025921-14-T TECHNICAL REPORT FOR NASA Grant NAG-2-541 NASA Technical Monitor: Alex Woo A NEW TECHNIQUE FOR SIMULATING COMPOSITE MATERIAL Institution: The Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, MI 48109-2122 Period Covered: Report Titl: Report Authors: February 1990- September 1990 Electronmagnetic Scattering from Two-Dimensional Thick Material Junctions M.A Ricoy and J.L. Volakis John L. Volakis Telephone: (313) 764-0500 Principal Investigator:

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Electromagnetic Scattering From Two-Dimensional Thick Material Junctions August, 1990 Mi. A. Ricoy and J. L. Volakis Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109-2122 Abstract-Because material junctions are commonplace on structures whose radar cross section is of interest, it is essential that their scattering properties be adequately characterized. The standard impedance boundary condition (SIBC) has been employed in the past along with function theoretic techniques to develop simple scattering models of material junctions with thin and/or high loss slabs. To extend these models to more general slabs, generalized impedance boundary conditions (GIBCs) and generalized sheet transition conditions (GSTCs) have been proposed. Unfortunately, the solutions obtained with these are usually non-unique in the form of unknown constants, and although the constants have been resolved for a few special cases, previous efforts were unable to determine them in the general case. This report examines the problem of the plane wave diffraction by an arbitrary symmetric two-dimensional junction, where Generalized Impedance Boundary Conditions (GIBCs) and Generalized Sheet Transition Conditions (GSTCs) are employed to simulate the slabs. In chapter 2, GIBCs and GSTCs are constructed for multilayer planar slabs of arbitrary thickness and the resulting GIBC/GSTC reflection coefficients are compared with exact counterparts to evaluate the GIBCs/GSTCs. In chapter 3 the plane wave diffraction by a multilayer material slab recessed in a perfectly conducting ground plane is formulated and solved via the Generalized Scattering NMatrix Formulation (GSMF) in conjunction with the dual integral equation approach. Various scattering patterns are computed and validated with exact results where possible. In chapter 4, the diffraction by a material discontinuity in a thick dielectric/ferrite slab is considered by modelling the constituent slabs with GSTCs. A non-unique solution in terms of unknown constants is obtained, and these constants are evaluated for the recessed slab geometry of chapter 3 by comparison with the solution obtained therein. Several other simplified cases are also presented and discussed. In chapter 5 an eigenfunction expansion method is introduced to determine the unknown solution constants in the general case. This procedure is applied to the solution of chapter 4, and scattering patterns are presented for various slab junctions and compared with alternative results where possible. Chapter six presents a short summary of this report and some recommendations for future work.

TABLE OF CONTENTS LIST OF FIGURES............... LIST OF TABLES................................ viii CHAPTER I. INTRODUCTION.......................... II. DERIVATION OF GENERALIZED TRANSITION/ BOUNDARY CONDITIONS FOR PLANAR MULTIPLE LAYER STRUCTURES....................... 2.1 Derivation of Infinite Order Conditions............ 14 2.2 Low and High Contrast Approximations for Matrix Element Operators............................ 22 2.3 Evaluation of the Boundary/Transition Conditions...... 27 2.4 Concluding Remarks..................... 43 III. DIFFRACTION BY A MULTILAYER SLAB RECESSED IN A GROUND PLANE VIA GENERALIZED IMPEDANCE BOUNDARY CONDITIONS.................... 44 3.1 Description of GSMF Procedure................ 48 3.2 Plane Wave Diffraction and Mode Coupling............ 51:3.3 Reflection and Launching of a WNaveguide Mode.... 56 3.4 Computation of Spectra for Material Insert in a Perfectly Conducting Ground Plane..................... 5S:3.5 Specialization to the GIBC Representation........ 5. ):3.6 Numerical Results........................ 62 3.7 Summary............................. 6!. i

1V GENERAL SOLUTION OF THE DIFFRACTION BY A M ATER - I. DISCONTINUITY IN A THICK DIELECTRIC FERRLIT ': SLAB.......................... 4.1 Dual Integral Equation Formulation.............. 7G. o lution of the Dual Integral Equations........ 4-.:3 Discussion of the Solution and Some Applications... 4.3.1 Diffraction by thin single layer discontinuous slabls.: 4.:3.2 Diffraction by a resistive-resistive junction.... 4.3.3 Diffraction by grounded metal-dielectric junctions. '.i 4.3.4 Diffraction by a thin dielectic/fetrrite half plane. 1 4.4 Conclusion............................ ' V. RESOLUTION OF NON-UNIQUENESS ASSOCIATED WITH THE GIBC/GSTC SOLUTION.................. 101 5.1 Modal Decomposition for the Symmetric Slab Problem...i03 5.1.1 Exterior Cross Section Functions........... l05 ~5.1.2 Interior Cross Section Functions................ 5.2 Recas.tin of the Dual Integral Ecuation Solution for a Material Discontinuity............................5.3 Determination of the Constants............... 122.5.4 Validation of the Solution........... 1 5.5 Summary.....................152 VI. SUM M AI RY...................... 15:3 A PPE N DICES.................................. 157 BIBLIOGRAPHY........................... 161 ii

LIST OF FIGURES Figure.[ SI t.ruct lies nmoelled by SBC........ 1.2 Resistive/conductive sheet simulation of thi n slab....... ~2.1 Infinite rmultilav er slab.........................1 2.2 (a) Multilaver coating on ground plane. (b) Equivalent opaque sheet. 1 2.3 (a) Zero thickness resistive and conductive sheet similation of multilayer slab. (b) Distributed resistive and conductive sheet simulation of m ultilayer slab.............................:3 2.4 Coordinate system for reflection and transmission coefficient derivation.......................... 27 2.5 E, polarization low contrast simulation of a single layer having relative constitutive parameters e = 2,,u = 1.............. 30 2.6 E, polarization low contrast simulation of a single layer having relative constitutive parameters e = 3.5,, = 2.............. 31 2.7 E, polarization high contrast simulation of a single layer having = 3.5. = 2............................... 32 2.S Ey polarization high contrast simulation of a single layer having e= 5, = 3................................. 33 2.9 Hy polarization low contrast simulation of a single layer having relative constitutive parameters e = 2,, = 1.2.............. 36 2.10 Hy polarization low contrast simulation of a single layer having relative constitutive parameters e = 3.5, /. = 2............ 37 2.11 H, polarization high contrast simulation of a single layer having 3.. = 3............................... *.. 111

'2.2 1 H polarization high contrast simulation of a single layer having -= ). =: 3......................... '. '.; L ir-:l:ation of a three layer slab using a 14' order condition.... 2. 'i l::ulation of a three layer coating on a PEC ground plane using a L4 order condition......................... 2.15 Simulation of a three'layer coating on a PMIC ground plane using a 14`. order condition...........................:3.1 ia) *Multilaver slab recessed in a PEC ground plane. (b) Representation of slab as surface with reflection coeftficcit R.......:3.' Illustration of recessed stub geometry [a) and associated subproblems: (b) direct diffraction, (c) mode coupling, (d) stub reflection. (e) mode reflection at the waveguide mouth, (f) mode launching. 46:3.:3 H: polarization backscatter echo width for a material insert with =.2A, = - - j.0001,,u = 1.2 modeled by low contrast GIBCs (see Table:3.1 for an explanation of the legend entries). (a) direct diffraction component (b) modal component (c) composite..... -;3 3.4 H. polarization backscatter echo width for a material insert with - =.4A. e = 3.5 - j.0001, / = 2 modeled by low contrast GIBCs (see Table 3.1 for an explanation of the legend entries). (a) direct diffraction component (b) modal component (c) composite..... 64 3.5 H: polarization backscatter echo width for a material insert with r =.4,\, e = 3. -j.OOO1, A = 2 modeled by high contrast GIBCs (see Table 3.2 for an explanation of the legend entries). (a) direct diffraction component (b) modal component (c) composite..... 66:3.6 H, polarization backscatter echo width for a material insert with r = A, 11. - j.0001, A = 7 modeled by high contrast GIBCs (see Table 3.2 for an explanation of the legend entries)........ 70 3.7 H, polarization backscatter echo width for a three-layer material insert with (r1 =.4A, e = 11. - j.0001, 1 = 7), (r2 =.4, 62 = 3.5 - j.0001, 12 = 2), (63 =.2 - 0001, 3 = 1.2). In the legend entry ab,c(d,e), a and b denote the approximation in c-1 of the high constrast layers 1 and 2, c denotes the approximation in T of the low constast layer 3, while d is the order of the composite GIBC and e is the total number of modes.............. iv

4.1 '" Dj 'i!-''ljedl -:ieett. t'V Distriluted siheet discontinuity...' (,e,:,i::L<ries mod;eled by a discontinuous distributed sheet. (a) Material half-plane. (b) Material-material join. (c) Grounded join... 75 r.;3 la< Illustration of C contour in the complex a plane and (b) complex A plane................... 4.4 Thin slab structures and their distributed sheet representation (ai Material-material junction. (b) Material-metallic join. (c) Material half-plane. (d) Equivalent distributed sheet.............. 4.5 (ia) Recessed slab (PEC stub) (b) Grounded slab wit t"u:icated upper plate (c) Recessed slab (PM.C stub' Ad) (T IBC sheet.... )4 4. I: polarization backscatter echo width for a material insert having =.04A. 6 = 2 - j.0001, tL = 1.2 modeled with O(w) low contrast GC IB C s.................................. 4.7 H: polarization backscatter echo width for a material insert having LU =.10OA, = 2- j.0001. 1 = 1.2 modeled with 0(w3) low contrast GIBCs; b1o = -.4.578+j.2593, bll = -.0408-j.0111, b20 = -.0401 -j.0:3S..................................... 5.1 Smmetric multilayer slab discontinuity illuminated by a plane wave 102 5.'2 Symmetric multilayer slab with irregular termination illuminated by a polarized field excitation....................... 103 5.3 (a) C contour in the complex A plane. (b) Deformation for region 1 integrals. (c) Deformation for region 2 integrals............. 115 5.4 Indexing scheme for constants...................... 124 5.5 H, polarization backscatter echo width for a recessed slab with r =.4, E = 2, [ = 1.2 modeled by O(r) second order low contrast GIBC (see Table 5.2 for constants)...................... 13:3 5.6 H: polarization backscatter echo width for a recessed slab with r =.1. t = 2. / = 1.2 modeled by 0(r3) fourth order low contrast GIBC (see Table 5.2 for constants).................... I: t V

5.7 H. polarization backscatter echo width for a recessed slab with =.4. 5.: = 3 modeled by O0(~-1) second order high contrast CGIB see Table 5.2 for constants).. 7. /f;f pjiolarization backscatter echo width for a material half-plane with - =.005. = 2, i = 1.2 modeled by 0(r) low contrast GSTC (see Table 5.2 for constants).....1............ H, polarization backscatter echo width for a material half-plane with =.05. e = 2. Ai = 1.2 modeled by 0(r) low contrast GSTC (see Table.5.2 for constants)....................... 137 5.10 H, polarization backscatter echo width for a material half-plane with =.10. e = 2, Ai = 1.2 modeled by O(r) low contrast GSTC (see Table 5.2 for constants)..................... 5. 11 H, polarization backscatter echo width for a material half-plane with r =.20. e = 2, 1 = 1.2 modeled by 0(r) and O(r3) low contrast GSTC (see Table 5.2 for constants).................. 139).5.12 H: polarization backscatter echo width for a material half-plane with r =.01, e = 5,.l = 3 modeled by 0(~O) (even symmetry) and O(^-1) (odd symmetry) high contrast GSTC. The free space side is modeled by O(r) (even symmetry) O(r2) (odd symmetry) low contrast GSTC. (see Table 5.3 for constants)............. 10 5.13 H. polarization backscatter echo width for a material half-plane with r =.05. e =.5, / = 3 modeled by 0(0~) (even symmetry) and (0-1) (odd symmetry) high contrast GSTC. The free space side is modeled by O(r) (even symmetry) 0(r2) (odd symmetry) low contrast GSTC. (see Table 5.3 for constants)............. 141.5.14 H, polarization backscatter echo width for a two layer.07A symmetric material half-plane with r1 =.005, el = 5, /i = 3 and r2 =.03, e = 2, I = 1.2. Layer 1 is modeled by O(K~) GSTC, layer 2 and the free space layer are modeled by 0(r) (even symmetry) 0(r2) (odd symmetry) low contrast GSTC (see Table 5.3 for constants).... 14.5.15 H, polarization backscatter echo width curves for different thickness material half-planes with e = 2, u = 1.2. All cases are modeled by O(r) low contrast GSTC (see Table 5.3 for constants)....... 143 vi

