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A spectral theory for planar dielectric waveguides Dods, Steven R. A. 1990

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o A SPECTRAL THEORY FOR PLANAR DIELECTRIC WAVEGUIDES Steven R. A. Dods B. Math. University of Waterloo M . Math. Universitjr of Waterloo A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF E L E C T R I C A L ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA June 1990 © Steven R. A. Dods, 1990 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 The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The problem of electromagnetic wave propagation across the junction of two similar pla-nar dielectric waveguides is analysed, within the Kirchhoff approximation, by expanding the field into transverse variations of all possible modes. It is proven that the expansion can represent any solution for any planar dielectric waveguide. The spectral function is introduced into the representation, and this helps resolve some of the theoretical problems in passing from the limit of closed waveguides to open waveguides. Using the spectral function and the Gel'fand-Levitan integral equation some new exact solutions to novel dielectric planar waveguides can be found. Examples of waveguiding by total internal reflection or by Bragg reflection (which are physically very different processes) can be generated by changing a single parameter in the formulation. Usually the representa-tion for an open dielectric waveguide requires the matrix spectral function. However the Gel'fand-Levitan reconstruction is defined for scalar spectral functions. A technique for constructing the spectral matrix and the scattering solutions from two spectral functions is demonstrated. This technique uses a variational formulation of a scattering experiment. The connection between a dielectric structure and the characteristics of propagation on it is obscure. However the connection between these characteristics and the spectral func-tion is much clearer. It is sometimes possible to make predictions about the properties of the waveguide by looking at its spectral function only. Since the connection between the spectral function and the dielectric structure is well established by inverse spectral theory, introducing the spectral function has been of help in establishing the desired connection between the dielectric structure and the characteristics of propagation on it. . • v ii Such considerations suggest one of the above waveguides is sensitive to small perturba-tions and could be used as an electro-optic modulator. Detailed calculations confirm the hypothesis. 111 Table of Contents Abstract ii List of Figures vii Acknowledgement viii 1 Introduction 1 2 Basic Theory 13 2.1 TE and T M Modes 14 2.2 Separation of Variables 17 2.3 Consideration of Losses 19 2.4 Transverse Representation 20 3 Closed Planar Dielectric Waveguides 26 3.1 Eigenfunction Expansion 26 3.2 The Spectral Function 29 4 Dielectric Waveguides Open on One Side 35 4.1 Limit of mb(E) 35 4.2 Spectral Function for the Open Waveguide 37 4.3 Eigenfunction Expansion 41 4.4 Completeness of the Expansion 43 5 Dielectric Waveguides Open on Both Sides 46 iv 5.1 Parseval Equality 47 5.2 Matrix Spectral Function 49 5.3 Eigenfunction Expansion 54 5.4 Aperture Antennas 56 5.5 Two Dimensional vs. One Dimensional Representation 58 6 Inverse Spectral Theory 65 6.1 The Povzner-Levitan Representation 66 6.1.1 Asymptotic Behaviour of ip 69 6.1.2 The Paley-Wiener Theorem 72 6.2 The Gel'fand-Levitan Integral Equation 74 6.3 A Solution Using the Spectral Function 77 7 Inverse Methods for Open Dielectric Waveguides 83 7.1 A Scattering Experiment . . . 85 7.2 Orthogonality Relations 87 7.3 A Bragg Reflection Waveguide 91 8 A Bragg Modulator 101 8.1 A Choice of Basis Functions 103 8.2 Spectral Function for the Problem . 108 8.3 Expansion Coefficient 110 8.4 Coupling Coefficient 112 9 Conclusions 115 References 119 v A The Uniqueness Theorem of Electromagnetics 124 B Zeros of the Transverse Resonance Equation 128 C The Completeness of the Expansion • 130 C.l Green's Operator . . 130 C.2 The Bessel Inequality 132 C.3 The Extremal Property 133 C.4 Completeness theorem 135 vi List of Figures 1.1 Discontinuity in a Dielectric Waveguide 3 1.2 Closed Dielectric Waveguide 7 1.3 Dielectric Waveguide Open on One Side 8 1.4 Dielectric Waveguide Open on Both Sides 9 2.1 Possible Values of the Longitudinal Wavenumber /3 24 2.2 Possible Values of the Transverse Wavenumber K 25 5.1 A Dielectric Interface 60 6.1 Potential V(x) with E0 = - 6 . 4 x I O 1 3 ™ - 2 and cx = 10 6 81 6.2 Potential V(x) with E0 = 2.35 x 1 0 1 4 m - 2 and q = 5.0 x 10 5 82 7.1 Relations between Quantities in the Half Space 84 7.2 Magnitude of Reflection Coefficient vs. Normalized Transverse Wavenum-ber n/k, 95 7.3 Example of a Bragg Reflection Waveguide 98 8.1 A Bragg Waveguide Modulator 102 8.2 Magnitudes of the Spectral Components 105 8.3 Coupling Coefficient cp vs. Change in Refractive Index 114 vn Acknowledgement I would like to thank Dr. E.V. Jull for his supervision and encouragement. Thanks are also due to Dr. D. Ludwig for help with the mathematics, to Dr. N . Jaeger for some useful input in the early stages of this work, and to Dr. R. Miura for his careful review of the thesis. I would also like to acknowledge the financial support of the Natural Sciences and Engineering Research Council, and the University of British Columbia. Finally many thanks to my wife, Anita, for bringing a breath of fresh air into my life. vm Chapter 1 Introduction The subject of integrated optics is currently at the center of efforts to use light as a car-rier frequency for communication systems. There are many forces influencing this trend. On the one hand there is demand for higher channel capacity and therefore greater bandwidth, and if the carrier frequency is higher the possible bandwidth is larger. Mi-crowave and millimeter wave frequencies have proven useful in satisfying these demands, and optical frequencies are the new frontier. On the other hand the laser supplies a coherent source of light at suitable frequencies, and the low loss optical fiber provides a suitable medium for long distance transmission. Any communication system also needs components that can perform signal processing tasks such as amplification, multiplexing, modulation, and demodulation. These processes are all familiar in integrated electronics and the fabrication technology of this field has been used to create optical circuits which perform some of these functions. A planar dielectric waveguide is the structure used to confine and guide light in an integrated optical circuit. Hence the planar structure is of interest in its own right. Beyond this the planar structure can be used as a model for more complicated structures, for example within certain approximations it can be used to model the cylindrical optical fiber [1]. Since this guiding is the most basic function performed, it is important to know everything about it. The effect of small discontinuities in the geometry is of interest for two reasons: first, because discontinuities are inevitable anyway, they will occur from small defects in fabrication or when the guide couples to other devices; and second, 1 Chapter 1. Introduction 2 discontinuities can be used as the basis for the integrated optical components themselves, such as transformers, grating couplers, and feed structures. Once it was established that the planar dielectric waveguide was important and that the effect of a discontinuity was of special interest, a series of papers devoted to this subject was published. The typical configuration is illustrated in Figure 1.1. One of the first comprehensive treatments of this subject was written by Shevchenko in 1969 [2]. He modelled dielectric and surface waveguide tapers and other continuous transitions by using a formulation based on local normal modes. At about the same time Marcuse published a series of papers [3] - [6] which addressed the same problem but modelled the tapers and continuous transitions as a series of small waveguide discontinuities. Both authors consider the electromagnetic field to be a sum of two kinds of solutions: guided modes and radiation modes. In the mathematical sense the guided modes correspond to proper eigenfunctions and the radiation modes correspond to improper eigenfunctions. A few years later Clarricoats and Sharpe [7] attempted a solution by applying the method of mode matching. This method can be used when there are many guided modes present and most of the field energy is found among these modes. This approach is well suited to the situation where the dielectric slabs have cladding layers and therefore many cladding modes. Hockham and Sharpe [8] solved the same junction problem including the radi-ation modes in the case when the reflected radiated modes can be neglected. In 1975 Mahmoud and Beal [9] approximated a solution by expanding the reflection coefficient and transmission coefficient functions as Laguerre polynomials. In the same year Rulf [10] found an asymptotic solution for a small discontinuity. Rozzi [11] solved the problem the same way as Mahmoud and Beal but also discussed the convergence properties of the solution. Brooke and Kharadly [12] [13] have found exact solutions by modifying the physical situation slightly. They put perfect conductors on x =constant planes above and below Chapter 1. Introduction 3 X 1: ^2 «1 £ 2 Figure 1.1: Discontinuity in a Dielectric Waveguide Chapter 1. Introduction 4 the dielectric waveguide. Since the closed waveguide has a discrete spectrum they were able to solve the problem by the mode matching method. The slow modes decay expo-nentially in the transverse direction and so it was expected the conductors need not be more than a few wavelengths from the dielectric. In the twenty years since this problem was introduced, no one has succeeded in finding a definitive solution, and the problem still appears in the literature from time to time. (See for example [14], [15], or [16].) This thesis does not present another method for solving the waveguide discontinuity problem. Instead the beginning of the thesis is concerned with addressing some issues the above papers leave unresolved. The second part of the thesis will show some new results which appear in the process of resolving these issues. One of these issues involves the generality of the representation used for the elec-tromagnetic field. The field is assumed to take a specific form which includes a sum of guided and radiation modes. It is easily verified that a field of this form must satisfy the wave equation, but it is not clear that every solution of the wave equation must be of this form. In the cases where the authors go so far as to obtain numerical solutions, the problem seems to have been solved and so the representation must have been adequate. On the other hand most of the detailed investigations have been concerned only with the simplest possible structure, the junction of two uniform dielectric waveguides, as shown in Figure 1.1. However, more recently, attention has shifted to more complicated structures, some involving many layers and some involving continuous variations of the permittivity (gradient index profiles) [1] [17] [18] [19]. This is because, generally speaking, a more complicated structure can give superior performance for a specific application. It would be nice to have some assurance that the representation commonly used does in fact represent the most general solution for all possible planar waveguides, regardless of their structure or excitation. Chapter 1. Introduction 5 The treatment of the waveguide discontinuity problem by Brooke and Kharadly [13] leaves another unresolved issue. This technique is physically well-motivated, and gives good results. However, critics have questioned its relevance to open problems [11], be-cause strictly speaking the problem has no radiation modes: the conductors keep all energy confined to a finite transverse range. It is at least a small leap of faith to as-sume that free space radiation modes can be adequately modelled by a finite sequence of waveguide modes. In some ways this problem is related to the previous question of the generality of the representation of the field. There it was asked if the representation for a finite (closed) system was sufficiently general and separately if the given form was adequate for the open case. In the above paragraph these two questions are important but it is also asked if a wide closed waveguide can model an open one. In other words it needs to be known if the representation of the closed waveguide can pass through the limit of infinite separation and become a representation for an open waveguide. This limit has been considered by other authors, but only in the Soviet literature [2] [20]. In this thesis the limit of infinite separation is considered first in Chapter 4, when one reactive surface recedes to infinity, and later in Chapter 5, when both walls are removed. In order to have the limit make sense it was necessary to assume a specific form of loss distributed throughout the medium. However these losses can be made arbitrarily small and a lossless system may be approximated. Since the questions considered so far have nothing to do with actually' finding the unknown field in the aperture plane z = 0, it will be assumed these are given. The unknown fields on the plane z = 0 may have been found by using one of the techniques in the above papers. Alternately, in the case where the discontinuity is very small, the field in the z — 0 plane may be assumed to be the same as the incident wave from the half space z < 0. This is known as the Kirchhoff approximation. With the fields in the Chapter 1. Introduction 6 z = 0 plane given, the problem of finding the field for z > 0 is a boundary-value problem. Rather than speaking of this as a boundary-value problem, it is sometimes described as a source problem, in which the sources of electromagnetic fields are given and the resulting fields are calculated from them. To describe the situation in this language, electric and magnetic surface currents are postulated to be flowing on the surface z — 0: where ri = z (by convention the inward normal is used). The surface is divided into infinitesimal surface elements and the electromagnetic field due to each element is found over the entire space. Then the total field is found by adding all these infinitesimal contributions. The source method is not used in this thesis. Instead the problem will be posed and solved as a boundary-value problem. However, the magnetic and electric surface currents are introduced because in describing the boundary-value problem it is useful to be able to distinguish between the supposed value of the electric and magnetic fields on z = 0 (written J„ and M s ) and the solution of the current boundary-value problem, written E(a;,j/, 0) and H(x ,y , 0). The nomenclature has just been borrowed from the source problem. Of course the objective is to satisfy the equalities (1.1) and (1.2), and so this is really a boundary-value problem. The first problem considered in this thesis is the closed dielectric waveguide, illus-trated in Figure 1.2. In Chapter 3 the generality of the expression for the electromagnetic field is established for an arbitrary permittivity profile and excitation. A representation of this solution using the spectral function and a Stieltjes integral is introduced. The advantage of this representation is that the same form may be used in the limit as the width of the waveguide tends to infinity (b —> oo). The result of taking this limit leads Me{x,y) = E(x,y, 0) X n (1.1) 3a(x,y) = -H(x,y, 0) x n (1.2) Chapter 1. Introduction 7 x elect r ic and magne t i c cu r r en t s Figure 1.2: Closed Dielectric Waveguide to a dielectric open on one side (Figure 1.3). In Chapter 4 this limit is examined in detail and sufficient conditions for its existence are given. It is proven that in the limit the representation for a closed waveguide becomes the same as the representation for an open waveguide. In effect this justifies modelling an open waveguide with a closed one. The error encountered in using a closed waveguide can be made arbitrarily small by making the waveguide arbitrarily wide. The open waveguide of Figure 1.3 is open on one side only. This finds immediate application at radio frequencies because in this range reactive surfaces are sometimes used for surface waveguides. At optical frequencies, however, it is more usual for the waveguide Chapter 1. Introduction x k elect r ic and magne t i c cu r r en t s e(x) ^|§§reactive surface Z,\', '"////////////////////////////////////////////, Figure 1.3: Dielectric Waveguide Open on One Side Chapter 1. Introduction 9 Figure 1.4: Dielectric Waveguide Open on Both Sides to be open on both sides, as in Figure 1.4. The representation must be modified for this case, and this is done in Chapter 5. In this problem the spectral function becomes a spectral matrix, and so the representation is more complicated. In the case where the dielectric is uniform (e.g., e(:r) = 1 for all a;) the problem must reduce to the usual Kirchhoff theory of aperture antennas, with z = 0 as the aperture (see [21]). This is demonstrated in Section 5.4 and the angular spectrum theory of aperture antennas [22] is shown to be a special case of the representation used in Chapter 5. In Chapter 6 the emphasis changes. The previous chapters resolved certain theoretical problems related to the representation of the field. In the process the spectral function Chapter 1. Introduction 10 has been applied and has proven to be useful, particularly in the transition from a closed to an open waveguide. Chapters 6 to 8 are more concerned with the physical interpretation of the spectral function: given a spectral function, what can be said about the corresponding dielectric structure and its propagation characteristics? The relation between a dielectric structure and its propagation characteristics is often obscure, for example it is not easy to find the propagation constants of a given dielectric structure, and it is impossible to find the dielectric structure (uniquely) given only the propagation constants. However, there is a well-established relation between the spectral function and the dielectric structure. Chapters 2 to 5 show how to construct the spectral function given the dielectric structure, Chapter 6 shows how the dielectric structure can be uniquely constructed from the spectral function [23]. Since this calculation goes the opposite direction from the original direction, it has been called an 'inverse problem' and the subject is called 'inverse spectral theory'. The most difficult part of inverse spectral theory from the point of view of compu-tations is the solution of the Gel'fand-Levitan integral equation, but this equation does admit some exact solutions [24], [25]. The interesting thing is that because this integral equation has nothing in common with the original differential equations of the forward problem (such as those in Chapters 2-5), the exact solutions of the Gel'fand-Levitan problem do not reproduce the exact solutions already known for the forward problem. Instead they form an entirely different family. Thus by reformulating the problem as an inverse problem new exact solutions can be obtained, and an example is shown in Section 6.3. The approach to this problem using the Gel'fand-Levitan theory is appropriate when the data is given in the form of the spectral function. Since the spectral function is used anyway in the representation of the electric field, this seems a logical approach especially when dealing with junction problems as described above. The Gel'fand-Levitan approach Chapter 1. Introduction 11 is related to another theory due to Marchenko [26], which is more appropriate when scattering data is given. This theory and extensions of it can be found in a series of papers by Kay and Moses [27]. The mathematical equivalence of these two approaches has been demonstrated by Chadan and Sabatier [25, page 79]. The Marchenko approach has been applied to planar dielectric waveguides already. Examples of waveguides obtained this way can be found in papers by Jordan and Lakshmanasamy [28] or Yukon and Bendow [29]. The objective of any analysis of propagation or waveguiding is to establish the connec-tion between the physical structure of the medium or waveguide and the characteristics or special features of the propagation. Having established the connection between the spectral function and the dielectric structure, the attention now turns to the connection between the spectral function and the propagation characteristics of the structure. The most obvious connection is that the structure has guided modes only if the spectral function has discontinuities. In the converse case, a discontinuity may lead to a guided mode, but only if the associated longitudinal wavenumber is real. In Section 6.3 the connection between discontinuities and the existence of guided modes is explored. At this point the depth of this approach is revealed: from a single example it is possible to demonstrate waveguiding by total internal reflection (the usual method) or by Bragg reflection (a more exotic method) merely by changing the location of the discontinuity in the spectral function. The physical process responsible for the first'kind of waveguide is different from the process responsible for the second, but the spectral theory shows a connection on a mathematical level. The inverse spectral theory of Chapter 6 applies to problems defined on a half space x £ [0,oo), and so it applies directly to dielectric waveguides open on one side (Figure 1.3). Since most optical waveguide problems will be two sided, like Figure 1.4, these methods need to be adapted. This is the subject of Chapter 7. The whole space (—oo, oo) Chapter 1. Introduction 12 is given from two half spaces ( —oo,0] and [0,oo). The two half spaces are handled separately using the methods of Chapter 6. The crucial point is to be able to construct the spectral matrix of Chapter 5, and to do this the input impedance of the half space must be found directly from the spectral function. The approach shown here is based on recent work [30] and is done by performing a theoretical scattering experiment and deducing the input impedance from the reflection coefficient. As will be explained in Chapter 7, for the example shown in this thesis the reflection coefficient is a rational function of the wavenumber and so the system need not be reconstructed by the above technique. I. Kay has shown another method for this case, which is based on the Marchenko integral equation [31]. In Section 7.4 the results are used to construct an example of a novel Bragg reflection waveguide. Bragg reflection waveguides have been studied before [32], based on the Bloch wave solution for periodic media [19], and are still being investigated for applications [33]. The Bragg reflection waveguide of Section 7.4 operates on the same physical principles as the other ones, but since it has not been based on the Bloch wave formulation the details of the solution are different. There is another property of the new Bragg reflection waveguide that can be guessed by looking at its spectral function. The discontinuity (representing the guided mode) is embedded in the continuous part of the spectrum. Nearby the guided wavenumber is a whole spectrum of radiation modes. It is reasonable to expect that small perturba-tions in the waveguide geometry will transfer energy to these other modes, leading to a propagation governed by diffraction, and this amplitude will decay at the rate l/y/r. If these perturbations were controlled in some way, through the electro-optic effect for example, then the amplitude of the guided mode would be modulated owing to the con-trolled transfer of energy to the radiation modes. In Chapter 8 a detailed calculation is performed to find the depth of the modulations. Chapter 2 Basic Theory In the introduction three possible planar dielectric structures were described: a closed planar waveguide and two open waveguides, one open on one side and the other open on both sides. In this chapter some basic equations which apply to all three structures are derived. Properties which are specific to each structure will occupy the next three chapters. The analysis is intended to model a waveguide with a small change in structure at the plane 2 = 0. The electromagnetic field on this plane is given approximately by the waves incident on the plane from the z < 0 half space (the Kirchhoff approximation). Maxwell's equations for a source free medium with a variable isotropic permittivity (e) and permeability (/x) are V • (eE) = 0 (2.1) V • (/zH) = 0 (2.2) V x E+jufiH = 0 (2.3) V x H - jueE = 0 (2.4) The source of the radiation is almost always harmonic in time, and the Maxwell equations are linear, so there will be no frequencies other than that of the source. For this reason the time harmonic form of the Maxwell equations has been used. A time dependence of e^ut is assumed. With the help of the vector identities V x V x A = V(V • A) - V 2 A (2.5) 13 Chapter 2. Basic Theory 14 V - ( / A ) = A - V / + / V - A (2.6) V x ( / A ) = V / x A + / V x A (2.7) and assuming E and H are sufficiently differentiable, the equations (2.3) and (2.4) can be uncoupled to yield differential equations in either E or H : V 2 E + V ( - V e • E) + - V / i x V x E + u;2e/xE = 0 (2.8) e u V 2 H + V ( - V ^ - H ) + - V e x V x H + w 2 e / i H = 0 (2.9) u e (2.8) and (2.9) are equivalent formulations of the problem, either (2.8) or (2.9) can be used to obtain an}' solution. If the boundary-value problem has been posed with the electric field specified, (2.8) is more convenient than (2.9). Similarly for a problem in which the magnetic field is given the reverse will be true. A simplifying assumption is now made: it is assumed the direction of the incident radiation from the z < 0 half space is parallel to the x — z plane. In effect, the fields in the 2 = 0 plane are assumed independent of y. Since the structure itself has no variations in the y direction, by symmetry it follows all the fields will be independent of y. Hence in all equations the derivatives with respect to y can be neglected. 