. Note that for the resonances shown in Fig. 4.3, tg « (0.07)27r/'d'^/ej, which is well within the limit that the wavelength of the radiation is much less than the thickness of the grating, tg <C 27r/tDy/e s". Also note that the "exact" code is actually only correct to within the size of the mesh steps. We conclude that when the slab thickness is small compared to the wavelength of interest, the Green's function approach works very well, even when the scattering strength of the grating is significant. This model has been tested against experimental results for both square and triangu-lar gratings. The details of the experiment can be found in the following references[17. Chapter 4. Accuracy of the Model 65 37, 38]. Here we simply show (see Fig. 4.4) a typical experimental plot, and a fitted the-oretical plot calculated with the Green's-function model for the square grating data[17]. The structure consisted of an 80nm GaAs (es = ec = 12.25) guiding core cladded with 1800nm of oxide (eox — 2.25), and semi-infinite GaAs below, and air above. The core contained air holes (e9 = 1.0, A = 500nm) etched entirely through the core yielding a filling fraction of 28%. It is clear that the two agree extremely well. Experimentally, there are uncertainties in the hole diameter, layer thickness, and in the value of the re-fractive index of the oxide. This uncertainties all lead to an experimental error which, at best, is greater than the error in peak location found when comparing the "exact" and Green's function models. We therefore conclude that, in the interest of comparing with experimental data from thin but strongly textured gratings, the Green's function based model presented in this thesis is effectively exact. 4.3 Thin Grating Approximation In the last section we concluded that our model, despite the approximations, is accurate to within the experimental error of reflectivity experiments. In this section we will look more closely at the approximation of a thin grating. The accuracy of this approximation can be harder to estimate and in general can be quite different for different structures and choices of the value of ec. As we will see, we can set up the calculation in such a way as to minimize the errors introduced by this approximation. Chapter 4. Accuracy of the Model 66 0.60 IS COL 0.35 \ 0.30 80 100 120 140 160 180 200 220 240 260 t [nm] Figure 4.5: A plot of the location of the upper and lower edge of the 2nd order gap in a I D textured free standing waveguide of thickness tg, es = 12.25, eg = 1.0, and a filling fraction of 25%. The dotted lines are for the calculation with the Green's function approach, the solid lines with the "exact" finite difference method. To address the limitations imposed by the thin-grating approximation we first consider a free standing waveguide (es = ec = 12.25), of thickness ts, textured with a ID grating (eg = 1.0, A = 500wn), with a thickness tg = ts, and filling fraction of 25%. Figure 4.5 plots the location of the upper and lower edges of the 2nd order gap calculated with the Green's function, and "exact" methods. It is clear that for thin gratings the two methods agree extremely well. When the waveguide is 250nm thick, a thickness greater then those that are usually of interest in the context of photonic crystal membranes, the two methods agree to within 4.5% for the upper edge, and 0.6% for the lower edge. This Chapter 4. Accuracy of the Model 67 demonstrates that it appears difficult to specify a simple relationship between the model accuracy and the grating thickness. For a given structure, the accuracy depends on the mode being studied, and different layer structures may result in different errors for a fixed grating thickness. Although it is difficult to find a simple relationship between grating thickness and the model accuracy we can further investigate the accuracy of the model by considering a subtle aspect of the calculation, the value of ec. We first investigate the thin grating approximation, and the effect of ec by considering an untextured waveguide. The solution of Maxwell's equations presented in Chapter 2 treated the grating in a slab waveguide as a polarization source term. In the limit of no grating our Green's function solution of Maxwell's equations should accurately model the reflectivity from a simple slab waveguide with no grating. In the absence of a grating there are no bound modes above the light line and thus the spectrum there simply consists of the Fabry-Perot background. However, with an incident evanescent field we can probe the guided modes of the untextured guide. Figure 4.6 shows the reflectivity from a 80nm, free standing, slab waveguide (es = 12.25) in a region below the light line of the cladding material. This is modeled by setting the filling fraction of the grating to zero. The solid curve is found when ec — 12.25. With this choice of ec the susceptibility