5.16 H, polarization backscatter echo width curves for different thickness c'....' 'ed [A'!', dwith e = -. Lt = 3. Tlhe Lnul)mber to the left of t,he ~- '. ',',l-,::the ()-,~ oft the I.c. GIBC for tlie tree space side. a t_:.:ini'te:' tio e.ig.t of the colon denotes the i0 1- of the L.c.:ai GIBC................................. 1 t..1,.i- polar'zation backscatter echo width curves for different thickness (rounded slabs with e = 11. t = 7. The number to the left of the colon denotes the O(r) of the I.c. GIBC for the free space side. and the number to the right of the colon denotes the O(~-l) of the h.c. GIBC for the right hand slab................... 14..5.1s H- polarization backscatter echo width curves for different thickness grounded junctions. with e6 = 2. yt = 1.2 and 62 = 5. A/L =:3. The number to the left of the colon denotes the 0(T) of the I.e. GIBC for the left hand slab and the number to the right of the colon denotes the O(-L) of the h.c. GIBC for the right hand slab......... 146 5.19) H. polarization backscatter echo width curves for different thickness grounded junctions, with c =- 5. 1, =:3 and 62 = 11. -2 = 7. The number to the left of the colon denotes the O(j-1) of the h.c. GIBC for the left hand slab and the number to the right of the colon denotes the O(~-') of the h.c. GIBC for the right hand slab.... 147.5.20 H- polarization backscatter echo width curves a grounded junctions, with '1 = 2 /^i1 = 1.2, r1 =.025 and 62 =, 2 = 3. The first number to the left of the colon denotes the 0(r) of the I.c. GIBC for the left hand slab and the first number to the right of the colon denotes the O(n-1) of the h.c. GIBC for the right hand slab. The second number on either side of the colon denotes the 0(r) of the l.c. GIBC for the free space slab needed to give the two sides the same thickness. 148 5.21 H. polarization backscatter echo width curves a grounded junctions, with e1 = 5, 1l = 3, 71 =.05 and 62 = 11, /2 = 7. The first number to the left of the colon denotes the O(n-1) of the h.c. GIBC for the left hand slab and the first number to the right of the colon denotes the O(c-1) of the h.c. GIBC for the right hand slab. The second number on either side of the colon denotes the 0(r) of the l.c. GIBC for the free space slab needed to give the two sides the same thickness. 149 vii

LIST OF TABLES Table 2.1 Low constrast approximations to q;j operators............. '2.2 High constrast approximations to qi operators.............':3.1 Low contrast approximation to waveguide modes for a layer with e = 3.5 - j.0001. = 2.0, and r =.4\. For each low contrast boundary condition. the three numbers of the column headings indicate the order of the approximation in thickness r, the order of the resulting boundary condition. and the total number of modes (see chapter two). 67:3.2 High contrast approximation to waveguide modes for a layer with = 3.5 - j.0001., = 2.0. and r =.4A. For each high contrast boundary condition, the three numbers of the column headings indicate the approximation in the index of refraction s-', the order of the resulting boundary condition, and the total number of modes. (S 4.1 Odd symmetry parameters for distributed sheet similation of right hand side material x > 0. See Appendix 1 for definition of M1+ split functions................................ 90 4.2 Even symmetry parameters for distributed sheet similation of right hand side material x > 0. See Appendix 1 for definition of M+ split functions................................ 91 5.1 Asymptotic behavior of integrand for the functions Fdiff (x = 0, y). 127 5.2 Values of solution constants for curves presented in Figures 5 to 12. 150 5.3 Values of solution constants for curves presented in Figures 12 to 1. 151 viii

CHAPTER I INTRODUCTION The use of non-metallic materials is now commonplace on airborne vehicles. andl frequently trwo material slabs of different composition will abut each other to form a junction. For example. thin radar-absorber material (RAM) strips of different composition - often joined end-to-end on a metal surface for radar cross section reduction. Also. in microstrip antenna configurations a discontinuity in substrate composition is used to suppress unwanted surface waves and in many cases a planar array is terminated at a metal-dielectric junction. The electromagnetic effectiveness of structures and devices such as RANI coatings or microstrip antennas is influenced by the scattering behavior of any material junctions present. It is therefore important to obtain a characterization of their behavior. The scattering behavior of a material slab junction is revealed by an examination of its plane wave diffraction, and this is the overall topic of the dissertation. For discussion purposes, it is useful to divide these into non-penetrable and penetrable slab junctions. The scattering problem associated with non-penetrable junctions was aided greatly by the introduction of standard impedance boundary conditions (SIBCs) [351. introduced to model thin metal backed coatings and layers of hihli loss (see Figure 1.1). Conceptually, SIBCs are first order boundary conclitioW. 1

Lossy half-space y 1 4y ' X '"I y X /SIBC shee SIBC sheet, impedance ql Material Coating Y.:i:ii:i.:i:.i:I:ii:i iiii:::::::.............................. /Ey + DEy + - - 'jkrlE= 0 on y=0 aH jk + -y H O on y=O ay rl Figure 1.1: Structures modelled by SIBCs. whlich replace the original material slab with an equivalent "sheet," on which the tangential electric and magnetic fields are related through a simple proportionalitfactor known as the "impedance"' of the sheet. Alternatively, the SIBC relates the normal field components and their normal derivatives by the same proportionality factor. The SIBC model thus eliminates the need to consider the field interior to the slab. reducing a two-medium problem to a one-medium one. Following this modeling scheme, grounded slab junctions and coated half-planes are represented as discontinuous SIBC sheets, which readily permit the application of function theoretic techniques such as the Weiner-Hopf method to obtain diffraction solutions. These techniques yield a unique solution upon application of the standard eclde condlition [49], which dictates that the stored energy in the vicinity of the 'the order of a boundary or transition condition refers to the order of the highest derivative present when the condition is cast in its normal derivative format

3 discontinuity must remain finite. This tswo step approach in computing the diffraction f:-ro.:n ~:'a J'::cons has been successfully exploited by many researchers;:341 'IS. -' 17';)i. Hrodwever. the approach is predicated on the validity of the SIBC. which is eStr ic'e(i to rnodelin, very thin and/or lossy coatings. More general slab junctions must therefore be characterized by other methods. W ith regard to junctions formed by penetrable slabs, a close analog to the SIB( sheet modlel is the -"resistive' and "conductive" sheet simulation 121]:3,1. These i:Cets are chiaracterized by simple first order transition conditions which relate the tanential fields across the sheets. In particular. a resistive sheet (see Figure 1.2). supports an equivalent electric current which produces a discontinuity in the tangential magnetic field across the sheet. The proportionality factor relating the tangential mnagnetic field discontinuity to the equivalent electric current is denoted as the 'resistivity" Similarly, the conductive sheet model is the dual of the resistive sheet and supports an equivalent magnetic current, with the resulting proportionality factor denoted as the "conductivity" of the sheet. Like the SIBC models above, resistive and conductive sheets models of slab half-planes and junctions are amenable to 'V einer-Hopf methods for the computation of diffraction solutions [1] [44], with the same comments given above applying here also. We remark, however, that the resistive and conductive sheet models are very restrictive (much more so than SIBC models) and cannot be used unless the modeled slabs are very thin and of high permittivity and/or permeability. Hence, as in the case of impenetrable slab junctions, alternative methods are needed to model more general junctions. One such exact approach was employed by Aoki and Uchida [3] to tackle the problem of plane wave diffraction from a penetrable single-layer slab junction. Their method involved rewriting the junction field components in terms of a Fourier o

4 Thin Slab tY I I..... I 1-......................l.: r ~:~:~' )~ * jkT(~-l) - 1/R jkT(-.1) - 1/> m Y Resistive sheet of _ 2, = x Ex+E- = 2,Zo(Hz-H), E:+Ez = -2R Zo(H+-H) + Conductive sheet of conductivity Rm A-ix Zo(H:+H,) = -2m(E:-Ez) Z(Hz+Hz) = 2,m(E>-E,) Figure 1.2: Resistive/conductive sheet simulation of thin slab.

ries representation. This approach subsequently led to the generation of \\einerHopf ecua':os e:pressed in terms of unknown spectral functions. However..-exp.:i ri:ui tese functions could not be obtained. and the resulting solltin,1ivoi'ed a rcumbersome iterative procedure requiring knowledge of rather complex,,e~r> an-]:'un:ctions. Another possible approach to modeling thicker and more penetrable layered discontinuities is to replace the SIBC and resistive sheet transition conditions -ST(C by seneralizecl impedance Lc-und!ary conditions (GIBC) [15j [1' `231 and generalized sheet transit on condlitions (GSTC) [59] [39], respectively. The GIBC and GSTC are equivalent one- and two-sided sheet representations which, unlike their SIBC and STC counterparts. display second and possibly higher order derivatives of the field components on the equivalent sheet. These higher -c:rder derivatives are responsiIble for the increased accuracy of the GIBC/GSTC relative to the SIBC/STC. and the effectiveness of the GIBC/GSTC is roughly proportional to their order. This. of course. implies that an adequate GIBC/GSTC modeling of any lavered material is possible, provided that GIBC or GSTC of sufficient order are employed. Many GIBCs and GSTCs have been derived to model all types of single and multiple layers [59] [39] [31] [431. However, most of these GIBC/GSTC are either limited in order (usually second) or else are only valid for specific geometries. This, of course, points to the need for developing more general GIBC and GSTC capable of simulating a wide variety of layered slabs and coatings. Once a slab junction is represented as a discontinuous GIBC/GSTC sheet, then function-theoretic techniques may again be applied to compute the plane wave diffraction. as in the SIBC/STC case. However, function theoretic solutions based on the application of GIBC/GSTC simulations yield solutions which are non-unique

6 even after the application of the standard edge condition [321 [41] [43] r361. AlK recipr ci'y i- not necessarily aat isfed i1] 44] [,7]. unless this condition is ex ''. it: eftorce(d. I'niqueness is required of any physical solution, whereas reciprocity w'it reaspe(t to the transmitter and receiver is necessary when the scattering body is electrically passive. As noted in [41], one may take advantage of this inherent non-uniqueness to force a solution which is at least reciprocal. if not unilque. Thy non-uniqueness of the solution. usually manifested in terms of unknown constants [41] for finite-order GIBC "GSITC and unknown entire functions [32] for infinite-order GIBC/1GSTC. Most GIBC,/GSTC diffraction solutions to date have been obtained using second order GIBC/GSTC. and some of these have put forth arguments dealing with the cancellation of non-physical poles to propose a unique solution [53] [54' r43] [5]. However. these previous efforts offer no method of determining the unknownr constants for more general GIBC/GSTC simulations. If left unresolved. this issue would seriously impede the practical utilization of GIBC/GSTC for a characterization of m erial junctions. In an attempt to resolve it. the unknown constant appearing in a second order GIBC solution was recently related to the field at the sheet discontinuity [42]. Unfortunately, the edge field is seldom known apriori and this relation is not therefore of practical use. Nevertheless, it demonstrated that a unique solution may be possible with a GIBC/GSTC simulation. An example where it was possible to obtain a unique solution is given by Leppington r21], who considered the surface wave reflection by an abrupt change in slab thickness. The slab was modelled using second order transition conditions equivalent to those given in [59] and [39]. Leppington was able to determine the reflection coefficient uniquely in the limiting case of vanishing thickness by matching the interior field far from the junction with a static representation of the interior field in the viciniitv of

the junc io1n This suggests the possibiltiy of working with internal fields to resolve r'^e n;, i(: l te i-ue. all approacth which to date has received little attention andl - Ti,:al f t' is dissertation is to develop a plane wave diffraction model for Lenerai symmetric thick multilaver slab junctions. Four main chapters follow dealing \vith the derivation of the GIBC/GSTC, the formulation and formal solution of tlhe plane *wxave diffraction by certain GIBC/GSTC approximated slab junctions. and tile subsequent resolution and explanation of the ron-uniqueness phenomenon described above. In chapter two. arbitrary order GIBC and GSTC are constructed for multilayeredl planar slabs of arbitrary thickness. Initially, recurrence relations are derived for the fieldls in adjacent layers and are then employed to develop infinite order boundary transition conditions that are conveniently expressed as a matrix product. Approximations to the matrix element operators for low and high contrast materials are subsequently employed to obtain finite order boundary/transition conditions. Finally, numerical results are presented in which the exact reflection coefficients are compared with those implied by the GIBC/GSTC to provide a measure of the conditions accuracy and utility. In chapter three, the plane wave diffraction by a multilayer material slab recessed in a perfectly conducting ground plane is formulated and solved via the Generalized Scattering Matrix Formulation (GSMF) in conjunction with the dual integral equation approach. This problem is significant in that a unique GIBC solution is obtained which can be used as a benchmark to test other GIBC/GSTC solutions. In the first part of the chapter we summarize the GSMF procedure. The dual integral ecuatior method is then employed to formulate each of the'subproblems and the necta.:'