2.1 T E and T M Modes Equation (2.8) consists of three differential equations each containing the three compo-nents of the electric field Ex,Ey, and Ez. The uniqueness theorem says that the solution will be unique once either the tangential components of the electric field are specified or the tangential components of the magnetic field are specified over the bounding surface of the volume in question. This volume and its bounding surface depends on the problem, but in all cases in this thesis at least part of the bounding surface is on the plane z = 0 and the remaining surfaces are either at infinity or are reactive planes. The uniqueness Chapter 2. Basic Theory 15 theorem shows that the field need not be specified on the portions of the bounding sur-face which are reactive in order to ensure uniqueness of the solution. In Section 2.3 some small losses will be introduced, and because of these losses it is safe to assume that as x or z tends to infinity the E and H fields will tend to zero. The losses vary directly with the square of the amplitude and therefore the decay is exponential with distance. Since the area of any bounding surface grows only algebraically with distance the value of the surface integral in (A.IO) over the surface at infinity will be zero. Hence the bounding surfaces at infinity need not be considered either. Therefore in all three problems consid-ered a unique solution is obtained once the tangential part of the electric (or magnetic) field is specified on the plane 2 = 0. In the electric case the tangential components on the plane 2 = 0 are Ex and Ey, and the solution of the problem must involve matching the solutions of (2.8) with the given tangential components on z = 0. In the case where e and p, are functions of x only and there are no variations in the y direction, equation (2.8) for Ex and Ey reduces to d*Ex , d2Ex + 8 /V \ + w 2 / t ( j , ) e ( i | . ) j B i ( ! | ! ) 2 ) = 0 ( 2 1 Q ) dx2 dz2 dx e a2 p a 2 p / Q p ^ + l f - ^ + ^ W " W £ ' ( l ' 2 ) = 0 ( 2 n ) If solutions for Ex and Ey can be found then the Ez can be recovered from (2.1). There is no coupling between Ex and Ey in (2.11) and (2.10) and so the two polar-izations can be considered independent modes, and can be analysed separately. If the incident radiation is polarized parallel to the y axis (perpendicular to Ve) the result will be transverse electric (TE) waves controlled by equation (2.11). To see this let the incident radiation be parallel to the y axis and therefore Ex is zero on the plane 2 = 0. A possible solution of (2.10) is the zero solution. The zero solution satisfies the bound-ary condition on z = 0, on the reactive surfaces at x = 0 and x — b if they should be Chapter 2. Basic Theory 16 present, and at infinity. The uniqueness theorem says this is the only solution and so if the incident radiation is parallel to the y axis Ex will be zero everywhere. Equation (2.1) says ^ + ^ 1 = 0 .(2.12) Ox dz and since Ex is zero everywhere then Ez must be zero everywhere too. The lack of a longitudinal component to the electric field is why this kind of mode is called 'transverse electric'. The T E mode is accompanied by magnetic field components Hx and Hz available from equation (2.3). On the other hand if the incident radiation is polarized parallel to the x axis (parallel to Ve) then transverse magnetic (TM) waves will result. To see this note that since Ey = 0 on the plane 2 = 0 and by using the uniqueness theorem again the zero solution of (2.11) is the only solution and Ey — 0 everywhere. Equation (2.3) can be used to calculate the magnetic field H = ^ V x E = ^ ( f i - ^ ) (2.18) JUJU JU1U \ ox oz j The lack of a longitudinal component to the magnetic field motivates the name transverse magnetic. In solving for a TM mode, it is possible to use equation (2.10) alone. However, if this is done it will not be easy to find the other TM field components, Hy and Ez. A more practical approach is to start with Hy instead. Since the magnetic field is parallel to y in a TM mode, equation (2.9) reduces to d (ldHv\ d2Hv , If the magnetic field Hy in the plane z = 0 is given (e.g., as a result of the Kirch-hoff approximation) then (2.14) together with the given electric surface current Js is a boundary-value problem. Once this problem is solved, the other components Ex and Ez Chapter 2. Basic Theory 17 can be found from (2.4). Since any polarization can be represented as a linear combi-nation of parallel and perpendicular polarizations, the complete solution will be some combination of T E and T M modes. This thesis is about dielectric waveguides. In a typical dielectric the permittivity e can be made to vary with position x according to the composition of the dielectric materials at this position. In a dielectric the permeability p, is close to the vacuum value p0 and so the derivative p' can be neglected. Thus (2.11) becomes d2Ey d2Ey , -feT + - j / + *'W(z)^ = o (2.15) and so this will be the basic equation used for T E waves. Note that the form of the basic equation for T M waves (2.14) does not depend on whether p is variable or constant. This fact will be useful in Section 2.4. 2.2 Separation of Variables To solve equations (2.15) and (2.14) it is possible to use separation of variables. For example, for the T E case (equation (2.15)) use the substitution Ey(x,z) = X(x)Z(z) (2.16) in equation (2.15) and divide by XZ: X" Z" — + y + e(s)u;Vo = 0 (2.17) The first and third terms are functions of x only and the second of z only. Hence the sum of the first and third terms must be some constant independent of both x and z. Let this constant be /32 and X" + u2p0e(x)X -{32X = 0 (2.18) Chapter 2. Basic Theory 18 Z" + {32Z = 0 (2.19) The general solution of (2.19) can be written Z(r3,z) = A{{3)e-j0z + B(/3)ej0z (2.20) If Ey(x,z) is to be a bounded function of z G [0, oo) then there are three possibilities: 1) A — 0 and (3 is in the upper half of the complex plane, 2) B = 0 and f3 is in the lower half plane, or 3) /3 is a real number. Possibility (3) will be excluded by an argument presented in Section 2.4. The choice of possibilities (1) or (2) is arbitrary and one or the other can be chosen without loss of generality. In the following the second choice is used, and the function A(f3) is absorbed into the X function. Solutions of the form (2.16) are therefore Ey{x,z) = X{p,x)e-** (2.21) where X is a solution of (2.18). Similarly for the TM case the substitution Hy(x,z) = XTM{x)e-^z (2.22) in equation (2.14) leads to e(^A%) -(32XTM+u2ueXTM = 0 (2.23) This equation appears quite different from the TE equation (2.18). However, a standard change of dependent variable [34] will reduce (2.23) to the form (2.18). Let X = ^=XTM (2.24) and (2.23) becomes X" + (je - ^ + a, V) A' - f32X = 0 (2.25) so (2.25) is the same as (2.18) with the expression in brackets replacing the factor u2u0e. Chapter 2. Basic Theory 19 2.3 Consideration of Losses This thesis is about the analysis of lossless dielectric waveguides, so it might be expected that loss of any kind need not be considered. However if loss is not included problems can appear even on the theoretical level. Some evidence of this is seen in Appendix A , in which it is demonstrated that without assuming at least small losses, the uniqueness of the solutions to the problems defined in a volume is not assured. If losses are not considered, difficulties related to nonuniqueness will appear in subtle forms throughout these problems. Loss usually appears as a small imaginary part in certain eigenvalues. In subsequent chapters it will be found necessary to bound some eigenvalues away from the real axis in order to ensure the convergence of the eigenfunction expansion. Therefore in this approach to the solution, the consideration of losses is necessary even on the theoretical level. Therefore loss is introduced into the equations by adding a small negative imaginary part to one or the other of the constitutive parameters e or fi0 '• e{x) = e1(x)-je2{x) (2.26) or r1 ~ Mo - J M z ) (2.27) The use of these substitutions does not represent an attempt to model a real lossy system. Rather, the substitution is used to bypass the theoretical problems associated with lossless systems. As will be seen in subsequent chapters, the choice of a loss distribution such as (2.26) or (2.27) can lead to a problem which is either easy or difficult to solve. In this thesis the strategy is to choose a loss distribution which yields a convenient (easily analysed) mathematical system. Once this somewhat arbitrary problem has been solved, the solution of the original (lossless) problem may be obtained by passing to the limit of Chapter 2. Basic Theory 20 zero loss: lim e2 —» 0 or l im/i 2 —• 0 (2.28) Implicit in this strategy is the assumption that, provided the losses (e 2 , / i2) a r e made sufficiently small, the solution of any lossy system will approach the corresponding lossless system. It is true that the losses introduced by substitutions like (2.26) and (2.27) are of an arbitrary and perhaps unphysical nature, but at the end of the calculation the limits (2.28) will be applied and we will obtain the solution to the problem e(x) = €i(x) u. = p0 (2.29) which was the original problem posed. 2.4 Transverse Representation On the plane 2 = 0 the electric field is taken to be a given function Ey(x,0) = M(x) (2.30) for T E modes and the magnetic field is taken to be given for TM modes Hy(x,0) = J(x) (2.31) The separated solutions (2.18) and (2.25) on their own are clearly insufficient because 'X(j3,x) (or A") satisfies the respective differential equation and is not an arbitrary func-tion. However, there is hope that M and J can be represented as some linear combination of A' functions with different values of/5. Chapter 2. Basic Theory 21 The differential operators could be defined d2 LTE = + u ; V ( s ) (2.32) L™ = i + Y e - \ ^ + ^ e { x ) ( 2 - 3 3 ) and A' and X could be thought of as eigenfunctions of the systems LTEX = f32TEX (2.34) LTMX = PTMX (2.35) Representing M or J as linear combinations of these functions would then be an eigen-function expansion in X or A'. There are some problems with this identification. Because of the consideration of losses uj2pe(x) is a complex-valued function and as a result the operators LTE,TM are not self-adjoint. It would be nice to have self-adjoint operators because this would allow us to take advantage of results from the spectral theory of linear operators. As the solution of (2.19) makes clear, /3 is a longitudinal wavenumber. Since we are contemplating an eigenfunction expansion of M or J A v h i c h are variations in the transverse direction, it makes sense to define a transverse wavenumber. The definition of this number is somewhat arbitrary, but for problems having a region in which e assumes a constant value or does not vary significantly there i s a natural choice. Suppose the constant e„ i s the permittivity in such a region. Here the field can be expected to be a sum of plane waves of the form e-j(nx+Pz) where K2 +{32 = k2 = u2pes (2.36) Chapter 2. Basic Theory 22 The square of the transverse wavenumber K has special significance and so a new variable is introduced for it E = K,2 (2.37) The fundamental differential equations for the TE and TM modes, (2.18) and (2.25), can be written in the same forni V>" + [E - V{X)\II>{K,X) = 0 (2.38) where V(x) is a real valued function. In the TE case the tp and V functions are denned II>(K,X) = XTE((3,x) (2.39) V(x) = k] - u2u0e{x) (2.40) and it is required to choose a loss distribution such that V(x) will be real valued. In the TE case the permeability is assumed real (fi = u0). If the imaginary part of the permittivity e2 is assumed constant such that iv2u0e2 is equal to the imaginary part of k2, the potential function V(x) becomes real, and the operator 2 L = -L + V(*) (2-41) dx will be self-adjoint. In the TM case the definitions will be ^ ( K , Z ) = X(0,x) = e-V2(x)XTM({3,x) (2.42) V(z) = kl-u,*pe{z)-££ + ^ (2.43) in order to make V(x) real in this case it seems easiest to let e(x) be real valued, and let u2 be denned , s 5(k2) " » < • > = ( 2 ' 4 4 ) Chapter 2. Basic Theory 23 and with these definitions the operator (2.41) will be self-adjoint. There are some im-mediate consequences of this: first, if eigenvalues exist, they are real; and second, non-degenerate eigenfunctions are orthogonal [35]. Also these eigenfunctions are solutions of a real equation (2.38), in which there is no coupling between real and imaginary parts, and so the eigenfunctions themselves can be taken as real without loss of generality. Hence if these eigenfunctions are used in an expansion of M(x) they are at least real and orthogonal, properties which the X(ft, x) functions are not known to have at the outset. There is also a justification for the use of the ifi functions on an intuitive level. In a region of constant e, the ifi's become harmonic functions of KX. Hence in this region an eigenfunction expansion is a Fourier expansion. M(x) and J(x) are transverse variations which are resolved into Fourier components and indexed by a transverse wavenumber. Thus M(x) or J(x) is resolved into plane waves in a way familiar from the theory of aperture antennas. This equivalence will be formally demonstrated in Section 5.4. Since the eigenvalues in (2.38) are real, the possible values for ft in the eigenfunction expansion are restricted: equation (2.36) implies the imaginary parts of k2 and ft2 must be equal. The discussion following equation (2.20) in Section 2.2 concluded that ft could be either real or in the lower half plane. Equation (2.36) shows ft can't be real for an eigenfunction, so ft must be in the lower half plane. Write ft = ft' + jft" and ft must He somewhere on the hyperbola 2ft'ft" = 5(k2e) (2.45) in the fourth quadrant, see Figure (2.1). If the spectrum of the system is discrete, ft will consist of discrete points on the hyperbola. If the spectrum is continuous, then ft may be any point along a section of this curve. From (2.20) it can now be seen that the electric field will consist of waves travelling from left tc right only and lim E(x,z) = 0 z—>oo (2.46) Chapter 2. Basic Theory 24 radiation modes ^ evanescent modes guided modes Pi Pi Pn complex (3 plane Figure 2.1: Possible Values of the Longitudinal Wavenumber [3 lim B.(x,z) = 0 (2.47) as would be expected of a lossy system. Note that the results (2.46) and (2.47) will help in applying the uniqueness theorem of Appendix A. For example the closed waveguide of the next chapter can be thought of as being closed at infinity with the fields equal to zero there. The surfaces at x — 0 and x — b have a known reactance, and the value of the tangential component of either the electric or magnetic field is known on the surfaces 2 = 0 and z = oo. This closes the boundaries of the guide and the uniqueness theorem then shows only one solution is possible. In a physical sense one could say that uniqueness of the solution is assured Chapter 2. Basic Theory 25 evanescent modes •\ »(«) radiation modes « 2 ^ > ) g u i d e d m o d e s < complex K plane i Figure 2.2: Possible Values of the Transverse Wavenumber K since it is now known there are no waves coming in from z = oo. Since possible values of /3 are restricted by (2.45), the definition of the transverse wavenumber (2.36) restricts the possible values for K as well. These possible values are shown in Figure 2.2. Chapter 3 Closed Planar Dielectric Waveguides In Chapter 2 the basic methods of analysis were introduced and the conditions required for the existence of a unique solution were outlined. Most of these remarks apply equally to the open and closed waveguide problems. In the next three chapters (3-5) the method of eigenfunction expansions is presented and the question of the generality of the method is addressed: does the eigenfunction expansion work for any choice of initial conditions M and J and any choice of dielectric structure e = e(x)l The answer to the question is easier to obtain for a closed waveguide than for an open one. Therefore the results will be presented for the closed waveguide first, and the discussion of the open problems will be postponed to Chapters 4 and 5. 3.1 Eigenfunction Expansion The closed waveguide is made by including reactive surfaces on the planes x = 0 and x = b. On a reactive surface the ratio of the tangential components of the electric and magnetic field is constant. It is assumed the reactance is pure imaginary corresponding to either pure inductance or capacitance (no resistance). The reactive surfaces will result in boundary conditions at the points x = 0 and x — b being appHed to (2.38). In the T E case the reactances are defined Ey(0,z) #,(0,2) (3.1) Z.2 Ey(b,Z) Hz{b,z) (3.2) 26 Chapter 3. Closed Planar Dielectric Waveguides 27 and since for T E modes Ey{x,z) = i>(K,x)ej&z (3.3) equation (3.2) results in the boundary conditions ^'(/e>0) + ^ ( K , 0 ) = 0 (3.4) i>'{K,b) + HiP{K,b) = 0 (3.5) where h = Zsi UJ   — — (3.6) H = ^ (3.7) In the T M case the reactances are = (3.8) >s2 f f v (0 ,2) Ez(b,z) (3.9) The substitution is Hy(x,z) = y/^)-ifi(K,x)e-ifiz (3.10) and the boundary conditions are of the same form as (3.4) and (3.5) with h and H replaced by h = ^-juje(0)Zel (3.11) H = Hb) + J U J e { h ) Z°2 ( 3 1 2 ) Hence both T E and T M modes can be written as solutions of a system of the form i>" + \E- V{X)]1>(K,X) = 0 (3.13) ll>'(K,0) + hll>(Kt0) = 0 Chapter 3. Closed Planar Dielectric Waveguides 28 1>\K,b) + Hi/){K,b) = 0 If the loss is chosen as outlined in Chapter 2, then h, H, and V(x) are all real valued and (3.13) is the familiar Sturm-Liouville system. This system always has an infinite sequence of eigenvalues En and an infinite sequence of real eigenfunctions associated with them OK,*)}~i (3-14) These functions are orthogonal and can be normalized to have unit length [34] / tjj(Kn,x)ijj(Km,x)dx = Snm (3.15) Jo For TE modes equations (2.39),(2.40), (3.4), and (3.5) are used in (3.13), and solutions to (2.15) can be written in the form oo Ey{x,z) = £<M/>K,x)e-^ (3.16) n = l Solutions of this form satisfy the boundary condition on the surfaces x = 0, x = b, and z — oo. The remaining surface is z — 0 and here Ey(x, 0) must be equal to the magnetic current M(x): oo M{x) = 5>nV>(K„,») (3.17) n = l Provided that M(x) can be written this way the qn's are chosen so that qn = fh M(x)r/,(Kn,x)dx (3.18) For TM modes equations (2.43), (3.11), and (3.12) are used in system (3.13) and solutions can be written Hy(x,z) = J2<lnyfe(x)i>(Kn,x)e-^ (3.19) 71 = 1 and by a similar argument qn = - -±=Li>{Kn,x)dx (3.20) Chapter 3. Closed Planar Dielectric Waveguides 29 and the closed waveguide problem is solved. The expansions (3.16) and (3.19) are solutions to the problem only if the series of (3.17) can represent the given function M(x), or in the T M case, the given function — J(x)lyje(x). In mathematics, this is known as the question of completeness. The possible values for Af(x) exist on an infinite dimensional space and if the sequence (3.17) is always successful in representing M(x) then the set {ijj(nn,x)}™=l is said to be complete on the space. There are some well-known theorems that apply to this case, and they are summarized in Appendix C. In this Appendix it is shown that a sufficient condition for completeness is M E C 2[0,fc]. Similarly the series (3.19) will reproduce the boundary condition on z = 0 provided a similar continuity condition for J(x) is met. These constraints are only mathematical in nature, and do not pose any real constraints in modelling the physical problem. By now it should be clear that although the T E and T M cases are separate problems with different solutions, the method of analysis is the same. If the solution for a T E polarization is shown, the same method can be used for the T M case after a few minor substitutions. Therefore from now on only the T E case will be presented for the sake of brevity. The method for the T M case can be easily constructed from the T E example at any point if desired. 