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DATE DUE DATE DUE DATE DUE 2/05 p:/C|RC/DateDue.indd-p.1 Development of Periodic Green’s Function Using Analytic Signals By Jun Gao A DISSERTATION Submitted to Michigan State University towards fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Electrical and Computer Engineering 2006 ABSTRACT Development of Periodic Green’s Function Using Analytic Signals By Jun Gao Periodically arranged radiating or scattering structures have plenty of important applications, which range from frequency selective surfaces, antenna arrays to meta materials etc. While the response of the structures has been well studied in the frequency domain, there is a plenty of analytical challenges in the time domain. This dissertation develops a purely time domain approach using analytic signals to find the field radiated by a periodic dipole array in free space, in the presence of a half space and over layered media. The field is represented as a series of time domain F loquet modes by using Poisson summation formulae. In free space, each mode is expressed as a zeroth-order Bessel function. With minor modifications, the same modes can be derived by using either the Causality Trick or the Whittaker formulation, and much more physical insight can be gained accordingly. At the presence of a halfspace, the reflected field is represented as a finite integral using Causality Trick and can be evaluated numerically. It is also shown that it can be expressed in a integral-free form by using the Discrete Complex Image Method. As a typical layered media case, the periodic dipole array over a microstrip is studied, and each mode is represented by a geometrical series of transient fields. The series can be further simplified and the mode is represented by five semi-infinite integrals, which can be either evaluated numerically or represented in closed-form. The analytic signal formulation is proven to be effective and elegant means to solving for the radiation/scattering from the periodic dipole array purely in time domain. To My Dear Parents iii ACKNOWLEDGMENTS I’d like to take this opportunity to express my gratitude to all my friends and colleagues who helped me to complete this dissertation and who have played so im- portant roles in my life at Michigan State University. First I’d like to thank my advisor, Dr. Shanker Balasubramaniam, for guiding me into the world of electro- magnetics, for your open-minded conversations and enlightening ideas, and for your continuous support during my Ph.D. It’s always enjoyable and relaxing to talk to you, and I know you are always there to help. Thanks to Dr. Edward Rothwell for the valuable suggestions in group seminars and private conversations, and for offer- ing me the Latex templates. Thanks to Dr. Leo Kempel for many humorous and fearful criticisms in group meetings that make me confident in facing any challenge during presentation. Thanks to Dr. Guowei Wei for exposing me to a wider world of computational mathematics and for introducing me to students in mathematics. A thank you also goes to my teammates, Gregory Kobidze, Chuan Lu, He Huang, Jorge Villa and Pedro Barba, Chris Trampel and Jun Yuan, for the inspiring conver- sations and professional help, and for so many pleasant and hard-working days and nights we spent together. A heartily acknowledgement to my parents, my brother and my sister for your unconditional support. With your encouragement, I take pride on every progress I have made and will make in the future. A special thank you to my girl friend Xiaobin, for taking care of my daily life during these hard days, for providing me with professional help, and for staying with me when I was submerged in oblivion. iv TABLE OF CONTENTS LIST OF FIGURES ................................ KEY TO SYMBOLS AND ABBREVIATIONS ................. CHAPTER 1 Introduction ..................................... 1.1 Literature review ............................. 1.2 Overview of the research ......................... CHAPTER 2 Preliminaries .................................... 2.1 Periodic dipole array ........................... 2.1.1 Periodic Green’s function ..................... 2.2 Analytic signals and transient Weyl’s identity ............. 2.2.1 Analytic signals .......................... 2.2.2 Transient Weyl’s identity ..................... 2.3 Frequency domain Floquet modes .................... 2.4 Discrete Complex Image Method .................... CHAPTER 3 Radiation from a Periodic Dipole Array in Free Space .............. 3.1 Using Causality Trick ........................... 3.1.1 A single Floquet mode ...................... 3.1.2 Mode pairing ........................... 3.2 Using time domain Weyl’s identity ................... 3.3 Using Whittaker formulation ....................... 3.4 Results ................................... CHAPTER 4 Radiation from a Periodic Dipole Array in the Presence of a Halfspace 4.1 Analytic signal representation for a single dipole ............ 4.1.1 Analytic signal representation for the periodic dipole array . . 4.1.2 Analytic signal expression for the F loquet mode ........ 4.2 Closed-form expression based on DCIM ................. 4.2.1 Mode pattern for the array source with a complex time factor 4.2.2 Complex exponential curve fitting ................ 4.3 Results ................................... 10 12 14 18 25 25 28 31 35 38 41 53 53 56 58 62 65 70 72 CHAPTER 5 Radiation from a Periodic Dipole Array over Microstrip Structures ...... 80 5.1 Mixed-potential formulation for a single dipole ............. 80 5.2 Analytic signal representation as ray series ............... 83 5.3 Discussion ................................. 86 CHAPTER 6 Conclusions and Future Work ........................... 89 6.1 Future work ................................ 90 BIBLIOGRAPHY ................................. 97 vi Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 5.1 Figure 5.2 LIST OF FIGURES Problem setting in free space .................... Field radiated under different excitation settings in free space Deform the path in kz plane ..................... Deform the path in k,, plane ..................... Deform the path into the complex plane to capture the pole singu- larity .................................. Deform the path into the complex plane to capture the branch point singularity Mode pairing on the Ewald sphere ooooooooooooooooo Enlarged circular path on the Ewald sphere ............ Synthesis of a causal mode from two non-causal component modes Synthesis of the PGF from component F loquet modes ....... Synthesis of the PGF from component Floquet modes ....... The waveform and spectrum of the Rayleigh Pulse Validation of the field radiated by narrow-band array sources in free space ............................... Validation of the field radiated by wide-band array sources in free space .................................. Problem setting at the presence of a halfspace ............ Deformed path over half space .................... Complex exponential curve fitting .................. The field due to the source with a complex time factor ...... Deform the path in K. plane ..................... Calculated Floquet mode oooooooooooooooooooooo Problem setting over a microstrip structure Multiple reflection within a microstrip structure .......... vii 21 22 23 24 43 44 45 46 47 48 49 50 51 52 74 75 76 77 78 79 87 88 KEY TO SYMBOLS AND ABBREVIATIONS FDTD: F mite—Difference Time-Domain MOM: Method of Moments PGF: Periodic Green’s function CT: Causality Trick MPM: Matrix Pencil Method DCIM: Discrete Complex Image Method viii CHAPTER 1 INTRODUCTION 1 . 1 Literature review Many structures of electromagnetic interest possess a periodicity in one or more di- mensions. A typical example is an antenna array, which is formed by replicating an element antenna in two dimensions, and many of its parameters can be adjusted if the array is large enough. Another example is the frequency selective surface (FSS), which consists of one or more layers of materials that are periodically arranged, and it can be used to control the energy that penetrates a radome. Periodic structures are also used to form a electromagnetic bandgap (EBG), that can prohibit the electro- magnetic wave in specific frequency region from transmitting. Novel electromagnetic materials, the so-called meta-material, can be obtained by using periodic structures. Due to the special electromagnetic properties of periodic structures, plenty of research has been conducted in this area. The frequency domain response of these structures is well studied, and there is a wealth of papers and books on this subject; see Refs [1, 2, 3, 4, 5] and references therein. For transient analysis, both finite difference time domain (F DT D) and integral equation based techniques have been investigated. In A typical F DT D approach, the periodic structure is truncated into one single cell by using periodic boundary conditions, and this approach has been applied to solve for scattering from a variety of periodic structures [6]. The challenge has been overcoming the need to know future values. Some methods to overcome this have been suggested in [6] and [7]. In using an integral equation approach to analyze these structures, the Green’s function for the periodic structure, or the field radiated by the corresponding periodic dipole array, plays a crucial role. Time domain integral equation based methods have been recently developed for the analysis of periodic structures [8, 9]. In this dissertation, our goal is the development of time domain periodic Green’s function in terms of transient Floquet modes for layered media using a purely time domain approach; these modes have recently been derived in a series of papers by Felsen and Capolino [10, 11]. They have presented a systematic investigation of transient phenomena of radiation by dipoles arranged as an array (both linear and planar). Their analysis details two different methods that can be used for deriving time domain Floquet modes and relies on either a Fourier inversion of the frequency domain expressions for F loquet modes or spatial synthesis via a Poisson summation. While the first approach is relatively straightforward (in the sense that it is a Fourier inversion) the latter is very interesting in that this expression is derived entirely in the time domain starting from the retarded potential Green’s function. Both these method are not easily generalizable to the analysis of periodic structures in a layered environment. It can be shown that in a layered environment, the Fourier inversion of frequency domain F loquet modes can result in non-causal modes [12]. On the other hand, spatial synthesis is difficult as one needs to know the Green’s function a prion, and this makes it difficult to extend the formulation to layered media. An alternative formulation that we have espoused in this dissertation relies on using the transient Weyl’s identity. This leads naturally to a formulation using the Causality Trick (CT). The use of time domain Weyl’s identity to analyze fields radiated by time depen- dent sources is far from being rampant in electromagnetics. More often, as opposed to the Weyl identity, an analytic counterpart of the Whittaker integral is used [14]. This representation entails a radon transform of the source distribution onto four dimen- sional space-time spectral representation. Outside the source domain, this spectral representation lends itself to interpretation in terms of the propagating homogeneous and evanescent inhomogeneous time domain plane waves [14]. Roles of these waves in contributing to the causal fields have been explored as well. Space-time spectral representations involving Weyl’s identity have been used extensively in geophysics applications [15]. Indeed, [15] introduced the concept of the CT wherein one can use the Weyl’s identity with a combination of a source and its time symmetric counter- part to recreate the true fields for time t > 0. The beauty of this representation is that it involves only the propagating time dependent analytic plane waves; the con- tribution of the inhomogeneous plane waves vanishes. It was also shown in [15] that similar ideas can be extended to the analysis of layered media. Interestingly, a similar representation was explored as early as Whittaker [16]. His integral representation involved only a superposition of real time dependent plane waves. We also note that the Whittaker representation is the cornerstone of the plane wave time domain al— gorithm [17, 18, 19, 20, 21] that has been used extensively for accelerating the time domain integral equation solvers. 1.2 Overview of the research The purpose of this dissertation is to find the transient field radiated from a periodic dipole array in free space, over a half space and at the presence of layered media using analytic signal formulation. The radiated field is expressed as an infinite sum of transient Floquet modes. In contrast to the existing transient analysis methods, this dissertation is unique in that each Floquet mode is regarded as a superposition of analytic transient plane waves and are expressed in closed-form over half space and at the presence of layered media. This dissertation begins with the literature review and introduces the important work that has been done in this field. Background material is briefly discussed in Chapter 4.2.2, including the configuration of a periodic dipole array, the Floquet mode in frequency domain, and the analytic signal formulation. The main part of the dissertation begins with deriving the periodic Green’s func- tion in free space. Although this problem can be solved by conventional inverse Fourier transform or by direct spatial synthesis, these methods depend heavily on the elemental Green’s function and are hard to extend to more complicated physical set- tings such as layered media. Chapter 3 first schematically analyzes the propagation of the field and limits our interest to only the propagating case, which corresponds to the scenario when the excitation signal travels faster than speed of light. Based on the transient Weyls’s identity, the radiated field is represented as an infinite double summation of transient Floquet modes. Each Floquet mode is expressed as a infinite double integral, and is reduced to a one-dimensional integral by applying Poisson summation. Three methods are attempted to solve the integral and give the same result. The first method convert the integral into a contour integral and further more, a branch-cut integration. It’s proven then that the radiated Floquet mode behaves as a zeroth-order Bessel function. Thereafter, the Causality Trick method is applied. The contribution to the Floquet mode is categorized into homogeneous plane waves and inhomogeneous plane waves, and Causality 'Ifick shows that the contribution of the inhomogeneous plane waves can be equivalently represented by another set of ho— mogeneous plane waves, which add much symmetry to the problem. The Whittaker formulation is used at last. It can gives the same result but is only limited for the simultaneous excitation case. Numerical examples are given to demonstrate the high- pass property of the transient Floquet modes and indicates that transient Floquet mode is especially efficient to calculate field generated by band-limited sources. Chapter 4 adopts the mixed-potential formulation as the starting point, and the analytic signal version of the mixed-potential representation is derived. The procedure to derive the periodic F loquet mode based on the transient Floquet mode is very similar to that of the free space case, but the reflection coefficient should be taken care of with caution. The Causality Trick applied in free space suggests that the mode with opposite mode indices will partly cancel each other, therefore it’s more appropriate to find the sum of the two modes. The synthesized mode is expressed as a semi-infinite integral but there is no direct way to evaluate the integral since the mathematical form of the reflection coefficient is too complicated. Accordingly, the reflection coefficient is approximated by a finite sum of complex exponential functions, and each function contributes a complex time factor to the Floquet mode. As a comparison to the single dipole case where a complex image can be incorporated into the Green’s function readily, the complex image cannot be inserted directly into the free space F loquet mode, which is a zeroth-order Bessel function, as it’s well known that the zeroth-order Bessel function blows up when the imaginary part of its argument increases with mode indices. Therefore, the field resulting from a complex time factor is obtained by deforming the path of integration, and finally the mode can be represented by a modified Bessel function with a complex argument, together with some remnant waves. Thereafter the transient Floquet mode can be expressed as the contributions of all these complex time factors and a closed-form expression is obtained. When the problem is extended to layered media, the derivation of the F loquet mode looks more complicated. For transient analysis in layered media, the field cannot be represented in terms of a generalized reflection coefficient, but the physical interpretation of geometrical optics series is still valid. Chapter 5 explains how the geometrical series are obtained and how they are synthesized. By expanding the reflection coefficient in frequency domain, the geometrical series is obtained and is immediately transported to the analytic signal representation. when the transient Floquet mode is expressed as the sum of plane waves, it’s surprising to find that the geometrical series can be represented in closed-form again, but with a different expression. Further analysis shows that only five two-level approximation is necessary to represent the coefficients of the kernel, and the field can be evaluated very efficiently instead of computing a infinite geometrical series. CHAPTER 2 PRELIMINARIES This chapter aims to provide the necessary background knowledge for the analysis of radiation from a periodic dipole array. First, the basic concept of a periodic dipole array is presented, and the time domain periodic Green’s function (PGF) is introduced to represent the radiated field. Second, analytic signals and the time domain Weyl’s identity are introduced to represent the PGF by a series of time domain Floquet modes. For better understanding of the time domain modes, the frequency domain counterpart is analyzed thereafter. Finally, in order to find a closed-form solution in a layered media setting, the matrix pencil method (MPM) is introduced to approximate the reflection and transmission coefficients by a finite sum of complex exponential functions, and the popular Discrete Complex Image Method (DCIM) is adopted to represent the field in an integral-free form. The following notation will be used throughout the text unless specifically em- phasized: the field and the Green’s function in frequency domain is denoted by cal- ligraphic letters, in analytic time domain by upper case letters, and in real time domain by lower case letters. Vectors are represented by boldface letters, and vari- ables with a tilde overhead may be used sometime to represent a temporary variable. The other variables, or constants, are represented by either upper or lower case letters for convenience and their specific meanings will be apparent from the context. 2.1 Periodic dipole array Many material structures of electromagnetic interest are arranged in a periodic way, i.e., it is derived from the same element that repeats itself periodically along a line, on a surface or within a volume. Physical periodic structures consists only of finite number of elements. However, an infinite periodic structure permits more elegant mathematical processing and still serves as a good estimate for the practical problems. Meanwhile, practical radiating or scattering elements may have very complicated geometrical properties, and for convenience of computer-based processing, they can be discretized into many dipole elements. Therefore, in this dissertation, our attention will be focused on the radiation from an infinite periodic dipole array. The periodic dipole array of our interest is placed in a surface, and the array may be located in free space, over a half space or over a layered media. Such a configuration in free space is shown in Figure 2.1. The array resides in the x—y plane, and the inter-element spacings along a“: and 3) direction are denoted by d1,- and (1,, respectively. Without losing generality, we assume that the origin of the coordinate system coincides with one of the dipoles. For a radiating structure, the manner in which the dipole elements are excited plays an crucial role in the electromagnetic performance of the structure. Many practical problems require the application of the sources sequentially, so we assume that the dipole elements are sequentially excited here. It’s simple to understand how to “sequentially” excite a dipole array along a line, but for the array located in a whole surface, we prefer to introduce the concept of a plane wave excitation signal to facilitate the understanding of the excitation pattern. As shown in Figure 2.1, assume that the dipoles are