s solutions are obtained for both E- and H, polarizations. These are given in terms s: s-mbtr-;ic split tunctions which are then evaluated for the specific case of a muitiia\.',grondecd slab byi casting the reflection coefficient in a form compatible with a GIBr;inLuia.ion of chapter two. A number of scattering patterns are presented and thb accuracy of the GIBC simulation is examined by comparison with known results f,, homogeneous slabs. In chapter four. the diffraction by a material discontinuity in a thick dielec. tric/ferrite slab is considered by modelling the slab as a distributed current sheet obeying generalized sheet transition conditions (GSTC). In the first section of the chapter, the GSTC representation of the distributed sheet discontinuity is used to develop dual integral equations in terms of the unknown spectral functions propor. tional to the sheet currents. These equations are then solved in the standard manner to yield expressions for the spectral functions in terms of unknown constants. The constants are dependent on the geometry and properties of the discontinuity, and are identified in this chapter for a few specific discontinuous layers whose diffraction solution is available. Chapter five deals specifically with the determination of the unknown constants for the solution presented in chapter four. This is accomplished by introducing a general eigenfunction expansion which is valid everywhere and subsequently recasting the solution obtained in chapter four into this format. This field is then analytically continued to the slab interior and continuity is applied at the material junction to provide the remaining constraints for determining the unknown constants. Specifically a point matching scheme is proposed in which an overdetermined system of equations is generated and solved for the constants using a least-squares teclhnilc1it Various diffraction patterns are given validating the obtained solution for cer~:

9 electric iat'-p ianes and metal-dictrctric junctions on a ground plane. Finally cirves are xe~:l::; a:OS t!i(k!.;-.aplanes and junctions to illustrate their Clitra, io;j e<- '..<': lt ir e>ic ss.

CHAPTER II DERIVATION OF GENERALIZED TRANSITION/BOUNDARY CONDITIONS FOR PLANAR MULTIPLE LAYER STRUCTURES A GIBC/GSTC diffraction coefficient can only be as good or as versatile as its constituent GIBC/GSTC. Therefore, before diffraction coefficients of any generality can be developed it is necessary to construct GIBC/GSTC which are valid across all ranges of slab composition and thickness, and this is the task of this chapter. The two configurations considered herein are the multilayered slab having symmetric or nonsymmetric material composition (about its center) and the multilayered coating on a ground plane, as illustrated in Figures 2.1 and 2.2(a), respectively. The derivation of the GIBC/GSTC is accomplished via the Taylor series expansion method, whose versatility enables the treatment of non-planar as well as planar layers. In effect, the resulting conditions allow the simulation of the multilayered configuration as an opaque or transparent sheet (see Figures 2.2(b) and 2.3(b), respectively). In proceeding with the development of the GIBC/GSTC. we initially derive recurrence relations for the fields in adjacent layers. These are subsequently employed to develop infinite order boundary/transition conditions that are conveniently expressed as a matrix product. Approximations to the matrix element opera'-: 10

II y 0 EU gu tU 1 1 ' 1 EL.~L tL_ 1 ' 1'1 0 EL,IL tL" NL NTL N. x z eL a.L, b' b' b Figure 2.1: Infinite multilayer slab.

12 N Y I x 2 " 2 ~ 1 1 " z A(a) Z Y (b) Figure 2.2: (a) Multilayer coating on ground plane. (b) Equivalent opaque sheet.

i3 N y 'u -T =u- - -- -=- - - -- - - m+e '- U ea -\ b b I1 y=yU= + y=O ^F / ~=~L, eq- b N Y=- r m311 (a) N u YU= m=1 y=yU y=O-a y=yL _ t,LCU i I i i II IiII) F+ - eq \ I N F=L,=L L...... b b -------- b b. --- —Y= ' rnm m-1 (b). l:3: a) Zero thickness resistive and conductive sheet similation of multila.er slab. (b) Distributed resistive and conductive sheet simulation of multilaver slab.

14 low and high contrast materials then lead to finite order boundary conditions, p: na,:-. A;'neri-al uesults are presented in which the exact reflection coefficients -cpared wir e -h ose implier by thle GIBC/GSTC to provide a measure of the co; 'i, cns acc:uracy and utility. 2.1 Derivation of Infinite Order Conditions Consiier the multiple layer slab with A\u upper layers (y > 0) and -\-L lower layes (y < 0) as illustrated in Figure 2.1. The physical parameters corresponding to thle mn upper layer are denoted by e,,t -.' < which refer to the relative permittiviit and permeability. the index of refraction and the layer thickness. respecti-vel. I, a similar manner, the physical parameters corresponding to the mh lower layer arr given by EL L, L,,. In the following we derive transition conditions to effectivel, replace their presence with a distributed current sheet. "We begin this derivation by first introducing a relation between the fields on the two sides of a single layer. This is generalized to relate the fields of distant layers and those at the upper and lower boundaries of the fictitious current sheet. For convenience, we may consider both polarizations simultaneously by introducing the definitions Ey, Ey polarization (Hy = 0) Fy = Hy, Hy polarization (Ey = 0) e, Ey polarization u = (2.1), Hy polarization Using a Taylor series expansion, the normal field components at the top and bottom of the mth upper layer may be related as tlmFyj= u+,U.. +(r +..-...++( 1 ) ( —) =0 (), +Z0 (1r=)!

1,5 w. here = = ~ - ';.,,, pi* i;s at o 'eator (-eti: ed as j a Cle arl. '2.2 - 2.3) provide relations of the fields within a single layer and if w e are ro Cerive a condition relating the fields at the top and/or lower surface of the slab. it is necessary to establish similar relations among the fields in different layers. As a dfir step towards this goal we may proceed to express the right hand sides of (2.2) and (2.3) in terms of the fields above the boundary y = r +.. + '. To do so. it is instructive to resolve the right hand sides of (2.2) and (2.3) into a summation of odd and even derivatives of Fy. Subsequently, the wavae equation may be inv-oked to rewrite the normal derivatives in tangential form. thus allowing the application of the field continuity conditions. We have =rn' U (2.6) with 82 =6+6 (2.7) where k ax ja _' i = [&. (2.s) Invoking now the continuity of uFy and 68Fy (including their tangential derivatives) across the layer boundaries, (2.2) and (2.3) may be rewritten as Fl =Tr +-u+...+(rT_ )+ = Qu(u, K T, 6t ) F. Iy=r+rU++...+( ()+ 1~~~~~~~ 2~ ";~ 1,) ~:' ~ (2.91 ' *? -) i

16 In this. F =.2'.i, L8<FJ C2( ZlT-.t) ==. 11^ Q t( u.:. r. ) = C' 2.1k q.-i. ~Z..r "! = q,2( —. --- -. sin) = sin kk U q22( l.. t ) = cos (ko - ( '2. / Expressions relating the normal fields in adjacent lower layers may be obtained in a similar manner. We have. F. gL'r2L.-(rL I) = QL(U L. r~L E ) ' FJ Y TL _ _, L )+ F=L,: -. = QL(Um m' ^,, t).' J=-L er-. (;&.L)* (2.1:3. u=t,-J ( _, m,'r _-.....,-. where now Qqr(u,) =, r 7, ) -q72(u, K" T t) QL(U r T. 6) = (2) 14) -q21(u, K, 7, 62) q22(u, K,, 6t)j Equations (2.9) - (2.14) constitute fundamental recurrence relations for developing multilayer GIBC/GSTC. Each of the ql1,q12, etc. is an infinite-order linear differential operator in even powers of S52. This is evident when the sin and cos terms are cast in their Taylor series representation (note that the square root functions appearing in (2.12) do not have branch cuts). The finite-order boundary conditions are then derived by truncating the Taylor series representation of qll, q12, q21, and q2 -Applying (2.9) and (2.13) recursively, we may establish a relation between the fields at the top and bottom of the layer. We have [i L( K r6')] 7 Y= —r2L -(TL )+ = FJ=o- -, -- ', - L

17 Jn^-'^-^? 0 L>.h.7K,, S*j-l _ '.'.= F1 e-, i- r * *'rt-'lusr across y = 0. i2.15) - (2.16) imply the GSTC condition QL(n,um Tr L t)L 2 - L"=' =[ QC(mt )] u L +..+(i_., )+ The part of the slab occupying y, < y < r- + r +.. +,. may now be replaced b!v the upper back!gro nid medium (with its geometrical and material parameters Ienotec bDy t1.e.uli)script b and the superscript U) while the part of the slab occupying /L > j > - -L - ~ is replaced by the lower background material (with geometrical and material parameters denoted by the subscript b and the superscript L). Using a Taylor series expansion the boundary fields may be related to the equivalent fields at y = yC and y = yL. In so doing, we obtain r n 1 LTr. = 2 = n[h QU~(U T[, 2) 0QL(Ub,.I b b t) FeqJy= (2.18) Lm=l w here NL L = ErL-(- yL) 1=1 IvU = Z -U (y) (2.19) 1=1 Although compact, the transition conditions (2.18) provide little insight into the )phlysics they represent. It is therefore instructive to reorganize them in a form that leads to its phlysical interpretation. To this end we introduce the definitions [a=~t L nz t ] f (Li(6t) Z~2(t:) Q L k 6r 1, K (n1 7 rn t )7

18 r -, II C)i r 1 Alrt t; L -L 62 I., - N M t jl F =1 - I I ~~~~1 ~~~~~J \t - I (, r, c >. _ 1! _= I - r -t, (( L L L.',) QL C- C- C-{ I cl 5 I:- r?a? [ (2 ') I ') 3 I ][ Q-.E. - C. r.6 ) (rr. =li' ' L, m? (2.20, Substituting these into (2.15) yields [ll + e lK ) [^12U + 2](<2) - I t6" ]^) [^C el 2) 2L21 + t21] () 't2) [L t - C2J (t 22J ) 0 L1 -Le] \Ct2).[^i-^W) h Uh 1 6 'L h) 1- el] ( 2 ) [Lll it SllJ(6t 12 + g 12]1 t rC L + U^h1](: 2) [f 8 + Zih ] (6 2 )J L [-C 11t-1L] (6t "21 U21](6t2) _rF - (Et):[C2 - [L2- 2h ]()) h J [12C 1- 2 ) A [f h - Uh ](62) h (2). 2-2 _; ( 2.22 ' In (2.21) -(2.22).AFJ and AFh. are given as AFeJ = E+Eh -aE. = y d e Ht t - AF- f s, [EY -E;] = -. [Et - ] FAF H+- H = YSt H [E Et -]: [H -H]: y =- [t ]- ],t whereas Mh^ and J,^ are defined as (2.23) MC = E + = -y [H + =! & Je =Y [E + E;= —, * [E+ + Et] sJ =,u H+ + uLH; = Y. 6t, x [E+ + jh = + + HY = -St [ + H lfh = sy: & FH~~H;]=- S.[H~+[i~ ~j.