3.2 The Spectral Function The main computational difficulty in carrying out the above method of solution is finding the set of eigenfunctions ijjn(x). In this section some connections of this set to other concepts are described. These connections will prove especially useful when dealing with the open problems of Chapters 4 and 5. Chapter 3. Closed Planar Dielectric Waveguides 30 Suppose we have two linearly independent solutions of Lijj = Etp such that </>i(/c,0) = sin a 4>2(K,0) = cos a: </>I(K,0) = — cos a c/>2(/c, 0) = sin a (3-21) Suppose a is chosen so that </>2(K,CC) conforms with the boundary condition at x = 0 (3.4). This means a is chosen such that tana = -h = ^ ~ (3.22) Zsi <J)2(K,X) is the solution of the initial value problem satisfying the boundary condition at x = 0, hence the functions at the eigenvalues (j>2(nn,x) will also satisfy the second boundary condition at x = b. They may be used as the basis for the expansion (3.17). These functions are not orthonormal however, so the normalization constants pn need to be defined Pn = <f)l(Kn,x)dx (3.23) ./o With this definition, the eigenfunction expansion will now appear qn = tM{x)(i>2(Kn,x)dx (3.24) Jo Ey(x,z) = £ ) (3.25) n = l P n The spectral function will now be introduced. For now it will amount to little more than a change in notation, and so it might seem somewhat contrived. Nevertheless it is an important concept and its meaning for dielectric waveguides will take on more substance in subsequent chapters. For the problem on the interval [0,6], let the spectral function pb(E) be a series of step functions. It is constant for all E except at the eigenvalues En where it jumps by l/pn Pb(E) = £ - (3.26) En<E Pn Chapter 3. Closed Planar Dielectric Waveguides 31 With this definition the pair (3.24)-(3.25) may be written q(n) = f M(xU2{K,x)dx (3.27) Jo q(K)4>2{K,x)e-^z dpb(E) (3.28) - oo where the sum in (3.25) has been replaced by the Stieltjes integral (3.28). There is another function of the spectral variable E which is sometimes useful in constructing the spectral function. It is called m and as will be shown, is related to impedance. Following Titchmarsh the general solution of the differential equation Lijj = Etp may be written as a linear combination of the solutions <&\ and (f>2 [36] <P(K,X) = </>I(K, X) + mb(E)(j)2(K, x) (3.29) mj, is chosen so that ip wiD satisfy the boundary condition at x = b: ip'{n,b) + H<p{n,b) = 0 (3.30) lence mb(E) _ -—— —j- (3.31) Note that mb is a ratio of entire functions, and hence mj, is meromorphic. The poles of mi, occur when the denominator of (3.31) is zero, that is when 4>2 satisfies the boundary condition at x — b. Since cj>2 also satisfies the boundary condition at x = 0 for all E, a pole in m£, will occur if and only if E is an eigenvalue En of the system (3.13). There is a connection between the function rnb(E) and the input impedance of a related system defined on a half space with respect to the x axis. Consider the half space x > 0 with e = e(x) there. Suppose the reactive surface at x = 0 is removed and plane waves are incident from the lower half space x < 0. Most of the incident power is reflected from the x = b reactive surface, and a small amount is lost in the dielectric. Chapter 3. Closed Planar Dielectric Waveguides 32 The situation in x < 0 can be summarized Ey(x,z) = Ey(x,z) + Ery(x,z) = e-i(™+M + Y(Ky(™-W (3.32) where T(K) is the reflection coefficient for the wavenumber K. From the form of equation (3.32) it is clear that in the following &(«•) < 0 in order for.the results to be physically relevant. The choice is analogous to the one made for /3 in the discussion following (2.20) in which either the upper or lower half plane can be chosen without loss of generality. The input impedance at x = 0 is defined as the ratio of the tangential parts of the electric and magnetic fields which, incidentally, is independent of z. The situation in the upper half space x > 0 must involve some linear combination of d>i and <j>2 which satisfies the boundary condition at x = b, i.e. <p(n,x). Matching up the tangential components of electric and magnetic fields at x — 0 yields A<p(n,0) = 1 + T (3.34) A<p'(n,0) = -JK+JKT (3.35) where A is a constant of proportionality that will soon disappear. Now Z i „ = i ± I = = «» a 3 1 —I <f (K, 0) cosa-mj, sma where the definitions (3.21) and (3.29) have been used. Conversely Zin cos a - JK sin a mb = — : — (3.37) Zin sm a -\- JK cos a Hence mi, can be found given Z{n and vice versa. Chapter 3. Closed Planar Dielectric Waveguides 33 The hypothetical scattering experiment of equation (3.32) leads to an interpretation of the waveguide as a transmission line in the transverse direction. The meaning of the pole in the m function has a simple physical interpretation. At the pole, the transmission line is resonant, and may support a non-zero field even if the entrance port at x = 0 is terminated with the reactance Zai. With transverse resonance, a non-trivial solution to the fields is possible and energy may travel in this mode. The concept of transverse resonance is often stated in terms of Z rather than m. This is a natural choice for metal waveguides because on metal boundaries Z = 0 and Y = oo. However in dielectric waveguides reactive boundaries are much more common and so a natural choice for the desired meromorphic function will be one that includes the reactance of the boundaries Z„\ and Zt2 at x = 0 and x = b. The function m,b(E) has the desired properties. In the remaining part of this section, two more properties of will be demonstrated. One is that the poles of are simple, the second is that the residue of the nth pole is the reciprocal of the nth normalization constant pn. This is proven in [37] and in the following these proofs are sketched. Because of this second property, the spectral function needed in the eigenfunction expansion can be found from any of the functions m^E), which is true for any solution f(n,x) of Lf — Ef. In particular this identity holds for ip(n,x) with Xi = 0 and x2 = b. Only the imaginary part of (3.38) will be important. To help split it into real and imaginary parts define Z(K), or r(/c). These results make use of the integral identity f(K,x)f'(K,x)]" - r\f'(K,x)\2dx+ H(E-V)\f(K,x)\2dx = 0 (3.38) E = X+jv (3.39) and m = m' + jm II (3.40) Chapter 3. Closed Planar Dielectric Waveguides 34 the first term in (3.38) is 'i = <p(n,by(K, b)-^,oW, o ) (3.4i) The first term on the right hand side of (3.41) is real because of the boundary condition (3.30). From the definitions (3.21) and (3.29) it can be shown that 5{^70>'(K,0)} = 5{mb(E)} (3.42) From the imaginary part of (3.38) S{mb(E)} = v t ^ ( K , aj) + mb{E)cj)2(K,x)\2 dx (3.43) Jo To show the poles of mb(E) are simple, let E = Xn + ju and consider the limit u —* 0. Suppose the pole is of order p. Then the right hand side tends to infinity at the rate i / 1 _ 2 p , while the left hand side tends to infinity at the rate v~p at most. This forces p = 1 and the pole is simple. Suppose the residue at E = A n is rn. Multiplying (3.43) by v and taking the limit v —• 0 yields - r n = r2n I" \<p2{*n,x)\2dx (3.44) Jo and so r" = rbj.2,~l w = — ( 3 - 4 5 ) $°(j>l(Kn,x)dx Pn To summarize, the spectral function is needed to carry out the eigenfunction expansion to solve the closed waveguide problem. Either the input impedance must be found directly or the reflection coefficient must be used to find it. The impedance is used to find the mb function, which has poles at the eigenvalues of (3.13). The spectral function is constant everywhere except at the eigenvalues, where it has step discontinuities. The size of the nth step is given by — r n , the residue of the nth pole of mb(E). Chapter 4 Dielectric Waveguides Open on One Side The open problem can be obtained by passing to the limit of infinite separation of the reactive surfaces at x — 0 and x = b, i.e. the limit b —» oo. Hence this problem is solved once the behaviour of the important quantities of the closed waveguide is known in the limit. This problem has been considered by many authors in the mathematical literature [38] [36] [39] [37]. In this chapter these results are summarized and their significance to the electromagnetic problem is discussed. 4.1 Limit of mb(E) We start with mb. It can be shown analytically that as the H in the second boundary condition (3.5) varies, mb(E) traces out a. circle in the complex plane [39]. Also if a sequence of increasing widths b is chosen this will define a series of nested circles. As b —> oo either the locus of possible mb will tend to a circle in the complex plane, or, in the case the radius of nested circles tends to zero, mb will tend to a single point. These two cases are known as the limit circle and limit point cases respectively. It can also be shown that if the limit point case holds for one complex value of E, then it holds for all non-real E. Likewise if the limit circle case holds for one value of E then it will hold for all values. Therefore this classification depends on the operator alone. All examples considered in this thesis will be in the limit point case, in fact there is a theorem which gives a simple sufficient condition for the limit point case to hold [40]: 35 Chapter 4. Dielectric Waveguides Open on One Side 36 If V(x) is bounded below, i.e. if V (x) > — k for all x in [0,oo), then the operator L is in the limit point case at infinity In the limit point case the result of the limit b—*oo Km rrib(E) = m(E) (4.1) is independent of H in the boundary condition at x = b (3.5). This is equivalent to saying that the input impedance is independent of the reactance Zs2 on the surface x = b. This situation only makes sense if the medium is slightly lossy. One can say that the input impedance is not sensitive to Zs2 because in order for Zs2 to influence the input impedance the wave must travel a distance of at least 2b. For any fixed level of loss, if b is made arbitrarily large the influence of Zs2 must become arbitrarily small. Systems which are identical except for having different reactances Zs2 on x = b must, as b tends to infinity, approach the same input impedance to conform with the limit (4.1). In practice, it is easy to find m(E) in (4.1). If the dielectric structure is viewed as a transmission line in the transverse direction, it is sufficient to know the input impedance of the line, and then m(E) follows from (3.37). Consider equation (3.29) in the limit b —> oo The rrib(E) function in (3.29) was defined so that the left hand side (p(n,x) satisfies the boundary condition at x = b. What can this mean when b —» oo? As in the above paragraph, the small losses play an important role in the answer. The boundary condition at x = b represents a reactive surface and no power crosses it. Because of the losses a wave which returns from it must have a smaller amplitude than the incident wave. Since only a finite amount of power is being delivered to the system, the total energy of the ip(n,x) = </>I(K,X) + rn(E)(j>2(K,x) (4.2) Chapter 4. Dielectric Waveguides Open on One Side 37 system must be finite, even if the width tends to infinity. Therefore one expects / \<P(K,X)\2 dx < oo (4.3) Jo of course this could also have been deduced from (3.43). This establishes that the solution represented by (4.2) is in £ 2 [0 ,oo) . In fact for operators L in the limit point case it can be proven that the tp. in (4.2) is the only solution in £ 2 [0 , oo) [39]. Therefore to obtain a definition of m(E) which is consistent, m is defined so that ip(n,x) satisfies the boundary condition (3.30) at x = b when b is finite, and when b —» oo this condition is replaced by the condition that ip(K,x) G £ 2 [0,oo) . Therefore in the case b = oo, m(E) could be found just by finding a solution of (2.38) in £ 2 [0 ,oo) and then using (4.2). An example of this approach will be shown in Chapter 8. 4.2 Spectral Function for the Open Waveguide To carry out the eigenfunction expansion (3.28), we need to know the spectral function. In Section 3.2 it was shown this function can be constructed from mb given the poles and the residues of the poles. As b becomes large, the normalization constants (3.23) become large and so the jumps in the spectral function become small. At the same time, the eigenvalues En become more closely spaced, so pb tends to grow with E by about the same amount but with smaller steps. As b tends to infinity the step functions pb(E) approximate a continuous function, and in the limit the poles blur together to yield a continuous p(E). Hence the method of constructing the spectral function shown in Section 3.2 is clearly inadequate for the limiting case. Some other method of constructing the spectral function will need to be used. A simple example will illustrate the inadequacy. Suppose V(x) = 0 and therefore <f>i(n,x) = sin ct cos KX sin/cx (4-4) Chapter 4. Dielectric Waveguides Open on One Side 38 , / \ sin a . 02\K,X) = COS Ct COS KX -\ sm KX (4.0) K Also for V r(x) = 0 , Zin(K) = 1, and hence-from (3.37) . _. cos ct — JK sin a . „. m ( £ ) = - (4.6) sm a + JK cos a where K = \/E~, and the root is taken so that K is in the lower half plane. This function has at most one pole at K = K1 where K1 — j tan a (4.7) Since K is defined to be in the lower half plane, this is a real pole only if tana < 0. The negative tangent means the surface is inductive, and the pole is responsible foT the well known surface wave an inductive surface can support. If tan a > 0 the pole is in the non-physical sheet of E and is an example of what Tamir and Oliner have called a non-spectral pole [41], so-called because this pole does not support a surface wave. In either case m has no more than one pole, and it is obvious that the single function (f>2(Ki,x) cannot represent a general function A4(x). To see exactly what is happening it is instructive to look at rnb(E) for large b. Since L is in the limit point case the value of mb(E) in the limit b —> oo is independent of H in the boundary condition at x = b. Take for example H = 0, (corresponding to a magnetic wall at x = b) and find mt, directly from the definition (3.31) cos ct cos Kb -\- K sin ct sin Kb mb(E) = - I — 4.8 sm a cos Kb — K cos a sm Kb The poles are found at the zeros of the denominator, the nth pole satisfies Kn tan Knb = tana (4.9) Hence mb has an infinite number of real poles, and the density increases linearly with b. The nth residue at the nth pole is 2K„(COS a cos Knb + Kn sin a sin Knb) 6(sin a sin Knb -f Kn cos a cos Knb) (4.10) Chapter 4. Dielectric Waveguides Open on One Side 39 since the quantity in square brackets is not, in general, zero, the residues are tending to zero as b —> oo. Hence as b becomes large the density of modes increases but the significance of any single mode to the expansion decreases. In the limit, any single mode has no contribution, but a group of modes may still carry a significant amount of energy. Under these conditions the spectrum is said to be continuous. No single mode, linear combination of modes, or countably infinite number of modes can represent the field, but it can still be represented as an infinite (Riemann) sum of infinitesimal contributions. The spectral function can be found in this case also, but obviously not by a direct evaluation of the residues of poles. Another method for calculating pf, on the finite interval generalizes more readily. A short description of the method found in [39] will now be given. Consider the Parseval equality for the function ip: [b\p(K,x)fdx = f ^ K ^ l M ^ x W ( 4 1 1 ) J ° n = l Pn which is a consequence of the Parseval equality (C.56) using the unnormalized eigenfunc-tions 4>2(Kn,x). Consider E — En times the inner product on the right hand side: (E - En) I ip(K,x)4>2{Kn,x)dx = [ [</>2L<p - <pL(f)2}dx (4.12) Jo Jo where L is the differential operator defined by (2.41). For such an operator the so-called Green's formula applies: for any / and g in an interval [a^ a^ ] f° r which L is defined and for which Lf and Lg make sense lX\gLf - fLg)dx = {f,g}(x2) - {f,g}(Xl) (4.13) W here {f,g}(x) = f(x)g'(x)-f\x)g(x) (4.14) This formula follows by integrating the left hand side of (4.13) by parts. In (4.12) <j>2 and L<j>2 are real so (4.13) applies and (E - En){^4>2) = WM(b) - {<fM{0) (4.15) Chapter 4. Dielectric Waveguides Open on One Side 40 In consideration of the boundary condition (3.5) the first term is zero and from the definition (3.21) the second term is one. Hence / W , | . * = IMlSM^K = r M^L (416) On the other hand, the left hand side of (4.16) occurs in (3.43), and so /- dPb(E') 5{mb(E)} J-co \E-E'\2 &(E) { ' ' From this, we want to calculate pb. The integral operator on the left can be undone by integrating along the real E axis: Let E = X 4- ju and consider Um [M'%{mb(X + jv)}dX = lim / * ' f°° , , (4.18) — hm , -! X2~ <T . 1 A i - O -tan tan dpb(a) = Tr[pb(X2) - Pb(Xi)] v v In the case where the limit is taken from below the axis (y —> —0), the result is the nega-tive of the above. The advantage of the form (4.18) is that it quantifies the contributions of poles in m without referring to them directly. The existence of the limit lim pb(E) = p(E) (4.19) b—> 00 is justified by the existence of the limit rrib(E) —> m(E). Hence if an input impedance for the system can be defined, the spectral function follows immediately from (4.18). For example, from the example of (4.5)-(4.10) -TB = - - h m 0 S{m(A + ju)} = . 2 ' — (4.20) ah* 7 r !/-»-o sm a + K / cos^ a and so 1 fE VXdX P(E) = ~ / • 2 \ * T~ 4.21 7 r Jo s i n a + . A c o s 2 a Chapter 4. Dielectric Waveguides Open on One Side 41 for E > 0. This is a continuous function, rather than a series of step functions. For E < 0, however, m(E) has no imaginary part, so dp/dE is zero and p is constant there. The only exception is when E = Ei . Ej = - tan2 a (4.22) At this point m has a pole, and more care must be taken in evaluating the change in p. Equation (4.18) can still be used to calculate the residue but the limit and the integral cannot be interchanged. If tan a is positive the value of the residue is zero, as is expected of a non-spectral pole. However if tan a is negative a non-zero residue is obtained which means p has a jump discontinuity and a guided wave is possible at this wavenumber. 4.3 Eigenfunction Expansion The eigenfunction expansion for the open waveguide is a straightforward generalization of the form of the expansion for the finite case shown in (3.27) and (3.28). The crucial point is that the limit (4.19) exists. Given that p(E) exists the solution is simply /•oo q(n) = \ M(x)4>2(hi,x)dx (4.23) Jo /oo q(rt)(j>2(ri,x)e-rfz dp(E) (4.24) -oo It is easily shown that Ey(x,z) in the above form satisfies the wave equation (2.15). The uniqueness of the solution (4.24) can be proven by applying the uniqueness theorem of Appendix A. The argument is similar to the one used in Chapter 3 except that both the upper and right hand boundaries are closed at infinity. The fields are zero there because of the small losses in the system. The surface x = 0 is reactive and the tangential part of the electric field is known on 2 = 0. This closes the volume and uniqueness follows. In other words, the solution is found by starting with the given transverse electric field M(x) in the space domain in £ 2 [0 ,oo) , and transforming to q(n) which is on a Chapter 4. Dielectric Waveguides Open on One Side 42 spectral domain C2(p). In transforming back to the space domain a complex exponential (which is unity on the plane z = 0) is inserted which gives the result Ey(x,z) the desired properties for a solution. In fact the transformation of (4.23) - (4.24) is a unitary transformation, which means the transformation is one to one, onto, and preserves the inner product. The invariance of the inner product is important and will be useful in Section 4.4, so this section will conclude by examining this invariance [39]. Parseval's equality plays an important role in defining the inner product. Consider (4.11) written in the form / » b / » o o / \f(x)\2dx = / \F(n)\2 dPb{E) (4.25) JO J-oo where F(K) is the transform of f(x): F(K) = [b cj>2(K,x)f(x}dx (4.26) Jo If pb{E) exists in the limit b —> co then a similar equation will hold with Pb{E) replaced by p(E) (using (4.19)). Note the identity = ! / i + / 2 ! 2 - | / i - / 2 | 2 + i l / i + i / 2 | 2 - i | / i - i / 2 ! 2 (4.27) Hence the Parseval equality implies too too / h{x)f2(x)dx = / FiWFtWdpiE) (4.28) J 0 J —oo Equation (4.28) is significant because it defines the inner product in both the space domain and the spectral domain. The spectral function is needed to give the spectral components F\ and F2 the correct weights. With an inner product defined, a norm can be defined: F(n)F(K)dp(E) (4.29) Chapter 4. Dielectric Waveguides Open on One Side 43 it follows that the function F exists on an inner product space. Therefore the Bessel in-equality applies to elements of this space: if {Gi(K)}?=x are a set of orthonormal elements in the inner product space then £!{F,GS)|2 < |!F||2 (4.30) i = l and the proof is identical to the one shown in C.2. A special case of the Bessel inequality which will be useful is the Schwartz inequality. It states that for F and G, members of an inner product space, it follows \{F,G)\ < \\F\\\\G\\ (4.31) This can be proven by letting n = 1 in the Bessel inequality and normalizing G so that Gi becomes an orthonormal set trivially: G l = l i i i i ( 4 ' 3 2 ) Therefore KF> lj§ii>l = - ( 4- 3 3 ) and (4.31) follows. If = 0, the equality in (4.31) is trivial. 