excited by an excitation signal, and the signal travels across the plane like a time domain plane wave travels. This plane wave is characterized by the parameter of s, which is defined as the ratio of the speed of light c over the speed of the plane wave 1). i.e., (2.1) a: H I This parameter is called the relative slowness of the plane wave. Accordingly, a vector 3 is defined with its components to be the ratio of c over the corresponding propagation speed of the wavefront. According to this definition, when the dipoles are excited simultaneously, or, the equivalent excitation signal travels at a speed of infinity, 3 = 0. When the dipoles are excited sequentially but the equivalent excitation signal travels at a speed less than c, we have s < 1; otherwise 3 Z 1. It can be readily understood that this plane wave defined for the excitation signal is not physical because none of the electromagnetic waves travel at a speed other than c in free space. The excitation parameter 3 determines whether the radiated field can propagate effectively. This can be demonstrated by a two-dimensional case when an array of line sources reside on the 1: axis in a 23—2 plane, as shown in Figure 2.2 (a). As is well known, the field radiated by a single line source has a cylindrical wavefront. When 3 = 0, all the sources are excited at the same time, and the wavefront should be the envelope of all these cylindrical wavefronts, forming two parallel wavefronts that moves along 2 axis in both directions. When .9 < 1, e.g. s = 1/2 as in Figure 2.2(b), the sources on the right are excited later than those on the left, or vice versa. Since the switch-on signal arrives earlier than the field radiated by the other sources, the wavefront appears as an acute angle and propagates forward. When .9 > 1, e.g. s = 3/ 2 as in Figure 2.2(c), the switch-on signal arrives later than the field, and that is the case for all the sources on the right. Therefore, a wavefront cannot be formed and the field cannot propagate. Thus, the case of interest in this dissertation is s < 1. This qualitative analysis offers some help to understand the problem, and a quanti- tative analysis is to be performed thereafter to find the radiated field. As we know, the electric and magnetic field radiated by a single dipole can be represented by Green’s function, accordingly a periodic Green’s function (PGF) can be defined to represent the field radiated by a periodic dipole array. The following subsection explains how the PGF is defined. 2.1.1 Periodic Green’s function For the configuration shown in Figure 2.1, the array source currents are denoted by J (r, t) = J J (r, t). Using notation similar to [11], the array currents are given by J(r, t) = f: f(t) * 6 (r’ — xmn) (5 (t — -:—s - xmn) (2.2) m,n=—oo where * denotes a temporal convolution, xmn = mdxi‘ + rtdygj, s = slain + 33,3) 2 33, c is the speed of light in free space, and f (t) describes the pulse shape. The electric field E and magnetic flux density B radiated by this array can be represented in terms of vector magnetic potential A as a —82 2 —E t = ——A t VV - A t at (r’ ’ at? (r’ ) + C (r’ ) (2.3) B(r,t) = V x A(r, t) and A satisfies the vector wave equation, 1 82A(r t) 2 ’ _ V A(r,t) — 67—h?— — —uJ(r,t) (2.4) A scalar PGF can be defined by Mr. t) = pint) * gar, t) (2.5) In this way, the original problem has reduced to solving the following scalar wave equation, 00 1 (92 r,t 1 V2gp(r, t) —— C—2——;q-p—(———) = —. 2: 6(1‘, — Xynn) 6 (t — 28 ° X77171) (2.6) at? m,n=—oo and it can be readily shown that 00 gp(r) t) — Z 6 (t — Ir _ x'Innl/C — S ' an/c) — 47rll‘ —an] (2.7) TTl,Tl=—OO This representation is elegant and easy to implement. Unfortunately, the sum- mation converges very slowly, therefore it’s computationally expensive to calculate the potential using (2.5) and the field using (2.3). In addition, it’s diflicult to apply this representation in a more complicated setting such as a layered media. As an alternative, the PGF can be expanded in the spectral domain using analytic signals, and the new representation converges much faster and can be readily applied to lay- ered media. The following section gives a brief introduction to analytic signals and spectral domain expansion, which is known as the transient Weyl’s identity [15]. 2.2 Analytic signals and transient Weyl’s identity It is well-known that in frequency domain, the field radiated by a dipole can be expanded in spectral domain and represented as a superposition of plane waves. The introduction of plane waves brings considerable physical insight and computational simplification to our problem. The field radiated by a dipole in time domain can be represented as a spectral superposition using analytic signals, which is the so—called transient Weyl’s identity. In what follows, we shall outline the basic properties of analytic signals and introduce the transient Weyl’s identity and a useful technique called Causality Trick. Further details may be obtained from [13, 14, 15]. 2.2.1 Analytic signals Consider a real function f (t) with Fourier transform .7:(w). The integral oo . F(t) = TIT/0 f(w)cjwtdw, for t E C (2.8) 10 is known as the analytic signal associated with f (t) Here C denotes the complex space. The concept of analytic signal should not be confused with analytic functions. However, we can prove that the analytic signal F (t) is an analytic function of t for every t in the upper half of the complex t plane. As is well-known that a real signal has a symmetric spectrum, we can readily prove that the real part of F(t) is f (t) Denote the imaginary part as f (t), and their Fourier transform as Q(w) and 3(a)). From 2.8 We can see that 2}"(w), w > 0 GM = (2-9) 0, w < 0 because 9(a)) = .7:(w) + jaw), it follows that . _ —j}'(w), w > 0 9(a)) = -J}'(W)Sgn(W) = (2-10) j]:(w), w < 0 it means that f (t) can be expressed in terms of f (t) Since the inverse Fourier transform of —jsgn(w) is 1/(7rt), we conclude from the convolution theorem that f(t)=f(t)*—1—=/OO f”) d7 (2.11) nt _00 7r(t — 7') which means that f (t) is the Hilbert transform of f (t) Thus, the analytic signal can be obtained from the real signal directly, which avoids frequency domain entirely. This important property makes it possible to formulate a purely time domain approach based on analytic signal representations. The preceding analysis shows that the analytic signal is closely related to Fourier 11 transform. Equivalently, we can define an analytic signal transform as follows .7:(w) = /00 dt exp(——jwt)§R{F(t)} {0°00 (2.12) F(t) = F/O dwexp(jwt).7-'(w), for %{t} > O and the real function is obtained simply by taking the real part of the analytic signal. Considering the importance of the convolution operation, a convolution operation between an analytic function F (t) and a real function h(t’) can be defined as, F(t) * h(t’) = [00 d0F(t — a)h(a) for S‘{t} > 0 (2.13) —'00 As is well known, when F012) = 1 the corresponding real signal is the Dirac delta function 6 (t) The associated analytic signal, or the so—called analytic delta function, is defined as, A(t) = i/Ooodwexpfiwt), for 8‘{t} > 0 (2.14) The analytic delta function A(t) plays a central role in the formulation presented herein and some useful properties are listed below, ~ 6(t)+j/(7rt), s{t}—_—0 j/(7rt), s{t}>o o A(t) = o Ift E R and SM} 3 0, then A(—t—K) = A(t+7{) o Ift,a€Rand 3M} 30, then A(t—a—rt)=6(t—a)*A(t—rt) where the over-line denotes the complex conjugate. 2.2.2 Transient Weyl’s identity Based on the analytic signals, the potential radiated by a time dependent dipole source in free space can be represented using the transient Weyl’s identity, rather than using the conventional retarded potential form (5 (t -— R/c)/(47rR). The transient 12 Weyl’s identity can be written in terms of analytic delta function using g(t) = W = ER(G(t)) (2.15a) gift/f: d2p~—1—A[t-pm($-IC’)— pyo— y’)— pZIz—z ’I] (2.151») _ __,2_.2 p2_\/012 pa: pg - 2 2 2 p2 lfp +1) S 1 C = I I I’ y / (2.150) —jlpz| up; +223 > 1/c2 where at denotes derivative with respect to time, d2p denotes (Ip-Idpy and R = Ir—r’l. This representation comprises of a superposition of both homogeneous and inhomo- geneous time domain plane waves. Alternatively, one can develop another representations that comprises only of prop- agating homogeneous time domain plane waves. As elucidated in [15], this can be developed using a superposition of plane waves due to a source and its time-symmetric counterpart. This superposition is also known as the Causality Trick (CT), and it ensures that the the field generated by these sources is identical to that generated by the original source for t > 0. Mathematically, these statements correspond to G303) = Git) - Cit-t) —3 : Sag/f (1213— A(t—PIIC—pyy— pzlzl) R2 pz _3_ 8W2 //122d2pp:( A( —t-px:v-pyy—pzlzl) Z _-—('3 1 - 48/] 6121) [- A(t—prx—pyy— pzlz |)+ :A(t+prw+pyy+pzlz D] R2 P2 P2: 2-3 1 1 48/] (12;; [— A(t—pxar-pyy- pzlz |)+ +:A(t'Pr$—Pyy+pzlzl)] R2 1’2 I) (2.16) 13 The subscript ’s’ in Gs(t) implies that the function 03(2‘.) is composed of symmetric sources. In deriving the third equation, the second property of the analytic delta function is used, and the last equation used the fact that the integration domain is symmetric. Note that the integrand vanishes when pz is purely imaginary (for 193,. + p32, > 1/c2). Consequently, the domain of integration is limited to 193,123, such that 12% + p32, 3 1/c2 and can be understood as a superposition of downgoing and upgoing analytic time domain plane waves. It is evident from the above expression that the overall integral is considerably simplified. In the next chapter, we will present different derivations of the time domain Flo- quet modes that are required to evaluate the time domain periodic Green’s function. These derivations will be predicated upon the time domain Weyl’s identity in (2.15). We will use three different approaches. The first will be a direct application of the Weyl’s identity. Next, we will use the Causality trick to do the same. Finally, we will use the Whittaker’s formulation as well. The first two derivations rely on the analytic delta function, whereas the last will be based on the Dirac delta function. It should be pointed out that the derivation of time domain Floquet modes is similar to the derivation of the frequency domain Floquet modes, and the properties of time domain modes can be better emphasized by comparing with the frequency domain counterpart. Therefore it’s very helpful to have a brief review of the frequency domain F loquet modes before we dive into the time domain analysis. 2.3 Frequency domain Floquet modes For the same problem setting as shown in Figure 2.1 in frequency domain, the array currents are given as J(r,w) = j Z N...) 2. 