19 E?=r EJ E+, -d. f E+ d Se? l Ey - Ey I- y=yIL def Hy H y=yL. (.) \We also note that the superscript notation e and h refer to Ey and H, polarized exci:ations. respectively. In view of (2.24). thie transition conditions given by (2.21) and (2.22) are now readily interpreted as a representation of.distributed" resistive and conductive sheets occupying the volume yf < y < yL and supporting equivalent electric and magnetic currents (see Figure 2.3a). These, of course, give rise to discontinuities in the fields (and their tangential derivatives) at the upper and lower boundary of the sheets. The equivalent electric and magnetic currents are denoted by Jeh' and M.l'h, respectively, and are defined in (2.24) above. Additionally, the subscripts J and M appearing in (2.23) denote field discontinuities traditionally associated with the presence of electric or magnetic currents, respectively. For convenience, the currents and resulting discontinuities fF are presented in terms of both normal and tangential fields. Wllen y = 0+ and yL = 0- (see Figure 2.3b), the distributed resistive and conductive sheets are "compressed' onto an infinitely thin sheet occupying the plane y = 0. Such thin sheet representations are attractive for the application of transform technicques in diffraction problems and are generalizations of the resistive-conductive

20 sheet simulations given in [59]. [i391 and [31]. In contrast to the referenced si,, lations. t1he oit,, given here exhibit coupled electric and niaonetic currents. \w,?'rese'lce complicates the application of analytical techniques. This coupling is at tril.utattie to the distributed nature of the polarization currents across the origin slab configuration. If. however, the multilayered slab is symmetric about y = 0 arn IL = -_L. then it can be shown that.l and J decouple. In particular, for th, special case ~11 = gill ~12 = -1-12 ~21 = -21, 22 = 122, (2.26 leading to 21(St ){AFJ} = -12(){J } U2(t){, F} = -AP(6t){lie} ^){11 ) {A } t= -lS12 t ){Xh} U22(){,F} = -21( t){ } (2.27) when substituted into (2.21) and (2.22). Evidently, the coupling of the current components depends on the degree of asymmetry in the slab and an assessment on the level of coupling can be obtained by comparing the magnitude of the operator coefficients in ~11 - U1,, ~12 + U/1, C21 + 121 and ~22- U22 relative to those in ~11 + '11, ~12 - /12, ~21 - t21 and ~22 + 122 -If a ground plane is inserted in the symmetric slab at y = 0, the resulting str ctture becomes opaque and its sheet simulation is further-simplified. For the case i. v

21 around plane is a perfect electric conductor (PEC). the electric currents are shorte(l out 1, ' se3ond and third equations of (2.27) become the boundary conditionIs,,:o-.sonding o a coated PEC. Conversely, a perfect magnetic conductor (PNIC):,-.- t ou!tr 'ie rr.arnetic currents. leaving the first and fourth equations of (2.27) as the corresp)onding sheet simulation. An alternative approach to deriving the GIBC corresponding to coatings is to employ image theory in conjunction with 2.27). Tihe fieldi components in the equivalent image configuration of a PEC grounded slab,ie related as E(y > 0) = E(y < 0) 6yE{(y > 0) = -yEyv(y < 0) H(y > 0) = -H(y < 0) 6yHy(y > O) _ yHy(y < 0) <.0 and when these are subsequently introduced in (2.27) we obtain the boundary conditions EU21(){E+} +Uh2(6){6E7} = 0 U,1( ){H} +U( ){6H+} = 0. (2.29) corresponding to a slab on a PEC. Similarly, the image fields for a slab on a PMC satisfy the relations Ey(y > O) = -Ey(y < O) 6EyE(y > 0) = 6yEy(y <O) Hy(y > 0) = Hy(y <0) 6,Hy(y > 0) = -yHy(y < 0)

sItl3 t( L{E}-I (ctKZ} = 0 Ub t {H } +U 2({ ',H} 0. (2. when substituted in (2.27). Expressions (2.29) and (2.31) represent opaque conduc tive and resistive sheets. respectively, and are the dual of each other. We note that if.u = i. ('2.29) reduce to those given in [31]. To summarize the above development, sheet simulations were derived that nuoc;! (or replace the presence of) the multilayer slab and coating. Equations (2.21) throug (2.27) are referred to as generalized sheet transition conditions (GSTC) for the transparent resistive/conductive sheet representation of the multilayered slab. On the other hand. equations (2.29) and (2.31) are described as generalized impedance boundary conditions (GIBC) for the opaque sheet representation of the coated ground plane. They are given in a compact matrix form and are valid for any arbitrary finite number of layers. Their versatility, however, is offset by the presence of infinite order derivatives as implied by the definition of the operators, thus, limiting their applica. bility to analytical and numerical treatments. It is therefore appropriate to consider finite order approximations of the operators leading to conditions of practical use. In the following we consider such approximations of the operators on the assumption of low contrast (small tc) and high contrast (large tc) layers or coatings. 2.2 Low and High Contrast Approximations for Matrix Element Operators Low contrast approximations to the matrix element operators may be derived by replacing the trigonometric functions in (2.12) by their Taylor series expansions

23,-'iicil are:en truncated. To 0( r f) these become Il r,( L! i nr t( jf ),' ' ' 1-,'-! =,l/~ntu,^) s - j~ al( '.7c./. ) (K7 ) i'=U m=i 'I- f i- IU. K. *.. *- ) (i i ll 6' ~~~~~~~~~~~~~~~~~~~~~~~~ ~t11 intf(( ) " (int(- ) ^ JU ^ ~, n12(h,7. /,m) |(<) /=0 m=l. nt(,. ) * nt( ) + ^:L,, a3(^j,r.l m) ( m O 1=0 m=) 2 '.:3' al(, '.. m) = a2(K. 7., lm) a3(K.r..m) (in I-2( n - 1) (- )1i-(m.)!( k,0- )2m 2(m-1) (m - )!(l)!(2m + 1)! (-1 )l+-m(m)!( kor)2m- l.2(m-) (m-1)!(1)!(2mr-1)! (2.:33 It is a simple matter to demonstrate that the substitutions 2 def - K - K - 62 = K c2 2 -62 t y (2.34) applied to (2.32) lead to a normal derivative representation of qll - q22. To obtain high contrast (large ic) approximations for qij, it is necessary to utilize the binomial expansion - 14 6 r/ -iz, 1/ = il _ (6t\4 13 ( 6t\6 1,M 3 (6t -; < (2.35) lSubstitLuting this into (2.12) and again employing a Taylor series representation for tlie trigonometric functions, we obtain to O(i-^) the approximations CO1J',K

24;- 1.!IL F=l II=mnaz1l,2p-.VI+1] f-. p=r l=ma={l.2p-M+l]2 i. " - - sin(kor'7) j..I1 ' - r+1 l.- m+11 f ILL —1 l=rmar{1,2p-A-1} t: lmax{il,2p-Af}.,,r /.o.p) ( )P K (. 7 (. 0. O) p) 2 ) K'2p t l-i r2(l- 1., )T (I - 1,1. p) (. P) ^':pt2-( ' t K2Pp-l T,(I,..-) T(li.l,.P) b2(p) X!! (_l)int(l/2)(k )l (cos(ko TK) I is even (l)! ssin (kot); I is odd J (-l)int(l/2)(ko )l sin(korK) I is even (I)! - cos (koTK); is odd J = Ebl(il)b(i2). bl(il,)b2(i11+1). b2(il+) (2p- 3)!! 2Pp! (2p-l)!! 2Pp! - X(X - 2)(X-4)...()(3) ( -1)!! - 1 (2.36, (2.37) The sum defining the function T (11, 12, 2) includes all product terms satisfying the relation Z, im = p. It shobld also be noted that 11 and 12 denote the number of bl and b2. respectively, comprising the product terms. For example, if 11 = 2, 12 = 1 and p = 3 then T(2,1,3) = b(l1)bl(l)b2(1); if 1 = 2, 12 = 1 and p = 4 then T(2.1.4) = bi(2)bi(l)b2(1) + b1(l)bl(2)b2(1) + b(l)bl(l)2(2) and if I =. 1 = and p = 4 then T((2.0.4) = bj(l)bl(3) + b1(2)bl(2) + b1(3)bi(1). Correspon;ukirt.L

i order Low contrast q,, coefficients: - '.' = ';';; = ' '/12 =;t1^k1. q21 = kr(1 (K9 - (} 1.3..= q. = I - -(9 -Et ), q12 =Jl tjlk [i 1 ( - )2 (2 -2 ()1:l= 1jki, 2 - l-t, 2?1 l i- -) (2 _- 2)] 2 q 1= i (- _')2l [1 - ( - ))1 /u] - k2 -) k Table 2.1: Low constrast approximations to qlj operators. ex-ressions in terms of normal derivatives can also be obtained bv employing (2.34) i ( 2.36). Expressions (2.32) - (2.37) represent finite order approximations to the operators q11.q?12.q2? and q22, and can be used to generate finite order GIBC or GSTC. To do so for a specific multilayered slab or coating, one first examines each constituent layer and approximates the corresponding matrix elements by their low contrast expressions (2.32) - (2.33) or high contrast expressions (2.36) - (2.37), as appropriate. As an example, basic low contrast and high constrast approximations to the qij operators are give in Tables 2.1 and 2.2. These finite order expressions are then substituted into the matrices Qu and QL given by (2.11) - (2.14). The resulting expressions for the U and C matrix operators are finite polynomials in even powers of E2a. In passing, we note that the simulations presented here-in may also be extended to model a longitudinally inhomogeneous slab once this is approximated as a layered slab such as that shown in Figure 2.1.

26 Order High constrast qij coefficients 0h~ ql = q22 = cos (NkTl). 1q2 = 0.q21 = u sin (Kkl) - - COS ( Tc) -1 qli = 22 = cos (- km!) + sin (ikrT) -, q12 = sin (~kT1) q21 =- Z- sin (K ks 1) {1 - s2 } - cos ( ) krl) tcl Vr$2l- _ -~ - cos(nkT1) hN | {t=, 22 = cos (K r1) {i - - } - sin (Kk-1) -' q21- 2K2 cs |12 = ^ sini 1 - 2Kk^1 cos (k-1'), q21 = sin (hk7) {1 - 2^2 ^ }2 q = q22= cos (kTl) {1 - } | +slflA Xk) {-2 + p3! - ' 7iYi * / 7 \ fi. 52rc k ( )22 1 ukr, 52 ql2 = ^ sin (ikrT) {1 + _, )} _ u"ki-,2, cos (Kk ) 212 -8_2 2r2?< * / i \ Ci 52 (fkr )2 64 54, (kr2* 14 q21 = sin (Kkrt) {1 - r (- \ - 5t + ( ^z } - cos krl) f{k - t-r - (k'r)3 } u_ ___2 2i 8cr3 48r3 ' Table 2.2: High constrast approximations to qj operators.

27 y Figure 2.4: Coordinate system for reflection and transmission coefficient derivation. 2.3 Evaluation of the Boundary/Transition Conditions To evaluate the accuracy of the derived boundary/transition conditions one approach is to compare the plane wave reflection and transmission coefficients implied bK the finite order sheet simulation with the corresponding exact coefficients. Consider the plane wave { ~} = { H } e:} k U (x cos a sin 3+ y sin a+zcosa (.3 8) H1; H1oy f incident upon the sheet satisfying a given GIBC or GSTC (see Figure 2.4). The generated reflected field can then be written as E{ E) = }~ ekzb (cosasin/3-y sina+zcos acos3) (2~>3', Hy \RHHyo

and in tile case of a GSTC. the transmitted field takes the form;' E 1H > == 7; e <k co' (r cos,' sin j ' -- sin i t cos' cos. )3') o ndnd( the refection coefficients RE.H and corresponding transmission coefficie nt TE.! e su bstitute (2.38) - (2.40) into one of the boundary or transition conditions given by (2.21). (2.22), (2.27), (2.29) or (2.31). By carrying out the differentiations in a straightforward manner we find -ic,-&crRc: multilaver slab pmc pec Rec + Rpmc multilayer symmetric slab RE = (2. Rpec multilayer coating on PEC R c multilayer coating on PMC Re, - Re f Recrc 1multilaver slab TE = < ec -Rmc (2.42) Rpec '- Rpmc multilayer symmetric slab rmc Rec -,mc, multilayer slab Rpec - R-p m Rh mc iRh multilayer symmetric slab RH = 2 (2.4:3) Rhec multilayer coating on PEC Rmc multilayer coating on PMC - M"-c'rc multilayer slab 7H mc-e,, (2.44) LR^mc - R^c multilayer symmetric slab. In (2.41) - (2.44), < sin a U2 [(KU)2 cos2a] - Ue [(CU)2 cos a] 2ky,4 sin a 122 [(ICU)2 e Co -^ [( ) pc - hg sin a U12 [(K)%2 cos2 a] + ebUtf1 [(~[')2 Cos2 a]e <.' sin ca — 2 [( )2 cos2 a] - [()2 cos2 y pRec * sin a U22 [(cU)2 cos2 C] + FU 1e [(KU)2 cos2 j] pec C 2,;, U)2cO e 3111U vr22 [(~ b b 21 RN] [)2

inn ct' C h C - 1'i - L )2 Cos2 & Re = ' ^___- - ___ y___________-_ -1 y sin oa' ' y sin, i ' pmc - ' ' ^ [ 4' 2 _ e2 ) cos2 2c *:,;'!, t ) - -cos2 ' 11 I cos2 rC; IiC1 1 (K c i;? - C'. '. k, tpnc ht sin a Lip^ [% )- cos al -i- b i [(i )2 cos2 a] ": ':CJK5 o- n,\ Cos( and from Sineil's law Lsmain =,/%)'t -: ) 2cos2Q (.46) To obtain a composite sheet simulation of a multilaer slab or coati it is e subsequently combined to ield the and operators of the composite boundar condition. Thus. the accuracy of the overall simulation can be assessed by examining that of the individual layers comprising the slab or coating. In the case of a single laer simulation (ith non-shifted surface) the C and operators reduce to = = 1 12 = 1 = 0 and j =. This simplifies the analysis and in the following we examine the accuracy of the proposed GSTC simulation as a function of the condition's order. Only the -polariation incidence is discussed but similar K5 sin ct- t(' C' 2 cos. a. results apply to the iy ee-polarization case as well. Figures 2.5, 2.6, 2.7 and 2.8 present the maximum error in \Rex - Rgstc\ over real.angles as the layer thickness is varied. The data in Figure 2.5 corresponds to cond;L,-.-.',on L Thus. the accurac.' of the overall simulation can be assessed b examin'ina a low contrast simulatinon wsith r suface) th= 1.2 as the order of the transition condition increases from 2c to 9. In the region Ghere sRex Riatin s sthan.15io reaiangYes as the laver thickness is varied. The data in Figure 2.5 corresponds to a low contrast simulation with E, = 2 and ~ - 1.2 as the order of the transition condition increases from 2 to 9. In the region where JR, - R,,tcl is less than.1.5