4.4 Completeness of the Expansion In the following it will be proven that (4.24) is successful in reproducing the boundary condition on the surface z — 0, at least in the sense of the mean: i.e. it will be proven that /•oo / \M(x)- Ey{x,0)\2dx = 0 (4.34) Jo In this section the proof presented in reference [39] is sketched. Chapter 4. Dielectric Waveguides Open on One Side 44 Let q(n) be defined from (4.23), and let M&(x) be the field on z = 0 reconstructed from (4.24) integrated over a finite domain A = (—u,u) MA(x) = j q(n)<j>2(K,x)dp(E) J A (4.35) We want to show that as u —» oo the expansion will tend to the original given function M(x), in the sense of (4.34). Let p(x) be a function which is zero for x > a and let P(x) be its transform Now P(K) = / P(X)</>2(K,X)dx Jo f MA(x)p(x) dx = f f q(K)(f>2(n,x)dp(E) Jo Jo UA (4.36) p(x) dx = / q(n) / (j>2(K,x)p(x)dxdp(E) J A JO = / q(KJP[K~)dp(E) (4.37) J A At the same time the inner product rule (4.28) can be appHed to M(x) and p(x): roc poo / M{x)p(x)dx = / q{K)P{n)dp(E) (4.38) Jo J-oo Subtracting (4.37) from (4.38), too foo / [M(x)~ MA{x)]p{x)dx = / R(K)P{n)dp{E) (4.39) •^0 J — oo where R(K) = 0 for E £ A q(n) for E G A c A c is compHmentary of the set A . Applying the Schwartz inequality to (4.39) yields 2 f O O ^ fOO fOO / [M(x)~ MA(x)}p{x)dx < / \R(K)\2dp / |P( K )| 2 dp JO J —oc J — oo = j \q(n)\2 dp J°° \p(x)\2 dx (4.40) J AC Jo Chapter 4. Dielectric Waveguides Open on One Side 45 Suppose the function p(x) is chosen to be j M(x)-MA(x) for x < a p(x) = < I 0 for x > a Then the above definition implies [a \M{x)-MA(x)\2 dx < f \q(K)\2dp(E) (4.41) JO J&' In this equation "a" can be arbitrarily large, and so (4.41) holds even in the limit a —> oo. The power crossing the surface z = 0 is proportional to the L2 measure of M(x). A physical source will produce a finite amount of power and so the C2 measure must be less than infinite. From Parseval's equality /•oo /•oo / \M(x)\2dx = / \q(K)\2dp(E) (4.42) Jo J-oo and so the right hand side of (4.41) must tend to zero as p —» oo. Therefore tends to M in the sense of equation (4.34). Chapter 5 Dielectric Waveguides Open on Both Sides The problems in the previous two chapters included reactive surfaces. While reactive surfaces are common at radio frequencies they are much less common at optical frequen-cies, because in this range even good conductors can show significant loss or other more complicated behaviour. The situation in Chapters 3 and 4 finds application immediately at radio frequencies, but for optical frequencies the best model will usually be one that is open on both sides. Unfortunately this is the most difficult of the three. In Chapter 4 a solution to the waveguide open on one side was found (equations (4.23)-(4.24)) by generalizing the solution of the closed waveguide. In this section a solution the waveguide open on both sides will be found by generalizing the results of both Chapters 3 and 4. A few observations will show why the solution of Chapter 4 will fail for the waveguide open on both sides. The transverse variations of Ey(x,z) in this problem are not subject to any apparent boundary conditions, and so only the differential equation Lip = Eil) (5.1) applies to the spectral components. It takes two linearly independent solutions of (5.1) to span the solution space. For example, the solution of the initial-value problem 4>I(K, X) and <f)2(n,x), defined in equations (3.21) will suffice. Therefore in equation (4.24) there must be terms involving ^ ( K ^ X ) as well as <j>2(n,x). At the same time, it is clear (4.23) is inadequate. Since {4>i,<j>2} has been taken as the basis for the solution space of (5.1), the inner product (4.23) calculates only the projection of M(x) onto the subspace (f>2. 46 Chapter 5. Dielectric Waveguides Open on Both Sides 47 Information is lost in a projection and so (4.24) cannot hope to reconstruct M(x) from only this projection. Hence the inner product of M with c/>i must also be used. As a first step in resolving these difficulties it may be pointed out that some of these same problems would have occurred for the closed waveguide if an inappropriate choice of axes had been made. In Chapter 3 the transverse axis x was chosen so that the problem was defined over the interval x £ [0,6]. The functions (f>1 and (f>2 were denned at the point x = 0, and a was chosen so that c/>2 would satisfy the boundar}' condition due to the reactive surface at x = 0. This greatly simplified the analysis. Suppose this intelligent choice had not been made and instead the problem was defined on the interval a; £ [a, b] for some negative a. Since the lower reactive surface is now at x = a, both <f>i, and c/>2,(and the a value that defines them) have no special significance. Both must be used to represent any eigenfunction, and the analysis is more complicated. On the other hand, it will be possible to let a tend to — oo in the same way as b tends to -foo. This limit leads to an adequate model for the waveguide open on both sides. Section 5.1 is taken from Coddington and Levinson [39] and Section 5.2 reproduces their results, but without assuming a fixed basis. In [39] it is assumed the angle a which defines the basis functions from (3.21) is 7r /2. In Section 5.2 it is shown the results apply for any a, in fact this a can even be a function of E, and the results still hold. This will be of use in Section 5.5 and Chapter 8. The solution to the open waveguide problem is presented in Section 5.3 and an explicit proof of the eigenfunction expansion which is omitted in [39] is shown. 5.1 Parseval Equality The problem defined on [a, b] has the same properties as the closed waveguide, there is an infinite sequence of real eigenvalues En and a complete set of real orthonormal Chapter 5. Dielectric Waveguides Open on Both Sides 48 eigenfunctions {^n{x)}^=1 • Also, Parseval's equality will apply: for any M(x) £ C2[a,b} rb °° / |Mf»|2<te = £|(M,e B) J a n=l (5.2) the functions <f>i and <f)2 will span the solution space and so it must be possible to write any eigenfunction en(:c) as a linear combination of them. en(x) = ^ ^ ( K ^ X ) + a2n4>2{KTl,x) (5.3) where aln and a2n are real constants. If (5.3) is substituted into any of the terms on the right hand side of (5.2), a quadratic form is obtained a l n l 9 l ( K n ) | + aina2n9l(Kn)g2(Kn) + ° l n a 2 n 9 l ( « n )?2 ( « n ) + ^ J ^ ^ n ) ! where q is a, two dimensional vector with components g;(/t) = / Ad(x)cj)i(K,x)dx (5.4) Ja q can be viewed as a function of M(x). (5.4) maps a function from the space domain to the spectral domain. The quadratic form indicates the Parseval equality (5.2) can be written rb too / \M(x)\2dx = / qT(K)dps(E)q(K) (5.5) J a J — oc where Pe(E) is a real valued 2 x 2 matrix. The components are step functions, having jump discon-tinuities at the eigenvalues En. The size of the jumps is given from a l n and a2n. Both ajn and a2n depend on the interval 8 — [a, b], so this dependence is indicated by the subscript S in the spectral function. According to (5.5), the magnitude of the jumps at the discontinuities is Ps(En + 0) - Ps{En - 0) a l 7 l a 2 n a2n (5.6) Chapter 5. Dielectric Waveguides Open on Both Sides 49 The Parseval equality (5.5) is used to define an inner product on the spectral domain. Using the identity (4.27) it can be shown /O O T O O h{x)f2{x)dx = / F*(n)dps{E)F2(K) -oo J — oo where F 1 2 are transforms of / 1 | 2 : = / fj(x)(f>(K,x)dx J — oo and <p{K,x) = (5.7) (5.8) <I>I{K,X) Hence the inner product in the spectral domain is defined /oo FT(K)dP(E)F2(K) -oo Note that the transformation (5.8) preserves the value of the inner product ( / i , / 2 > = ( F i , F 2 ) (5.9) (5.10) In the following section the existence of the limit as b —• oo and a —> — oo of the matrix spectral function will be investigated (5.11) and it will be shown how the matrix spectral function can be calculated from the prop-erties of the upper and lower half spaces. 5.2 Matrix Spectral Function In both Chapters 3 and 4 it was demonstrated that the spectral function could be con-structed from the transverse input impedance 2Tin. This impedance is defined as the Chapter 5. Dielectric Waveguides Open on Both Sides 50 surface impedance seen by plane waves crossing the surface x = 0 from the lower half space x < 0. Since the whole space x £ ( — 0 0 , 0 0 ) consists of the two half spaces, it might be expected that the matrix spectral function p could be constructed from the two input impedances: one from the upper half space and one from the lower half space. This can be done and is demonstrated in this section. It is worth noting here that a matrix spectral function also appears in the analysis of the matrix Schrodinger equation [25, pages 135-137]. The method used in Chapter 4 can be generalized. This method begins by considering the inner product of (p(n,x), the general solution satisfying the boundary condition at x = b, with one of the eigenfunctions en(x). In this section there are boundary conditions at both x — a and x = b on either side of the origin, and we want to involve in the calculation the input impedances of the upper and lower half spaces. Therefore define ifA(K, x) = </>!(«, x) -f ma(E)d>2(K., x) (5.12) ipB(K,x) = <I>I(K, X) + mb(E)<j)2(K,x) (5.13) where ipa satisfies the boundary condition at x = a, <p'a(K,a) + h<pa(K,a) = 0 (5.14) and (fb satisfies the one at x = 6, (3.5). Here the input impedances of the upper and lower, half spaces appear indirectly through the m functions. mb is related to the input impedance of the upper half space through (3.37) and ma is related to the lower impedance through a similar formula / r - , v Za cos a + ju sin a ma(E) = -J-, 1 (5.15 Za sm a — JK cos a The change in sign results from the change in orientation when denning an input impedance into the lower half space. It is possible to write formulas like (4.12) for the intervals [a, 0] Chapter 5. Dielectric Waveguides Open on Both Sides 51 and [0,6]: (E - En) ( tpb(ri,x)en(x)dx = / (enLipb - <PbLen) dx Jo Jo = {<Pb, en}(x = 6) - {y>b, en}{x = 0) = -{<pb,en}(x=0) (5.16) for [0,6] and (E - En) ipa{n,x)en(x)dx = / {enL<pa - <paLen) dx Ja J a = {<Pa, en}{x = 0) - {if a, en}(x = a) = {<Pa,en}(x = 0) (5.17) for [a, 0]. From (5.12),(5.13), and (5.3) {<Pa,en} = fflin{^i,^»i} + a . 2 n { ^ i , ^ 2 } + m a a ' i n { ^ 2 ) ^ i } + " i 0 a 2 n { ^ 2 , ^ 2 } (5.18) {<fb,en} = ain{<t>i,<j>i} + a2n{</>i,02} + " i b a i n { 0 2 , <£i} + rnba2n{<i>2>4>2} (5.19) At the point x = 0 the definition (3.21) simplifies the above since {&,M(o) = o {^, <f>2}(0) = 1 i>2,<M(o) = - l i>2,<M(0) = 0 (5.20) Therefore (E - En) f tpb{K,x)en(x)dx = -a2n + mb(E)aln (5-21) Jo ,0 (E - En) I <pa(K,x)en{x)dx = a2n - ma(E)aln (5.22) J a Chapter 5. Dielectric Waveguides Open on Both Sides 52 Consider the function defined AC*) = <Pa(K,x)[mb(E)-ma(E))-'i for z < 0 (5.23) <Pb(n,x) [mb(E) - ma(E)) 1 for x > 0 where E is a parameter not equal to En. The co-ordinates of the function / x (the inner product of it with the eigenfunctions) can be evaluated with the help of equations (5.21) and (5.22). fah{x)en{x)dx = (5.24) Applying Parseval's equality E-En [b\fi(x)\*dx = £ r J — c dP6n{E') (5.25) \E-En\2 J-eolE-E'l2 since a\n is the 1—1 component of the spectral matrix. The above equation is useful because the left hand side can be evaluated directly in terms of the m functions (and therefore the input impedances). rb 1 / IA0OI5 Ja dx = \ma{E) - mb(E)\2 / \tpa(n,x)\2 dx + / \<pb(n,x)\2 dx Ja Jo From (3.43) / | P b («,aOI Jo and a similar argument leads to / |Va(K,a!)| 2 J a From (5.25)-(5.28) dx = dx — S[mb(£)] - 3[E] S[ma(E)} (5.26) (5.27) r. dpSn(E') _ 1 1 (5.28) ma{E)-mb{E) (5.29) \E-E'\2 5{E] If the impedances of the upper and lower half spaces are known, then the right hand side of the above equation is known. This is analogous to the situation in equation (4.17), Chapter 5. Dielectric Waveguides Open on Both Sides 53 and the integral operator on the left hand side can be 'undone' in the same manner: 1 psiifa) - psn(^i) = km - / . , . , d\ (5.30) . " i a ( A + j i > ) - m b ( A + Another component of the spectral matrix can be found by choosing a different func-tion from fi(x). Equations (5.21) and (5.22) yield Hnear combinations of a i n and a2n and so it is possible to isolate a2n instead. Take mb(E)<pa(K,x)[mb(E) - ma(E))-1 for x < 0 ma(E)ipb(K,x)[mb(E) - ma(E)}~1 for x > 0 with £ ^ £„. Equations (5.21) and (5.22) imply ^ / 2 ( a j ) e n ( a j ) ( i a ! = ^ ^ Repeating the steps from (5.24) to (5.29) yields (5.31) (5.32) dP622(E') 5[E] ma(E)mb(E) ma(E)-mb(E) (5.33) - c o | £ - E'\2 To obtain the off-diagonal elements of the matrix spectral function, multipfy (5.24) by the complex conjugate of (5.32) l n & 2 n / h(x)en(x)dx / f2{x)en(x)dx = Ja J a ]Hi — En (5.34) The left hand side is the nth term of the inner product ( / i , / 2 ) , so if a sum over aU n is taken dpsn{E) </i,/») = / fi(x)f2(x)dx = r As before the left hand side can be evaluated directly \E~Er, (5.35) rb [" ,0 ,b / fi{x)f2{x)dx = \mb-ma\~2 mb | <pa \2 d x + ma / \<pb\2 dx Ja Ja JO 1 fha5(mb) - mb5(ma) $ ( £ ) | m b - m Q | 2 1 $s[(ma -f mfc)(mQ - mb) 25(E) (m a - m b ) (m a - mb) 25(E) \ma-mb (5.36) Chapter 5. Dielectric Waveguides Open on Both Sides 54 These results can be summarized by the matrix equation where M is a 2 x 2 matrix denned by 1 M12 = M 2 a = ma - mb lma+mb 2 ma — mb M22 = (5.38) rna - mb This is the analogue of the half space equation (4.17). As in Section 4.2, the existence of the spectral function in the limit as h —> oo and a —• —oc is established by the existence of the limits m+{E) = lim mb(E) (5.39) b—*oo m_(E) = Km ma(E) (5.40) a—> — oo Applying this limit to (5.37) will replace ps by p. The integral equation in (5.37) can be undone in the same way as before. Let E = A -f jv and by (4.18) p ( A 2 ) - p ( A i ) = ^ K m ^ 3 ' 5[M{\+jv)}d\ (5.41) If the limit is taken from below the real E axis, the right hand side has the opposite sign. In this way the matrix spectral function can be found from the input impedance of the upper and lower half spaces. 5.3 Eigenfunction Expansion Once the matrix spectral function has been found, the representation for the waveguide open on both sides can be written by generalizing (4.23)-(4.24) /oo M(x)d>(K,x)dx (5.42) -oo Chapter 5. Dielectric Waveguides Open on Both Sides 55 /O O e~jl3MzqT(n)dp{E)(p(K,x) (5.43) -oo As in Chapter 4, it can be verified that Ey(x,z) satisfies the wave equation (2.15), and the uniqueness of the solution is assured if Ey(x, 0) is the given tangential electric field on the surface z — 0, M(x). In this section it is shown that Ey(x, 0) tends to M{x) at least in the mean. The proof is similar to the one outlined in Section 4.4. In Section 5.1 an inner product on the spectral domain was defined /oo F^(K)dp(E)F2(K) (5.44) -oo and if the norm in this space is taken to be (F,F) (5.45) then the transform (5.42) of £ 2 (—oo,oo) functions defines a Hilbert space. Therefore the elements of this space satisfy the Schwartz inequality, which according to the current definitions appears as /oo 2 »oo poo F*(n)dp{E)F2{K) < F^(K)dp{E)F1(K) Fl(K)dp(E)F2(K) (5.46) -oo J —oo J — oo The proof now continues in the same way as in Section 4.4. Consider an MA(x) defined MA(x) = / qT{K)dp(E)cp{n,x) (5.47) J A for the interval A = {—p^p), and see what happens as ^ —> oo. Let p ( x ) be zero for x < —a and x > a, and let P ( « ) be its transform P ( K ) = I" p{x)tp(K,x)dx (5.48) J — a Then / MA(x)p(x) dx ' — i f qT(n) dp(E) (p(n, x)p(x) dx J —oo J — a J A = J qT(n)dp(E) J <p(K,x)p(x)dx = I qT(K)dp(E)P{K~) (5.49) J A Chapter 5. Dielectric Waveguides Open on Both Sides 56 and /oo roo M(x)p(x)dx = / qT{n)dp(E)P(K) (5.50) -oo J — oo In the same way as before it can be shown /oo r oc [M(x) - MA(x)]p(x)dx = / R T ( /c)dp(£)P( /t) (5.51) -oo J — oo where o for E € A R(K) = < q(K) for E e A c Applying the Schwartz inequality (5.46) to (5.51) yields (5.52) /oo 2 /• /• oo [M(a.) - M A ( z ) ] p ( x)<te < / qT(/c)dpq(K) / \p(x)\2 dx (5.53) -oo ^ A c J — oo Letting p(a;) = M(a:) — ¥ 4 ( 1 ) for —a < x < a and zero otherwise will lead to [°° \M(x)-MA{x)\2dx = I ciT(K)dp{E)~qjK~) (5.54) J-00 JAC and since the C2 measure of M is finite the Parseval equality implies the right hand side of (5.54) tends to zero as p —> 0 0 . 5.4 Aperture Antennas In this section it is shown that in the case of a uniform dielectric, equations (5.42) and (5.43) of the previous section reduce to the familiar formulas used in the theory of aperture antennas. See for example, Jull [21]. In the first step we must choose suitable basis functions and (f)2, they are defined from (3.21). The choice of a is arbitrary and any value will do. Suppose a = 0 and therefore Chapter 5. Dielectric Waveguides Open on Both Sides 57 <))2(K,X) = cos KX (5.56) and the vector q is defined from (5.42). To find the spectral function we need to know the m functions for the upper and lower half spaces. The input impedance of both upper and lower half spaces is unity and using (3.37) and (5.15) cos a - j /cs ina j . m+(E) = — = (0.57) sin a -\- JK cos a K = c oBa+ jVc r i n a = i sm a — JK cos a K and (5.38) imphes M 1 2 = M 2 1 = 0 M 2 2 = -±- (5.59) 2K Carrying out the limit and integration of (5.41) dpn = —K2dK 7T dp22 — —dK 7T P12 = P21 = 0 (5.60) Since the m functions in (5.58) were defined for K values in the fourth quadrant, the E values are in the lower half plane and so a negative sign appears in the limit. Equations (5.60) hold for positive values of E. If E is negative, K is pure imaginary and so M is real. Therefore = o for E < 0 (5.61) dE Now (5.43) takes the form 1 t°° Ey(x,z) = — g~Jfi(K)z [g2(/t) cos KX — (JI(K)K sin KX] dK (5.62) 7T JO Chapter 5. Dielectric Waveguides Open on Both Sides 58 writing the harmonic functions as complex exponentials and rearranging The functions qi, q2, and ft are all symmetric with respect to n, and so Ey(x,z) = — T F(n)e ZTT J— OO -J{KX+/3Z ) £ (5.64) where F(K) = q2(n) - jnq^K) / • O O = / M{x)eJKXdx (5.65) — oo Thus in the case of a uniform dielectric the general transform solution (5.42) - (5.43) reduces to the usual plane wave and angular spectrum of aperture antennas. The solution (5.64) - (5.65) looks much simpler than the general form because it takes advantage of some symmetries found in this special case. In this case the transform doesn't need to be a two dimensional vector, instead it is a scalar complex function. The two dimensional nature of the solution exists as the independent real and imaginary parts of F. Similarly, the spectral matrix happens to be diagonal, and the rules of complex multiplication in (5.64) substitute for the matrix quadratic form in (5.43). 5.5 Two Dimensional vs. One Dimensional Representation In the beginning of this chapter the matrix spectral function was introduced because it is clear that equation (5.1) has two linearly independent solutions and, in general, both will be needed in an expansion like (4.24). However for a solution rp to be part of the spectrum it must be bounded over the domain x 6 ( — 0 0 , 0 0 ) [2]: (5.66) Chapter 5. Dielectric Waveguides Open on Both Sides 59 If a few simple examples of equation (5.1) are examined, it becomes clear that the number of independent bounded solutions depends on E. There are ranges of E for which two hnearly independent solutions of (5.1) can be found, ranges for which only one can be found, and others for which no bounded solution exists. When the derivative of the matrix spectral function is constructed, it is found the rank of the matrix is two, one, and zero for each range respectively. This reflects the fact that, depending on E, the spectrum can be two dimensional, one dimensional, or non-existent. Consider a range of E for which the spectrum is one dimensional. In this range the rank of the spectral matrix is one and so if the basis {<^ >1,^ >2} is chosen properly it will be possible to reduce the two dimensional equations (5.42),(5.43) to a simpler one dimensional form. This form will be identical to the eigenfunction expansion (4.23) and (4.24) of Chapter 4, with shghtly modified definitions of 4> and p. Of course this reduction will be exact only if the entire wave is contained in the given spectral range of E. However, even if the wave is not completely contained in this range, the suggested reduction is still useful as an approximation. An approximation of the same kind is often used in communication theory. Sometimes it is assumed a signal has a finite bandwidth, so the signal is adequately represented by spectral components from a certain range (bandwidth). In both communication theory and in the application considered above, the approximation is appropriate when it is known in advance that most of the energy of the signal (or wave) will be represented by spectral components within the specified bandwidth (spectral range). This technique will be used to solve a problem in a later chapter. In the following a simple example is shown to illustrate the above ideas. Consider the dielectric interface shown in Figure 5.1. Suppose ej > e2 and the obvious choice for transverse wavenumbers is made K\ +/52 = a;2/xoei (5.67) Chapter 5. Dielectric Waveguides Open on Both Sides x £2 t Figure 5.1: A Dielectric Interface K\ + /32 = u>2(i0e2 Since the expansion should work for any choice of a let 7T a = 2 and so the (j) functions can be written down , C O S K\X X > 0 <I>I(K,X) = C O S K2X X < 0 s i n ^ x > Q (f>2{n,x) _ J ^ x < 0 K.2 Chapter 5. Dielectric Waveguides Open on Both Sides 61 Let K i be associated with the spectral variable E: E K, and let so that Kn <*>Vo(ei - e2) (5.72) (5.73) From the form of the solutions, (5.70)-(5.71) it is clear that if E > A 2 both <f>i and <f>2 are bounded and so they are both part of the spectrum. When E < A 2 both functions are divergent in a; < 0. However some linear combination of (f>1 and (f>2 may still yield a bounded function. If this can be done then the spectrum is one dimensional and essentially the same as the spectrum in Chapter 4, within the specified spectral range (range of E). To make further progress it is helpful to construct the matrix spectral function. From (3.37) and (5.15) m+(E) = -JKT m-(E) = JK2 and from (5.38) 1 'J \(K2-KI) ~JKiK2 M \{K2 — Kl) Ki + K2 If E > A 2 both and K2 are real and (5.41) yields 1 0 For E < A 2 both K i and K 2 can be imaginary, writing (5.74) (5.75) (5.76) dp 2~E 1 7T (5.77) (5.78) Chapter 5. Dielectric Waveguides Open on Both Sides 62 K2 — K 2 ~ t ~ JK2 we find for the case 0 < E < A 2 dp d~E and for E < 0 dp d~E K. «2 (5.79) (5.80) In mathematics functions like fa(n,x) and </>2(/c, a:) can be regarded as basis vectors. Since they are hnearly independent, they span a two dimensional vector space. Matrices like (5.77), (5.79), and (5.80) are therefore linear transformations on this space. The special properties of the cases (5.77),(5.79), and (5.80) can be described in the language of matrix algebra. For example (5.80) is a zero matrix, so all the domain is mapped to zero. Therefore for E < 0 neither fa nor fa nor any combination of them is associated with the spectrum. In (5.79) the dimension of the null space and the range is one, so that only one Hnear combination of fa and (j)2 contributes to the spectrum. (5.77) has a nullity of zero and so any vector in the range is aUowed. The reason for introducing this mathematical notion is that in the case 0 < E < A 2 the equations can be simphfied. The matrix (5.79) represents a Hnear transformation and so the components of the matrix depend on the basis vectors. The matrix is simplified if a new basis is used, 6 say, with 6I(K,X) a vector in the range and #2(K,ai) a vector in the null space. Let 6I(K,X) = 4>I(K,X) — K/2'</>2(K, a:) 92(n,x) = n2'fa(K,X) + fa(K,x) (5.81) To find the coordinates of the derivative of the spectral matrix in the 6 basis, multiply Chapter 5. Dielectric Waveguides Open on Both Sides 63 the matrix by B\ and Q2 in turn to obtain the first and second columns. dp dE dp 1 dE 0 II 2 - ^ ^ [ ( l + K f ) < P l - K 2 ' ( l + K f ) < P : dp JEW :6», = dp K2 dE 4> 1 = o (5.82) (5.83) Where the subscript indicates the <j> basis is being used. Since o is the same in all bases, the second column of the matrix