Axg is zero and thus the entire structure (all the material) is described by the homogeneous solution. Recall equation (2.5), when Axg — 0 the particular solution is zero, and when ec (the dielectric constant used to calculate the homogeneous field) is Chapter 4. Accuracy of the Model 68 Re{5/pj F i g u r e 4.6: Reflect ivi ty spectra from a 80nm untextured free standing waveguide (e s = 12.25). Poles are due to coupling to guided modes wi th incident evanescent radiat ion. T h e solid curve is for e c = 12.25, and the dashed curve is for e c = 1.0. W h e n the latter calculat ion is done by subdiv id ing the grating into two and four thinner gratings and using the multi- layer code we find the dotted and dash-dot curves respectively. equa l t o the a c t u a l d ie lec t r ic constant of the m a t e r i a l the homogeneous s o l u t i o n descr ibes the a c t u a l s t ruc ture . Therefore , the peak i n th i s p lo t is d e t e r m i n e d b y the pole i n the homogeneous t e r m of the so lu t ion . S ince the homogeneous so lu t i on is independen t of the a p p r o x i m a t i o n we expect th i s peak to e x a c t l y co r respond to the u n t e x t u r e d g u i d e d m o d e energy. T h i s was verif ied b y so lv ing the t ranscenden ta l e q u a t i o n 1 for s lab waveguide modes . W e found a mode energy of Co/6g = 0.2873, w h i c h , as expec ted , is e x a c t l y the l o c a t i o n of th i s peak. x T h e transcendental equation, and dispersion curves for untextured slab waveguides can be found i n reference [22]. Chapter 4. Accuracy of the Model 69 However, if the calculation is done with ec = 1.0 then the homogeneous solution effectively "sees" no material, the DC component of Axg is non-zero and the structure is described entirely by the particular solution. This calculation gives the dashed curve. This peak is determined by the pole in the particular solution and thus depends on the approximation. The approximation that the field of a guided mode is constant over the width of the guiding core is, of course, a poor one, and thus we get the wrong answer. The dashed curve is wrong by a factor of ~ 1.1%. If we split the guide up into two (dotted curve) and four (dash-dot curve) layers using the multi-grating formalism (keeping ec = 1.0), the approximation becomes better and better, and the answer approaches the correct one. Figure 4.6 clearly illustrates the effect of the approximation on the answer we obtain. As all the material in the guide is described by Axg, and thus subject to the approximation, we can take the factor of 1.1% as a worst case scenario for how severe the effect of the approximation may become, in this type of structure. For untextured layers it is therefore clear that ec should be set to the dielectric constant of the layer. However, when texture is present the choice of ec is not so clear. Consider the same free standing waveguide (es = 12.25) as above but with a square grating of air holes (A = 500nm, eg = 1.0) etched entirely through the core such that the filling fraction is 50%. Unlike the untextured case where one choice of ec gave the correct answer and the other produced a better and better answer as the gratings got thinner, here both methods feel the effect of the approximation in equal amounts. Figure 4.7 plots the poles associated with the TE-like mode at the upper edge of the 1st order Chapter 4. Accuracy of the Model 70 Re{c5/pg} Figure 4.7: Reflectivity spectra from a 80nm free standing waveguide (es = 12.25) textured with air holes extending entirely through the guide yielding a filling fraction of 50%. Poles are due to coupling to guided mode at the upper edge of the 1st order gap with incident evanescent radiation. See text for the differences in the calculation for each curve. gap at the X point of the Brillouin zone. Curves 1 and 2 were obtained using the single grating formalism. Curve 1 and curve 2 correspond to the choices ec = 1.0 and ec = 12.25 respectively. Curves 3 and 4 were obtained by subdividing the 80nm layer into two 40nm thick gratings, each with ec = 1.0 and ec = 12.25 respectively. It is evident that as the gratings get thinner the two methods shift towards each other an equal amount. As the grating thickness goes to zero each method will converge to the correct answer. As a further example, in Fig. 4.8 we look at the p-polarized mode slightly off in the r — X direction from the zone-center at the 2nd order TE-like gap. This structure is a lOOnm thick free standing guide (es = 12.25) textured with a square array of air holes Chapter 4. Accuracy of the Model 71 0.48 0.50 0.52 Re{o5/f3g} F i g u r e 4.8: Reflect ivi ty spectra at 0/0g = O.Olx from a lOOnm free standing waveguide (e s = 12.25) textured w i t h air holes extending entirely through the guide yie lding a fi l l ing fraction of 25%. Resonances are due to the p-polarized mode at the T E - l