6(1" — xmn) exp(—jks - xmn) (2.17) m,n=—oo 14 where .7:(w) describes the frequency spectrum of the excitation, xmn = mdxzi: + ndyt) indicates the location of all dipole elements, 5 = 512: + 3ng = 35‘ suggests that the excitation currents have fixed phase differences, and k = wflE is the wavenumber. The dipole array is assumed to be located in z = 0 plane. The electric field 8 and magnetic flux density 8 radiated by the array currents can be represented in terms of vector magnetic potential A as £(r,w) = —ij(r,w) + t—l—VV - A(r,w) quc (2.18) B(r,t)) = V x A(r,w) and A satisfies the vector wave equation, (v2 + k2).A(r,w) = —/1.J(r,w) (2.19) Denote the vector magnetic potential by A(r,t)) = pjf(w)gp(r,w) (2.20) where Qp(r, w) is the scalar frequency domain PGF that is associated with the poten- tial. The original problem has reduced to solve the following scalar wave equation, 00 (v2 + k2) gp(r,W) : —' Z 6 (1" — X77171) €Xp(-ij ' X171") (2.21) m,n=-oo As is well known that the field in free-space radiated by a point source at r' = (23’, y’, 2') and observed at r = (3,2 , z), or the free-space Green’s function, is exp(—ij) g r,t.) = ( ) 47rR (2.22) 15 Therefore the frequency domain PGF can be represented as a series of retarded po— tentials, gp(l‘, w) —— Z expl—jk(R7rtn — S - an)] m,n=—oo 47TRmn (2.23) R771” 2 \/(I — flld$)2 + (y — ”Liz/)2 + Z2 This derivation is straight-forward, however, the summation converges very slowly, and is not practical for real-world applications. As an alternative, the free-space Green’s function can be expressed by Weyl’s identity as follows, H” (x— w)+k. (y —'.yi+kIz— zl) R21... (2.24) m:=\/E2—kg—k;fl ImszO which represents the Green’s function as a linear superposition of time-harmonic plane waves. Therefore the periodic Green’s function can be rewritten as, . ___ (T- md I,,)+k (y— ml yz)+k |z|) -js,km,,d —js kndy QPOW’ max 872 Mf/R.‘—ke k, e (2.2 = [W j(k Hx+kvy+k Iz [)ej(k I —s 110de +j(ky —s,,k)ndy m, n=—oc R2 Let f(u, v) = exp [j(kg; — srk)u + j(ky — syk)v] (2.26) It can be proven that its two-dimensional Fourier transform is, Fem) = 4.250., + s..k — way + 8k — kg) (2.27) 16 5) Using the two-dimensional Poisson summation formulae, 00 Z f(771dx,ndy) = m,n:—oo 1 i0: F(27rm 27m) (2 28) dlfdy m, dfil‘, ’ dy . -n=—:x: and applying it to Eq.(2.25) results in gp(r, w) = Z an(r,w) m,n=—oo exp [“j(k;rm$ + kyny ‘l‘ kzmnlzll] 9171110301) = - . J2drdykzmn 27rm 2.29 krm : 51k + d ( ) .1: 27m kyn :- Syk "l' T y kzrmt : \/k2 — k3", — kg); 1m kzmn S 0 Each field component gmn(r,w) is designated as a Floquet mode. When the mode indices m and n increase, kzmn becomes purely imaginary, and the mode is evanescent. Therefore only a finite number of modes are propagating, and the PGF represented as the summation converges rapidly for z > 0. This is the major advantage of using F loquet mode representation. It can be seen that the spectrum for every single mode is not symmetric, so the resulting transient response for the single mode is not real. However, the sum of two modes with opposite indices gives a real signal, therefore the final response is still real. The advantage of a spectral expansion representation is that it can be readily adapted for the layered media setting. The representation is basically identical to the free space case, and the principal difference is that the reflection or transmission coefficient should be included. Finally, we birefly introduce the Discrete Complex Image Method (DCIM). The 17 technique is popular in frequency domain analysis when approximating the field due to a dipole in the presence of layered media. We will find the technique useful in approximating time domain PGF in the presence of layered media. The method is briefly introduced in the following section, and the approximating functions are obtained by using the matrix pencil method (MPM). Those who are interested with MPM are referred to the literatures [27, 28]. 2.4 Discrete Complex Image Method Considering a single dipole in presence of a layered media, the Green’s function can be expressed accurately by many approximation methods. Among these methods, the most popular one in recent years is the Discrete Complex Image Method (DCIM), in which the basic idea is to fit the curve of reflection coefficient by a finite sum of complex exponential functions, and each such a exponential term corresponds to a complex image and the term can be incorporated into the classical Green’s function by using the Sommerfeld integral identity. Naturally, we can think about the popular approach of fitting the reflection coefficient with respect to the vertical wave slowness, unfortunately this may not work for a periodic setting, because the expression may not satisfy the radiation condition. This will be explained later in the following chapters. As an alternative, the reflection coefficient can be fitted with respect to the horizontal wave slowness, and such an approach is later proven desirable for our purpose. For better understanding why the second approach is preferable than the first, the basic ideas for these two approaches are presented as follows. As we know, for a single dipole over half space in frequency domain, the field in the region where the dipole resides can be denoted as the sum of the direct field and the reflected field, and the reflected field can be represented as a Sommerfeld integral 8.8 fl...) = A00 R(kp),70(kpp)kpdkp (2.30) 18 where R is the reflection coefficient represented as a function of kp, and the constant coefficient before the integral sigh is ignored for convenience. The most popular approach is to approximate the reflection coefficient by a series of complex exponential functions of k2, i.e., N R(k,,) = Z AieB‘kz, §R(B,-) < 0 (2.31) i=1 It is well-known that the Sommerfeld identity 0" exp(—jkzlzl) exp(-jkr1) . J k k dk = ___—— 2.32 ./0 3’92 0( p) p p T1 ( ) is also valid when the vertical location of the source is complex. Therefore, .7:(w) can be represented as N . f(w) = Z aiflafl i=1 1 (2.33) r,=\/p2+(z+z’+jb,-)2 The pole of the reflection coefficient lies on the imaginary axis in complex kz plane. Accordingly, in order to capture the singularity of the integrand accurately, the orig- inal path of integration L0 is deformed to L1 in complex It; plane as shown in Figure 2.3, where L1 also consists of two portions of L] and L2, and if L? is long enough, the spectral kernel can be accurately approximated. L? is chosen to be parallel with the imaginary axis such that the path keeps equal distances from the pole no matter where the pole is located. This feature is especially useful for a layered media case when there are many poles located on the imaginary axis. For electrically large prob- lems, the path L? should be very close to the imaginary axis such that the variations of the kernel can be more accurately captured. 19 On the other hand, the reflection coefficient can be approximated in terms of kp rather than kz as suggested in [29], N B(kp) = Zaieb‘ki’, an» < 0 (2.34) 221 Then, using 0° bk —b "’J k k dk =——— 2.3 accordingly, .7-‘(w) can be rewritten in closed-form as N -—a.ib,- The next task is to accurately approximate the reflection coefficient, which is not a continuous function and has singularity on the real axis in the complex kp plane. Therefore, the approximation cannot be performed along the original path, or, the real kp axis. This problem can be alleviated by deforming the path of integration as shown in Figure 2.4 and perform a two-level approximation. Since the integrand is analytic in the upper half complex kp plane, the original path of integration L0 can be deformed to L1, which consists two portions of L] and Lg. If the path L? is chosen to be long enough, the integral can be approximated accurately. 2O 7474747474747474747474 72 + 74% 74 747474 +7474 7474+ 7474+ 74 X Figure 2.1. A planar periodic dipole array with the inter-element spacings of dx and (Ly. The waveform of the excitation signal is marked out by thick curves, and the excitation wavefront travels like a plane wave does. The excitation parameter s is normal to the wavefront. 21 (a) (b) (c) Figure 2.2. Field radiated under different excitation settings. Assume any source is excited earlier than or simultaneously with those on its right and the source at the origin is excited at t = 0. The field is observed at t = d/c, where d is the inter-element spacing. (a) Simultaneous excitation. The thick lines stand for the wavefront, the arrows for the propagation direction, and the black dots for the sources. (b)Sequential excitation when s=1 / 2. The black dots represent active sources, while the white dots are not excited yet. (c)Sequential excitation when s=3/ 2. For convenience of observation the axes are scaled down. 22 w] Figure 2.3. Deform the path in kz plane 23 X X Figure 2.4. Deform the path in kp plane 24 CHAPTER 3 RADIATION FROM A PERIODIC DIPOLE ARRAY IN FREE SPACE Again, consider the problem shown in Figure 2.1. The PGF is obtained by using three different approaches. The first approach is to include a series of ghost signals based on CT and results in a finite range integral. This integral can be evaluated in closed-form by performing hyperbolic transformation. The second approach starts from the time domain W'eyl’s identity directly and results in a infinite range integral that contains branch point singularity. This integral can be converted into a contour integral and evaluated in closed-form. The last approach is to develop a Whittaker representation and find the closed-form expression by performing changing of variables. These three approaches are examined next. 3.1 Using Causality 'Ii‘ick The starting point of the analysis is the time domain PGF in (2.7) that is represented as a spatial summation of retarded potentials. Using this equation together with (2.15) results in the analytic periodic Green’s function that can be written as, GP“): 00 zgjg‘t/fz'f—zp m,n=-oo 33; sy A [t — px(:r: — mdx) — p.2y(y — ndy) — pzlzl — :mdx — :ndy] (3.1a) — .. :31 W], m, nz—oo R2 P2: 31:23 + s y s l S A [(t — —8—c——y—)— (p. — ~31) (cc — md.) - (pg — 7") (y — ndy>— szzI] 00 —8 = __t8[42:12A((T—pr—de)_p;(y—ni(1y)—p2lz|) (3-1b) ',"l IT'S—008 25 where T = t — (311' + syy)/c, pi. 2 pl- — sx/c and p; = py — sy/c. The redefinition of time is tantamount to shifting the time axis, and can be interpreted as being equiva- lent to analyzing simultaneously excited dipole array. This representation introduces considerable simplicity to the ensuing analysis as well as lends itself into physical interpretation of the results. As mentioned earlier, CT may be used to obtain the actual field due to a dipole at a point for all time t > 0 (assuming that the observation point is sufficiently separated and the source has ceased to radiate). This is a consequence of the symmetry of the expression (2.16). However, the same ideas do not translate readily to a periodic array, largely due to the phasing of the dipoles, i.e., additional delay terms in (3.1a). However, shifting the time axis as in (3.1b) has a very desirable effect; the expressions resemble (2.16). This implies that the response obtained using CT and the actual time response are identical for T > 0 rather than t > 0. Let’s have a closer look of the ghost signal. As we know, the field radiated by the original signal from an arbitrary dipole element located at (mdmndy) can be expressed as _—8 Gan) ”Sf/:2 2 2pA( (7’ — pI(:r - mdx)— pZJ(y — ndy) —pz|z|) (3.2) here we used Gm" rather than Gmn in order to avoid getting confused with the transient F loquet mode Gmn. Denote the field radiated by the corresponding ghost signal as Gin”, then it can be written as Glmn (t) = — Grim. ( _ t) = 19‘ ”Mfg; 2 (3.3) Sq?! + s y A 8(t “l" “'I—C—ZL‘ +1910?) — 7716132) +nylj —"dyl + lez I) 26 =‘5t ”2.6;“ 2 (3.4) s s A 8(t + pl.