in the 6 basis is zero. To find the first column, it is necessary to write (5.82) in terms of the 9 functions. Inverting (5.81) and substituting into (5.82) leads to = + (5-84) dE T(K1 + K 2 ' 2 ) V ' Therefore the matrix in the 6 basis is dp K l ( i+ K ' 2 ' 2 ) •0 dE e 0 0 (5.85) and the eigenfunction expansion (5.42)-(5.43) reduces to the single-sided expansion (4.23)-(4.24) of problem B, with <j)2(n,x) replaced by 91(hi,x) and p relaced by plx of (5.85): /oo M(x)61(K,x)dx -oo Ey{x,z) = / A q(K)61(K,x)e-jl3z dPll(E) Jo (5.86) (5.87) where the approximation is due to the finite bandwidth 0 — A 2 . The form of the function 6i(n,x) in the range 0 < E < A 2 is calculated from (5.81) to be ' C O S K J X - ^ S i n ^ x for x > 0 e - K 2 x for x < 0 6I(K,X) (5.88) Chapter 5. Dielectric Waveguides Open on Both Sides 64 In the range x < 0 the function 6X is a decaying exponential. This is necessary because any solution independent of this must contain an increasing exponential, which is not allowed in the eigenfunction expansion. Hence the form (5.86)-(5.87) could have been guessed at without the preceding analysis: it is obvious that in the range 0 < E < A 2 there is only one solution of (5.1) that is bounded over the entire x axis. This example serves to illustrate that the analysis can be greatly simplified by an intelligent choice of basis functions. The eigenfunction expansion in the new basis 6 also has a physical interpretation. In the lower half plane x < 0 the fields are all decreasing exponentially. There is no flow of power into the lower half plane and the surface x — 0 could be regarded as reactive (as defined in Section 3.1). This is not a reactive surface in the strict sense, however, because the reactance depends on the spectral parameter E. The reactance is calculated from the ratio of the derivative of 6X to its value at x = 0, as in (3.6): Z.i = -~- (5-89) which depends on E. Hence 6 differs from (p in that it is not tied to a fixed a in the definition (3.21). Instead a is chosen according to cot a = K ' 2 ' = - V A 2 - E (5.90) so that a follows the only bounded solution of (5.1) as E goes from 0 to A 2 . The eigenfunctions are thus the transverse variations of plane waves at various angles of incidence in which total internal reflection occurs. Letting a be a function of E is not allowed in Chapter 4, because it disturbs the self-adjointness of the operator L through the boundary condition at x — 0. However in (5.86)-(5.87) there is no explicit boundary condition at x — 0 and the operator remains self-adjoint because the inner product is defined over the entire x axis, with no special conditions at x — 0. Chapter 6 Inverse Spectral Theory So far the spectral function has been viewed only as a weight function necessary to normalize eigenfunction expansions. In this chapter it is shown that the spectral function actually contains all the information about the dielectric structure. In fact the dielectric structure can be constructed uniquel}' knowing only the spectral function. In other words the mapping described in Chapters 2-5 (in which a spectral function is found from a given dielectric structure) is injective, and an inverse mapping exists provided the given spectral function does indeed correspond to a real system. This chapter is about the inverse mapping, in which the dielectric structure is constructed from a given spectral function. The crucial step in this method is establishing the Povzner-Levitan representation, in which a connection is drawn between a known simple system, called the auxiliary system, and a more complicated unknown system. The connection is in the form of an integral transform. In Section 6.1 this form is motivated in light of the asymptotic behaviour of solutions of the basic equation (2.38) and the Paley-Wiener theorem. Use of the Povzner-Levitan representation allows a comparison of the spectral functions of the auxiliary and unknown systems, and leads to the Gel'fand-Levitan equation (Section 6.2). The solution of this equation gives the Povzner-Levitan transform explicitly and the complete solution of the unknown system follows. An example is shown in Section 6.3. 65 Chapter 6. Inverse Spectral Theory 66 6.1 The Povzner-Levitan Representation In inverse spectral theory the usual approach is to establish a relationship between the known solutions of a simple system, called the auxiliary system, and the unknown solu-tions of a more complicated system. The Povzner-Levitan representation is one way to establish this relationship. In Chapter 4, the problem was reduced to the system . j," + [E-V(x)]ip = 0 (6.1) V>'(0) + hip{0) = 0 (6.2) but explicit solutions can be found only for certain simple choices of V(x). Consider the asymptotic behaviour in the spectral variable. In passing to the limit E —> oo, the value of V becomes negligible compared to E, and (6.1) approaches the simple harmonic oscillator equation, y" + Ey = 0 (6.3) y'(0) + hy(0) = 0 (6.4) The solution to this system is s i n K x y[K,x) = cos KX — h (6-5) Therefore lim TJ}(K,X) — cos KX . . (6-6) E—.oo In some ways ip can be regarded as a perturbation of the simple solution cos/ex, the perturbation is stronger for smaller values of E. Define F(K,X) = tp(K,x) — cos KX (6-7) Chapter 6. Inverse Spectral Theory 67 (6.6) implies that as K -> oo, F —> 0. ip is symmetric in the variable rc: ip(K,x) = ij>(—n,x) (6.8) Since cosine is also symmetric, F is symmetric in rc as well. Since F is symmetric and tends to zero as rc —> oo it is possible to take the Fourier cosine transform in rc 2 f°° K(x,t) — — I F(K,X) cos ntdn (6-9) 7T Jo and from the inverse Fourier transform, /•oo F(K,X) = / K(x,t) cosntdt (6.10) Jo (6.7) and (6.10) imply /•oo ip(n,x) = cos nx + / K(x,t) cosntdt . (6-H) This equation expresses the solution of the differential equation (6.1) in terms of a solution of the simpler harmonic equation, cosrcx. The relation is an integral transform involving some unknown kernel K(x,t). The only thing known about the kernel at this point is that it is not a function of rc, by definition (6.9). Note that with if; written in the form (6.11), the limit (6.6) is automatically satisfied, because in the limit the integral in (6.11) becomes oscillator}'. So far we have not departed from the usual Fourier transform theory. The main result to be shown in this section is that for any bounded potential function \V(x)\ < C1 Vx£ [0,oo) (6.12) the kernel K(x,t) is triangular: K(x,t) = 0 for t > x (6.13) Chapter 6. Inverse Spectral Theory 68 so in equation (6.11) the integration need only go up to a;: tp(K,x) — cos KX + / K(x, t) cos Kt dt (6-14) Jo Equation (6.14) is the Povzner-Levitan representation and it seems to have appeared for the first time in 1948 [42]. Povzner arrived at (6.14) by applying the method of Riemann invariants to a related system of hyperbolic partial differential equations. In his derivation the triangular nature of K is a result of the fact that in wave theory, a point cannot be affected by events whose characteristics never intersect the point. In short the triangular nature of K is connected to the notion of causality. However there is a simpler and more direct route to (6.13). It can be shown directly that F(K,X) has certain asymptotic features which allow the application of the Paley-Wiener theorem. The Povzner-Levitan representation then follows immediately. This approach was used by Chadan and Sabatier [25] and they arrive at the formula sin KX fx . . sin Kt , . ip(K,x) - +/ K(x,t) dt (6.15) K JO K which is a form appropriate for a system with the boundary condition ip(K,0) = 0 (6.16) This is the desired boundary condition in quantum scattering theory, because in this application ip is a radial electron wave function which must vanish at the origin. However this boundary condition is not general enough for the current application, the boundary condition (6.2) is more appropriate for the analysis of dielectric waveguides. In the following (Section 6.1.1 and 6.1.2) the representation (6.14) is derived for the boundary condition (6.2) using the same approach as Chadan and Sabatier. Chapter 6. Inverse Spectral Theory 69 6.1.1 Asymptotic Behaviour of ip As suggested in the previous section ip can be regarded as a perturbation of a simpler harmonic system. Suppose (6.1) is rewritten ip" + Eip = Vtp (6.17) The left hand side is now the harmonic equation (6.3). Let two hnearly independent solutions of (6.3) be sm KX K y2 = cos KX We can attempt to solve (6.17) by the method of variation of constants, let ip(x) = a1(x)y1(x) + a2(x)y2(x) In this method one lets the a's be defined such that and substitution into the original differential equation yields the hnear system 2/i 0 v*. a 2 ViP which is solved to give a'j = cos KX V(x)V'(K,x) sin KX a, = —- V (x) ip(n,x) (6.18) (6.19) (6.20) (6.21) (6.22) (6.23) To find di(x) and a2(x) it is necessary to know (^0) and rp'(0). The system (6.1)-(6.2) is a boundary value problem, but solutions can be found by choosing initial values for ip Chapter 6. Inverse Spectral Theory 70 consistent with the boundary condition (6.2), for example let 4>(K,O) = i V>'(K,0) = -h (6.24) with this choice aAx) — — h + / cos nt V(t) ip(n, t) dt Jo a2(x) = 1- f ?E^V(t)if>(K,t)dt (6.25) JO K and from (6.20) , / v sin KX fx sin K(X — t)^,..,, . , , n„. ip{K,x) = cos nx-h—— + / i J-V(t)rp{K,t)dt (6.26) K JO K Thus the original differential equation (6.1) is transformed into a Volterra integral equa-tion. Provided the kernel in the integral is bounded, Volterra integral equations can always be solved by iteration [43]. We will perform these iterations in order to obtain estimates of the asymptotic behaviour of xp. Let T be the integral transform defined by = f'^LJlv(t)f(t)dt (6.27) Jo K and let the inhomogeneous term in (6.26) be h sin KX g(x) — cos KX (6.28) so (6.26) can be written (I-T)7f>=g (6.29) where I is the identity operator. The solution of (6.26) is Tp = {I-T)-'g (6.30) Chapter 6. Inverse Spectral Theory 71 where ( / - T ) - . 1 = I+ T+ T2+T3 + ... (6.31) Of course the solution represented by (6.30) only makes sense if the series in (6.31) converges. In order to investigate the convergence of the series we set up bounds for the functions involved: \g(x)\ < C 2el""l" (6.32) sin K(X — t) rt where K = K' +jn" (6.34) and the bound for V(x) has already been assumed (6.12). With these bounds it is easy to show that the second term of (6.31) is bounded: \Tg\ < C1C2Czew'\xx (6.35) It can also be proven by induction that, for the nth term, \Tng\ < C2eV>(ClC*x)n (6.36) n! Therefore MK,X)\ < f ; \Tng\ < C 2 e ' - " ' a f ; ( C l ^ ) n = C 2 e ^ e c ^ * (6.37) n=0 n=0 n -(6.37) guarantees the convergence of the series (6.31), pointwise in x. There are two important conclusions that can be drawn from the above. The first is that, because both g and the kernel of the integral transform T are entire functions of K , and because the series for ijj converges, the solution ip(n,x) is an entire function of K. The second is that equation (6.37) sets asymptotic limits on the growth of rj) with < C3e k " l ( * - 0 (6.33) Chapter 6. Inverse Spectral Theory 72 |rc|. ip cannot increase faster than e^x, therefore as a function of rc, ip is a function of exponential type x: In \IP(K, X)\ .„ „„. Um — ' 7 , ' n < x (6.38) |(t|-.oo \K\ 6.1.2 The Paley-Wiener Theorem The Paley-Wiener theorem draws a connection between entire functions of exponential type and properties of the Fourier transforms of such functions. Two classes of entire functions of the complex variable z can be identified. In the first class, the function F(z) is of exponential type A, meaning F{z) = o{eAW) (6.39) and it is a. member of £ 2 (—oo,oo) on the real axis: /oo \F(x)\2dx < oo (6.40) -oo The second class consists of entire functions of the form F(z) = fA f(t)ejztdt (6.41) J—A where fe£2[-A,A} (6.42) and / is the Fourier transform of F. The theorem states that these two classes are equivalent. In other words if a function can be shown to be a member of the first class then it must be possible to represent it in the form (6.41). A corollary of the Paley-Wiener theorem is that for functions of the first class F(z), the Fourier transform is zero outside the interval [—A, A}: 1 r°° /(*) = 7T / n*>~' zt dz = 0 (6.43) Chapter 6. Inverse Spectral Theory 73 for any t such that | i | > A. This follows in view of the uniqueness of Fourier transforms. Consider the function of the previous section F(K,X) = I/J(K,X) — cos KX (6.44) the results of the previous section together with the known properties of cosine show that F is an entire function of K of exponential type x \F{K,X)\ < C4ew'lx (6.45) If it can also be shown that F is a member of £ 2 (—oo,oo) on the real axis then the theorem could be applied. F is bounded over the real axis by exponential functions (6.37). Since F is symmetric in K it is sufficient to show /oo \F(K,X)\2CLK < oo (6.46) Also (6.26) in the definition of F (6.44) gives another definition of F: „, h sin KX fx sin K(X — t) F{K,X) = -f / }-V{t)ip(K,t)dt (6.47) K JO K Using the bounds (6.37), (6.12), and | sin | < 1 (for a real argument) \F(K,X)\ < ^ + ^ £ I reclC3tdt (6 4 8 ) K K JO It is now easy to show (6.46) is true and therefore F £ £ 2 ( —oo,oo). The conditions of the theorem are met and as a result F can be written in the form F(K,X) = [' f(t,x)ei**dt (6.49) where / is the Fourier transform of F in the variable K. Since F is symmetric in K it must be that /X f(t,x)smKtdt = 0 (6.50) J — X Chapter 6. Inverse Spectral Theory 74 and so F(K,X) = / cos nt / ( £ , x) dt = 2 / / ( £ , x) cos Kt dt J—x Jo Identifying K(x,t) = 2f(t,x) in (6.51) leads to ip(K,x) — cos KX + / K{x, t) cos Kt dt Jo the Povzner-Levitan representation. 6.2 The Gel'fand-Levitan Integral Equation In this section the connection between the kernel K(x,t) of equation (6.53) and the potential function V(x) of equation (6.1) is shown. The kernel may be obtained from the solution of a hyperbolic equation involving V(x). However the problem can be approached from another angle, it is shown that K(x,t) can be constructed from the spectral function alone. Hence the spectral function is sufficient to find the solution to the problem through (6.53). It is also shown that the spectral function can be used to calculate the potential function V(x) in equation (6.1) and the value of h in the boundary condition (6.2). Hence if the spectral function is specified the entire system and its solution is given all at once. The method that will be shown in this section is due to Gel'fand and Levitan [23], they were the first to show a general method of reconstructing the potential function from the spectral function. The conditions under which this method can be used are quite specific. It is assumed the solutions of the auxiliary system are the harmonic functions and a specific homogeneous boundary condition is satisfied at x = 0. In fact, although these details are needed for the Gel'fand-Levitan approach, the ideas can be generalized and the whole subject can be approached on a more abstract level, as has been done by Kay and Moses [27, pages 65-191]. They show the inverse problem can be (6.51) (6.52) (6.53) Chapter 6. Inverse Spectral Theory 75 solved under more general conditions, and the}' eliminate many of the unnecessary details (or special circumstances) which are required for the Gel'fand-Levitan approach. The Gel'fand-Levitan approach will be described here because it is sufficient for the purpose of this thesis. However it should be borne in mind that the Gel'fand-Levitan approach is only one of many related approaches, and so this thesis provides only one example of the possibilities within inverse spectral theory. The Povzner-Levitan representation (6.53) suggests a relation between the simple system y(rz,x) = cos K E , and the more complicated system Tp(K,x), the solution of (6.1). To get more information about the kernel K(x,t), (6.53) can be substituted into the original differential equation (6.1). After some manipulation the following results are obtained [23]: 8>K(x,t) T„ d*K(z,t) V(x)K{x,t) = ^ 1 (6.54) =0 (6.55) dx2 w v > ; d t 8K(x,t) dt t=o dK(x,x) 1, . . . Hence K(x,t) is the solution of a hyperbolic boundary value problem, and the method of characteristics could be used to find it, given V(x). Equation (6.56) is important because if V(x) were initially unknown, and K(x,t) could somehow be found by another means, then (6.56) could be used to find the unknown potential function V(x). The above equations at least estabhsh the existence of K(x,t), with this it is possible to consider a variation of (6.53) which will be useful. Instead of considering a mapping from C O S K X to ip(n,x) as (6.53) does, one could consider the inverse mapping, from ip to COSKX. Ii ip is regarded as the known function in (6.53), and cos a as the unknown, (6.53) becomes a Volterra integral equation with TP(K,X) as the inhomogeneous term. This equation can be solved by iteration as was done in Section 6.1.1. The iterated Chapter 6. Inverse Spectral Theory 76 kernels can be gathered into a single kernel K and the solution written as COSKOI = xp(K,x)-\- / K(x,t)xp(n,t) dt (6.57) Jo The eigenfunction expansions used in Chapter 4 and elsewhere can be summarized by /oo xl>(K,x)if>(K,x)dp(E) = 8{x-t) (6.58) -oo This equation can be derived from the pair (4.23) and (4.24), letting z — 0 and elim-inating q(rt). Equation (6.58) is called a completeness relation and it shows how the eigenfunctions can be normalized. Multiply (6.57) by ip(n,s) where s > x, and integrate the result with respect to p(E): ( • O O / cos Kxx/j(n,s)dp(E) = 0 (6.59) J — oo for s > x. Multiply (6.53) by cos KS and integrate both sides with respect to p{E). Let s < x and in light of (6.59): /O O [X [OO cos KS cos KX dp(E) •+ / K(x,t) / cos Kt cos KS dp[E) dt (6.60) -oo Jo J — oo The cosine functions satisfy a completeness relation of their own. From the Fourier cosine transform it is easily shown 2 f°° — / COS KX COS Kt dK — 5(x — t) (6.61) 7T JO or equivalently f°° / cos KX cos Kt dpanxiE) = 6(x — t) (6.62) J — O O where paux is the spectral function of the uniform system , ^VE E>0 Paux(E) = { w (6.63) 0 E < 0 Chapter 6. Inverse Spectral Theory 77 The function cr(E) is defined as being the difference between the spectral function of the new system with that of the auxiliary system: cr(E) = P(E) - Pa,ux(E) (6.64) with the substitution (6.64), (6.60) reduces to K{x,s) + G(x,s) + f K{x,t)G(t,s)dt = 0 (6.65) Jo where roo G(x,s) — / cos KX cos KS do(E) (6.66) J — OO This is the Gel'fand-Levitan equation. It is useful if the spectral function of a new system is specified. If the spectral function of the new system is given, and the resulting Fredholm integral equation can be solved for the kernel K(x,s), then the potential function of the new system is given from (6.56) and its solutions are given from (6.53). Even the value of h in the boundary condition (6.2) may be recovered from the spectral function, provided (6.65) has been solved for K(x,t). With x — 0 in (6.53) V>(K,0) = 1 (6.67) and taking the derivative of (6.53) with respect to i and setting x = 0: tp'{K,0) = if (0,0) (6.68) The above equations hold for all rc, and so h = -W^ = ~ K ^ (6.69) 6.3 A Solution Using the Spectral Function Solutions of (6.65) may be used directly in Chapter 4. The function I/J(K,X) of this chapter is a multiple of the function fa(K,x) of Chapter 4. Hence the solutions (6.65) Chapter 6. Inverse Spectral Theory 78 may be used in Chapter 4 with only a trivial renormalization. On one hand xp satisfies the initial conditions (6.67) and (6.68), while on the other hand (f>2 satisfies the initial conditions (3.21) 4>2{K, 0) = cos a ^>' 2 (K , 0) = sin a Therefore ( ^ 2 ( ^ , 2 : ) = cos aip(K,x) (6.70) <J)2(K,X) = cos aX/J'(K, X) (6-71) which means a must be chosen such that tana = if (0,0) (6.72) Since the waveguide open on one side was solved in terms of the expansion function $2, the equations of this chapter will be normalized this way. Instead of using cos nx as an auxiliary system, the scaled auxiliary system >^aux is used <j>a,ux(K,%) = cos a cos KX (6.73) and so the Povzner-Levitan representation is <P2{K,X) = </>aux(«,a:) + / K(x, t) ^aux(«, t) dt (6-74) Jo The spectral function of the auxiliary system has been renormalized to 1 2 E E > 0 Paux(E) = { c o s Q X (6.75) 0 E < 0 the Gel'fand-Levitan equation (6-65) has the same form except the Q function is renor-malized /oo cos KX cos Kt da(E) (6.76) -0  Chapter 6. Inverse Spectral Theory 79 A number of exact solutions to (6.65) are known [25] but in this thesis we will examine only the simplest one. Suppose the new spectral function p differs from the auxiliary one Paux by a step function p{E) = pa,ux{E) + c0U(E - E0) (6.77) where U is the Heaviside step function. As can be seen from the expansion relation (4.24), the discontinuity in p will result in a guided mode in the field expansion (provided /3(n0) is a real number). Also from (3.45) the normalization for this mode is seen to be r°° 1 / 4>\{HQ,X) dx = — . (6.78) JO C 0 The equation (6.77) simplifies the Stieltjes integral in (6.76) to a single term and leads to the solution of (6.65)[23] •rsr *\ c ° c o s 2 a c o s \/rEox cos y/Eot , M K{x,t) = - — — ^ (6.79) 1 -f c0 cos a J 0 cos^  \/E0s ds For this solution A"(0,0) = -co cos2 a (6.80) and so a is a solution of the transcendental equation tana = — c0 cos2 a (6.81) Also the sequence (6.81),(6.80),(6.69), and (3.6) implies that the surface x = 0 has the reactance Z.. = f ^ (6.82) tan a To find the waveguide corresponding to the given spectral function (6.77), equation (6.56) is used, followed by (2.40) in order to recover the permittivity profile e(x). In (6.77) E o can be either positive or negative, both yield a guided mode with the transverse wavenumber no = JEO (6.83) Chapter 6. Inverse Spectral Theory 80 but the physical processes responsible for the guiding are different in the two cases. If E0 is negative, let K0 = jr/o and from (6.56) 70C! sinh270^(1 + -p- sinh27 0x + ^ ) — c? cosh470a; V(x) = - 2 ^ — (6.84) ^ ' [l + ^ s i n h 2 7 o ^ + £ f ] 2 ^ * As i —> oo, V'(a;) —> 0 and e —> e„. K0 is an imaginary number, so as x —» oo the solution tends to e 7 n a : . The longitudinal wavenumber is A, = sjK - E0 (6.85) an example is shown in Figure 6.1. This describes a typical guided mode in a dielectric (although the profile in this case is more typical of those obtained by inverse scattering methods than by conventional methods). The longitudinal wavenumber /30 is greater than the substrate wavenumber ks and the waveguiding occurs because there is a region of high refractive index that permits transverse propagation within the region, but outside the region the transverse wavenumber becomes imaginary, causing the power to be reflected back into the guide. If 0 < E0 < kl a guided mode is formed in the system defined by V l s m 2 v ( l | ^ s i n 2 V + f ) + c ; c o s 4 V  1 ( X ) = [l + - s m 2 « 0 * + - p ( 6 ' 8 6 ) This time V(x) —> 0 as x —> oo again, although