i k e 2nd order Bragg condit ion, (a) is for e c = 1.0 and (b) is for e c = 12.25. Not ice the difference i n scale on the horizontal axis. Shifts i n the locat ion of the peaks are due to the th in grat ing approximat ion. See text for how the calculat ion differs for curves 1, 2, 3, and 4. Chapter 4. Accuracy of the Model 72 (tg = ts, A = 500nra) yielding a filling fraction of 25%. A l l the curves in Fig. 4.8a were calculated using ec = 1.0. The solid curve (curve 1) is the result of the calculation done with the thick grating split up into 3 gratings of equal thickness. As these gratings are thin this solution is very close to the correct answer. When the calculation is done with 2 equal gratings (curve 2), 2 gratings where one is 80nm thick (curve 3), and one grating (curve 4), the answers shift slightly due to the thin layer approximation. When the calculation is done using ec = 12.25 (see Fig. 4.8b) we find the 3 grating calculations to be almost the same, since the gratings are thin. However, now the corresponding shifts, when we model the structure with multiple gratings as above, are smaller. This is directly due to the approximation. For the calculation shown in the latter plot (Fig. 4.8b), 75% of the material within the grating is described by the homogeneous solution, and only 25%, the amount within the holes, by the particular solution. In the former plot (Fig. 4.8a) the opposite is true, 75% of the material is described by the particular solution and is thus subject to the approximation. We therefore conclude that to maximize the accuracy of our solution we should model filling fractions less than 50% by setting ec equal to es. For filling fractions greater than 50% we should do the opposite; set ec equal to eg. This minimizes the amount of material described by the particular solution and thus subject to the approximation, and maximizes the amount described by the homogeneous solution. In addition, we now postulate two ways in which the approximation can be made even more accurate. Chapter 4. Accuracy of the Model 73 From the above discussion it is clear that the accuracy of the approximation depends on the filling fraction of the grating. The filling fraction dependence of the Fourier com-ponents of the grating is dominated by the DC component since it is directly proportional to it, however, the important information, the shape of the grating, is entirely described by higher order components. We therefore postulate that a very accurate solution could be found if one chooses an ec such that the DC Fourier component is zero. The value of the higher order components would be appropriately adjusted to describe the entire structure with higher order components alone. This is effectively equivalent to defining a average index of refraction for the textured region. We postulate that this method would maximize the accuracy of our Green's function based model. The postulate above is however still limiting when it comes to multi-grating calcu-lations as it still requires one unique ec. If the structure one wishes to model contains gratings with different filling fractions and/or different material, the chosen ec may im-prove the approximation for one grating but not for another. The most general way to overcome this problem is to define each grating with its own ec, such that the D C com-ponent for each is zero, and then describe the coupling between the gratings by deriving Green's functions which couple each grating to all the other ones. This would require the derivation of £ 2 individual Green's functions for equation (2.44) and £ individual Green's functions for equation (2.46), instead of using the single cavity Green's function employed herein. A general algorithm could be used to generate these Green's functions automatically. Chapter 4. Accuracy of the Model 74 In summary, an in-depth investigation of the thin layer approximation and the role of ec has yielded the following details: i) in order to maximize the accuracy of the model, structures with a filling fraction less (greater) than 50% should be modeled by setting e c = e s (ec=e3)- ii) with the use of a filling fraction dependent ec in the grating region(s) it may be possible to increases the accuracy of the model even further. One should note that the subtle effects discussed in this section produce shifts in the resonances on the order of < 2%. These shifts, in most cases, are much less than the experimental uncertainty, and uncertainties in other theoretical solutions, and are therefore inconsequential to the quality of the model for most of its applications. Chapter 5 Applications In previous chapters we gave a theoretical development of the model, a physical interpre-tation of the results, and a discussion on the accuracy of the model. Given this thorough description, we are now in a position to discuss some examples of the many applications of the model. 