(:r — mdx) +py(y — ndy) + fizlzl + fmdx + gndy) change the variable of pa; into —p3, and pg into —py, and using the symmetry of the integration domain, the ghost field can be rewritten as _—a anll(7—) —t Tag/f}? Bi? (3.5) s s A (t — p$(a: — mdx) — py(y — ndy) + lezl + fmdx + 23172113,) It should be noted that p2 is not affected by these sign changes as it is an even function with respect to either pa; and py according to the definition (2.15c). The expression of the ghost field shows that if the original signal is time-delayed, then the ghost signal should be time-advanced. This symmetry explains why the CT is still applicable when the additional excitation delay is present. Denote the expression due to the ghost signals as 01,9, and the sum of these two as Gps. Using (2.16) within the context of a planar periodic structure, the resulting periodic Green’s function can be expressed as, G123“): l+ 0119(1) =:’—a’ My}, _ (3.6) [8: NT - p215 - pyy— PzIZI) + p— :TIA( -p§c$ - pLy +sz2|) Z 00 a. Z 6 (t +pgmdx) * Z 6 (t + pandy) m=—oo n=—oo For inhomogeneous plane waves, pz is purely imaginary, therefore if; 2 —pz. From the expression above, it is apparent that the two inhomogeneous plane waves cancel each other. Therefore, the periodic Green’s function can be constructed using homogenous plane waves only. Mathematically, the integration domain is reduced to 1);» E R, which 27 is a circular region with the radius l/c. This means that the original infinite range integral can be reduced into a finite range integral by using CT. For convenience, we focus on the original signal 0,, first, and investigate Gps at the end of this section. 3.1.1 A single Floquet mode Using the convolution property of the analytic delta function, and interchanging the summation and the integration (this is permissible as the function converges) results in _—at 2P :87? f/miz -——A(T—p§:$—pyy— WM) (3.7) X 6( t + pgmd, + pgndy) m,n=—oo Denote the summation as [1, i.e. 00 Z (5(t + pfvmdx + pgndy) (3.8) m,‘n=—OO Here I 1 can clearly be regarded as an array factor. It describes the modulation effect of the array on the transient plane wave. Since this array factor is dependent on p3; and py, different plane waves are modulated differently. Therefore, the summation of all of these plane waves results in a mode pattern different from that of a single dipole. The summation over m and n can be separated by using the convolutional property of the Dirac delta function, and 11 is rewritten as 00 00 Z 6(t + pgmdm) * Z 6(t + gag/nay) (3.9) 7722—00 122—00 Each infinite summation can be simplified using Poisson summation formula 00 00 1 27m 6 t — mT = ex t— 3.10 2( )ZI—Tlp(jT) () m=——oo n=~oo 28 and [1 can be rewritten as I i 1 exp (jt 27ml) * 1 exp jt 27m 1 : —— ———- _— _— m,n=—oo ”)9:de Finds: lPQdyl Pgdy OO _ Z 27r (5 27rm _ 27m exp (jt 2am) (311) 00 lpépzdzdyl pfrdx dig Pgda: . m,n=— OO _ Z 1 6 ,_ ,ndx ex ,t27rm — {Imldy py medy p ‘7 pévdx -m.n=—oc In the above equation, indices m, n have been reused to denote modes. Its distinction will be apparent depending on the context. Also, the final step was derived using the fact that 6 [f (17)] = Z, 6(x —:1:,-) / I f ’ (2:2)I where 1105’s are the zeros of the function f (:13) The presence of the delta function in the above equation implies that the integral w.r.t p’ in (3.7) can be readily evaluated, and Gp can be written as y 00 021(15): 2 if {1&3 eXI) Jig-:3 * A(T — p356 - p’ y — szZI) _ 87T2lmld R P2 13’ y "Ln—-00 y x (3 12a) __ {2 —at / «112.12 _ __2___ __ m,n=—oo 8” lm'dy R ”2 where 0’ p; = pig—Z I I P, p, 293:3? +Pyy = game: + Hay) = a?“ ' P) (3.1%) In the above equations, 0;; = 27rm/dx, cry = 27m/dy, a = a$i+ayyj, p = xi+yy, a = 01% + 0132;: 8 = C/(1_ 32)(52:01x + 531011), and fl : C/(l — 32)\/a2 _ (5'ny _ Small/)2- The next step in evaluating the Green’s function is to perform the convolution. Mak- 29 ing use of the closed-form expression of the analytic delta function, the convolution can be expressed as, ' 1 [2 = l/ (10 I I exp (jag—f) (3.13) W R (T—pxrvayy-szZI)-0 p3; Extending a into the complex plane, we find that the integrand is analytic everywhere except at a pole located at up = ’r — pin: — pay -— pzlzl. Since 8‘{pz} S 0, then %{0p} 2 0, which means that the pole is always located in the upper half plane (the case of 8{0p} = 0 is regarded as the limiting case when 0,, is approaching the real axis from the upper half plane). Meanwhile, the integrand vanishes either on Poo or P_oo depending on the sign of alt/pg. Consequently, the integral can be closed by either P+oo or P.3o as shown in the Figure 3.1 and converted into a contour integral. When Olav/P55 < 0, the integrand vanishes on P.00, and the integral can be enclosed by P.00. Since the integrand is analytic everywhere within the contour, the integral vanishes. When (rm/p; > 0, the integrand vanishes on P+oo, and the integral can be enclosed by P+oo. Consequently, 12 = 327rResg_,ap — I I exp ]0’—- LW(T—px:v-pyy-pz|2|)-0 P3: (3.14) - I I 0:1: = Qexp [J(T - pair - pyy —pzlz|)-,-] pa: where ” Res” refers to the residue. Therefore 12 can be expressed as, , a, a 12 =2exp [J(T—péw-pLy-pz|3|)~,£] U (71:) (3-15) pm pa: Substituting this expression into (3.12a), G'p can be written as 00 CW) = Z Gmn(t) (3.16) m,'n.=—oo 30 Gmn(t) = sign(a$)Ae'—ja°p/ dz); . Cf; , exp (38,137" ’ J'PlzTO) U (SI/E) R Jpa: pz p.13 p11: , 2 we) pa: where A = c/(27rdxdyx/1 —— 39), T0 = |z|v1 —— 32/0, Gmn is the time domain Floquet (3.17) mode, and sign(a$) denotes the sign of (ax). To simplify this expression, change pg: into pfr/ax, and Gmn can be rewritten as _ja.p 00 (1p;- . T . / Gum“) = A5 . ,2 , 93(1) .7—, _ 919270 0 31% 1’2: pa: (3.18) Similar to the derivation of Gp, the ghost signal Gps can be derived. Accordingly, the sum of these two signals denoted as Gps in (3.6) can be written as cps“) = Z Gps,mn(t) m,n=—oo _jap dpfc .T . , .7" .— Gps,mn(t)=Ae -—- exp JP ~JpzTo +exp 317+.7p’z7'0 II? p;>0,p;ER jpfzrzp’z {1: (3.19) Therefore, a single F loquet mode is expressed as a finite range integral. 3.1.2 Mode pairing It can be seen from (3.19) that the F loquet mode is constructed by the upgoing homo- geneous plane waves of the original Floquet mode Gmn, together with the downgoing ghost plane waves of the ghost mode 0%,". Each of these plane wave components can be characterized by a (193,, pg], 197,) triple, and it can be shown that the corresponding triples for all the wave components of all the modes define an Ewald sphere as shown in Figure 3.3. For mode Gpsmn, p3, and p3 are linearly related (3.12b) in pic—pg, plane as shown by the thick line segment EB along line P. This line corresponds to a circle 31 on the Ewald sphere and an enlarged version of this circle is shown in Figure 3.4. Accordingly the component waves Gm." and 0%", correspond to the arc in the first quadrant and that in the fourth quadrant respectively, and are therefore differenti- ated. It can be shown further that the arcs in the second and third quadrants, or the dashed line segment 9, is associated with G(_m)( and GL1”) (_n). This suggests ~71) that the mode Gpsmn and G PS.(—m)(——n) are closely related. Denote the integrands of Gmn and G(_m)( as 01 and G2. i.e., —n) Grim: G1, G(—m)(—n)=/ 02 (3.20) (B 6 It can be readily verified that 09...: £902, Gim,_,,= f9 01 (3.21) Thus, the ghost field complements the opposite mode, which suggests that a simpler expression can be obtained by adding this two modes together. :t g- smm t = g s,mn t +9 3 -—m —n t = ER{/ G + G } 3.22 p, (l p (l p,( )( )() GBUJI 2)_ ( ) This observation is to be verified in the following derivation. The expression (3.19) is desirable because it is a finite range integral, and this characteristic is especially important when dealing with more complicated problem settings such as layered media. For free space problem, the expression (3.19) can be written in a more elegant form by changing of variables H: = l/pfv, and a closed-form expression can be obtained 32 accordingly. The resulting expression is _- . (1h: . . . . Gwynne) = Ae 3" P f 0 R B,— [exp (JET - who) + (W (W + 110270)] rc> ,p’zE . z — Ace—ja'p 1&— ex ('KT) c0"( ' T ) — K>B p’ER jplz I) J b Pz 0 p2: (re—@2432 (3.23) For the case of s < 1, it can be verified that the integration domain can be simplified as K. > 3 + 3. Perform change of variables K = B + fl cosh 6, and Gpsflrm is rewritten as . ,- 00 _ ‘ Gps,mn(t) = AeJ(’3T'—a'p)/ d6(—j)efl37 (305119 cos(,370 sinh 0) (3.24) V 0 Since the signal of interest is the real part of the analytic signal, it can be obtained by taking the arithmetic average of the signal with its conjugate. Thus, the real Green’s function can be written as Gps,mn (t) + Gps,mn (t) 2 _ 00 = 2A cos(flT — a ' p)/ d6 sin(/3T cosh 6) cos(fiTO sinh 9) 0 9ps,mn (t) = (3.25) _ 00 + 2A sin(fi7' — a - p) / d9 cos(fiT cosh 6) cos(,870 sinh l9) 0 = 2A cosh/37 — a - p)13 + 2A sin(BT — a - p)I4 As shown in the Appendix, the integrals can be evaluated in closed form, and are 13 2 322,0 (3 ,2 _ T3) U(T2 - g)sign(»r) 1r (3.26a) I4 = ——§Y0 (fl 7'2 — 73) U('r2 - 73) + K003 7'2 — Tg)U(Tg — 7'2) 33 Therefore, the F loquet mode for the time 7' > 0 is, g...,.,..,.