more slowly V(x) = 0(x~1) rather than exponentially as before. For this guide ^ (KO^X) € £ 2 [0 ,oo) . This result is somewhat counterintuitive, because a mode with wavenumber K0 < ks and /30 < ks should be a radiation mode. It should be a plane wave propagating in the substrate and therefore not guided. However the plot of (6.86) in Figure 6.2 reveals the physical process responsible for the guiding. The most salient feature of Figure 6.2 is the almost periodic variations in the permittivity. The wavenumber of these variations is 2n0, which is the condition for Bragg reflection. Hence we can conclude that waveguiding in this case occurs by a mechanism similar to that explored by Yeh et al [19]. Chapter 6. Inverse Spectral Theory Figure 6.1: Potential V(x) with E0 = -6.4 x 1013m~2 and cx = 106 Chapter 6. Inverse Spectral Theory x (pm) Figure 6.2: Potential V(x) with E0 = 2.35 x 1014m~2 and ca = 5.0 x Chapter 7 Inverse Methods for Open Dielectric Waveguides In the example of Section 6.3 the surface x = 0 has a fixed reactance according to (6.82). Therefore this approach is appropriate for problems like the waveguide open on one side, which have a surface of constant reactance. However most optical waveguides are open on both sides, because in optics the waveguides are usually made from dielectrics alone, without making use of reactive surfaces. Section 6.3 showed how inverse spectral theory maj' be used in Chapter 4, and the object of this chapter is to show how it may be used for waveguides open on both sides, by connecting it to the theory of Chapter 5. In this thesis the following approach is used: it is assumed two half spaces with respect to the x axis are given, one is the upper half space x > 0 and the other is the lower half space, x < 0. The spectral functions of the upper and lower half spaces are given. There is a reactive surface on the plane x = 0, and this reactance is generally different on the upper side of the plane from the lower side. The reactive surface is then removed, to leave a waveguide open on both sides, and the two half space spectral functions are replaced by a matrix spectral function. In Chapter 5 it was shown that the matrix spectral function, which includes informa-tion about guided modes, was given by the m(E) functions of the upper and lower half spaces. Therefore the theory of Chapter 5 will be of use if the m(E) function of a half space can be calculated from the spectral function of the half space. By now there have been many quantities denned and their interconnections are com-plicated, so for the benefit of the reader the situation is summarized in Figure 7.1. By 83 Chapter 7. Inverse Methods for Open Dielectric Waveguides 84 Relations, in Order of Appearance: 1. Initial value problem (Chapter 3) 2. Initial value problem (Chapter 3) 3. Definition of m, equations (3.31),(4.1) 4. Conventional methods, with definition (3.33) 5. Equation (3.36) 6. Equation (3.37) 7. Equation (4.18) 8. The Povzner-Levitan Representation (6.53) 9. Equation (6.56) 10. Equation (6.65) 11. Equation (6.66) 12. Equation (6.72) 13. Equation (7.31) 14. Equation (7.32) Figure 7.1: Relations between Quantities in the Half Space Chapter 7. Inverse Methods for Open Dielectric Waveguides 85 following the arrows in the diagram it is clear that given p{E), it is easy to find the system e(x) and most of the quantities associated with it, provided the Gel'fand-Levitan equation (6.65) (path 10) can be solved. However it is not easy to reverse path 7, and so it is not yet clear how to calculate m(E) from p{E). The object of this chapter is to introduce the last paths (13 and 14), so that m(E) may be found. m(E) is related to the input impedance Zin(n), and both of these can be deduced from a scattering experiment. The method shown in this chapter is based on recent work [30] but it is not the only way this problem could be solved. The m(E) function can be found from the spectral function [37] and the relation between this function and the reflection coefficient has already been shown in Chapters 3 and 4. If this is done for the examples of Section 6.3 it will be found the reflection coefficient is a rational function of the wavenumber n. The potential function and scattering solutions may then be found by the method of Kay [31]. 7.1 A Scattering Experiment Suppose the spectral function of a problem in the half space x > 0 is given together with the solution of the associated Gel'fand-Levitan integral equation. Suppose the reactive surface at x — 0 is removed and the space x < 0 is filled with a uniform dielectric with e = e„, so that V (x) = 0 for a; < 0. If a wave is incident from below it is expected some energy would be reflected because of the variations in e starting at x = 0. A reflection coefficient could be defined on the surface x = 0, and if it were known the normalized input impedance could be deduced from l + r(ic) , x z" = <" ) Solutions (p2(K,x) to the wave equation are already known, but these solutions refer to the case where the surface x — 0 has reactance (6.82). In the scattering problem this surface is merely an interface between two media. Let the solution of the scattering Chapter 7. Inverse Methods for Open Dielectric Waveguides 86 problem be written $(K,X). In particular let $ be the solution resulting from an incident wave with amplitude a $(K,X) = a(K)e-JKX +b{K)ejnx (7 .2) for x < 0. The form of $ is, at this point, unknown. However $ and its first derivative must be continuous on the boundary x = 0. Hence (7.2) implies, at the point x — 0, *'(K.0) - J / 6 $ ( K , 0 ) = -.2JKCL(K) (7 .3) Also from physical considerations it is known this function must be square integrable on [0,oo) and hence it must have an eigenfunction expansion in (p2(ttyx): /o o g(K,K')<l>2{K',x)dp{E') (7 .4) - O O The coefficients g(n,K') must now be chosen such that the boundary condition (7.3) is satisfied. It is not advisable to substitute (7.4) directly into (7.3), as the derivative of an eigenfunction expansion may fail to converge at certain points. Instead it is better to recast the scattering problem in a variational form to avoid the occasional convergence problems. The original differential equation (6.1) and the boundary condition (7.3) associated with it can be obtained as stationary values of the following functional f O O f o o = / {($')2 + V$2}dx+JK$2(K,0)-4JK${K,0)a(K)-E / $2 dx (7 .5) Jo Jo It can be verified that 8F — 0 yields (6 .1) and (7.3). The requirements on the convergence of the expansion (7.4) are now more relaxed. In the next step (7.4) is substituted into (7.5). The coefficients g(n,n') are varied until a stationary value of F is obtained. The problem is made tractable by defining orthogonality relations for the (p2 functions. With the coefficients thus obtained, $ can Chapter 7. Inverse Methods for Open Dielectric Waveguides 87 be calculated from (7.4). It should be noted that, because of the completeness of the eigenfunction expansion this procedure will provide the exact solution rather than an approximation. 7.2 Orthogonality Relations The substitutions of the previous section will involve a number of inner products of the cf>2 functions and their derivatives. It is therefore necessary to investigate the orthogonality of these functions. Insight may be gained by first examining a system of finite extent in the x direction that would result by including a reactive surface on the plane x = b, and then passing to the limit as b —> oo. Suppose the spectral function of the infinite system is known and is the same as the function of Section 6.3: •p{E) = sec2 ot-^E U(E) + c0U{E - E0) (7.6) where U is the Heaviside step function. This p(E) corresponds to the potential given by (6.86), and to obtain the finite system a reactive surface can be placed on any plane x — b. The function 4>2(K,X), as well as being given by (6.74), is the solution of the initial value problem </>'2'{K,x) + [E-V(x)]<f>2{K,x) = 0 (7.7) 4>2(K, 0) = cos a ^>' 2 (K, 0) = —Co cos3 a The function <p2(n,x) is fixed and known because the initial value problem is well posed. If we add the additional constraint due to the reactive surface at x = b: $ U , 6 ) + #<MK,&) = 0 ( 7- 8) it is found this boundary condition is satisfied only for a countably infinite set {Kn}£Li, the eigenvalues of this Sturm-Liouville system. Chapter 7. Inverse Methods for Open Dielectric Waveguides 88 The form of the functional F(<p) (7.5) suggests defining the following bilinear forms operating on arbitrary functions u(x) and v(x) in [0,6]: 7ib[u,v} = / u(x)v(x)dx (7-9) Jo Vb[u,v) = / {u'v' + Vuv}dx - hu(0)v(0) + Hu(b)v{b) (7.10) Jo The orthogonality properties of these forms when they operate on eigenfunctions are well known [34] rHb[<p2{Kn,x),4>2{Krn,x)\ = pn6nm (7-11) The object of this section is to find relations similar to these, but which apply to a system of infinite extent in the x direction. It can be expected that in the limit 6 —> oo the result of the operations Ti and T> will become generalized functions of the spectral variables. Define the distribution x: X(K , K ' ) = / (p2(K,,x)(p2(rt'\x) dx (7-12) A distribution is defined from the effect it has on test functions. Let 0(rc) be a test function in the spectral domain (i.e. 0 is an arbitrary C2 function with compact support). 0 is related to another function 8 in the space domain through the expansion relations /oo fa{K,x)Q{n)dp(E) (7.13) -oo and too 0(K) = / (p2(K,x)8(x) dx Jo Now consider /oo x{K1K,)Q{K,)dp{E') (7.14) -oo Chapter 7. Inverse Methods for Open Dielectric Waveguides 89 after interchanging the order of integration /•oo ^oo yoo - = 4>2{K,X) \ \ <p2(n', x)(p2{rt\ X') dp(E') 0{x') dx' dx J0 Jo J-oo The integral in the center is known to be a 6 distribution in the space domain [25], therefore fOO /»oo I = / <p2{n,x) S(X - x')0(x')dx'dx Jo Jo f°° = / <fr2(rt, x)6(x) dx Jo = Q{K) (7.15) Hence the x distribution is very similar to the S distribution. In fact it is tempting to write X(K,K') = 6(P(E)-P(E')) (7.16) however the expression on the right becomes ambiguous when E — E' and E is part of the discrete spectrum, since the function p — p(E) is not well defined there. The operator T>b will also tend to a distribution in the limit b —> oo. Define Tf(K,*c ' ) = lim Vb[4>2{K,x),4>2{K\x)] (7.17) o—>oo To determine this distribution it is better to consider first the integrals of the test function in a finite domain, we have /oo too Vb[(j)2(Kn,x),(p2{K\x)}Q{K')dpb{E') = / Ennb[<p2(Kn,x),(P2(^^)M^)dpb{E') (7.18) In the limit b —» oo the left side of (7.18) will become /oo V(Kn, «')©(«') dp(E') -oo The right side, on the other hand, tends to /oo X{nn, K')Q(K') dp(E') = ENQ{Kn) -oo Chapter 7. Inverse Methods for Open Dielectric Waveguides 90 in the limit b —> oo the sequence {K„} becomes dense on the spectrum so KN can be any point in the spectrum. Therefore T,(K,K') = EX(K,K') (7.19) In this section we use the identities developed above to solve the scattering problem posed in Section 7.1. When the eigenfunction expansion (7.4) is substituted into the functional (7.5) inner products of the (p2 functions result. It is convenient to rewrite (7.5) in terms of the forms 7J? and ri: F{ip) = $] + (JK - tan a ) $ 2 ( « , 0) - 4 J K 0 ( K ) $ ( K , 0) - EH[&, *] (7.20) Equation (7.4) is substituted into (7.20) and the order of integration is interchanged to obtain F as a function oi g(n, K'): /OO fOC / g{K,K')g{n,K',)V^{(p2{K\xl(l>2{K\x)}dp{E,,)dp{E^ - o o J—oo /oo r oc / g(K, K')9{K, K^H^IM"',*), dp(E")dp(E') - o o J — oo + (JK - t a n a ) $ 2 ( « , 0 ) - 4 J ' K O ( K ) $ ( K , 0) (7.21) (7.22) In (7.21) the function <p2 is fixed and given by (6.74), the spectral function is given by (6.77) and the coefficients g(n, K') are to be varied. Note that $ ( K , 0) also varies. The first variation with respect to g(n, K') may be calculated: \SF = r(E'-E)g{K,K')dp{E') Z J — oo /oo 8g(K,K')<j)2(K\0)dp{E') -oo - 2jna(K)J™6g(K,K')<l>2{K'10)dp{E') (7.23) Setting 8F — 0 and recalling that (p2(K, 0) = cos a 9{K, K') = * Jj^l J — ± J - cos a (7.24) Chapter 7. Inverse Methods lor Open Dielectric Waveguides 91 Substituting this back in the original expression (7.4) r°° dp(E') * ( K , 0 ) = [2J'KO(K) - (JK - tana)$(/c,0)]cos2a / v ' (7.25) J-oo E' — E Define 7)(K/\ = mn2 /-v / E V K)  cos2 a r - ^ T ; (7-26) V 1 J-oo E' — E v ' Equation (7.25) can be solved for $(«,()):• i / '' r\\ 2jria{rl)v{rl) .„ „ „ , 1 - f - (j« — tan a)v{K) therefore , — 2JKa(n) cos a 1 9 ^ K } = 1 + ( , K - t a n « ) , ( * ) £ - £ ' ( 7 " 2 8 ) From (7.2) it is evident that $ ( K , 0 ) = O ( K ) + 6(K) (7.29) and since T = h/a then = (JK + tan a)v(K) - 1 ^ ; ( J K - tana)v(/c) + 1 ^ ' ^ From (7.1) the input impedance is _ JKV(K) (7.31) 1 — tan a V(K) and the function m(E) may be deduced using (3.37) m(E) = J^^l-tana (7.32) 7.3 A Bragg Reflection Waveguide In this section the methods introduced in this chapter are appHed to the example iUus-trated in Figure 6.2, (equation (6.86)), to obtain a novel Bragg reflection waveguide. It wiU be shown that with a suitable choice of parameters a guided mode wiU occur with transverse wavenumber /tO) an d that this mode is the only T E mode possible. Chapter 7. Inverse Methods for Open Dielectric Waveguides 92 Suppose the spectral function of the upper half space is given by (6.77) with 0 < E0 < k2. In order to find the matrix spectral function it will be necessary to know m+(E) (where the + subscript refers to the upper half space). Equation (6.77) is used to calculate V(K) directly ~ 2 - 2 r°° dn1 c0 cos a I f°° dK V(K) = ——~ + ~ n—i K?, — K 7T JO K — K .2 VQ  f\ ii u u rv — rv c0 cos2 a 1 f°° dn' = —2 7 + ~ TI w , ' N 7 - 3 3 KQ — K 7T J-oo {K'— K)[K + K) In Chapter 2 it was established the dielectric must be at least shghtly lossy in order for the theor}' to be self consistent. In view of this and equation (7.2), it is clear that K must have a small negative imaginary component. The integral in (7.33) can be evaluated by closing the contour in either the upper or lower half planes. In either case the result is the same , . c0 cos a j . . V(K) = —2 72 - r. (7-34) ; 2 ( The m+(E) function is now found from (7.32). Strictly speaking, it is necessary to find a first by solving the transcendental equation (6.81). However it is almost the same thing to assume that the quantity c0 cos2 a was given instead of c0 alone. Hence it is convenient to define C i = c0 cos2 a (7.35) and take c\ as the given. As will be demonstrated, the only place where c0 occurs in the equations without the cos2 a factor is in equation (6.78). Therefore the transcendental equation (6.81) only needs to be solved if the numerical value of the normalization integral (6.78) needs to be known. It is instructive to find explicitly the reflection coefficient T(K). Equations (7.35) and (6.81) imply tana = - c x (7.36) Chapter 7. Inverse Methods for Open Dielectric Waveguides 93 and (7.36),(7.35),(7.34) .in (7.30) gives 2 ' 2 [ K ) ~ CI(K20-2K2)+JKC2 + 2JK{K2-K2) { ' a plot of the magnitude of the reflection coefficient versus the transverse wavenumber K is shown in Figure 7.2. Note that at the point K = K 0 , r ( « o ) = (7-38) JK0 + Ci and so jr(/c0)| = 1- This means the mode with K = K0 is completely reflected: no power escapes to infinity, as is expected. This is seen clearly in Figure 7.2. Note also that the condition due to the conservation of energy |r(«)|2 < 1 (7.39) for real K is satisfied. The equation (7.37) is a rational function, having one zero and three poles. This indicates the above example may also be generated from the Marchenko theory using the method of Kay [31]. At this point this analysis becomes related to other investigations which use rational reflection coefficients [44] [28]. Equation (7.37) will now be factored so the location of the poles may be seen. The zero is at the point K = j^- (7.40) C l The poles are found from the solutions of the cubic equation 2s3 + 2c l S 2 + (c\ + 2K20)S + cj/s2, = 0 (7.41) where s — JK. Since this is an odd order real polynomial with positive coefficients, there is no non-negative solution, but there will be at least one negative solution. Let this solution be a = -s1 (7.42) Chapter 7. Inverse Methods for Open Dielectric Waveguides 94 where S i is a positive real number. The other solutions to (7.41) may be found by dividing the polynomial (7.41) by (s + s-y), and solving the remaining quadratic equation. The result is * 2 = - cO + y(Cl - Siy + 2 .^4- 4«g (7.43) s3 = \{s, - ci) - y(c1~siy + 2sl + iKt (7.44) The quantities under the square root signs are positive, and so the remaining roots are always the complex conjugate pairs represented by (7.44). The position of the poles of T(K) has been found for the example of Figure 7.3, by first solving (7.41) numerically for Sy. The positions of the poles are K J = J2.49934 x 105 K2 = 1.53314 x 107 + jl.25033 x 105 K3 = -1.53314 x i o 7 + jl.25033 x 105 These poles are all in the upper half plane, and as such they are non-spectral poles and contribute to the continuous part of the spectrum only. Hence the structure in the region x > 0 with a uniform dielectric in the region x < 0 would have no guided modes. The Bragg mode which is described in this section comes only through the interaction of this structure with another structure placed in the other half-space x < 0. Hence in the following we must include contributions from both half spaces. The ra+ function is associated with the upper half space x > 0 and is given by c0 sec2 a m+(E) = ° - j tana (7.45) If if* 2 if ' ' Using the trigonometric relations (7.35) and (7.36) this can be written m + ( £ ) = ^ i ± 4 ^ i ( i ± i ) + C ] ( 7 , 4 6 ) Kin tv rt Figure 7.2: Magnitude of Reflection Coefficient vs. Normalized Transverse Wavenumb KJ' ka Chapter 7. Inverse Methods for Open Dielectric Waveguides 96 In this section we want to investigate the existence of guided modes. From equation (5.41) of Chapter 5 it is clear that guided modes occur only at the poles of ~M.(E), since only at the poles will the spectral matrix be discontinuous. From the definition of M(£ l) (5.38) it is clear the necessary and sufficient condition for a pole in M is m + ( £ ) - m _ ( £ ) = 0 (7.47) If a dielectric structure is chosen for the lower half space, m_(E) could be calculated and used with (7.46) in (7.47) in order to look for guided modes. Rather than proceeding in this straightforward way, however, a digression will be made in order to show that the condition (7.47) is really the same thing as the condition Z+{K) + Z.(K) = 0 (7.48) where Z+ is the input surface impedance looking into the upper half space and is the input surface impedance looking into the lower half space. Condition (7.48) is the well-known principle of transverse resonance [45]. Using (3.37) and (5.15), (7.47) can be written in terms of Z+ and Z_: Z+ — jn tan a Z_ + JK tan a Z + tan a -f JK Z_ tan a — JK JK{1 +tan2a)(Z+ + Z_) ( ? 4 Q ) m_L — m_ = (Z+ tan a -f JK)(Z_ tan a — JK) so the conditions (7.47) and (7.48) are equivalent (with the possible exception of the case K = 0). In fact the theory of Chapter 5 can be used as a derivation of the principle of transverse resonance. Z(K) is a more physically relevant quantity than m(E), so in the following (7.48) will be used instead of (7.47). A dielectric structure needs to be chosen for the lower half space such that (7.48) is satisfied for one value of K. Note from (7.38) and Figure 7.2 that the mode with K = K0 is completely reflected in the upper half space. Therefore if this mode was also reflected in Chapter 7. Inverse Methods for Open Dielectric Waveguides 97 the lower half space a guided mode would be possible. The lower reflection need not be Bragg reflection but could be the more conventional total internal reflection. The lower dielectric could be just a uniform slab, with thickness xg and permittivity eg, below the point x — —xg, the dielectric could end in air, as shown in Figure 7.3. The example of Figure 7.3 was generated by choosing values for A, n,, n a , and /3o-From these fca, ks, and K0 may be calculated. The choice of ct is arbitrary, it may be any non-negative number, but an effort was made to make cx small enough so that the required permittivity variations can be manufactured with real materials. Cho et al [32] demonstrated Bragg reflection based on the Bloch solution due to Yeh [19]. They used the material Al3.Gaj_3.As, and they indicate the index of refraction may be made to vary within the range 3.24 to 3.43 at the given frequency. In the example of Figure 7.3 c0 was chosen so this range would not be exceeded. It is convenient to make the index of refraction continuous at the point x = 0. Therefore we let the index of refraction of the guide section [—xg,0] be equal to the value the index of refraction of the Bragg reflector assumes at the point x = 0. It is easily shown that V(0) = 2c20 (7.50) and so kg (and hence eg) may be calculated from the relation kg = y/k* - 2c2 (7.51) A guided mode may occur at the point when total Bragg reflection occurs (when and f3 = /30). If the width xg is suitably chosen, and if total internal reflection occurs at x = — xg, then a standing wave in the guide section will be set up and a guided mode will be formed. To calculate a suitable xg, we first find the surface impedance in the region x < 0. The impedance at the discontinuity at x — — xg will be, normalized to Chapter 7. Inverse Methods for Open Dielectric Waveguides 98 Index of Refraction (n) 4 2 0 ng -na 1 1 1 I -1.5 -1 -0.5 0 0.5 1 1.5 X (fim) A = 1.15/im, n, = 3.35 ( Al.2Ga.8As), na = 1.00 0o = 1.00 x 107 rn"1, ka = 5.46364 x 106 m"1, *, = 1.83032 x 107 m"1 , K0 = 1.53299 x 107 m"1 kg = 1.82895 x 107 m"1, c, = 5.0 x 105 m"1 ng = 3.3475, xg = .855415 fim Figure 7.3: Example of a Bragg Reflection Waveguide Chapter 7. Inverse Methods for Open Dielectric Waveguides 99 the level of the guide section, ZS = -^ (7.52) K„ where K9 = Jk2 - f32 and na = ^/kl - /32 (7.53) The surface impedance Z-(K) will be the value this impedance assumes at the point x = 0. This can be found from conventional transmission Une theory 2-{K)= S . + i t a a * , * , ( ? 