0.40 0.35 c. I E 3 0.30 0.25 0.20 0.15 0.10:~ ~ '.. '''...''', i;-: —:l r.-i'. / /: t; ~!,, /,, / I I i' 1 / ' ' ' ' I I / I; 'i I I jI / i,: *, I. ' - /,i t - 'lii I / i;:,i ~~~~~I/,~~~~~~! /', t: ' i, ',*.1 ' / ', ' ' \ ' ' I: I **l~I It - /;.-. I:I ' I /~~I,/ ' /:.,., -. I,: / i ' ' ' I /: ~~ ~~/ /';,' xt /,i * / ', ' I',' /i / I, / /,' I //, I / /,,. / i!,;,',i -- / / " " /, /: ~ / / I / I - /, ',: *''I, ^ ". r: ^ ^ ^ ^ ^ 1.... i...I: ii I -. '-I - I.~J 4 - m~ ~'t// tt..'-:'':...-r...o:,. /..,L..,r-.....t.,,... 0.05 nnI 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Thickness (X) L -- 2nd order......... 3rd order ------ 4th order...... 5th order - - - 6th order.. ---.- 7th order - - - - 8th order. --- —. 9th order Figure 2.5: Ey polarization low contrast simulation of a single layer having relative constitutive parameters e = 2, u = 1.2

-1r o5 F:=S~~~~~~~~ ~Maximum IRex - Rgstcl o (b 0 t. 0. 0 0 * - 0n 0 o k) o " 0 0 o '-10 ' Ct O;-_ ^ ~'", a; *- -; --- X X^~ ~?ro I _ _ — _ _ II: "B~~~~~V C |; I;: - '::::::: 'K' 5i 0- 0-' pII ~ E o9 o t-,.J o i-i. o i,. (1) C~ '"".........~~~~i i.1,,....,............

:32 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 1-v^~~~~~~~~~~~~~- t/'^ tI /;; * I! / / I v ' ( k / ' ~ r i '~ I'...~, /. \ I\.< (. * lI':\,,,: / ',~i, *..., I ', / ' i... I / ' ' i u~~~~~.ut:, —...,,/,!......~,. '.,V'..:... ~r... n ran L 'V -., L.. "-ZF:-. U.UU 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Thickness (x) 1.00 4th order......... 6th order. —..- 8th order - - 10th order - - - 12th order Figure 2.7: Ey polarization high contrast simulation of a single layer having e = 3.5. p = 2

33 0.40 0.35 - 0.30 0.25 - 0.20 -0.15 - 0.10- ) 0.05 - 0.00. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Thickness (X) 1.40 1.60 1.80 2.00 4th order -...... 8th order ------ 12th order...... 14th order Figure 2.8. Ey polarization high contrast simulation of a single layer having e = D. p = 3

:34 ito be considered as an acceptable error) each curve displays a quadratic siha posSiVl" ).0.i" s- mail "kinks" or perturbations superimposed'. In general. the ran,. - i:dl ci'n improves as the order increases. although in certain instances it A, actua'l deteriorate slightly. In Figure 2.6 the tnaterial parameters are increased to Or = 3.5 and / = - We now observe that the maximum allowable thickness to maintain the same error, as in Figure 2.5 is smaller whlen employing the same order transition condition. I.; particular. the degradation is such that a 17th order condition is required to eqla the performance of the 10th order condition in Figure 2.5. As before the maximurm allowable thickness or performance of the condition increases with the order. To compare the performance of the low and high contrast approximations, the. curves in Figures 2.5 and 2.6 were recomputed using high contrast transition conditions. The results are given in Figures 2.7 and 2.8 where the curves now follow an oscillatory behavior unique to the high contrast conditions. In general, the high contrast conditions provide an improved simulation for this choice of constitutive parameters. For example, when the order of the condition is increased from 4 to 12, the allowable thickness that can be accurately simulated increases 7-fold, significantly better than the performance of the low contrast conditions. Most importantly, the high contrast conditions allow the simulation of much thicker layers with the same error criteria. A typical example are the 11th and 10th order simulations in Figure 2.6 and 2.7, respectively; whereas the low contrast simulation allows a maximum thickness of only 0.28A, this increases to beyond 1A when employing a high contrast condition of comparable order. Inherent with their derivation, the high contrast conditions are expected to provide improved simulations as the refractive index in'As a reference, when t\Re - Rgstc =.1743, the corresponding phase error is 10~ when R,,tt = |IRex =l.

3.5 crease. Thi iS indeed observed in Figure 2.S where the laver's relative constitutive.ara:lete's a:e. =.5 arid Aor = 3. As shown. a layer of up to '2 thick can ie,-;a:el ^iaied:'.! tla 12'' order condition. Figures 2.5. 2.6, 2.7 and 2.S are,ui(',-t;t e'1I.:)I olarization in Figures 2.9 2. 10 2.11 and 2.12. respectively. The eS;ult, are very sil ar to the Ey polarization case. and the same remarks,.uade aitove are applicable here. LUsing the single layer data, such as those presented in Figure 2.5-2.12. it is possible to synthesize a multilaver simulation. As an example, consider an Ey polarized plane wave incident on a three laver slab having e = 5- jO, [' = 3-jO r7 = 4 for the first layer, E 3 = 3j. O, p = 2 - jO, 4 4 =.4 for the second layer and - = ' - j0. / t = 1.~2- j0,,r' =. for the last top layer (see Figure 2.13). To select the individual layer models for Ey polarization we conjecture that the maximum error of Rex - Rgstc for the composite sheet will be bounded by the sum of the maximum errors of IRe: - Rstc of the constituent layers in isolation. Examining Figures 2.5-2., we observe that the sum of the errors of the 6th order high contrast representations of the bottom and middle layers and the 5th order low contrast representation of the top layer amounts to.062 as required by the design criteria. This suggests that the composite sheet employing these representations will have an acceptable performance when the total thickness is 1A. The actual maximum error of the designed simulation is compared with the sum of the isolated layer errors in Figures 2.13. The corresponding errors with a PEC and a PMC inserted at y = 0 are also plotted in Figures 2.14 and 2.15. We note that the sum of the isolated layer errors serves as a reasonable upper bound for the new simulation design.

36 0.40..i..... ~/ I: "I ' ' i '; '. 0.35 / /! '' 0.30 / / ' / / i I I / - -0 ---/ a 5 t' I or '; /!: i\ _ 0.250 -, - /: ^ ~0.250 - /!: i ~ I, t Figure 2.9: polarizatonlow contastsit/o /, /; si l e h r ai s0.20t- i/ /: p: / a 1 I 0.20 / / a / 5 / a a: / I:l i,,...... 0.10 - a / / / I / / / / 0.0 0.05 0.10 0.15 0.20 0.25 030 0.35 0.40 0.45 0.50 - - 4th order./ 5 order / //,)~.' II -I - 6th order 7th order.5- - - 8th order.....-........ 9th order Figure 2.9: Hy polarization low contrast simulation of a single layer having relative constitutive parameters e = 2, u = 1.2 I,II,:. 2n orde..... 3rdre...... 4th o~rd 0~~~~~~~~.... 5hor

o5* FS~~~~~~~~~~= ~Maximum IRex - Rgstcl to ~ —C. o o O o o o o o o,-^~~~~~~ I r --- 0 --- - t6 6 O 00 0 A 0 0 aII 11 i E a: o P~:-)- - - — C *-'^v 0 -II I ~ ST^v,~1 -- I t,,.- ', S o o -..

38 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05, ~ ~ 'I I.?I I~~~~~~~~~~~~~ " I a ~~~~~~~~~~~~~~ a.,, I, a / a a \\, a, I 't ' ' 7 t...-/1,, ~,.~,,.:'~J: /J:',I,~... / \!,\ i \ \!; \ '; '* ^ *'7^ I 0.00 L ' L.. " L '- ' -'" 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Thickness (X) 4th order......... 6th order ------ 8th order...... 10th order.- - - - 12th order Figure 2.11: Hy polarization high contrast simulation of a single layer having e = 3.5. P =2

:39 0.40 0.35 0.30 0.25 _ 0.20 ~ a0.15 - 0.10 0.05 ": 0.00 02 04 6 0 1 0.00 0.20 0.40 0.60 0.80 1.00 1.20 Thickness (.) 1.40 1.60 1.80 2.00 4th order......... 8th order. ---.- 12th order...... 14th order Figure 2.12: Hy polarization high contrast simulation of a single layer having e = 5, A =3

40 0.40 0.35 r, Ca E) x 0.30 0.25 0.20 0.15 0.10 0.05 0.00 L 0.80 0.90 1.00 1.10 1.20 Thickness (X) Composite sheet........ 3 single sheets I - ---- -- I~~~~~~~~~~~~~~~~~~~~~~~~ Figure 2.13: Simulation of a three layer slab using a 14th order condition.

0.40 0.35 C.).0 C4.4 r~ k 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.80 0.90 1.00 1.10 1.20 Thickness (X) --- Composite sheet. ---.... 3 single sheets Figure 2.14: Simulation of a three layer coating on a PEC ground plane using a 14th order condition.

42 01) 11 E E >4 M~ 0.40 0.35 0.30 0.25 0.20 0.15 7 0.10 0.05 0.00 0.80 0.90 1.00 1.10 Thickness (X) 1.20 Composite sheet.. ---... 3 single sheets i. Figure 2.15: Simulation of a three layer coating on a PMC ground plane using a 14' order condition.

43 2.4 Concluding Remarks It '. -!-' (-a-.At -ith the development of generalized impedance bounlary a> is:rau'v c-.n conditions for multilayer coatings and layers. A major effort was also, ' Ipi *Ast)revent t:liem in a form which is compact and convenient for further usage. Because of the generality of the derived conditions they are ideally suited for use in rhe folloing c. -eters where general diffraction coefficients are developed.

CHAPTER III DIFFRACTION BY A MULTILAYER SLAB RECESSED IN A GROUND PLANE VIA GENERALIZED IMPEDANCE BOUNDARY CONDITIONS The canonical geometry formed by two semi-infinite planar slabs joined end-toend is a difficult one to model properly, particularly if the slabs are thick. For this reason very few computed results exist to verify GIBC/GSTC diffraction solutions obtained for thick slab junctions. This need is addressed herein by computing a unique alternative GIBC/GSTC solution for a specialized thick slab geometry. This solution provides a benchmark which can be used to partially verify more general GIBC/GSTC diffraction solutions. The special problem considered in this chapter is the plane wave diffraction by a multilayer slab recessed in a ground plane in Fiaure 3.1, and the alternative method used is the generalized scattering matrix formulation (GSMF) [25]. Related but simpler geometries have been studied in the past [2] [6] [20] [11] [26] [8] [48]' [9] [f52] [58] [28], and most of these solutions involved the interior and exterior fields. This is alleviated herein by modeling Xhe slab as a surface characterized by a plane waave reflection coefficient R(cos <0) (see Figure 3.1). where <o can be extendecl tlhrough analytic continuation in the complex plane. This enables us to cari:,t 44

-t material layers A~ """ PEC (a) releci ng surface, Fcos (PJ ' / (b) Figure 3.1: (a) MIultilaSer slab recessed in a PEC ground plane. (b) Representation of slab as surface with reflection coefficient R. the analysis in a symbolic manner regardless of the inhomogeneity profile of the laver. Hence, although our focus in this chapter is the multilayer recessed slab, the derivations will be applicable to any vertically inhomogeneous slab. The problem herein is formulated via the dual integral equation approach [10] in conjunction with the GSMF. The GSMF is applied to the recessed stub structure of Figure 3.2(a), depicting a perfectly conducting half plane elevated a distance 6 above a reflecting surface with a perfectly conducting stub recessed a distance d away from the half plane edge. This formulation requires the solution to a number of indiv-idual ubproblems. As illustrated in Figure 3.2(b)-(f), they correspond to the problems of direct diffraction, mode coupling, mode reflection, and mode radiation.