(t) = vrAcosmr — a - pug (Ia/r2 — 3) w? — 2)sign(T) + m4 sin(57' — a - p) [—YO (fl 7'2 — 7'3) U(T2 — 7'3) + :KO ()3 T2 — 7'3) U(Tg — 72)] (3.27) As is evident that the sine term is not causal, therefore this F loquet mode is not causal. However, when both m and 72. change signs, the only change in the expression of gmn(t) is that the signs of B and a are flipped, and accordingly the sine term changes the sign. Therefore the superposition of gpsmn with gps,(_m)(_n) results in a causal mode. Consequently, it is more convenient to define the causal Floquet mode as 913mm“) = gps,mn(t) + gps,(—m)(—n)(t) (3.28) = 27rA cos(BT — a - p)J0 (5 7'2 — 73) U(T2 — filSiEMT) It is found that this expression is identical to the paired mode in [11] for T > 0. These concepts are illustrated in Figure 3.5, where the modes for both positive and negative time are depicted. The mode 91,1 is depicted in Figure 3.5(a), and the small tail during 0 < T < T0 indicates that the mode is not causal. The mode g(_1)(_1) is depicted in Figure 3.5(b), which is also non-causal but the tail has the opposite sign. Their sum, depicted in Figure 3.5(c), is causal. The final signal with the negative time part truncated is shown in Figure 3.5(d). It is interesting to note that for p = 0, the modes and the total field are anti—symmetric as shown in Figure 3.6. When p ¢ 0, some of the modes might be anti-symmetric but the radiate field is not, as shown in Figure 3.7. In the figure, only the modes with the mode indices 34 up to 1 are summed together for the chosen normalized Rayleigh source [11], and the results obtained by using element-by-element summation is also provided. From the figure, it is apparent that the agreement is excellent. 3.2 Using time domain Weyl’s identity The same transient Floquet modes can also be obtained from (3.18) without the aid of the ghost signal. The integral for the analytic signal in (3.18) is not easy to evaluate directly, however, it is easier to deal with the associated real signal. Similar to (3.25), the real signal can be obtained by taking the arithmetic average of the analytic signal and its complex conjugate. It can be found that a mode is closely related to the complex conjugate of the mode with opposite mode indices by extending the definition of K. into the whole real axis. Define the extended square root function p’z for Gmn as ‘si Inc-B p’, '2252 p2“) = g ( )l 2| P2 (329) -J'lp’z|, 1022 < [32 Denote the corresponding square root function for the opposite mode G _m,-n as p2”. Similarly its extended definition is ,, sinn+3 p", p'l22fi2 192043) = g ( )l 2' z (3.30) _jlpzli p22 < fi2 These new definitions are consistent with the previous ones. It can be readily verified that PTA-H) = PAH) (3-31) 35 and the real signal associated with mode Gmn is written as _ 077172“) + Gum“) 917m“) — 2 _ é foo dnexp [J(h‘" - P270 - 0 ° 0)] 2 0 II); ' (3-32) 0 exp [Jim - plro + a - m] +/ dn . ,, —oo Jpz Similarly A 0 exijiT—p’z'rO—a-p g—m,—n(t) = 2 [/ d5 [ ( . , )] —oo .7172 (3.33) ee exp [Jim — pine + a - m] + / d»; . x 0 .7192 Note that the expressions for these two modes are the same with the integration limits are different. Obviously the expression can be greatly simplified when these two modes are added together. Denote the sum of these two modes as Gitnn, and rewrite it as 97in“) = Gum“) + G—m,—n(t) g {/00 am We - pgxe — a . m) 2 —oo .7sz dIs: ex exe(xee—e:e+e.e>] +/ . II —oo .7172 (3.34) dn — 2 —oo j, /K2 _ 3‘2 [Crew—am + e—j(fir-a-p)] 00 exp [j(fi7' — Wm” =AcosB‘r—a- / d5 . ( P) —00 j H2—32 _ A foo .exp [j(n'r — \/ x2 — [3270)] 36 Note a change of variable for K. - B into E is performed in the first integral in (3.34), and a change of variable for K + B into It is performed in the second integral. This equation can be converted into a contour integral by extending K into the complex plane and specifying the Riemann sheet 3{p'z} < 0; a condition that is consistent with (3.29) and (3.30). The resulting integral has two branch points at $5. The original path of integration can be shifted slightly below the real axis to avoid branch point singularity as shown in Figure 3.2. When 7" < lg, the integrand vanishes on P_oo. Consequently, the path of integration can be enclosed by P.00. The integrand is analytic everywhere within the contour, and therefore, the integral vanishes. This implies causality. When 7' > 1?, the integrand vanishes on P+oo, so the path of integration can be enclosed by P+Oo, and the integral can be converted into two branch cut integrations. This integral is readily evaluated [11], resulting in the time domain Floquet mode being expressed as, gitnnfl) = 27rA cos()'37' — a - p)J0 (fl 7'2 —— 7'3) U(T — To) (3.35) This expression is identical to that obtained by inverse Fourier transform in [11]. The approach presented in this section is very similar to that in [11], and such techniques as Poisson summation, mode pairing and branch out integration are used in both formulations. However, there are some subtle differences. In an analytic signal formulation, the signal associated with the analytic signal is always real, but not necessarily causal; mode pairing is performed to make the signal causal, i.e., to make the signal physically meaningful. In [11], the signal can be made causal a prion, but it’s not real because the spectrum is not symmetric as shown in (2.25); mode pairing is performed to make the signal real, i.e., to make the signal physically observable. 37 3.3 Using Whittaker formulation For the simultaneous excitation (s = 0), the CT can also be applied in the spherical coordinate and the periodic Green’s function can be represented as a Whittaker- type integral. This is because the up-going and down-going homogeneous plane wave components are totally symmetric and a spherical integration domain can be used in p—space. Also, as all the modes are homogeneous, the CT expression can be expressed in terms of real signal directly. Therefore, the CT expression can also be represented as: _ 6(t— 5) _ 60+?) R R _ 7r 27r . 87r c 0 0 c In this equation p = sin6cos (.65: + sin63in (by) + cos 02. Note that here p is defined (3.36) slightly differently than before, and this gives the parameter a clear geometrical optics meaning; it indicates the propagation direction of the component plane wave. Using this expression for each dipole in the array, the total field can be written as 8t 7r , 2” p - r 00 n-xmn gp3(t) = — /(‘) sm 6(16/0 6 (t — ——C——) * Z 6(t + ) (3.37) 87r2c c mTL=—OO The convolution can be rewritten using Poisson summation formulae, .an A F.- I "U OX 3 s v I md nd {330-1% 036).:30—1), 69) m Tl S”: c2 exp (j 27rmct) * exp (—j 27rnct) lPxPy da: dyl Pa: drz: Pydy mnz-oo 00 c ,27rmct nd, : Z Wexp (jT)6(tan¢_de) mn=—oo Pr 3; pa: :1: y 38 Substituting (3.38) into (3.37), and carrying out the time derivative and convolution, Gps is expressed as ‘ 3n/2 Jc 7’ 1nn- oo I(y 0 '_fl/2 ( ) HR 3.39 sign(m cos (6) 27mm (t _ c ) (5 t , ndm ex ' an _ sin 0 cos2 <35 D J kxdx (p mdy Denoting every single mode as gnm(t ), and letting tanq§= C, (3. 39) can be written as a sum of two integrals, sign(m)c /" 1 t = —————— d19— gmn( ) ' 0 sin0 347rdxdy {/ dCGXP|:— j27r (m_. + myC) +j27rm\/1 +.C2 (ct - zcosfl)] R dr C11; SlIl 6d]: mix 6 — . (C mdy) (3 4O) 2 ._ —/dCexp[— J27? (n_1_.r+myC)_j27rm 1+.C (ct 20086)] 1? dr dz SUlfidx mix 6 _ (C Indy) } This equation now reduces to .9an) = Slgn(m):::1:(d;](a7; p) f0” ddm sin [sign(m )a (ctcosecB — z cot 6)] cexpl— j( a p) )1 = 0———Si 6 — 6’ 27rdrdy 7rd n6 sin[a (ct cosec zcot )] (3.41) Perform change of variables as cosecO = cosh 7 (3.42) cot 0 = sinh 7 39 and let z/(ct) = tanh 70 (3.43) Following the similar derivation in the Appendix, for t2 > 22/c2, (3.41) can be rewrit- ten as, gmn(t) : cexggrdigiy P) Ad’V Sin CO t2 — :—2 C05h<7 — ’70) Sign(7‘) (3.44) cexp —j(a - p) 22 . -_- 2dxdy ]J() ca t2 — c—2 SlgIl(7‘) Similarly, for t2 < 32/02, (3.41) can be rewritten as, _ ' . 2 gmn(t) = cexp( fa p) / (17 sin ca” 2—2 — t2 sinh(7 — 70) =0 Therefore, (3.41) can be rewritten as ceX) —'a- 22\ Z2 . This mode is not real, and it becomes real when summed together with mode g(—m)(—n)(t)a i-e-a 97in“) : 977271“) + g(_,,,)(_.n)(t) . 2 dxdy c Again this expression is identical to that obtained before. In a Whittaker-type representation, the field can be represented in terms of real 40 signals directly, which simplifies the analysis greatly. However, for the general case, or the sequential excitation, the origin of the spherical integration domain is shifted away from the origin, which makes it impossible to write the up—going and down- going plane waves in a generalized form. Therefore, the Whittaker-integral is no longer applicable. Meanwhile, Whittaker-integral cannot deal with inhomogeneous waves, which appears in many problem settings, e.g., computing fields due to periodic sources embedded in layered media. Consequently, the Weyl- and CT-based solutions are preferred. 3.4 Results To validate the PGF represented by transient F loquet modes, two tests are performed to calculate the Green’s functions using different source excitations. As discussed in [11], higher-order modes have higher local cutoff frequencies, and whether a higher- order mode can be excited or not is determined by the spectrum of the source. There— fore, in order to examine the convergence of the Floquet mode series, a band-limited source is preferred to excite the preferred modes. Here, we choose the normalized Rayleigh pulse as in [11], whose waveform and spectrum can be represented in closed- form, f0” = ’R ((j + 0.353M1)5) - 4 7r ]4K. 4|I(—n) O 9 1- mn causal O l ((1) Figure 3.5. Synthesis of a causal mode by two non-causal component modes. The parameter are chosen as d;- : 1,dy = 1,33; = 0.3, sy = 0.4,m = 1,31 = 1,2 = 0.5. The vertical axe represents the mode magnitude and is normalized to c/(2dxdy); The horizontal axis represents time T and is normalized to 1/c. (a) Mode 91,1 is non-causal; (b) Mode g_1,_1 has the same non-causal part except the sign flipped; (c) The sum of these two modes; (d) The final mode is causal. 47 15 10 I Figure 3.6. Synthesis of the PGF from component Floquet modes. The modes with the mode indices up to 3 are summed together. The parameters are the same as in Figure 3.5 except that the observation point is located at a: = 0, y = 0, z = 0.5. 48 gp(t) O Figure 3.7. Synthesis of the PGF from component Floquet modes. The modes with the mode indices up to 3 are summed together. The parameters are the same as in Figure 3.5 except that the observation point is at x = d;- / 3, y = dy/3, z = 0.5. 49 .— -4 -3 -2 -1 0 1 2 3 4 Radian frequency (m/wM) Figure 3.8. The waveform and spectrum of the Rayleigh Pulse. This pulse is char- acterized by its finite bandwidth and finite temporal support. The finite bandwidth helps to filter out higher-order Floquet modes and the finite temporal support makes it computationally inexpensive to implement. 50 0.15 I I I I I I I — - -Floquet 01 Element _ I I 0.05 _ FHELD(A) _0.