5 4 ) 1 + j Z„ tan Kgxg The normaUzed impedance of the Bragg reflector ma}' also be found. <7 ( \ Z+(K) • K2p- K2 + JKCj v For the wave with K = rt0 Z+(K0)=J^ (7.56) Co where ng0 — ^ jk2 — {3$. This impedance has no real part and so a guided mode for this wavenumber is at least a possibiUty. Define Zt(K) = Z+(K) + 2_{K) (7.57) It is now required to choose xg such that Zt(n0) = 0. Using (7.54), and (7.56) tan Kg0Xg = KG0 2C° + J K A 0 (7.58) Kg0 ~ 3c0Ka0 and so xg is any of the possible solutions of (7.58). Having found one guided mode, the existence of all other T E modes can be estabUshed by looking at all possible values for the transverse wavenumber K. From Figure 2.2 of Chapter 2, this includes the real axis and the negative imaginary axis. On the real axis K only needs to be investigated in the range | K | < ks because for n greater than this f3 Chapter 7. Inverse Methods for Open Dielectric Waveguides 100 is imaginary and cannot result in a guided mode. In the range 0 < K < ks, K0 is the only possibility for a guided mode because for any other wavenumber in this range the reflection is not complete and the mode must loose energy. On the other hand K may be on the negative imaginary axis n = — jf where 0 < 7 < 0 0 . (This second case includes the mode n = 0) . For these modes it is possible to use the transverse resonance equation (7.48) directly. The calculation is tedious, so the details are shown in Appendix B. In the resulting expression all terms are non-negative and at least one is always positive. Therefore the expression is never zero, (7.48) is never satisfied, and there are no guided TE modes in this range. It is interesting to note that, if the given value of xg makes a zero of Zt, then any number of transverse guide half wavelengths can be added and the zero still remains, as the value of ZT(K0) is unchanged: , T 1 7 T x'g = xg + — . (7.59) Kg Hence, single mode operation is maintained regardless of the width of the guide section xg, as long as xg is maintained at precisely the right number of transverse wavelengths. In this respect this guide is unusual, since for waveguides of almost all kinds, the larger the transverse dimensions, the more guided modes will be supported. The mode structure of this single mode will be unusual also. If total internal reflection were employed on the plane x = 0, and a large xg were chosen, there would be many modes due to the many possible ray angles between x = — xg and x — 0. The effect of using Bragg reflection at x = 0 is to ehminate all but one of these modes. The guided mode is a multiply reflected plane wave in the interval \—xg, 0]. Chapter 8 A Bragg Modulator There are a few features of the Bragg waveguide of Section 7.4 that suggest it would make a good electro-optic modulator. The waveguide was derived from a system with the spectral function (6.77) with E0 a positive number. Hence the guided mode is embedded in the continuous part of the spectrum. Nearby the guided wavenumber, K.O, the spectral function is continuously increasing, which means there are radiation modes nearby n0. Only a small change in the transverse wavenumber is required to transfer energy to the radiation modes which will carry energy away to x = oo. The plot of the reflection coefficient in Figure 7.2 tells the same story: the slope of the curve leading to the peak is quite steep, so small deviations in the angle of incidence of the reflecting plane waves have large effects. In this chapter a detailed calculation will be performed on the Bragg waveguide of Section 7.4 to estimate the size of the modulations, and thereby take a step towards assessing the feasibility of the design. The structure to be studied is shown in Figure 8.1. In the region z < 0 the structure is the same as shown in Figure 7.3, and it is assumed the structure carries a fully developed guided Bragg mode at n — n0 and /3 = {30. In the region 0 < 2 < I, the structure is identical except for the guide section — xg < x < 0, which has an applied electric field and is assumed to be sensitive to electro-optic modulations, and so the permittivity here varies according to the strength of the externally applied field. The region z > I is identical in structure to the region z < 0. The field at z — I includes the guided mode and some radiation from the mid section 0 < z < I. However 101 Chapter 8. A Bragg Modulator 102 Waveguide parameters the same as in Figure 7.3. Figure 8.1: A Bragg Waveguide Modulator as z increases the radiation will disperse and the field will return to the well developed Bragg mode, although its amplitude may be less than the incident mode in z < 0. The electro-optic effect in most materials is very small. Even if an electric field approaching the breakdown strength of the dielectric is applied the change in the relative dielectric constant is still only a fraction of a percent. Since the change is so small there will be very little reflection of the incident wave from the surface 2 = 0 (or from the surface z = I). The main effect of modulating the index of refraction will be to change the propagation characteristics of the waveguide in the section 0 < z < I. These considerations justify the use of the KirchhofF approximation, in which the electric field Chapter 8. A Bragg Modulator 103 in the plane z = 0 is taken to be the same as the incident wave in the section z < 0. It is equivalent to suppose there is a magnetic surface current on the plane z = 0: Ms(x,y) = K nc(x,y,0)xh (8.1) The problem of finding the electromagnetic field for z > 0 was the subject of Chapter 5, so these methods will be used. 8.1 A Choice of Basis Functions An analysis could be done in a straightforward manner. An arbitrary value of a could be chosen to fix the basis functions <j>i(n,x) and (p2(K,,x)] and the coefficients q(«) could be found by taking inner products of these functions with M(x). The matrix spectral function would then be constructed and the field at z — I found from the eigenfunction expansion solution (5.43). The amount of light coupled into the waveguide in z > I is found by taking the inner product of the field at z = / with the waveguide mode. The above analysis is possible but rather messy. There is another approximation which greatly simplifies the situation. The index of refraction of air is much lower than the GaAs material shown in Figure 8.1. For all transverse wavenumbers such that E < k] - kl (8.2) the field solution in x < — xg is a decaying exponential. Any solution to the wave equation that is linearly independent from this involves an increasing exponential, and is therefore not part of the spectrum because it violates the condition of boundedness (5.66). Therefore the spectrum in the range (8.2) is at most one dimensional, and the situation is similar to the case examined in Section 5.5. In Section 5.5 it was demonstrated that, provided spectral contributions outside the range (8.2) are ignored, the problem can be reduced to an eigenfunction expansion involving only a single eigenfunction and a Chapter 8. A Bragg Modulator 104 scalar-valued spectral function (5.86)-(5.87), as in Chapter 4. This simplification occurs naturally provided the right choice of basis functions is used. It is easy to guess what this eigenfunction must be: it is the solution that decays exponentially in the region x < — xa. As pointed out in Section 5.5, the approximation is valid provided there is not a significant amount of energy in the spectral components outside the range where the spectrum is one dimensional. In this problem this range is 0 < E < k] - k2a (8.3) The incident waveguide mode E0 is significantly far from the edges of this range and even when the spectral components in the modulated section are calculated, most of the energy still lies near E0 (see Figure 8.2). Hence even a smaller range than (8.3) can be used with little error. Following Section 5.5, assume a new basis 0 defined 0i(ie,O) = sini? .2(/s, 0) = cos i9 0((/c,O) = -costf 0_(K,O) = sintf (8.4) similar to the definition (3.21). Unlike the equations (3.21) however, the angle i? may be a function of E. Let 6x(K,X) be the bounded solution. As (5.86) and (5.87) show, 82(tt,x) need not be calculated since it will not enter into the eigenfunction expansion. 8I(K,X) in the range — x9 < x < 0 is . . COS I? . • 6 4 ( K , X ) = smu cos K . F F _ s i n K j i (8-5) K9 where = A2-/32 (8-6) and in the range x < —x9, Qi must be 61(K,X) = Cej^x (8.7) Chapter 8. A Bragg Modulator q(K)fe A n g / n g - 0.05 A n 3 / n 9 = 0.10 An9/ng = 0.15(%) 0.8374 0.8375 0.8376 0.8377 0.8378 0.8379 0.838C RELATIVE TRANSVERSE WAVENUMBER «/*, Waveguide parameters the same as in Figure.7.3. Figure 8.2: Magnitudes of the Spectral Components Chapter 8. A Bragg Modulator 106 where jna = sjp - kl (8.8) The continuit}' of &i and Qx' at x = — xg results in the condition K„ sin i? sin n a x a — cos i? cos Kaxa .„ J*a = • q £ T L - ^ g • (8-9) Sin V COS K g X g + rtg 1 COS 17 S i l l KgXg and i? can be found in terms of E, (or K, Ka, and rcff): tan* = - «*™«**9+3*a*inKg*. ^ tZg Kg S 1 H ttgXg J' K,a COS KgXg 6I(K,X) in the region x > 0 can be found from the scattering solution $ of Chapter 7. This solution is given since the the coefficients g(n, n') of equation (7.4) have been found in equation (7.28). For simphcity, suppose the input (incident) amplitude CL(K) is chosen so that the coefficient of (E — E') is cos a, then #(*,*) = cos a j^—rrjME') (8.11) and since (p2(rt,x) is given by (6.74) $(rc,x) = U(K,X) + / K(x,t)u(hi,t)dt (8.12) Jo where /oo COS Av'x jdp(£') (8.13) - OO rv /*C and since p(E) is given from (6.77) c 0 cos 2 a cos n0x 2 f 0 0 COSK'X , "<"•"> = — ^ 5 — + ; / „ ^ a r * " ' (»•") The integral in the second term above is evaluated as in Section 7.4, by writing the cosine as two complex exponentials and closing the contour in either the upper or lower half planes. The result is . , C i COS KQX j • U{K,X) = — 2 °- - J-e->™ (8.15) Chapter 8. A Bragg Modulator 107 the substitution (7.35) has also been used. The kernel K(x,t) in (8.12) is given by (6.79), and (8.15) in (8.12) gives *(«,*) = -^ -*»+. ClrCa°ST , - 2 A - 2 + l re-'" cos nosds K 1 + C\ J 0 COS-2 KQS ds KQ — KZ K JO (8.16) As expected, in the region far away from the waveguide, x —> o o , the scattering solution $ reduces to a plane wave travelling away from the waveguide. The form of is e~ J K X, it is the transmitted portion of the incident wave in the scattering experiment of Section 7.1. $ i s similar to the Jost solution /(/c,_) of quantum scattering theory [46], in fact it is easy to show f(K,x) = JK$(K,X) (8.17) This is interesting because in the example the potential V(x) is not in the Faddeev class L\ as described in reference [25, page 324]. Equation (8.17) is therefore a counterexample to show that the condition V(x) £ L\, while being a sufficient condition for the existence of a Jost function, is not a necessary condition. Some useful information can be obtained by borrowing a few tricks from quantum scattering theory. Consider the function $( — «,,_). This function is also a solution to the wave equation (2.38) because only the square of the wavenumber K appears in the equation, so $(K ,_ ) and $( — K,X) are solutions to the same equation. Therefore the Wronskian determinant W[$(K,X), $( — «,_)] is independent of x and it can be evaluated at any convenient point on the x axis. It is most convenient to evaluate W in the limit x —> oo. The result is W[$(K,X)^(-K,X)} = ^ (8.18) K Therefore $(«;,_) and <_( —K , _ ) are linearly independent. The function 6J(K,X) is another solution of (2.38). Any solution of a second order equation can be represented as a linear combination of two linearly independent solutions. Chapter 8. A Bragg Modulator 108 Since $(«,,_) and $( — K,X) are known explicitly (8.16) then an explicit form for 6J(K,X) is available from them. Let 6I(K,X) = a 1 (K)$(K, x) + a2(«)$(—K,X) (8.19) and Wm^x)^^^)} = CI2(K)— (8.20) W[^(K , X ) , $ ( - / C , X)] = ax(K)— (8.21) To calculate the left sides of (8.20) and (8.21), it is convenient to use the point x = 0, since Q\ and 6[ are given by (8.4) there. The result is fll(K) = ~ [$ ' ( -« , 0) sin + $(-ic,0) COST?] (8.22) O 2 ( K ) = - A [ * ' ( K , 0 ) s i n . » + $(ic,0)cosi?] (8.23) and from (8.16) $ ( K ' 0 ) = " ^ + ^ i ^ ( 8 2 4 ) % 0 ) = - l - - ^ L + ^ (8.25) Equations (8.10) and (8.22) to (8.25) are used in (8.19) to obtain #a in the region x > 0. 8.2 Spectral Function for the Problem In the next step, the spectral matrix is calculated. As demonstrated in Chapter 5 this ma-trix can be found from m+ and m_. These quantities are related to the input impedance of the upper and lower half spaces respectively, and so m + and m_ can be found from the impedances. In this case, however, there is a more efficient way to calculate m. In Chapter 4 it was mentioned that for any non-real value of E in the wave equation (2.38) there is only one solution in £ 2 [0 ,co) , and it is represented by (4.2). Although Chapter 8. A Bragg Modulator 109 neither of the basis functions 8X or 82 is square integrable, m times 82 and added to 61 does yield a square integrable function. Since m depends on the choice of basis functions, in this chapter (4.2) becomes <p(n,x) = 01(K,x)+m±(E)82(K,x) (8.26) If ip e £ 2 [0 ,oo) then (8.26) defines m+(E), and if <p € £ 2 ( - o o , 0 ] then (8.26) defines m_(E). From the asymptotic value of $(« , a;) in (8.16), it is easy to see that for 5(E) < 0, $(K,X) is in £ 2 [0 ,oo) , and therefore tp(n,x) in (8.26) must be some multiple of $ ( K , X ) . Let this constant of proportionality be h(n): <p{K,x) = }I{K)§(K,X) = 81{K,x) + m+(E)82{K,x) (8.27) N ow h(K)W[6u$] = W[6u<p] = m+W[euB2] = m+ (8.28) h{K)W[62,$] = W[82,p] = .W[fl 2 > ^] = - 1 (8.29) and so W[fli,'$] _ sini9$'(*>°) + C Q S T 9 $ ( K , Q )  m + ~ ~W[82,§] ~ ~ C O S I ? $ ' ( K , 0 ) - s i n i ? * (K , 0 ) ^ ' For m - , recall that ^ ( K , sc) was defined to be the solution of the wave equation with an exponentially decreasing value as x —> — oo. Therefore 8X is already in £ 2 (—oo,0], and therefore m.(E) = 0 (8.31) This result emphasizes the departure from the usual m(E) theory made in equation (8.4). Allowing the angle $ to be a function of E in (8.4) permits (8.31) for the correct choice of i ? ( £ ) . To construct the matrix spectral function equations (5.38) are used. Since m_ = 0, (5.38) reduces to the special form m. Chapter 8. A Bragg Modulator 110 M12 = M2i = - \ M22 = 0 ( 8.32) Only the imaginary part of M contributes to the matrix spectral function. Hence the only non-zero term in the matrix is the 1 — 1 component, given from (5.41) * 2 l _ > _ ! bm (8.33) dE 7r _(_:)-—0 m + 8.3 Expansion Coefficient The approximate value of the electric field in the half space z > 0 will be given by (5.87) once the expansion coefficient q(n) is calculated. From (5.86) the expansion coefficient is the inner product of the magnetic current at z = 0 with the basis function Q\. Consistent with the Kirchhoff approximation, the electric field on the plane z = 0 is assumed to be the same as the fully developed waveguide mode in z < 0. The magnetic current, defined Ms = E x n (8.34) is taken as a source term for the field in z > 0, and M s is used in the integral (5.86). Since E is parallel to y, then M g is parallel to x and only the x component need be considered. M(x) therefore (with the x subscript dropped) is equal to the transverse variation of the electric field in the region 2 < 0 : M(x) = <p2(hi0,x) for x > 0 (8.35) and M(x) = cos a cos ng0x -f smKgox (8.36) Kgo for — xg < x < 0, where " P O = y/kl-Pl (8.37) Chapter 8. A Bragg Modulator 111 Finally M ( x ) = (cosa cosKg0x - — s i n ^ J eJ«.«(*+*.) (8.38) V KgO ' J for _ < — xg, where J K a O = - * _ (8-39) The forms of M(x ) and ^ (K^X) are different in the upper and lower half planes, so the integral (5.86) will be split into two halves [OO g+(#s) = / M(x)61(K,x)dx (8.40) Jo ,o q_(n)= / M(x)61(Kix)dx (8.41) J — oo and q(n) = q+(K) + q_(K) (8.42). Consider From (8.35) and (8.19) [OO l-OO q+(n) = aj(/t) y <£2(KO, £ )$(•«, + _ 2 (K) J ^ 2 ( K 0 , x )<_(-«, x) _x (8.43) going back to the definition o f ( 8 . 1 1 ) , and interchanging the order of integration I = / 4>2(K0, X )$(K, X) _X Jo = cosa / — / <P2(K0,X)4>2(K ,x) dx dp(E') J oo K — K J 0 J —OO rf — ri and from the orthogonality relation (7.15) cos a . • 1 = ( 8 ' 4 5 ) rv n rv Using (8.45) in (8.43): cos a q+{K) = [aj(/t) + a2(/c)]-/CQ /C C O S G ! [ C O S T ? - C L sini9]-2 (8.46) ttr\ rx, Chapter 8. A Bragg Modulator 112 The evaluation of g_ is straightforward but cumbersome. (8.5) and (8.7) are used for 0i and (8.36) and (8.38) for M(x) to obtain sin(KFF0 - Kg)xg sm(ng0 + Kg)xg 9_(/s) + cos a sin d sin a sin i9 cos a cos i? 2(K.G0 — Ug) 2(/tp 0 + K S ) C O s ( / C p 0 - Kg)xg - 1 C O s ( / t p 0 + Kg)Xg ~ 1 2(K.s0 — Kg) + 2(/C f f 0 + Kg) 3 COs(/Cp — Kgo)xg — 1 C O s ( « s + Kg0)xg sin a C O S T ? 2 (K G -K s o ) 2(Kso + Kg) sin(/tso - K g ) _ p sin(KFF0 + K g ) x g 2(KSO — Kg) 2(KgO + Kg) (8.47) sm a . cos a cos Kg0Xg sm KgoXg KgO ' . o C O S 1? smv cos KgXg H sn\Knx, 9 9) i (K A 0 + K 0 ; The product of q(n) with the rate of change of the spectral function (8.33) is shown plotted for the example of Figure 8.1 in Figure 8.2. It is assumed that a voltage is induced across the waveguide section, causing a perturbation in the refractive index there because of the electro-optic effect. As expected, the greater the voltage, the greater the coupling to the modes adjacent to K0- Note that even for the largest perturbations considered, almost all the energy is confined to a narrow band around K0. This justifies the truncation of the integral in the expansion (5.87). 8 . 4 Coupling Coefficient After the radiation has travelled length / beyond the plane z = 0, a significant amount of energy may have shifted to the other wavenumbers. To find the amplitude of the wave coupled into the waveguide at z = I, (i.e. the transmission coefficient), an approximation similar to the one in the previous section is used. It is assumed that the reflection from the surface z — I is negligible, and the electric field there is given from an expansion like Chapter 8. A Bragg Modulator 113 the one in (5.87): Ey{x,l) = f q(K)e,{K,x)e-jpi^-dE (8.48) JA dE This function is equated to a magnetic surface current on the surface z = I, which is used as the source term for finding the field for z > I. In the region z > I, there will be a new basis function and expansion coefficient q(ht). The amplitude of the guided mode in z > I is given by the inner product of the magnetic surface current with the waveguide mode (/>2(KO, X): f°° q(n0) = / Ey(x,l)(p2{K0,x)dx (8.49) J — oo = f q(K)e-j01 f°° 6l(K,x)(P2(Ko,x)dx^§dE (8.50) JA J-OO dE and from (8.42) the integral in the center is the original expansion coefficient q(n), so q(no) = J^q\K)e-M*£j±dE (8.51) This amplitude is compared with the amplitude of the incident wave g(/t0): /oo (pl(ri0,x)dx (8.52) -oo to obtain the coupling (or transmission) coefficient, defined q{K0) The coupling coefficient indicates the depth of modulation of the modulator. Figure 8.3 shows the coupling coefficient of the structure of Figure 8.1 as a function of the perturbation of the index of refraction, for a few choices of the length I. Chapter 8. A Bragg Modulator COUPLING COEFFICIENT length I = 5 m m 10mm 15 m m 2 0 m m 0.8 0.6 0.4 H 0.2 i 1 1 1 1 1 r 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 RELATIVE CHANGE IN INDEX (%) Waveguide parameters the same as in Figure 7.3. Figure 8.3: Coupling Coefficient cp vs. Change in Refractive Index Chapter 9 Conclusions A standard problem in integrated optics has been introduced, together with the standard method of solution. The problem consists of finding the electromagnetic field in the half space z > 0 which contains a layered dielectric structure, given the electric or magnetic field is known on the plane surface 2 = 0. This problem has been solved by other authors by using generalized eigenfunction expansions. They solve the problem without dwelling on the theory of eigenfunction expansions in great detail, their objective being simply to solve the above problem. However in this thesis it is shown there is some advantage to considering the theory in greater depth. The first advantage the theory affords is to estabhsh the generality of the eigen-function expansion solution. Some assurance is sought that the expansion solutions (3.16),(3.19),(4.24), or (5.43) can represent any possible solution to the above standard problem. It was noted that the uniqueness theorem (Appendix A) shows the solution is unique once the electric or magnetic field is specified on the plane 2 = 0 and so the question of generality reduces to the question of the completeness of the eigenfunction expansion on this plane. Standard arguments from the literature suffice to prove the completeness, and this establishes the generality of the eigenfunction expansion solution. The second advantage is in obtaining new exact solutions to novel waveguides. It can be shown that instead of starting with a given system and finding the spectral function, the direction of the calculation can be reversed: from a given spectral function one can find the corresponding system. This is called an inverse problem. The formulation of 115 Chapter 9. Conclusions 116 the inverse problem is quite different from the original forward problem. Hence when the inverse problem admits exact solutions it generally does not reproduce one of the already known exact solutions but is more likely to belong to a new class. Hence solutions to the inverse problem lead to new exact solutions of the forward problem. Third, the connection between a dielectric structure and its propagation characteris-tics is often quite obscure. For example, given a dielectric structure it is sometimes not easy to tell even if guided modes exist let alone what their propagation constants may be. However the connection between a planar dielectric structure and its spectral function is well established, and in some cases the connection between the spectral function and the characteristics of propagation is clear. For example guided modes exist only where there are discontinuities in the spectral function, and conversely a discontinuity will lead to a guided mode, provided the corresponding longitudinal wavenumber is real. This is only one example of the connection between the spectral function and the characteristics of the propagation. It is hoped others will follow. To the extent this connection is made it can be said that introducing the spectral function is a worthwhile step in establishing the connection between a dielectric structure and the general characteristics of propagation on it. For planar dielectric waveguides the possible modes can be classified into T E and TM modes. It is found that although the basic differential equation in each case can be reduced to the same form (2.38), the potential function V(x) in this equation will be different for the TE and TM cases. Therefore the spectral function for TE modes is different from the spectral function for TM modes. As yet, no direct connection between these spectral functions is known and the parallel and perpendicular polarizations must be handled as separate problems. Consider the condition listed above: a discontinuity in the spectral function leads to a guided mode provided the corresponding longitudinal wavenumber is real. This Chapter 9. Conclusions 117 is interesting because it suggests that the inverse approach must be able to generate examples where the waveguiding is due to Bragg reflection as well as examples of the more usual waveguiding by total internal reflection. If a discontinuity is placed at a point in the spectrum where both longitudinal and transverse wavenumbers are real, then if there is waveguiding it must be due to Bragg reflection. An example was constructed and the corresponding dielectric did have periodic variations with wavenumber 