5.1 Resonant Filter One of the most promising applications of 2D textured waveguides in the near future is the idea of a notch filter. To satisfy the increasing demand for more wavelengths per optical fiber (WDM) one needs to generate and control the flow of light over quite a large bandwidth. Current technology in the field does most of the optical filtering with bulk optics, in free space. The filters themselves are typically multi-layer dielectric stacks. An alternate, waveguide-based optical filter has been suggested[31, 33, 36]. The formalism developed in this thesis allows for the direct modeling of these filters. Below we emphasize 75 Chapter 5. Applications 76 the novel perspective and additional understanding that the Green's function approach provides. The basic idea is to take advantage of the unity reflectivity observed when the fre-quency of near normal incidence light is close to that of one of the leaky zone-centre modes attached to the textured waveguide. The advantage of 2 D over ID gratings is their polarization insensitivity, which in the Green's function approach is directly trace-able to the fact that the degenerate eigenstates at the zone centre must have opposite polarizations when detuning away from zone centre along high symmetry directions in the lattice. Using the insight provided by the present formalism, we can sketch a design strategy for controlling the centre frequency, linewidth, and background reflectivity from a 2 D periodic grating embedded in a planar waveguide structure. To obtain the desired centre frequency, one designs a planar waveguide with an effective refractive index for the lowest order T E mode, TITE, a n d a corresponding 2 D grating lattice and pitch such that the 2 n d order T E slab modes are zone-folded to the Brillouin zone centre at the desired centre frequency. The background reflectivity is controlled by the non-resonant Fabry Perot resonances of light reflected from the air/top-cladding layer interface, and hence the overall thickness of the guiding region. The linewidth of the resonance can be tuned by adjusting the phase of the radiative component of the leaky mode, at the grating location, as it undergoes multiple reflections within the cladding layers, and hence it can be varied, for a fixed background reflectivity, by varying the position of the grating within Chapter 5. Applications 77 • H \ £ 0.5 • o 0.354 0.356 0.358 0.360 0.362 Re{S/pg} Figure 5.1: Normal incident reflectivity spectrum of a polarization insensitive notch filter based on a honeycomb grating. See text for structure's geometry. the slab. Figure 5.1 demonstrates a polarization insensitive resonant filter designed with a honeycomb grating. The structure is a 200nm free standing guide (es = 12.25) containing a 25nm wide grating of cylindrical oxide holes (eg = 2.25) with a filling fraction of 25% and real space hole spacing of 333.3nm. Due to the symmetry of the grating the resonance in Fig. 5.1 is independent of the polarization of the incident field[33]. As mentioned above, buried gratings make it possible to control the linewidth of the filter. We demonstrate how the linewidth of the filter changes with the location Chapter 5. Applications 78 i i Grating Location [nm] Figure 5.2: Linewidth of the notch filter shown in Fig. 5.1, as it varies with the location of the grating in the slab. In this structure the linewidth can be altered by a factor of 6 by just shifting the location of the grating by 70 nm. Recall that z0 — 0 refers to a grating centered about the center of the cavity. of the grating in Fig. 5.2. The linewidth of the filter oscillates, as the location of the grating is varied, with the frequency of the oscillation determined by the dimensions and refractive indices of the waveguide. This oscillation in the lifetime (oc 1/linewidth) of the eigenmode is a result of an oscillation in the strength of the zeroth order component of the polarization at the grating. The phase shift associated with the multiple reflections of the radiation within the cladding layer causes the radiation, reflected back to the grating location, to either enhance or degrade this polarization strength. This effect, unique to Chapter 5. Applications 79 buried gratings, alters the linewidth of the filter by a factor of 6 in this structure. 