Bu!qpunie apow (j) 'Ilnotu ap!nSa^AM a\ I e uo14pal 9potu (a) 'uoit:)aUa qnls (p) 'SuTldno: apouI (:) 'uoitl:)ejjp palop (q):smlalqojdqns paleOoss? pue (e) ilamuoa qnjs passa:9a jo uo[!jlsnlII:F'' a:n2ij -- = \V i i (P) (q) =W~~~~~~~~~~~~~ ii (e) VrJ1JJJ#,J',j'JJIj,,,,,, ff I II II II' II~JT1 II0 11 03d I//////////// //////// / /. _ f -. \. _~~N 0 G 6

47 Once solutlons to each of these subproblems have been obtained via the dual integral ecquation r me:l.. e (_can be combined in accordance with the GSNIF prescriptio(n to:"eld'i 'e i. a"acion for the original structure in Figure 3.1(a). I;:fort tntely. the conversion of the symbolic solution into one of practical use proves to be a formidable task when the reflection coefficient of the grounded slab is obtained in its exact form. The fundamental difficulties are related to: * the factorization or splitting of the associated \Weiner-Hopf functions into components regular in the upper and lower half complex plane, and * the extraction of the complex zeros (i.e.. the waveguide modes) associated with the split functions. The pertinent Weiner-Hopf functions cannot be factored analytically and one must therefore resort to a numerical scheme (e.g., see [281). Also, in solving for the complex roots of the pertinent split functions, it is necessary to employ a search algorithm in the complex plane, a process which is numerically intensive. bWie circumvent these difficulties by replacing the grounded slab by an opaque sheet satisfying a GIBC of the form given in chapter two. Under a GIBC approximation, the approximate reflection coefficient is cast as a ratio of polynomials in cos X or sin X, making the determination of the complex poles and zeros of the reflection coefficient a simple task. As a result. the required WVeiner-Hopf factorizations can be obtained analytically leading to computationally efficient solutions. In the first part of the chapter the GSMF procedure is summarized. The dual integral equation method is subsequently employed to formulate each of the subproblems and the necessary solutions are obtained for both E: and H, polarizations. These are given in terms of symbolic split functions which are then evaluated fort -i

4S speci.nc case of a multilaver grounded slab by casting the reflection coefficient is fot:'.. i-,::.'?a "i>'e w-ith a GIBC simulation of the slab given in chapter two. Res-', are i ven and the accuracy of the GIBC simulation is examined by comparison W\t known results for homogeneous slabs. 3.1 Description of GSMF Procedure In this section. the generalized scattering matrix formulation (GSNIF) is app'ii,: to the geometry given in Figure 3.2(a). This consists of a perfectly conducting ha>l. plane located a distance 6 above the grounded slab, with a perfectly conducting stijrecessed a distance d away from the half-plane edge. To concurrently treat both the E~ and H. polarizations of incidence, the quantities F: and F, are introduced. The, are defined as Ez, Ez polarization, F, = o:3. ZoH, Hz polarization. ZoH,, Ez polarization, F~ = (3.2 E, H, polarization. and from M.axwell's equations jvl aFZ F j v3F (3.3) k ay where -1, E, polarization, v1 = (3.4) 1, H, polarization. The individual problems to be considered in the GSMF prescription [25] are as follows: 1. Evaluation of the direct diffracted field by the substructure in Figure 3.'2(b) due to a plane wave incidence. This field can be expressed as Fh (Po) = Pda (cos a cos o: 6) e -P da

wvhere P i is the spectrum associated with the currents induced on the hailf l'a-e a'i',._* are htde usual cylindrical coordinates of the observatlon: oirt. A1':. aixl C i; t! e cornplex contour composed of the directed line segments ' - _ X _- jl. -70- jO. - jO]. - jO. T - j3c in the complex oa plane. 2. Evaluation of the field coupled into the waveguide due to a plane wave incidence (Figure 3.2(c)) as illustrated. We denote the field corresponding to the nr' coupled mode as Ff (o) = C, (cos O; S) e-kn (:3.6, xwhere C, (cos o',; ) is usually referred to as the coupling coefficient and kn is the propagation constant associated with the nth mode.:3. Evaluation of the modal field reflected at the stub (Figure 3.2(d)). This can be expressed as Fr,,en: wh ere Fn,, is the stub reflection coefficient of the nt mode to the nzth mode. 4. Evaluation of the reflected field at the waveguide mouth due to the nth mode (Figure 3.2(e). This can be expressed by Rmn (6) e- km where Rmn (6) is the reflection coefficient of the nth mode to the mth mode. 5. Evaluation of the radiated field attributed to the mth mode incident at the waveguide mouth (Figure 3.2(f). This field can be expressed as Fm () = J Pm (cos a; ) e - S(a-)da (3.7) where Pm (cos a. 6) is proportional to the spectrum of the currents induced on the half plane due to the incident mth mode.

.50 Accordingly, the scattered field by the recessed stub geometry in Figure 3.2"ai g';ver v ' tor 7 > ') F o. o,.. di = Pdd (co a. Cos 0 )+Pod (cos a. cos o,; 6. d)] e kOPcos(-)dQ (33.s where PmRd (cos ca. cos Oo; 6, d) is associated with the presence of the stub and includes the contribution of the waveguide modal fields. It can be written in a matrix forr as:25! P,,.i (cos a. cos o: b. d) = [Pm (cos a; 3)IT {[I] - [ln (d)1[mn[W mn (d) [Rmn ()]} imn (d)]Tr mn][Vmn (d)][Cn (CS aOo: 6)] in which the brackets signify column or square matrices depending on whether one or two subscripts appear, respectively. In addition, [I] denotes the identity matrix and 'llV,, (d)l is the modal propagation matrix whose elements are given by ejknd, m = n V, mn(d)== (3.10) 0, m#n. To obtain the field scattered by the recessed material slab it is only required to set d and 6 to 0 in (3.8) and (3.9). In this case, [Wmn (d = 0)] becomes the identity matrix and [rmn] reduces to [I] or -[I] for Hz and E. polarizations, respectively. Thus. Prmod becomes Pmod (cos a, cos 0o) Pmod (Ccos, cos o; 6 = 0, d = 0) = [Pm (cos c) ] {[I]-v[Rmnl}- [v1C (cos O)] (3.11) where Pm (cos c) l Pm (cos a; 6 = 0)

51 C' (cos,) -= C, (cOS 0o: = 0) Rr, RMr (= -0). 3.1i -Vce. 'th1.e Lei - sca tere'd by the vertically inhomogeneous recessed slab can be expressectd a F' (oo ) = / P cs(c ososo)e-Jkopco(-O)da y > 0 (3.1:3) where P,, (cos ct. cos o0) = Pii (cos a. cos 0) + Pmo (cos a. cos.). (3.14) TIhe steepest descent method can then be employed to evaluate (3.13) and ob-tain rhe (iffracted field. 3.2 Plane WNave Diffraction and Mode Coupling Consider the plane wave F- = ejko(rcosoo+ysino~ ) (3.1.5) F = vI sin p0ejko(zcos0o+yin0o) (3.16) incident at an angle Ce upon the structure depicted in Figure 3.2(b). In the absence of the perfectly conducting half-plane, the total fields may be written as (for y > 0) FPw = Fz + F (3.17) FP = Fj + F (3.18) \vi t hF = R (cos o)ejko( csO~-ysinO~) (3.19) F; = - lR (cos Oo) sin Ooejko(zcos oo-y sinO ) (3.2

52,,Ihere R (cos o.,) denotes the plane wave reflection coefficient of the grounded sd reterred o 7 = )0. For the general case of a plane wave incident at an angle a Q vert cal in'horrogeneous ground plane coating, R(A = cos a) may be represented a. R 'A(\)=-T~ B(\)3 - A (A)+ ~/1-~2B(A) ' where. (A) and B (A) are even functions of A. with any branch cuts in.4 (A) ais appearing in B(A) and vice versa. The introdulction of the perfectly conducting half plane at y = 6 generates, additional scattered field component Fs so that the total fields become F, = Ff +FS:3.22. F, = F + F (3.2:3 This scattered field is due to induced currents on the perfectly conducting half-plane. and can thus be represented by an angular spectrum of plane waves. A suitable representation is [10] i P (cos a) eiko sin ae-jko p cos (-O)d > 6 F? = fc Q (cos a) e - kosin [e-Jkoco(+) (324 +R (cos (7 - C)) e-Jkopcos(-)-) dca 0 < y <, implying - c v1 sin a P (cos a) eJkinae-jkOpcos (-Q)d > 6 F = fc 'v sin a Q (cos a) e-jko~ina [e —kpcos(C) (3.25) -R (cos (7 - )) e-jkocos(-a)] dCa 0 < y <., in which P (cos a) and Q (cos a) are the unknown spectral functions to be determined from the boundary conditions at y = 0. These are

.53 B3 (Co:'lj::'; ot.:e tangential electric fields at y = <. -x < x < 13': 'otri::' i of tne tangential magnetic fields at y = 6. x < 0 13; \c: i'iil. L tangential electric field on the perfectly conducting halfplane at y = -. x > 0 and 1 we note that the boundary condition at the slab surface is implicitly taken into account by the reprcsenta.i'on (3.24) and (3.25). The application of (B1)-(B3) in conjunction wiun (3.24) and (3.25) is straightforward. It results in the set of equations p (A) = 'Q (A) ri - - R (A) e-J2kos ] (3.26) J Q (A) -2 (A) e-Jkod,\A =; x < 0 (3.2) Q Q(A,3 (A) ) - iR(A) e-. ~ ] e-J'd = | Q- - (,\() e> k 2iko&VlJ eikoSd x -t3(A,) IlAe ioS/ [1 - lR(A,) e2kVS"/] ejkOxO:x > 0(3.28) where we have set A = cos a, Ao = cos 0p and 1 —, E. polarization, v2 (A) = (3.29) 1/x/1- A2, Hz polarization. f 1/1-AJ2, Ez polarization, 3 (A) = ' (3.30) 1i1, Hz polarization. These are sufficient to obtain a solution for Q (A) and P (A). However, before proceeding, it is necessary to rewrite certain terms in the integrands of (3.27) and (3.28) as products of 'upper" and "lower" functions, that is, functions free of poles, zeros, and branch cuts in the upper and lower half A planes, respectively. In the process of doing so. we introduce the definitions U2(A) - V2 (A) V (x) (3.31 ',

.54 L'3 (,\ U (') ) - (\),:3 (,) I-('I (,\) e~- 2jkv f L, (A:,) A: (A: )3: I IL-I C 'ILA L, (A: 6d) UC (A; 6) = A (A) (1 + A) e-2iko ^) Sv'U~B (A) (l-tl e2k ) (:3.:3 Ls (A) s (A\) = A (A) + /1-2B (A). 3.3. In these. L, (A ). L, (A: ), '7 (A), t'3 (A) are lower functions while U;8 (A),,L' (A\ ^. Substituting (3.31)-(3.35) into (3.27) and (3.28), we have J Q(A) t () v (A)ejkorAXdX=0; x<0, (3.36) / Q (A) v' (A) v (A) L ( ) (; 6) e-jkod - (3 vo) 3 (o 3k6 /; w(6) U (; ) ekox; X > 0(3.37) L, (Ao) U, (Ao) These coupled dual integral equations can be solved for the unknown spectra by examining the analytic properties of the integrands, and the reader is referred to [10] and [58] for a more explicit description of this process. From (3.36) and (3.37) the unknown spectra are determined to be L. (A; 6) U, '(A) v (Ao) v2 (Ao) 1+.) - 27j w (A; 6) L2 ( o v (A) (A) ( + A) (:3 -vLL (A; 6) L (A; 6) - (A) (A) ek 2-jLs (A) L2L 3 (A) ) -v (A),v (A) (A + A