2K0 as would be expected of a system based on Bragg reflection. In Chapter 7 a waveguide was designed based on the Bragg properties of the upper half space. This example illustrates the relationship between the solution in the half space, which is represented by a spectral function, and the solution over the whole space (two half spaces) which is represented by a spectral matrix. Because of the completeness of the representation, it is possible to prove that the demonstrated mode is the only guided mode possible for this polarization. It is also shown that the guide can be made arbitrarily wide and still support only one mode. This is contrary to the behaviour of most other waveguides, in which increasing the transverse dimensions will increase the number of guided modes. In Chapter 7, the spectral function was used to generate an example of a single mode waveguide. In Chapter 8 this new waveguide is examined in the spirit of the fourth paragraph above: what can one guess about the propagation characteristics of the structure by looking at its spectral function? This waveguide differs from others in that its guided mode is embedded in the continuous part of the spectrum. There are many radiation modes nearby, so one might expect that the guided energy may be easily coupled to radiation by small perturbations in the waveguide geometry. Such a structure might find application as an electro-optic modulator. In Chapter 8 a detailed calculation is performed to confirm the above hypothesis. To make the calculation easier an approximation is introduced. In the case where the Chapter 9. Conclusions 118 index of refraction of one half space is much lower than the index of refraction of the other, a convenient approximation has been developed in which the guided modes can be treated as surface waves over a reactive surface. The reactance of the surface may be a function of wavenumber. The nature of the approximation is that it ignores the radiation modes escaping into the lower index medium. If the radiation can be expected to exist mostly on the side with higher index of refraction, the approximation is valid. This calculation is performed on a slightly perturbed waveguide. It is found that even for very small perturbations (0.15%) a significant amount of energy can be transferred out of the guided mode over a moderate propagation length (1.0cm). This confirms the hypothesis. The future of this approach lies mainly in strengthening the connection between the spectral function and the characteristics of the propagation. It is the author's belief that many more connections remain, and that the full significance of the spectral function in this application is not yet known. References [1] T. Okoshi, Optical Fibers (Academic Press, New York, 1982) [2] V . V . Shevchenko, Continuous Transitions in Open Waveguides (Golem, Boulder Colo., 1971) [3] D. Marcuse, "Radiation Losses of Tapered Dielectric Waveguides," Bell Syst. Tech. J. 49, 273-290 (1970) [4] D. Marcuse, "Mode Conversion Caused by Surface Imperfections of a Dielectric Slab Waveguide," Bell Syst. Tech. J. 48, 3187-3216 (1969) [5] D. Marcuse, "Radiation Losses of Dielectric Waveguides in Terms of the Power Spectrum of the Wall Distortion Function," Bell Syst. Tech. J. 48, 3233-3242 (1969) [6] D. Marcuse and R. M . Derosier, "Mode Conversion Caused by Diameter Changes of a Round Dielectric Waveguide," BeU Syst. Tech. J. 48, 3217-3232 (1969) [7] P. J. B. Carricoats and A. B. 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Fikouris, and N . K. Uzunoglu, "Scattering from an abruptly terminated dielectric-slab waveguide," J . Lightwave Technol. 3, 408-415 (1985) [15] H. Shigesawa and M . Tsuji, " Mode propagation through a step discontinuity in dielectric planar waveguide," IEEE Trans. Microwave Th. Tech. 34, 205-212 (1986) [16] S.-J. Chung and C. H. Chen, " A Partial Variational Approach for Arbitrary Discon-tinuities in Planar Dielectric Waveguides," IEEE Trans. Microwave Th. and Tech. 37, 208-214 (1989) [17] K. Okamoto and T. Okoshi, "Computer Aided Synthesis of the Optimum Refractive-Index Profile for a Multimode Fiber," IEEE Trans. Microwave Th. and Tech. 25, 213-221 (1977) [18] Y . F. Li and J . W. Y . Lit, "Effective thickness, group velocity, power flow, and stored energy in a multilayer dielectric planar waveguide," J . Opt. Soc. Am. A 7, 617-634 (1990) References 121 [19] P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976) [20] V. I. 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SSSR 104, 695-698 (1955) [Math. Rev. 17, 740 (1956)] [27] I. Kay and H. Moses, Inverse Scattering Papers: 1955-1963 (Math Sci Press, Brook-line, Massachusetts, 1982) References 122 [28] A.K. Jordan and S. Lakshmanasamy, "Inverse scattering theory applied to the design of single mode planar optical waveguides," J. Opt. Soc. Am. A 6, 1206-1212 (1989) [29] S. Yukon and B. Bendow, "Design of waveguides with prescribed propagation con-stants," J. Opt. Soc. Am. 70, 172-179 (1980) [30] S.R.A. Dods, "Bragg reflection waveguide," J. Opt. Soc. Am. A 6, 1465-1476 (1989) [31] I. Kay, "The Inverse Scattering Problem When the Reflection Coefficient is a Ra-tional Function," Comm. on Pure and Appl. Math. 13, 371-393 (1960) [32] A. Y. Cho, A. Yariv, and P. Yeh, "Observation of confined propagation in Bragg waveguides," Appl. Phys. Lett. 30, 471-472 (1977) [33] J. Salzman and G. Lenz, "The Bragg Reflection Waveguide Directional Coupler," IEEE Photonics Tech. Lett. 1, 319-322 (1989) [34] R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962) [35] A.W. Naylor and G.R. Sell, Linear Operator Theory (Holt, Rinehart and Winston, New York, 1971) [36] E. C. Titchmarsh, Eigenfunction Expansions, Part I (Oxford University Press, Ox-ford, 1962) [37] E. Hille, Lectures on Ordinary Differential Equations (Addison-Wesley, Reading, Mass., 1969) [38] I. Stakgold, Boundary Value Problems of Mathematical Physics, Vol. I (MacMillan, New York, 1967) References 123 [39] E.A. Coddington and N . Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955) [40] E. Hille, "Green's transforms and singular boundary value problems," J . Math, pures et appl. 42, 332-349 (1963) [41] T. Tamir and A. A. Oliner, "Guided complex waves," Proc. I.E.E. 110, 310-324 (1963) [42] A. Ya. Povzner, "On differential equations of Sturm-Liouville type on a half-axis," Mat. Sb. N.S. 23, 3-52 (1948) [Am. Math. Soc. Transl. no. 5 (1950)] [43] F .G. Tricomi, Integral Equations, (Interscience, New York, 1957) [44] A . K . Jordan and S. Ahn, "Inverse scattering theory and profile reconstruction," Proc. IEE 126, 945-950 (1979) [45] R.E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960) [46] R. Jost, "Uber die falschen Nullstellen der Eigenwerte der S-Matrix", Helv. Phys. Acta 20, 256-266 (1947) [47] R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961) Appendix A The Uniqueness Theorem of Electromagnetics In this appendix sufficient conditions for the uniqueness of the solution of Maxwell's equations are shown. It is shown the solution within a closed volume is unique once the tangential components of the electric or magnetic fields are specified on the bounding surface of the volume. Maxwell's equations for a sourceless isotropic medium are V x E = -jufiH. (A.l) V x H = juieE (A.2) where e and fi are scalar functions of position. If the medium has losses then one or the other (or both) of e and fi must be complex e = ea - je2 (A.3) fi = fi0- jn2 (A.4) Consider the inner product of E with the complex conjugate of (A.2) minus the inner product of H " with (A.l): E - V x r T - J T - V x E = jw/iH" • H - jue'E • E* (A.5) Application of a vector identity leads to the appearance of the Poynting vector on the left hand side V • (E x H") = ju>e*\E\2 — ju!fi\H\2 (A.6) 124 Appendix A. The Uniqueness Theorem of Electromagnetics 125 The proof of the uniqueness is by contradiction. Following Harrington [47] two dif-ferent solutions to the equations are assumed, (E a ,H Q ) and (E^Hj,). From these two solutions a difference is formed SE = E Q - E 6 SH = H a - H f c (A.7) Because of the linearity of the equations, equation (A.6) applies to the differences as well as the original fields V • (SE x SB.') = ju>e'\SE\2 - jujp\SH\2 (A.8) Suppose the fields are defined in a volume V which has the bounding surface S. Integrating (A.8) over the volume and applying the divergence theorem leads to fs(8E x SH") • ri dS = ju J (e"\SE\2 - p\SH\2) dV (A.9) The real part of (A.9) is 3R f{SExm~)-hdS = -oj (e2\SE\2 + p2\SH\2) dV (A.IO) Both e2 and p2 are positive and so the right hand side is either zero or negative. If the left hand side is zero, then £E and £H are zero everywhere and the solution is unique. From equation (A.IO) a few sufficient conditions for uniqueness can be established by showing that under these conditions the value of the surface integral on the left hand side is zero. It is helpful to resolve the fields into a component normal to the bounding surface of V and a component parallel to it £E = SE± + SEl{ (A.11) SH = SK± + SH{, Appendix A. The Uniqueness Theorem of Electromagnetics 126 It is easily shown (8E x SH") • n = £E|| x £H[j • n (A.12) and this leads to three possible sufficient conditions 1. £E|j is zero everywhere on the bounding surface 5. In other words if the tangential component of the electric field is given over the entire bounding surface S, then only one solution to the Maxwell equations exists. 2. £H|| is zero everywhere on the bounding surface S. 3. <£E|| is zero on some part of S, and £Hy is zero on other parts, and the union of these regions covers the entire surface S • There is a fourth sufficient condition which applies to the closed waveguide of Chapter 3 and the waveguide open on one side in Chapter 4. Both these problems have reactive surfaces which are planes perpendicular to x. The volume V may now be taken as the volume bounded by the surface (or surfaces) of constant x and the 2 = 0 plane. Neither E nor H is specified on the x =constant surfaces but the components of the tangential parts E|| and Hy are related by ZTE = ± ^ ZTM = ±%- (A.13) The signs depend on whether the upper or lower surface is considered. In practice reactive surfaces are sometimes made from corrugated conductors having sharp corners. This can lead to field singularities at the corners. However from physical considerations, the corners cannot act as sources and so the volume integral on the right hand side of (A.9) must remain finite. This puts limits on the growth of E and H near the corners and so the assumptions of the uniqueness theorem are not violated even in this case [45]. Appendix A. The Uniqueness Theorem of Electromagnetics 127 As before equatiqns (A.13) apply to the differences as well as to the original fields SEy = ZTESHZ (A. 14) 8EZ = ZTM6Hy (A. 15) The surface integral on the right hand side of (A.IO) must now include the integral over the x =constant surfaces, such as » { / (SE x _ _ _ " ) • ids} and with the help of (A.14) and (A.15) this simplifies to »{/ (ZTE\HZ\2 + ZTM\Hy\2)ds} but ZTE a n d ZTM are by definition pure imaginary, so the contribution to the integral on the left hand side of (A.IO) is zero. Therefore in the third sufficient condition above, the surface can also be closed with areas having a. pure imaginary surface reactance. Appendix B Zeros of the Transverse Resonance Equation In this appendix zeros are sought for the equation ZT{K) = 2+{K) + Z.(K) (B.l) where (3 > ks. First multiply the expression for Z- by cosine in the numerator and denominator to remove the poles from the denominator: nr ( v • J(«o ~ *2) ~ ci*- . z * cos Kgxg 4 j sin Kgxg MK) = -JK9~2- T7 • w 2 2~^  + , - 7 • i B - 2 ) C ^ K 4 ( K — J C I ) ( K Q — K T J C O S KgXg 4 J Zs S i l l KgXg where ZS — Kg/Ka . In order for 2 T to be zero it is necessary that 0 '= t p ( « - o — K 2 ) cos KgXg 4- JKgCyK cos « sa; 5 + J K . g Z S ( « . Q — K 2 ) sin Kgxg -\-ZSC\K C O S KgXg 4" ZSK(K\ — K. 2) C O S K g X f l — J C 1 Z „ ( K . Q — / t 2 ) C O S KgXg -\-JC\K sin KgXg 4 JK(KI ~ K2) sin rcsxg 4- ci(«8 ~~ K 2 ) sin Kgxg (B.3) In the range f3 > ks the wavenumbers are all negative imaginary quantities: « = -31 K 9 = ~hg «a = ~ J 7 a ( B -4 ) where 7a - - fc2 128 Appendix B. Zeros of the Transverse Resonance Equation 129 If these are substituted into (B.3) 0 = -31g(Ko + 7 2 ) cosh lgxg - J 7 s 7 C i cosh7 P _ g - j^gZS{K20 A 7 2 ) sinh l g x g -jZsc\i cosh fgxg - JZSJ(KI + 7 2 ) c o s l i 7 P _ g - JC1Zi,(KI + 7 s ) cosh ^gxg -jc\-r sinh 7 g_ f f - J7(KQ + 7 2 ) sinh -ygxg - j c^/c 2 + 7 2 ) sinh igxg (B .6) A l l the above terms are of the same sign, and some terms are never zero. Therefore there is no solution to (B .6) in the range (S > k„, and therefore no guided modes in this range. Appendix C The Completeness of the Expansion To prove that the series (3.17) will always be successful for any choice of e(_) and M(x) £ C2[0,6], it is first necessary to introduce the Green's function of the problem and prove two theorems from the theory of linear operators: Bessel's inequahtj'- and the extremal property of Green's operator. The operator L as defined in (2.41) is not very useful for the purpose of proving the completeness theorem because it is not a bounded operator. Since it is not bounded it is not necessarily continuous or compact and these properties are needed in the proof. The proof of the completeness theorem shown in this appendix is adapted from Coddington and Levinson [39]. C . l Green's Operator The series (3.14) results from the solution of the eigenvalue problem together with the boundary conditions (3.4), and (3.5). It is well known an inverse operator to L, called Green's operator, Q, can be defined Lf = Ef (C. l ) LQ - QL = I (C.2) Green's operator is written (C.3) 130 Appendix C. The Completeness of the Expansion 131 and G(x,£) is the Green's function. Definitions (C.3) and (C.2) together imply that G should be the solution of the equation G"+ V(x)G(x,£) = S(x-£) (CA) with the boundary conditions (3.4) and (3.5). This solution is readily constructed from two linearly independent solutions of Lg = 0: Lgi = Lg2 0 (C.5) G(*,0 = (C.6) g[(0) + h9l(0) = 0 g'2(b) + Hg2(b) = 0 The solution of (C.4) is 9i{x)g2(()IW(gug2) x<£ . 9i(052(x)/W(gi,g2) x > £ where W is the Wronskian determinant. This function is obviously continuous in x and £ . This implies the operator Q is compact [35]. G will be well defined provided the Wronskian between the solutions gi and g2 is not zero. If the Wronskian is zero, this implies gx and g2 are linearly dependent and hence each satisfies both boundary conditions. As such they are eigenfunctions with eigenvalue zero. Hence the Green's function is defined provided the operator L does not have zero as an eigenvalue. L could have zero as an eigenvalue, but this would only be as a result of an inappropriate choice of e, in defining L. If a given L had zero as an eigenvalue the above problem could be resolved by choosing a different ee, since this choice was arbitrary anyway. Applying Green's operator to the eigenvalue problem (C.l) results in / - EGf (C.7) Appendix C. The Completeness of the Expansion 132 This can be rewritten Qf = Pf (C8) where V = | (C9) Hence the old eigenvalue problem can be written in terms of Green's operator, which has the form of an integral equation instead of a differential equation. The eigenvalues of the Green formulation (C.8) pn, are the reciprocals of the original differential form, En. C.2 The Bessel Inequality The C2 norm is defined as ' b 1 1/'2 ll/H = / f(x)W)dx (C.IO) and the inner product as if,9) = / f{x)gjx)dx (C.ll) Jo Note that the definition of the norm and inner product are such that 11/11 = [(/,/)]1/2 ' We can let the inner product be so defined in both the domain and the range of the operators L and Q. With this definition, L and Q are operators from one Hilbert space to another. Bessel's inequality states that, on an inner product space m n/ir> E K / ^ ) ! 2 (c.i2) n = l where ipn(x) = ip(nn,x) (C.13) Appendix C. The Completeness of the Expansion 133 are vectors in the inner product space which are orthogonal in the sense of the inner product (C.ll). The inequality indicates that if a series is truncated at m, there may be something missing in the expansion (3.17). The inequality can be demonstrated in a straightforward way. The norm of a function is always non-negative, in particular 771 l l / - E ( / ^ n ) ^ | | 2 > 0 (C.14) 71 = 1 the right hand side is expanded using the inner product and the bilinear property of the inner product: o < ( / - E ( / . i ) i , / - E ( / . i W = (fJ) - E(/.V>n)'(^n,/) - E(7^0 ( / .^)+El( />n) | 2 = ll/ll2-£!</> ^ >l2 (C15) and Bessel's inequality follows. C.3 The Extremal Property By this point it is known that Q is a compact self-adjoint operator mapping one Hilbert space to another. Under these conditions, it is known that Q will have at least one eigenvalue, given by the norm of the operator. The norm of an operator is defined by IISII = sup H0/H (C.16) ll/ll=i In other words the norm of the operator is the maximum value of the norm of the image, with the functions in the domain normalized to have unit length. Under the above conditions it can be shown that at least one of the values ±||£?|| must be an eigenvalue of Q. The proof begins by considering the sequence of functions used in Appendix C. The Completeness of the Expansion 134 defining the supremum in (C.16) Urn ||_7_|| = \\G\\ (C.17) n—too because Q is compact the sequence Qfn must admit a convergent subsequence Gfnk-Suppose this subsequence is chosen as the original sequence and it converges to g(x): lim Qfn = g (C.18) n—>oo This alone does not imply the sequence fn converges, but if Q is self-adjoint fn will in fact converge. To show this convergence, consider the following inner product (G2fn ' \\Gfn\\2fn,G2fn ~ \\G/nil'/n) (C.19) a straightforward calculation shows that, for self-adjoint G, the above equals \\G2fn\\2-\\Gfn\\" (C.20) but this is less than or equal to .HSirilS/nll2 - US/nil4 = (ll^ ll2 - ||£/n||2) \\Gfn\\2 (C.21) The quantity in brackets tends to zero as n goes to infinity, and so we can conclude G2fn-\\Gfn\\2fn^ 0 (C.22) Si nice lim G2fn = Gg (C.23) and Then (C.22) implies lim \\Gfn\\ = \\G\\ (C.24) Gg = \\G\\2 Hm fn (C.25) Appendix C. The Completeness of the Expansion 135 which indicates fn itself tends to a hmit, since it is defined b3r (C.25): Pass to the limit n —> oo in (C.22) and factor the operators to obtain (G+ \\G\\I)(G-\\G\\I)f = 0 (C.27) Hence either ||_?|| is an eigenvalue with eigenfunction / , or —\\G\\ is an eigenvalue with eigenfunction (G-\\G\\l)f A corollary to the above result is that ||_?|| must be the largest eigenvalue (in the sense of absolute value). Suppose there were another eigenfunction of G called / 0 with an eigenvalue p0 whose absolute value is greater than \\G\\- Then one could write W = W = l ? o 1 y 11511 ( c ' 2 8 ) but this is impossible since this contradicts the definition of \\G\\-CA Completeness theorem This theorem states that if f(x) _ C2[0,i] and satisfies the boundary conditions (3.4) and (3.5) then /(*) = E(/,i)i(») (C29) n = l and the series converges uniformly. From Section C.3 it is known that the largest eigenvalue is ||_<||, called pi and the corresponding eigenfunction, called ip1} is the solution of the integral equation fb io [bG(x,t)MOdti = PlM*) (C30) Jo Appendix C. The Completeness of the Expansion 136 such that = 1 (C.31) (the solution has been normalized to have unit length). Consider the new operator Gi defined from the kernel dOc.O = G(x,t)-PlM*)MO (C32) Qi is also a compact self-adjoint operator. Hence the extremal property applies and Gi has an eigenfunction ip2 and an eigenvalue p2 satisfying W = HSill (C.33) SiV>2 = 7>2V>2 (C.34) Let u(x) be an arbitrary continuous function on the interval [0,6] and consider Q\U. This function will be orthogonal to (Siu,V>i) = {Gu - pai/>1(u,V'i),i/,i) = {u,Gfa) -PI{UM = 0 (C.35) The function ip2 is continuous over the interval and so by (C.33) and (C.35) 0 = (GupiM = PiifaAi) (C36) and so the eigenfunction of Gi, ip2, i s orthogonal to the eigenfunction of G, ipi-In fact ip2 is an eigenfunction of Q as well, since Gip2 = Gii>2 +p^\(i>i,fp2) = V2^2 (C.37) Appendix C. The Completeness of the Expansion 137 Also from the extremal property we must have |p2| < \Pl\ (C.38) since if the opposite were true this would contradict the definition of Pi: |Pi! = H0II CC.39) In a similar way a sequence of operators can be defined from m Gm(x,() = G(x,£)-Y,PnM*Wn(t) AO) n=l By repeating the argument from (C.32) to (C.38) for each m we find a sequence of eigenvalues for the original operator Q and from the extremal property ||_<m|| = |pm|-Consider m \\Gmu\\ = \\Qu - ^ Pn(u,1pn)tpn\\ < |Pm|||«|| (C.41) Tl = l \Pm\ is n ° t an increasing sequence, in fact the sum is known from the Bessel inequality. Consider the eigenvalues written in the Green's function form: [bG(x,£)rl>n(t)d£ = p„Vn(z) (C.42) Jo For a fixed x, the right hand side could be regarded as the Fourier coefficients of G viewed as a function of (. Applying Bessel's inequality leads to n = l J ° for any m. G is a continuous function on a bounded interval and is therefore itself bounded. Let m rb ZPI\M*)\2 < / \G(z,l)\ad( (C.43) J n 7 = sup (C.44) *,£e[o,6] If (C.43) is integrated with respect to x over [0,6] m E ? n < 7 2 * > ( C 4 5 ) Appendix C. The Completeness of the Expansion 138 Therefore we must have lim p n = 0 n—»oo this means the left hand side of (C.40) converges and m GU —r ]P pn(u,l])n)l}}n (C.46) (C.47) n=l as TTX —> oo, at least in the C2 sense. In fact the series converges uniformly, and this can be seen by examining the tail of the series: for q > p, n=p and for any continuous function / , \Gf\ < iVb H/ll Therefore .71—p 1/2 (C.48) (C.49) (C.50) If the right hand side of (C.50) is viewed as a sequence indexed by q, the Bessel inequality (C.12) shows it must be a Cauchy sequence, regardless of the value of x. Therefore we can write oo Gu = ^2pn(u,if>n)ipn (C.51) n=l where the series converges uniformly. To obtain the completeness theorem in the form (C.29), let u = Lf. Since by as-sumption it wras found that it was sufficient to let u £ C°[0,fc] then it will be sufficient to let / £ C2[0,fc]. Then Qu = GLf = f (C.52) (u,v>„> = (L/,vg = (f,Lipn) , = En(fM = - ( / , V n ) Pn (C.53) Appendix C. The Completeness of the Expansion 139 and (C.51) implies oo f{x) = £ ( / , i M i M x ) (C.54) Tl=l which is the desired form of the completeness theorem. Equation (C.54) can be regarded as a generalization of Fourier series. With Fourier series the basis functions i p n are solutions of the harmonic equation ip" + E i p = 0 (C.55) with the appropriate boundary conditions at each end. Equation (C.54) is the same expansion but based on the more general equation (2.38). As with Fourier series, the Parseval equality holds, since from (C.54) and the orthogonality of the eigenfunctions it follows that / \f{z)\*dx = E l ( / ^ n ) | 2 (C.56) *f 0 i 71 — 1 

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