5.2 Textured Waveguide Properties Ultimately, in the interest of achieving integrated optical devices, one needs to control the in-plane flow of light through the waveguide. These planar waveguide based 2 D textured photonic crystals may offer one of the most attractive ways in which to achieve this control. In order to design integrated optical devices is 2 D P B G waveguide material one needs to understand the dispersion, lifetimes, and polarization properties of guided E M modes attached to these structures. Our model is a powerful tool with which to study these properties. The formalism developed in the thesis accounts for an arbitrary number of planar gratings located anywhere within, or on the surface of, an arbitrary stack of planar-layers. Therefore, practically any planar structure one could envision fabricating can be theoretically studied and optimized with our model. Cast as a reflectivity calculation the formalism directly models a powerful experi-mental method of characterizing any device, once fabricated. Therefore, by comparing measured to calculated spectra one can probe the properties of the device directly. This model has been used to compare with experimental specular reflectivity from both a square, and a triangular grating on a semiconductor waveguide. Figure 4.4, in Chapter 4 showed a typical reflectivity spectrum and the corresponding fitted model. It is clear from Chapter 5. Applications 80 that figure that the comparison is excellent, illustrating the power of the model. Given the physical interpretation of the reflectivity spectra in Chapter 3 it is then possible to understand the details of a device's bandstructure through these reflectivity measure-ments. A detailed study of the bandstructure and eigenmode lifetimes extracted from the reflectivity spectra of both the model and the experiment has been carried out, the results may be found in references [17], and [37]. In the past, modeling techniques, for mapping the band structure, have been com-putationally expensive[15, 32], limited by approximations[27], or even restricted to the modes below the light line[21]. This model provides a computationally efficient tool with which to study the entire bandstructure, both real and imaginary, above and below the cladding light lines. As mentioned above, in addition to simply studying the physical properties of devices, the model may also be used to optimize their properties for a specific application. In the previous section we demonstrated how the grating location may effect the resonance width; an effect which may be used to optimize the notch filter. In what follows we use the model to study the effect of the grating shape and filling fraction. Either inadvertently or intentionally it is possible for the grating's holes to have a slanted profile. It is important to understand what effect this slant might have on the guided mode eigenstates. For this example, consider a realistic waveguide structure with a 80nm GaAs guiding core (es = ec — 12.25) clad with air above, a 1500nm layer of oxide {eciad = 2.25) below, and an effectively semi-infinite GaAs substrate below the oxide. The Chapter 5. Applications 81 square grating (eg = 1.0, A = 500nm) is etched into, and penetrates entirely through the 80nm core. This structure corresponds to that studied in Ref. [17]. Figure 5.3 shows the p-polarized modeling results slightly off in the T — X direction from the zone-center at the 2nd order TE-like gap. The solid curve is the result when the air holes have a perfectly vertical side wall. The filling fraction has been taken as 25%, which corresponds to a hole radius of 141.5nm, as the pitch of the square grating is 500nm. When the radius of the upper opening of the hole remains at 141.5nm but the hole side wall is given a 14 degree slope, making the bottom of the hole have a radius of 121.5nm, then we get the dashed curve. The 14 degree sloped side wall was modeled with 4 gratings, each with a progressively smaller radius. It is clear that this is a significantly different result. However, this shift is in some sense not due to the sloped side wall but rather to the different filling fraction of the grating. The dashed-dot curve is for a grating with a 14 degree sloped side wall but with a top hole radius of 149nm and a bottom of the hole radius of 134nm. This profile retains the air filling fraction of 25% used when the side walls were vertical. It is clear that the effect of the side wall slant is actually small. However, this calculation points out the importance of the filling fraction on the result. In order to realize a specific desgin objective one should very accurately control the filling fraction of the grating. This is one of the sources of the experimental error mentioned in section 4.1. If one estimates the filling fraction from images of the top of the hole, this may lead to discrepancies between experimental and theoretical calculations. To further study the effect of a slanted grating profile we now calculate the flux Chapter 5. Applications 82 1.0 O 0.0 \ / \ 1 1 / I / 1 / 1 ( ! / 1 / 1/ 0.46 0.48 0 .50 0.52 0 .54 Re{S/pj Figure 5.3: P-polarized specular reflectivity at 0/0g = O.Olx. Resonances are due to the p-polarized mode at the 2nd order TE-like gap. See text for the structure's parameters. Solid curve is for a texturing with vertical side walls and a filling fraction of 25%. Dashed curve is with a texturing whose side wall has as 14 degree slope, and the