55 Recogniziing t!at P (,i\ = P(A\). we may then set 6 = 0 and substitute for the plariat iion ',[itL:L(',en2t t'tnc ionsr to obtain P ) = 2jL, (A) L, (A0) (A A) (40) ~L, (A) L) (Ao) )/T 7 (~.) P`^) = -/A, ),,\ (3.41) 2, jL (A) L3 (\0) (A r A ) w\here the superscripts e and h refer to the spectra associated with E. and H. polarizations, respectively. To solve for the field coupled into the;eg>,. /0 y < 6, x > 0, (3.21), (3.33) and M:3.3S) are substituted into (3.24) to obtain the integral expression J' CU (A.3),3 (A,) 1 L(A ) -eA F:- J-, 2jt.; (A),3 (A) (A L (,) L (A) L, (A) C, (A; &) sin (kyl A) 1 [2j.4 () + 2B (A)cos (koyv/~ )) e-kdx(3.42) for 0 < y < 6. This can be evaluated by closing the path of integration -via a semiinfinite contour in the lower half A plane. The sum of the residues of the captured poles then yields F. - c, ( f,\) A (,sini (ky/1 - ) ') FZ = E C (A0) J A (A,) _ + B (A,) cos (koy - A e-)jkon r=1 (3.43) where {A\} are the zeros of Uw (A; 6), C ( f\ )-A) = eiko6X/7;'-XA L. (Ao; 6) v+ (Ao) v3 (Ao) 'i -A Ls (An) Ls (o) U/ (A; 6) v (An) v +(An)(A, + Ao)' (3.44) and C; (,; 6) (3.43) dA A=\n Substituting (3.29) and (3.30) into (3.44) with 6 = 0, we obtain the more explicit

.56 forms for C! \)A as ('* ' = ~LA,,(Aj,. 1A - A. 1 -r -: L ' Ls(),)n)Ls(,X, L. (n) Ao tA) ~ =; - s 2A j L,.(A 1) I A\1 * Ar C 2 L,~, (A,\) V/f"-1- A A Ls (A,\) Ls (Ao) L,, (A\) A\,-, ' A4 for E: and H. polarizations, respectively. 3.3 Reflection and Launching of a Waveguide Mode Consider now then nhh waveguide mode field (for 0 < y < 6) {-, = C sin (b --- F.= j.4 (\n) ( k, +B (,) os (oy\ '-A) ejkon [rn = 'l(A(,n)cos (kycos - A +1jv Bl +jB (Anl - An sin (ky 1 F incident at the waveguide opening (Figure 3.2(e)). The radiated fields due to this excitation may be again represented by (3.24) - (3.25). Subsequent application of thle boundary conditions (B1)-(B3) then yields the dual integral equations (with the usual transformation to the A plane) (A) (A)L"(A)U(A () e 0 dA =O; x >0 (3.49) J-'^Q 3( 3 Ls(A)Us(A) /ooy30 ~ ~ jx - ejkox~n Q (A) v (A) v (A) e dA-e; x < 0. (3.30) 2cos (k. l-A where A (An), E, polarization V4 = (3.51) B (An), H, polarization. The solution of those proceeds in a manner parallel to the previous case. The resulting spectra are determined to be ~Q =^\ ^-V4L w(An; 6)U, (A) V3 (A,) (3.2 4lij cos (k,,8 I ) ~-(An)U,, (A; 6) v2 (A) v- (A,) v' (A) (A + \n) p()\ =v vIIU4Ll,(An; 6)L, (AX; ) v3 (A n,) ( 4-rj cos (k/ 1- A Ls(An)Ls (A) v- (A) v- (An) v+ (A) (A + An)

.57 with L!.. L,. =. etc. as defined in (3.34), (3.35) and (3.31), (3.32). v:iti-:;."*::":3.-:. 3.2 3 3.:3.:30),:3.5i) and (3.:3:3 into (3..53) and setting O = 0 \-e '.ave r n P, (.) = L(A. (A)L,. (A.n) v/TI\V"- (:3 5 /1-,j, r.(A) L~, (A,) A + A corresponding to the spectra for the Ez and H, polarizations, respectively. Th!1 modal field reflected back into the guide may be computed by stubstituting (3.52. (:3.21) and 3:3.3) into (:3.24) and employing the usual transformation to the A plane to obtain (for 0 < y < s) J 4J cos ( (A) (A) v ) (A + \n) L (.\.: i) e-:kSV/-T( L, s)(,)in (k,\; ) 1 2jA(( ) sin (, -J +2B(A)cos(kyV')~ e -jkorAd 3.56) A.s in the case of coupling, this integral can again be evaluated by closing the path of integration in (3.56) via a semi-infinite contour in the lower half A plane to obtain Fm = E Rmn LJA (Am) sin(ky + B (Am) cos (koy } e-iko x\ (3.57) where Rmn are the mode reflection coefficients given by R mn= t'3V4V3 (An) e-jiksiV/1 Rmn -; - = - Cos (ko6l - v) (A ") V2 (A) V3 (Am) (Am + An) L, (A,; S) Ls(A,)L, (AnX ) U'l(Am; 6)' (' When 6 is set to zero, this reduces to A(A\) L, (A,) V/i - / 1m - ^' Ls (Am) ~L (A,) UC 'Am) Am + A,

58 Lhw (A,) V'TT A7/FTt R =- B (\,) L, t\,) L (An) UL (\m) Am + An, ~ ur t:i E ~ an,. It polarizations. respectively. 3.4 Computation of Spectra for Material Insert in a Per fectly Conducting Ground Plane We now have all the necessary components required for constructing the spectr P^ associated with the multilaver slab recessed in a ground plane as defined in (3.14). Substituting (3.40), (3.46), (3.54), (3.59) and (3.11) into (3.14) we obtain th, E~ polarization result L,.(\A)L,,(\.,) v/1\ r n=\ n - " '~-Ai Pr; (A. \x) = 'jL (A,) L (,) v ~ A [ _ )_ __ (:3.61, where f:e ~ _.A(Am.) L, (Am) 1 /-m1(Ame! n / L, L-(A) X(An) Ls (n) n) m mn (3 tn = {[] + [Rmn]}-. (3.63) For Hz polarization, the spectra may be obtained by substituting (3.41), (3.47), (3.55), (3.60) and (3.11) into (3.14) to find p (, x -Lw (A) Lw(A,)vTTTV7 IV m V', (\A + A Pr$ 0 -2. L sL (LA)L, (A) A+A0 L *.A+An A+Am, (3.64) where n = B (A) L (Am) L, (An) UW (An) V+ ^An (3.65) lt:~ = {[I]- [R,,n]}, ( 3.66) Expressions (3.61) and (3.64) can now be substituted into (3.13) and the re'suLir integral can be evaluated by the method of steepest descents to yield the.':

.59 non-unitorm diffracted field,- /. e jkOp 3 (:i Os. C cos 0,) Ui -e@'/4 Prs \ COS C. COS 00) V-'~o.p w-hee,x. 0, denote the usual cylindrical coordinates. In (3.67). F3 and Pr, refer to fE and P~P in the case of E: incidence and to ZH3, Ph for H, polarization. Although not apparent, (3.67) is reciprocal Nwith respect to cos 6 and cos oQ. as it shlould. We also note that Prs (A,,\) is a combination of an inhomogeneous solution (direct diffracted term) and a sum of homogeneous solutions (modal contribution). It may also be easily shown from the asymptotic behavior of (3.61) and (3.64) tha, the homogeneous terms do not affect the edge condition. 3.5 Specialization to the GIBC Representation To obtain numerical results, we must first provide expressions for the multilayered grounded slab reflection coefficient (i.e. A (A) and B (A)), as well as the associated split functions and corresponding complex roots (waveguide modes). To accomplish these tasks in a simple manner, we consider the general GIBC approximation to R (\). This amounts to setting NA A(A) = (1-A2) n=O NB B(A) = Bn(1-A2), (3.68) n=O where A, and Bn are constants specific to the multilayered slab and are given in chapter two. Introducing (3.68) into (3.21) yields R(, -.=NA (1 - A2) - v- EN B(1 - 2) (3.69) L-ZE0A (1-. 1v= R(A) V~~~A / ) (3.69) l =o. 'n (1 - A2) + - T 'R,= o Bn (1 - 2)n

60 and note that for a given order of approximation, NA and -N' are finite and in en.A = -\' or.A4 = -.V\ 1. with the order of the condition equal to S = -max (2)-A, 2^B +1). To evaluate the split functions L, (A; 6) and U', (A; 6). (3.68) is substituted in, (3.34) and by setting 6 = 0 we obtain 2 v"T - 1n B, (1 - A2)a E. polarization L, (A), (A) = (3. ' 2 =o An (1 - A2) H. polarization. The split functions are then trivially obtained as r 2B.v3 -TT7 i [vA 1 -VrT77 - A] E polarization. (A) = L (-A)= A ] 2A.n A [HI A~ -v A Hz polarization. (3. 72 where Imv/1 -, < 0 and {an} are the NA zeros of E o Bnx( for E. polarization or =4_,Any,7 for H, polarization. It is apparent from (3.44) and (3.72) that the pertinent waveguide mode propagation constants are given by koV/1 -T. In a similar fashion, we may substitute (3.68) into (3.35) to obtain NA n NB Ls ()U() U = ZAniV )2 + BnV/ l 2n n=O n=O Ns = SniT_ n (3.73) n=O = ( + (3 74) n=l 'Tn where (S An/2 n is even Sn = B(n-1)/2 n is odd, Ns S NS I {7n} = (zeros of the polynomial E St (-1) }.- (3.X75 1=0

61 The factorization of:3.74) is again trivial upon making use of the well known splitting rePr maine to e e 'ripedance half plane problem:34'. Noting that I T /1vi-\2 (:3.76, \we ha&ve, ~-, (,~)- c (-,- (, - (A: U3(A) = L(-A) = '!:3.7 ) where fA (A\ /) Re(q) > 0. A: r) = IA (-A: ) = r, - -- ' /3 ' A t) (- - i/ )K (A; q) Re(7) < 0. and._7 = /'2 sina/2\^ [[ (3-/2 - a - ) (,/2 - a + )]2 T, (cos a: r) = - (3.79) I' 2/2) '[1 cos [l(/2t )1/ + cos (3i/-' )] In the above Re(r > 0). Im (</ - /i) < 0, = sin-l(ri),0 < Re(0), and T,, is the Maliuzhinets function [221, whose evaluation in algebraic form has been given in [50]. Whereas the zeros of CU (A) represent the waveguide modes, the zeros of L. (A) (which are the poles of K_-) correspond to the surface waves supported by the material layer. Although not required in this analysis, these are easily extracted from (3.78) and (3.79). The expressions (3.68) through (3.79) provide a complete description of a GIBC implementation and permit the simulation of any multilayered coating. We remark that a unique GIBC modeling of a given coating does not exist; in fact one may employ GIBCs of substantially different character to simulate the same configuration. This point is discussed in the following section and some numerical results are provided for illustration purposes.

62 3.6 Numerical Results n 'his section GIBC simulations of various material inserts are presented compared with exact results available for the case of a single layer. Due to! greater interest. data is presented only for the Hz polarization case. The Glir employed here-in are given in chapter two. These are valid for arbitrary multila.e coatings and are therefore suited for this application. In particular. these multila;p. GIBC are synthesized by combining the component-layer GIBC in an appropriat, manner, pointing to the necessity of understanding single layer simulations in orde, to construct multilaver ones. Figure 3.3 shows the far zone pattern of a single layer insert (E = 2 - j.0001. = 1.2,. =.2A) modeled by various "low contrast" GIBC (i.e. those GIBC which improve as the layer thickness or index of refraction decreases). To illustrate their relative contributions to the far zone pattern, the direct diffraction and modal contri. butions have been isolated in Figures 3.3(a) and 3.3(b), respectively, with the overall result presented in Figure 3.3(c). We note that for this low contrast GIBC, an 8th order simulation provides a reasonable approximation to the diffraction pattern. In Figure 3.4, both the thickness and the index of refraction have been increased in a low contrast simulation of a (single layer) material insert with e = 3.5 - j.0001. / = 2.0, and =.4A. In contrast to the previous figure we now observe that a 20th order simulation is required to obtain a converged result. This degradation with increasing index of refraction proves typical of low contrast simulations and illustrates the need for other types of GIBC whose performance improves in this range of material parameters. In Figure 3.5 the same material insert corresponding to the data of Figure:3.4

63 C: g 20.0 E 08 o o-20.0 C:.o.o o-1 0. O - -0. -$0.0 _=-40.0 4 b (a) (b) ' 20.0 c 10.0 C-30.C -40.0 Exact(3 modes) LC.(3.t2).lC.(7.8...4).......:.. I aI2 I.. Iaaa.22 0 30 60 90 120 150 180 Angle in degrees 0 30 60 90 120 150 180 Angle in degrees (c) 20.0 3' 10.0 I1 0.,- 10.0 m _-20.0 b-30.0 -40.0 0. ~.....i --.... I- I -.\\ \ \x\\\\\x\\\xxxi 2^. E2-j-OO0,.1T2W -.2~~~11 q-M ~ PEC I '% xk N1 1NN -\ \ \' r Exoct(3 modes).L.-C3.A 2).... LC.,(,8.4)........... I....I......... i. 0 30 60 90 120 150 180 Angle in degrees Figure 3.3: Hz polarization backscatter echo width for a material insert with r =.2A, E- 2-j.OOO1, I = 1.2 modeled by low contrast GIBCs (see Table 3.1 for an explanation of the legend entries). (a) direct diffraction component (b) modal component (c) composite.