‘ 1 1 1 1 1 1 1 1o 20 30 4o 50 60 7o 30 Tqu(UT) Figure 3.9. Validation of the field radiated by narrow-band array sources. The central wavelength of the source is 2dx. The Floquet modes with indices up to 1 are summed together and the reference field is calculated through element-by—element summation of the field radiated by 6400 dipoles. 51 r I v I r I ° F loquet Element _ I 0.06 I o g 0.04 0.02 00 rnELD(A) 20 25 30 35 4o TumE(UT) Figure 3.10. Validation of the field radiated by wide-band array sources. The central wavelength of the source is d9; / 2. The Floquet modes with indices up to 3 are summed together and the reference field is calculated through element-by-element summation of the field radiated by 6400 dipoles. CHAPTER 4 RADIATION FROM A PERIODIC DIPOLE ARRAY IN THE PRESENCE OF A HALFSPACE The introduction of the concept of the transient plane waves make the spectral expan- sion technique readily adaptable to a setting when a halfspace is present, as shown in Figure 4.1. The basic procedures such as the shifting of the time axis and the spectral axes, Poisson summation, mode pairing and singularity removal can be repeated for the halfspace problem. This chapter first gives the analytic signal representation for a single dipole at the presence of a halfspace, then applies it to the periodic dipole ar- ray to obtain the Floquet modes for the radiated field. The resulting expression for a Floquet mode can be expressed as a finite integral when CT is utilized. Alternatively, the Floquet mode can be interpreted as the superposition of the radiated field by a finite number of image sources with the aid of MPM; the image field is analytically derived and represented as integral-free form, and the final Floquet mode is repre- sented as a finite sum of integral-free expressions. This approach can be regarded as the periodic version of the popular DICM method. 4.1 Analytic signal representation for a single dipole For the problem shown in Figure 4.1, assume that the media in region 1(z > 0) and region 2(2 < 0) are non-magnetic, and are characterized by relative dielectric constants 61 and 62 respectively. The complex index of refraction N is defined as, N2 = 6—2 (4.1) 61 The dipole is located at point (:13',y' ,z’ ) in region 1, where the wave slowness is p1 = l/cl, and the wave slowness in region 2 is p2 = 1/02. 53 First, let’s consider the radiation from a single vertical dipole above the interface. As we know, for a vertical electric dipole, the field can be described by the Hertz potential vector 11', which has only a non-vanishing z—component and in region 1 satisfies the scalar wave equation, 2 (V2 — ii)7r21= —I—)£ (4.2) C where P2 is closely related to the source current by J2 = BPZ/Bt. For simplicity, in what follows we just address it as the current. In region 1, the potential «21 can be represented as the sum of the primary field WP and the reflected field me, 7rd = 71",; + 7r,» (4.3) and these fields are written as, -0 1 77p: W—ztf/j: d1)$dpy19_1A [t — [DICE _ 17”) py(y IV") pzllz — Z I] _—(9 7r,» —2t—87r//: dedppr I32- — f/C—12_px—’Py In the expression for 7n- above, R is the reflection coefficient. The absolute value in [t — p142: - 1")- pyty - y’) - 1921(z + z’)] (4.4) the primary wave represents the singularity at z = 2’, and the reflected wave has no singularity in region 1. The field in region 2, 322, or the transmitted wave, is denoted as 7rt. The transmitted wave has no singularity and satisfies the following homogeneous Helmohotz equation, 54 And it can be written as, :9. 7ft _—_ 8N2 00 T // dpx(lpy1‘9—A [t — 19.1.43: - :r’) — My - y’) — pelz’ + 19224 (46) _m z Where T is the transmission coefficient. The expressions within the analytic delta function can be regarded as a “complex time”, and they are written this way such that they coincide at the planar interface z = 0. This is similar to a phase matching on the interface in frequency domain. Such a representation can greatly simplify the expression of the reflection and transmission coefficients. Because of the symmetry of the problem, the only tangential electric field is E , and the only magnetic field is Hg. Note that 62 Ep = W71“; 82 _Eatap’” (4.7) 11,, = So the boundary conditions are given by, 0 7r 8 7r _ 21 = — z2 62 (92 (48) 2 7le = N ”22 Substitute the expression of 1rd and 7722 into the boundary conditions, and we obtain the reflection and transmission coefficients R : N2p21 “ 1322 N2le + p22 (49) T _ 2p21 _ 2 N (p21 + p.22) These expressions look similar to those in frequency domain except the variables of p21 and ng as functions of wave slowness replace those as functions of wavenumber. 55 Similarly, the expressions of the field radiated by a horizontal electric dipole can be obtained. In general, when the dipole elements are arbitrarily oriented, the radiated field can be expressed in terms of dyadic Green’s function as follows, E1 = (713;; + vv-)g(r|r’) . PSI) 9(I‘lr') = Qp(r|r') + 9"(rlr') gpmr’) = IGp(r|r’) = I——————6(t;rg/C) G;(r|r’) o 0 9"(rlr’) = 0 0;;(r1r') 0 _a%0£(rlr’) 5?,Garlr') Gar-Ira, "Gram , I R. ' G;(r|r’) = $5 // 2 61219 Rn p—l—ZA [t - pm: - a") - My - y’) - 1921(2 + Z')] _ 02(rlr’) . l C . P Rt - — (p21 - pew/(1921 +pz2) 1 Rn = (1)72le ~Pzz)/(N2P21+Pz2) _ C . _ 2W2 - 1)Pzi/ [(19.21 +Pz2)(N2Pz1 +pz2)] j (4.10) where I is the unit dyadic. In the above equations, the upper case bold letters represent transient field vectors, and the calligraphic bold letters represent transient dyadics. These notations shouldn’t be confused with the analytic signals and fre- quency domain quantities as used in the previous chapters. 4.1.1 Analytic signal representation for the periodic dipole array The introduction of the concept of the transient plane waves make the spectral expan- sion technique readily adaptable to the physical setting when a halfspace is present. The basic procedures such as the shifting of the time axis and the spectral axes, 56 Poisson summation, mode pairing and singularity removal can be repeated for the halfspace problem. Mathematically, the major difference is that there exists two dif- ferent z direction spectral variables pzl and p22, which results in the insertion of the reflection and transmission coefficients. Naturally, the definition of p21 and ng needs to be presented first. As defined earlier (3.12b) in 3, the vertical slowness p21 can be written as 01" (4.11) Similarly p22 can also be written in such a form. Due to the application of the transient Weyl’s identity, p32 is defined as, 1 p22 2 \/—2‘ _pz2z: ‘13-?) 62 (4.12) CI 02:37 Similar to (3.1b), pa; and py are shifted to p3, 2 pg; — sax/c and pg] = py — sy/c, and pf; and pg] obeys the same linear relationship as shown in (3.12b) after Poisson summation formulae is applied. Accordingly p22 can be rewritten as 1 C (4.13) N2—s2 2( + ) a2 = ———-——sa so ——- 0% cm xx yy n2 N2 s 5. P22: J27“(sz+f)2—(PL+—y‘l2 When N 2 = 32, p22 can be written as 1 2 1722 = _\/_’$_(3x0~’x + Syay) — (12 (4.14) K Cl 57 When N 2 < .92, p22 can be written as (H-fi2)2 - clsal cosy 52 = W (4.15) (:10 , 52 = ___—N2 _ 32\/N2 — 82811127 192:2: (521—2 ___\/32_( where "y is the angle between the vectors a and 3. When N 2 > 32, p22 can be written as ___M\/(H _ 32)2 _ p22 2 cm - clsa cos'y (32 = W (4.16) cm: . 52 = __N2 _ s2 \/N2 — 82811127 The complexity in the expression of p22 suggests that a periodic dipole array may exhibit very different electromagnetic performance when the material property and the excitation pattern are given differently. 4.1.2 Analytic signal expression for the Floquet mode 2 For mathematical simplicity, consider the case when N 2 > s . Following similar derivation procedures to the free space case, the reflected field is rewritten as .r 0° R' K: . ,. 077m“) : A/O fir; eXP [J(KT — p:1(H)T0 _ a ' P)] xp’zfin) - 192200 XIJQIW - 1922:) P21( )=\/(K _ 302 (4.17) p;,(,.=) (fie—82) w: _N2\/1—s2 X‘ 77177:? R’(I—p.e (418) plan) = (/(,. +3112 — (7% P2200 = \/(H + 32? - fig It has been shown in (3.31) that the reflection coefficients for these two companion modes satisfy fled-re) = pate) (4.19) and it follows that R/(—I€) : XP%1(_’€)-P:22(—K') szl(_’€) + p22(_h’) = X15107) - peg“) (4.20) XPz1(K) + 19,200 : R"(h) Therefore, the real signal gmn(t) associated with the analytic signal Gmn(t) can be written as great) = ; [Geese + Team] A 00 R’(I€) . I r =5/0 1p21re—a-p>1 (4.21) II A 0 (H) -. ” - T 59 Similarly the real signal g(_m)(_n) (t) associated with the opposite mode G(_m)(_n) (t) can be expressed as, ,. 1 9(-m>(—n)(t) = § lG<—m)(—n>(t) + G<—m)(—n>(t)l _ A 00 R”(I€) . " r — 2/0 77:1(x)e"pij(”"’z1(‘)70“”01 (422) A 0 R, h: . I r + 2 /-00 1221(2) exp [30” — 10.210070 ‘ a ' ‘0’] Therefore, a more compact form can be obtained if these two modes are summed together as follows, 91771171“) : 977271“) + g(—m)(—n)(t) A 0° R’ It , ,. = e 1.. 7.1:. w _. ~ ,, A m R”(K,) . II 7, + — 17—— 8 ET — z K3 7' + a ' 2 _00 Wed") xp [2( p 1( ) 0 0)] — 2 -oo j, 062 __ [32 MN) = RIM + .31) €XP(J'(5T - a '10)) + R”(e — a) exp(-J‘(BT — a - p» — A foo Rexp [Jim _ ,—————H2 _ 431276)] (4.23) and R(-K) = R’(—F~ + .31) 8744-7137 - a - (0)) + R"(—r~: - B1) exp(7’(BT — a - 10)) = R” eXP(-I(BT — a - p» (424) + R’(K + 31) exp(j(BT — a '10)) = B(K.) 60 It can be readily found that the integrand is an even function and can be further simplified to a semi-infinite range integral as exp [)(m — ,Ix2 — [3?7'6)] Ti 00 gum“) : A iR / dHR(I€) 0 j‘/I-i2 — [3% (4.25) 12(5) 2 cos(617' — a - p)Rp +jsin(1317‘ — a - p)Rm where the variables RP and Rm are defined as RP=R’(x+B)+R’(x—I§) _ X n2-fif— (/(K+B1~B2)2—fl§ x 32—fi¥+(/(n+51 —B2)2 —H% + Xy/HQ—flf- \/(K—B1+B2)2 ’33 We? —13f+ (/(n -Ii1+f32)2 —II§ (4 26) R... — me + B) — R” —61, close the path of integration by P1 and P+oo. The integrand is analytic within the closed path, then the integral along the closed path vanishes, and we have H®=h+h+hm an) where 11,12, [+00 denotes the integral along the path P1, P2 and P+oo respectively. Since B(t) —> T — T0+61 — j 6 R on P+oo, it’s readily verified that the integrand vanishes on P+oo, then [(6) = 11+ 12 (4.36) The objective here is to find the integral along the path P1 and P2. On P1, let K = jy, then the integral along P1 can be rewritten as T + 61) — Vy2 + ,1327'0) 1/y2 +52 0" . . ex — 11=/0 dyJexp(J5Ry) p( y( , 0° ex — T+6 -\/ 2+£32T = 3 / dycos(6Ry) p( 3‘“ ’2) 2y 0) (4.37) 0 y +3 °°d ,. eXP(-y(T+61) - x/y2 +fi2T0) — ys1n(6Ry) 2 _2 0 y +0 These two integrals can be regarded as Fourier sine transform and Fourier cosine transform with respect to 63. It seems that the integrand cannot be found from the Fourier transform tables directly, but the integrand can be written as the product of two different functions, whose Fourier transform is known. Then the integral can be represented as the convolution of