dashed-dot curve the texturing has a 14 degree slope but the filling fraction of 25% is retained. radiated upwards and downwards. The flux upward is given by the first of equations (4.1). In order to calculate only the radiated power we take ER to be given by equation (2.38) but with no homogeneous field added on. The flux downward is given by the second of equations (4.1) where again E T , being analogous to (2.38), is derived from (2.28) and includes no homogeneous field. Plotting these fields, for the same structure, and location in the Brillouin zone as in Fig. 5.3, we find the curves in Fig. 5.4. The solid (dash-dot) curve is the field radiated upwards (downwards) by the zeroth order component then the grating has vertical side walls. The geometry of this structure is Chapter 5. Applications 83 o.O 4= , , ^ 2 = ^ 0.50 0.51 0.52 0.53 Re{S/pg} Figure 5.4: Flux radiated upwards and downwards for a vertical, cone, and pyramid grating profile. See text for description of each curve. The straight lines are an aid to the eye, a (b) showing the difference between the amount radiated down to that up for vertical (cone) grating profiles. For the pyramid profiles the two peaks are very close to the same intensity. It is possible to control the ratio of radiation radiated upwards to the amount downwards by changing the grating profile. such that the flux radiated downwards is greater in intensity than upwards. When the side walls are slanted by 14 degrees (retaining the 25% filling fraction) with the larger end of the hole at the upper surface (cone shaped) the calculation yields the long-dashed (dotted) curve for the upwards (downwards) radiated field. This grating profile increases the ratio of flux radiated upwards to that downwards. If the grating is turned upside down so that the larger end is at the bottom surface (pyramid shaped) of the guiding core then the calculation yields the short-dashed (dash-dot-dot) curve for the upwards (downwards) fluxes. The ratio between the two has shifted to approximately unity. We Chapter 5. Applications 84 therefore conclude that by adjusting the profile of the grating it is possible to control the relative amount of power radiated upwards and downwards from the structure. The ratio of up to down varies, in this example, from ~ 0.78 for the cone shaped profile to ~ 1.0 for the pyramid shaped profile. In conclusion, the model presented in this thesis is an extremely powerful tool for both studying and optimizing optical devices based on 2D textured planar semiconductor waveguides. Chapter 6 Conclusions Using a Green's function technique, we have self-consistently solved Maxwell's equations for the reflectivity of planar dielectric waveguides that contain an arbitrary number of optically thin layers with 2 D periodic texture. Features in the specular reflectivity spectra reveal the dispersion and lifetimes of remnant slab-modes that are renormalized through scattering from the 2 D grating; these are the photonic eigenstates that are "attached" to the 2 D textured waveguide. Using the reflectivity calculation presented in this thesis it is possible to study the resonant eigenmodes of these structures throughout the Brillouin zone. It basically treats the leaky modes above the cladding light lines on an equal footing with true bound modes that exist below the cladding light lines. Through the use of example calculations for both square and honeycomb lattices we establish a physical interpretation of the reflectivity spectra in terms of the disper-sion, lifetime, and polarization properties of photonic eigenstates attached to the textured waveguide. In particular, we determine that the polarization of these renormalized eigen-modes, which in general is complicated, can be uniquely labeled as s- or p-polarized along 85 Chapter 6. Conclusions 86 lines in the Brillouin zone that exhibit reflection symmetry. The accuracy of this simple model, when the thickness of each textured layer is much less than the wavelength of interest, was verified both by comparison with an exact solution of Maxwell's equations, and by comparing with experimental reflectivity measurements on 2D square gratings etched into GaAs slab waveguides [17, 38]. We found excellent agreement with both comparisons and conclude that the model is accurate to better than ~ 5% for individual grating layers on the order of one tenth the wavelength of the radiation of interest. We also gave examples of some of the many applications of this model. We demon-strated the independent control of the linewidth, center frequency, and background reflec-tivity of a polarization insensitive notch filter. A study of the effects of the grating profile revealed that the grating filling fraction has the largest effect but asymmetric slopes can influences the ratio of power radiated upwards to that downwards. As well, we identified eigenmodes away from the Brillouin zone boundaries that consist of superpositions of TE-like and TM-like slab modes. This T E - T M coupling phenomena is unique to 2D tex-tured planar waveguides and may form the basis of waveguide-based polarization mode converts. Finally, we have suggested two future improvements that could be carried out to the model presented in this thesis. By including a filling fraction dependent ec one could improve the thin grating approximation by setting the D C Fourier components of the grating to zero. Second, modifying the code to calculate an arbitrary number of Green's Chapter 6. Conclusions 87 functions describing coupling from grating to grating would further generalize the model. There are a number of possible directions to take the research of textured planar-waveguides in the future. From an applications point of view implementing the ideas of notch filters, polarization mode converters, and optical chips discussed in this thesis may soon lead to vast improvements in technology. From a physics point of view there are also many interesting questions still to investigate. The work in this thesis has dealt entirely with optical resonances and infinitely periodic structures. One logical next step is to introduce defects into these P B G waveguides thus introducing gap states, as discussed in the introduction to the thesis. This is an area of intense study and in fact a modified version of the present formalism has already been used to study these defect states[37]. As well, it is natural to ask what happens when electronic resonances are included. Most of the work on these ideas to date has been restrict to artificial isotropic photon density of states[8], 1st order (linear) and second order nonlinear response[39], and most notably to bulk P B G material. A great deal of work is still needed to determine whether such effects are possible in P B G waveguides, where, if possible, they may be more easily realized experimentally. Furthermore, the regime of nonlinear resonances is still one with very little work. 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Astratov et al., "Resonant Coupling of Near-Infrared Radiation to Photonic Band Structure Waveguides", J. Lightwave Tech. 17 2050-2056 (1999). S. Tibuleac and R. Magnusson, "Reflection and transmission guided-mode resonance filters" JOSA A 14, 1617-1626 (1997) D . M . Whittaker, and I.S. Culshaw, "Scattering-matrix treatment of patterned mul-tilayer photonic structures", Phys. Rev. B 60, 2610-2618 (1999). Song Peng and G. Michael Morris, "Resonant Scattering from two-dimensional grat-ings" JOSA A 13, 993-1005 (1996) D. Bedeaux and P. Mazur, "On the Critical Behanviour of the Dielectric Constant for a Nonpolar Fluid", Physica 67 23-54 (1973) J . E . Sipe, "The Dipole Antenna Problem in Surface Physics: A New Approach" Surface Science 105 489-504 (1981) S. Peng and G. M . Morris, "Experimental investigation of resonant grating filters based on two-dimensional gratings," in Diffractive and Holographic Optics Technol-ogy III, I. Cindrich and S. H. Lee, Eds. Proc. SPIE 2689, 90 (1996) V . Pacradouni, W.J . Mandeville, A .R . Cowan, S.R. Johnson, and Jeff F. Young, "Dispersion and Lifetimes of Electromagnetic Modes attached to Strongly Textured Slab Waveguides", submitted to Optical and Quantum Electronics. Bibliography 91 [38] V . Pacradouni, A . R. Cowan, J. Mandeville, P. Paddon, Jeff F. Young, "Dispersion and lifetimes of leaky modes attached to 2D waveguide-based photonic crystals: experiment and theory", post deadline conference proceedings, 1999 OSA Annual Meeting, Santa Clara, California, Sept. 26-30, 1999. [39] K . Sakoda, and K . Ohtaka, "Optical response of three-dimensional photonic lattics: Solutions of inhomogeneus Maxwell's equations and their applications", Phys. Rev. B 54, 5732-5741 (1996). [40] N.W. Ashcroft, and N.D. Mermin Solid State Physics (Saunders College, Montreal, 1976). Appendix A Derivation of the Green's Function *—+ —* In this Appendix we give a detailed derivation of the Green's function gc (B; z, z'). The —* Green's function gc (@;z, z') is a propagator that propagates the fields from the the plane located at z' to the plane at z. Consider p-polarized radiation that leaves z' traveling upwards; polarized as pc+. If z is above z' then the first term in the sum, the direct term, describes radiation emanating from z' and propagating directly to z, this radiation experiences no reflections and undergoes a phase shift of e l W c ( - z ~ z ' \ This radiation arrives at z still polarized as pc+. This wave generates polarization at z, which at the microscopic level is described as causing the dipoles, making up the material at z, to oscillate at the frequency of the plane wave. This plane wave then continues on until it comes to the upper interface of the cavity. Here some of the wave is reflected and some is transmitted into the multi-layered