two functions. Rewrite these integrands as exp(— \/ 312 + 32T0)) OO 11 = If dy 008(6Ry)exp(-y(T +61))- 2 2 0 y +fi 0" ex —- T —— 2 2T —/0 dysmmm p< y< +61))),(yexp< \/y +73 0)) y 66 Some useful Fourier sine transform pairs are found [31] and listed below ffil’) 9(9): fowf (:L')sin( :ry)d:1:, for y > 0 e—a‘f/x for §R(a) > 0 tan—1(a“1y) (4 39) mam—We? + ewe? + 42‘ aeKaw—yz + 132W—y2 + 02 for §R(a) > 0, 31(6) > 0 and some useful Fourier cosine transform pairs are also listed f(:1:) =f0°°f c)os (:1:y)d:r .73, for y > 0 e"‘“’ for§Ra >0 a '2+a2 ( ) / (y ) (4.40) “New? + 03/7ch + a? K0(a\/W)/\/W for §R(a) > 0, 32(6) > 0 As we know from the frequency domain convolution theorem, the Fourier transform of two functions is the convolution of their inverse Fourier transform divided by 277. Therefore the integral can be represented in convolutional form as follows , T+61 2 2 ) __1_ [1: 3(62 +(T+612) 2)*(K0(,L3 6R+T0) 2 —tan_l( 5R )* H5RK1(3 5%;+T3) 1 /2 2 % 6R+T0 Note here the convolutions are performed with respect to 63. As we know that the (4.41) modified Bessel functions of either zeroth order or first order decay very fast with the increase of the absolute value of 63, the terms containing modified Bessel functions have only finite support, therefore, it’s not expensive to evaluate these convolutions. On P2, the integral is a branch cut integration. Let 6 = \/ H2 — 62, then the 67 integral can be written as _ 0° eXp(J(~:T - n2 - (32m) 12 — /0 exp(dn) j n2 _ 32 _ 0° . exP(- {2 +fi2(-5R -j(T +51») — [00 dram—m) 42+ fi, 00 _ 2 2 _ __ - =—j dgcosmfxm 6 +5( 512 J(T+51))) 00 2 +182 _ /00 d€Sln€ToeXp(—V€2 +I32(—5R -J'(T + 51») 00 {24-52 (4.42) m The second term vanishes since the integrand is an odd function with respect to 5. The integrand of the first integral is an even function, so the expression can be written 35, W2 + 52(-5R - j(T + (51») W This can be regarded as a Fourier cosine transform, and we can use the following 12 = —2j [:0 dg cos({7'0)exp(_ (4.43) transform pair [31] f (It) 9(y) = f6” f (:13) 008(wy)d$, for y > 0 exp[“5\/x2 + 02]/\/112 + 042, KMaW) (4-44) for §R(a) > 0, §R()3) > 0 Therefore the integral of our interest can be written as 12 = -2J'K0(fl T3 + (512 +j(T + 51))2) (4-45) When 7' < —61, close the path of integration by P3 and P_Oo. The integrand is analytic within the path and the integrand vanishes. Meanwhile the integrand vanishes on P_OO. Therefore [(6) = —13 (4.46) 68 Similarly let 5 = —jy, we have . 00 - exp(—\/y2 + (3270) 13 = —J / 4y 605(5331) exp>2)U] 2 p20 +82% ”22(12ij pzo Wl — )21i Z (ROTIuE)1A it — p.112: - 23') - py(y - y') - pzo(z - 2') - pz1(2ud)] u=1 (5.4) p02 (5.5a) .M/ W1 R2 sz 01 A [t - p.111: — x')- py(y- y ') — 1020(2 - 2')l (5619) twig/rd 2311313241 (R3. ’3“) A [t — m — 2%) — my — y’1 — pzo+ 32(511Ke — 81W 0 95m = /0 ”(1235.) + 124(5) + 12501111101 —- mm R101) = Rm + B) exp(jwv — a ~ p)) + 12,-(1. — B1exp(—j(Br — a - p1) = cos([3T — a - p) [R+(K + [31) + R—(H + 81)] (5-16) +jsin(BT — a - p) [R301 — Bl) — R—(n — 31)] And R205) to R5052) are defined in a similar way to R105). To evaluate these five 85 integrals, we can apply the two-level approximation technique to the coefficients us- ing matrix pencil method again, and the closed-form expressions can be obtained accordingly. 5.3 Discussion The field radiated from a single dipole over a microstrip is obtained by using geomet- rical series expansion and complex exponential approximation, and it’s interesting to compare the fields due to a single dipole and a periodic dipole array. For the single dipole case, the transient radiated field is represented as the sum of quasi-dynamic field and image field, and the former can be expressed in closed-form, but the latter can only be obtained by inverse Fourier transform. For the periodic dipole case, the field can be represented as an infinite sum of transient plane waves based on the ana- lytic signal formulation, and it enables the representation of the field by a convergent geometrical series, which can be summed together and expressed in closed-form. This again reveals the advantage to use plane wave analysis in analyzing electromagnetic scattering problems. 86 [7. (601/130) 11 (Gum) Figure 5.1. Problem setting over a microstrip structure 87 R12 (611 M1) (627 #2) Figure 5.2. Multiple reflection within a microstrip structure 88 CHAPTER 6 CONCLUSIONS AND FUTURE WORK In this dissertation, the fields radiated by a periodic dipole array in free space, in the presence of a halfspace and over a microstrip structure are investigated, and the field solution is expressed as an infinite summation of transient Floquet modes. This work aims primarily at at finding physical interpretations to the wave behavior and deriving expressions for faster implementation. The free space problem is a reformulation of the problem in [11]. Unlike the inverse-Fourier-transform-based approach, the solution is expressed by spectral ex- pansion using analytic signals. With the aid of analytic signals, the contributions of each homogeneous and inhomogeneous component plane waves can be conveniently analyzed. It also enables the application of CT technique to replace the contribution of inhomogeneous plane waves by homogeneous plane waves propagating along op- posite directions opposed to the original homogeneous plane waves. Mathematically, this replacement reduces the expression of the Floquet mode into a finite integral and can be evaluated in closed-form. Alternatively, another approach based on the transient Weyl’s identity is also presented, and the resulting infinite integral can be evaluated in closed—form by converting it into a contour integral. The third approach is based on the Whittaker-integral representation and the formulation is based on real signals directly. The first approach results in simpler expression and more physical in- terpretation, and the second approach is more general and is more adaptable to other physical settings. The third use real signals directly and therefore mathematically elegant, but is limited to simultaneous excitation at present. When a half space is present, the transient Weyl’s identity can be conveniently applied, and the reflection and transmission coefficients are inserted into the spec- 89 tral expansion expression. There are at least two ways to deal with this additional complexity. The first approach is to express it as a finite integral or semi-infinite integral, and evaluate it accordingly using an appropriate numerical integration tech- nique. The second is to utilize the DCIM to represent the solution as a finite sum of closed-form expressions. For the microstrip case, the analytic signal expression can be obtained by per- forming a ray expansion first. Since each ray corresponds to its transient analytic signal expression explicitly, the F loquet mode can be assembled by summing up all of these ray components. Finally, the field can be represented by a finite number of integrals, and each of them can be evaluated by either using a numerical integration technique or using the DCIM. 6. 1 Future work This dissertation focuses primarily on the theoretical derivation of the transient F 10- quet modes, and the performance of the periodic dipole array under different material settings and excitation patterns is not fully explored. Also the field radiated by the periodic dipole array over a general layered media is not studied yet. To be more specific, here are some suggestions for future work: In free space, the spectrum of the Floquet modes are affected by many factors. It’s theoretically interesting and practically useful to find out how the observation point, the mode orders and excitation pattern etc affect the mode spectrum. In the presence of half space, the refractive index and the excitation pattern can shape the Floquet mode deeply. At present only the case of 62/61 > 1 is discussed. More work should be done to investigate how to affect the Floquet modes by choosing different material properties and excitation patterns. At the same time, a better numerical technique should be developed to evaluate the modes. This numerical technique can be applied on the finite integral form or the semi-infinite integral form 90 respectively. The major challenge in the former is to find a semi-analytical semi- numerical approach since the integral contains two different kinds of singularities; the latter is more may place the emphasis on how to evaluate a highly-oscillating function. In a layered media environment, the problem is even more difficult since the re- flection coefficient becomes more complicated and the material setting permits more options. In this case the CT technique may face of the danger of becoming ineffec- tive. It would be even more difficult to find an inhomogeneous plane wave to cancel another. 91 APPENDICES 92 This appendix presents analytical evaluation of integrals encountered in (3.25), namely 00 I3 = / d0 sin(37 cosh 9) cos (370 sinh (9) 0 oo (1) I4 = / d0 cos(37 cosh 0) cos (370 sinh 0) 0 These equations can be rewritten as 1 OO 13 = 5/ d6 [sin (3(7 cosh 6 + 70 sinh 9)) + sin (/3(7 cosh 0 — 70 sinh 0))] 1 000 (2) I4 2 §/ (16 [cos (3(7 cosh 0 + 70 sinh 6)) + cos (3(7 cosh 6 — 70 sinh 6))] 0 When 72 > 73, let (1') = tanh“1 (If), then 1 OO 13 z —/ d6sgn(7) 2 0 [sin (3 72 — 73 cosh(0 + 56)) + Sin (3 72 — 73 cosh(9 — (1)0] = 1 [00 + [00) d0 sin (3 72 — 7g cosh 6) sgn(7) (3) 0‘) -¢ 00 (16 sin (3 72 - 7g cosh 0) sgn(7) J0 (3m) sgn(7) The third equation used the property that the integrand is an even function, and the II o\. “3' Nlfi last equation used the identity [26] 0° 17 / d6 sin(t cosh 6) = EJ0(t), t > 0 (4) 0 93 Similarly 14 can be simplified as 1 00 00 I4:—(/ +/ )d6cos(3 72—73cosh6) 2 «12 -¢ 00 :/ (16003 (3 72—7gcosh6) (5) 0 —7T The second equation used the property that the integrand is an even function, and the last equation used the identity [26] 0° 77 / d6 cos(t cosh 6) = —§Y0(t), t > 0 (6) 0 When 72 < 73, let (25 = tanh—1 (7L0), then 13 = 1/00 d6 [sin (3 72 — 73 sinh(¢ + 6)) + sin (3 72 — 73 sinh(q§ — 6))] 0 2 1 00 00 =—(/ —/ )d6sin(3 72—7028inh6)=0 2 4> —a> The last equation used the property that the integrand is an odd function. 1 OO 00 142—(/ +/ )d6cos (3 72—7gsinh6) 2 —¢ ¢> 00 =/ d6 cos (3 72 — 73 sinh 6) (8) O = K0 (3 7g — 72) (7) The second equation used the property that the integrand is an odd function and the last equation used the identity [26] 00 / d6 cos(t sinh 6) = K0(t), t > 0 (9) 0 94 In summary, 13 and 14 can be expressed as 13 _ ——JO (52 — 73) U (72 — 73) sgn(7) [4: ——Y0 (37 2—70>U(7-2_70) +K0 ((3 95 2_ 7' 7'0 2 BIBLIOGRAPHY 96 [1] l2] [3] [4] [6] [7] [8] [10] [11] [12] BIBLIOGRAPHY B. 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