slab above the cavity. Once in this slab the wave undergoes an infinite number of multiple reflections and phase shifts before finally being either transmitted out the top of the 92 Appendix A. Derivation of the Green's Function 93 structure or transmitted back into the cavity. A l l the multiple reflections and phase shifts are described with Fabry-Perot-like equations which, for a single interface, reduce to Fresnel reflection coefficients, and for a multi-layer structure, are composed of Fresnel reflection coefficients for each interface. For example, if the multi-layer above the cavity consists of a layer of thickness D and dielectric constant £j, and then a semi-infinite layer with a dielectric constant €j then tr> 7rij <itj f>e iWiD r — r • -4- °'1 1 , J l'° (A 1 ' 'Pup — 1 c,i i 4,,,-n V-^-^y where r and t are the Fresnel reflection and transmission coefficients for p-polarized radiation at an interface. They are rci= w ^ ~ w ^ (A.2) Wc6i + Wi6c and 2wcJec6i tci= c v c t (A.3) Wcei + Wi6c The subscript, c, as before, refers to the cavity. The transmitted wave propagates away, it is lost forever and does not generate any-more polarization. The amount reflected back into the cavity, which is a factor of rPup less in magnitude than the wave that hit the surface propagates back to z and generates more polarization. This wave has now experienced (in addition to the phase shifts included in r P u P ) a phase shift of e - l w ° ( z - L + z ' ) t a n c j has the polarization p c_. The wave then continues on and reflects off of the layered slab below the cavity picking up a factor of rPdown, and a phase shift of e ™c(2+2L-z'). T j p 0 n reflection the polarization Appendix A. Derivation of the Green's Function 94 changes back to pc+. For the upward going p-polarized wave emanated from z' we have the terms gPc+(z,z>) = c[pc+pc+6(z - z'yw^ + pc^pc+rPupe-iw^-L+^ + v _i_r r Jwc{z+2L-z') I FC+PC+1 Pup ' Pdown C + P c - P c + r P u p r p d o w „ e c<~ + ' -I- rj .f) ^ r 2 r 2 iwc(z+4L-z') , T Pc+Pc+1 P u p r P d o w n e ( A . 4 ) where C is a 8 dependent constant. Note that all 3 dependence has been suppressed. The terms can be sorted by tensor component as follows: 9Pc+ = C\pc+pc+9{z - z')e™^-*') y c + ^ C + I pup 1 Pdown C ^ ' Pup 1 Pdown C ^ ' Pup Pdown T • • • J + P c p c + r P u p e - ^ ^ - ^ ' ) [ l + r p „ / p _ e ^ 2 i + r ^ p r 2 d _ e ^ 4 L + ...]} ( A . 5 ) The infinite series in the square brackets are simple geometric series whose sums are given by l + x + x2 + ... = —— ( A . 6 ) 1 — x Summing these series we find the final expression for the terms in the Green's function which describe p-polarized radiation emanating from out the top of the plane at z' to be: gPc+(z,z') = C\pc+pc+6(z - z ' y w ^ Appendix A. Derivation of the Green's Function 95 + pc_pc+rPupe-iw^-L+zVDp (A.7) where Dp is given by equation (2.19). A l l other terms in each Green's function and homogeneous solution can be derived with analogous arguments to those presented above. Appendix B The Reciprocal Lattice and Fourier Components Here we describe the calculation of the reciprocal lattice vectors and Fourier coefficients needed in the calculations. Calculating reciprocal lattice vectors and taking the Fourier transform of Bravais lattices are standard practices in solid state physics where, due to the periodicity of infinite systems, much of the physics is described in reciprocal, or momentum space instead of real space. A discussion of these topics can be found in reference [40]. Here we present the topics in the context of 2D periodic lattices. Any 2D periodic structure can be describe by a repeating cell, or unit cell, with the following principle, real space, lattice vectors; ui = ax u2 = bx + cy u3 = U (B.l) 96 Appendix B. The Reciprocal Lattice and Fourier Components 97 where a,b,c are real constant. The reciprocal space lattice vectors are given by; U2 x ^3 2TT A 2nb „ V\ = 27i——— — = —x y Ui • (U2 x u3) a ac u 3 x 2TT „ V2 = 2 ? r - — — — = —y Ui • {u2 x u3) c ^ = 2rr _ =2TTZ (B.2) M l • [U2 X U 3 ) These vectors define the reciprocal space unit cell. The corners of this cell mark the location, in reciprocal space, of the Fourier components of the grating. The perpendicular bisectors of these vectors trace out the 1st Brillouin zone of the lattice. The zeroth order Fourier component being at the center of the zone and the 1st order components, being the nearest neighbours to the zeroth component, are located outside the 1st Brillouin zone. The magnitude of each Fourier component is given by the Fourier transform of the real space unit cell with the above reciprocal lattice vectors: Xnm = j uLL dxdyAX9(x, y)ein^+i™^ (B.3) where A is the area of the unit cell, and f = xx + yy. The function Axg(x, y) depends on the index contrast, and is different inside and outside the holes of the grating: 

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