64 o 20.0 o 100o - 0 10.0 O,n 0 -20.0. 5-.: _=-40.0,0< 0 b (a) (b) Exoct \ LC..(3j 4.2. I LC.(11,12,6) \ L.C.(19.20.10) \.. I....I..... I....I. I.. 20.0 c 10.0 0 0. o 0.0 m-20.0 — 30o.0,<. -40.0 I I I I I I I t 30 60 90 120 150 180 Angle in degrees 0 30 60 90 120 150 180 Angle In degrees (c) 20.0 'S 10.0 0. a. 0.0 0 *o -..10.0 c-20.0 ~-30.0 -40.0 Exoct(8 modes) - 1 i I C. (342)' LC.(,12,) LC.(19,20.10) 7 PE-t 0 30 60 90 120 150 180 Angle In degrees figure 3.4: H. polarization backscatter echo width for a material insert with r =.4\. = 3.5 -j.001, / = 2 modeled by low contrast GIBCs (see Table 3.1 fr an explanation of the legend entries). (a) direct diffraction component (b) modal component (c) composite.

65 is simulated with high contrast GIBC (i.e., a GIBC which improves as the index of re-lac:'I.': ic.:~':"'st:,L.' a t e larer thiickness decreases). In contrast to the data inl F.:re:3.-. 3 ' e. \' ol cjbserve tliat only a second order high contrast GIBC simulation is requii eed o accurately evaluate the far zone scattering. This difference in performance between the low and high contrast GIBC stems from the type of approximation employed in their derivation and the reader is referred to chapter two for a more in depth discussion. Some insight on the type of simulation provided by the low and high contrast GIBC may be gained through an examination of the wateguide modes predicted by the different simulations. These are presented in Tables 3.1 and 3.2 for the single layer simulations corresponding to the data in Figure 3.4 and 3.3. The exact modes given in the table are generated by the equation A, = \/ (-, ), n=0,,.... (3.80) V \koT/ \We observe that as the order of the low contrast simulation is increased, the data in Table 3.1 reveal that the waveguide modes are "picked up" in a sequential manner corresponding to increasing n in (3.80). On the other hand (see Table 3.2), the high contrast GIBCs pick up the n = 2 exact mode immediately and then "branch off" to pick up the other modes. The discrepency in pattern convergence between Figures 3.4 and 3.5 clearly suggests that the n = 2 mode is the most significant in terms of diffraction (for this particular configuration). We explain this physically by noting that the n = 2 mode may be resolved into its constituent rays which strike the interface at a characteristic angle (say 902od) which is greater than the critical angle'ig of the material insert. On the other hand, the n = 0 and n = 1 modes are associated with characteristic angles less than the critical angle. This implies that upon coupling into the slab, the n = 2 waveguide mode is partially transmitted into free space while the lower order waveguides modes remain bound.

66 o 20.0 a. o 10.0 0 _ 0 0 _-0.0 - 10. 0 -20.0 -30.0 Im _.-40.0 e4 b (a) (b) l 20.0 = 10.0 a E 8 0.0 V. 10.0 ~-20.0 — 30.0.-40.0 m~~~~ Exacf(S modes H.C..(, 1~r~4........)...... 0 30 60 90 120 150 180 Angle in degrees 0 30 60 90 120 150180 Angle in degrees (c) 20.0 ' 10.0 0 a. 0.0 0,10.0 m =-20.0,< b-30.0 -40.0 0 L\s\s\\\\\\\\\\ \\\\\ 4^:a.XE X3.51XX,0L=2 /1 [kxXN'\'\ NNNN-X^ - ^ \p\\ 7 0 30 60 90 120 150 180 Angle in degrees Figure 3.,5: H, polarization backscatter echo width for a material insert with r =.4,\ 6 = 3.5-j.0001,, = 2 modeled by high contrast GIBCs (see Table 3.2 for an explanation of the legend entries). (a) direct diffraction componernt (b) modal component (c) composite.

E1xact Modes L.C.( 1,2,1) [.C.(5,6,3) ~I L. C.(9, i o rj 1, - ( I" - ( 1 3,1 4,7) 1, - C - ( 1 7,18,9) I 1'. c.. ( -2 1, 2 -2, I I ) L Ex.. M e,.. 0. --- _l., _8_..._._._. _... 2.6 '(58 - O 2.6458- - jO 2.6158 -O 2.6458 - jo 2.6.158- 2.6158 - jO 2.6 15 jO 2.3318 - j0 2.3324 -.1518 2.3303 - 2.3318 - 2.3318 - 2.3318 jI} 0.8660 - jo.0001 -2.3324 - j.1517 I 1.7608 - j.0001 1.1583 - j.0001 0.8782 - j.0001 0.86612 - j.000( 0 - j2.6575 2.3156 - j.1518 1.5228 -jl.4058 0.8198 -j. 9935 0.3937 - j2.6289 0 - 4.2464 -2.3156-j.1517 -1.5227 - j.4058 -0.8197 -j.99I, -0.3936 - j2.6289 0 - j5.6624 3.0782 - j2.1315 2.2444 - j2.6318 1.6576 - j3.1906 0 - 7.0178 -3.0782 -2.1344 -2.2444 -2.6318 -1.6576 - -j3.190 0 - j8.3404 4.1090 - j2.8858 3.16:12 - j3.5300 0 - j9.6437 -4.1090 - j2.8858 -3 161'2 - j3.5300 0 - 10.93- 8 5.21!91 - j3.5008 0 - jl2.2168. -5.2491 - j3.5008 Cr-i -1I Table 3.1: Low contrast approximation to waveguide modes for a layer with e = 3.5 - j.0001,px = 2.0, and r -.4A. For each low contrast boundary condition, the three liumbers of the column headings indicate tile order of the approximation in thickness r, the order of the resulting bolundary condition, and the total number of modes (see chapter two).

Exact Modes H. C.(0,2,1) H.C.(1,4,2) H.C.(2,6,3) IH.C.(3,8,4) II.C.(4,10,5) 2.6458 - jO 7.9636 - j.0032 3.2862 - j.0001 2.3318 - jO 5.5825 -j.0023 2.2571 - j.0001 2.2813 - jO 2.2714 - j0 0.660 - jO.0001 0.8667 - j.0001 0.8649 - j.0001 0.8660 - j.0001 0.8660 -.0001 0.8660- j.0001 0 - j2.6575 0 - j2.1309 0 - j2.0694 0.6002 - j2.2920 0 - j4.2464 -0.6002 - j2.2920............~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0c) CAD CY~ Table 3.2: High contrast approximation to waveguide modes for a layer with t = 3.5 -j.0001,/A = 2.0, and r =.4A. For each high contrast boundary colndition, the three numbers of the column headings indicate the approxillation il the index of refraction ~-', the order of the resulting boundary condition, and the total number of modles.

69 The above hypothesis may be tested by computing exact solutions in which the,eieCtion of t he incil.:udecd waveguLcide modes parallels the order in w vich they are icked "up edepe Zing on whether a low or high contrast GIBC simulation is employed. fi!re:3.6( depicts a high contrast simulation of a single layer having e = 11 - J.OOO1./L = 7. and =.4\. The 10-mode result is a pattern obtained by adding in modes sequentially as determined from (3.80), thus paralleling a low contrast mode selection scheme. On the other hand. the single mode result contains the contributio! of only the n = 7 mode (the mode with Om > 0c), thus, parallelingl thle high contrast mode selection criteria. This clearly verifies that the most significant waveguide modes are those that are "visible', i.e. those with O8 > 0c. Finally, Figure 3.7 provides a simulation of a three layer insert composed of two ih contrast layers (e = 11-j.0001. / = 7, and r =.4\ and e = 3.5-j.0001,[ = 2.0, and r =.4A) placed beneath a low contrast layer with e = 29 - j.0001, = 1.2 and =.2A\. These are precisely the layers considered earlier in isolation. One might. therefore, expect that the order of a GIBC which provided a converged result for the single layer simulation will also provide an equally acceptable simulation when the slab is part of a multilayer stack. For the case at hand this is indeed true, as evidenced by the converged 9th order result. We also remark that the presence of the two high contrast layers enhances the modal contribution to the total diffraction when compared with the single layer data given in Figures 3.3(b) and 3.3(c). 3.7 Summary In summary, the scattering from a vertically inhomogeneous slab recessed in a ground plane was obtained through application of the generalized scattering matrix technique in conjunction with the dual integral equation approach. The solution was

70 20.0 10.0 0.0 m *3 00< bD -10.0 -20.0 - -30.0 - -40.0 - 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees U S \,\ ' a_ "-PE Fi ure 3.6: H, polarization backscatter echo width for a material insert with T =.4,. e 11.- j.0001, u = 7 modeled by high contrast GIBCs (see Table 3.2 for an explanation of the legend entries).

20.0 10.0 0.0 m 03. - e< b -20.0 - -30.0 -40.0 - 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees _ - TICr' --- I IC I I."\ v N \ \ \\ \ \ \ \, N' N N N\ \ \N r /.2/^=, ' 2i.000 L, =l.2 p3././-i..'000,..12.< i R-ti.4A3= 1.0001 =1 7 _//////,^^//^^^^ ~-PE( Figure 3.7: Hz polarization backscatter echo width for a three-layer material insert with (r1 =.4A, = 11. - j.0001, /i = 7), (r2 =.4A, E2 = 3.5 - j.0001, /2 = 2), (r3 =.2A, t3 = 2. - j.0001, 13 = 1.2). In the legend entry a.b,c(d,e), a and b denote the approximation in c-1 of the high constrast layers 1 and 2, c denotes the approximation in r of the low constast layer 3, while d is the order of the composite GIBC and e is the total number of modes.

72 specialized to the case of a multilayered slab simulated with a generalized imped, bo:ndary condition (GIBCi. Results were given for various single layer insert. it was seen that in the case of materials having sufficiently high index of refracti Li'.; contrast GIBC simulations converged more rapidly (with respect to the otr of the GIBC) and performed better than low contrast simulations. Finally. res1, were presented for a 1A thick lossless three-layer insert containing both high and r., contrast layers. It was shown that the simulation converged at the point predict. by the individual layer simulations, suggesting a method for constructing multila. simulations.

CHAPTER IV GENERAL SOLUTION OF THE DIFFRACTION BY A MATERIAL DISCONTINUITY IN A THICK DIELECTRIC/FERRITE SLAB In this chapter and the following one it is demonstrated that a GIBC/GSTC sheet characterization can yield a unique solution when supplemented with certain conditions at the sheet discontinuity which do not require an apriori knowledge of the edge fields. As a vehicle for presenting this solution procedure we employ the dual integral equation method to consider the plane wave diffraction by a discontinuous distributed sheet (see Figure 4.1(b)). This very general model is capable of representing material half-planes, material junctions, and material discontinuities on grounded structures, such as those shown in Figure 4.2. In addition, a distributed sheet model typically renders the same degree of accuracy as the usual infinitely-thin sheet. but with a lower order condition. It is, therefore, of much practical interest. In the first section of the chapter, the GSTC representation of the distributed sheet. discontinuity is used to develop dual integral equations in terms of the unknown spectral functions proportional to the sheet currents. These equations are then solved in the standard manner to yield expressions for the spectral functions in terms of unknown constants. The constants are dependent on the material and geo 73

74 y................................. td /F Odd GSTC: x 2 F -F I"tF ( k2 ') + +- U 2 I Tay[F =0 k J Even GSTC: ax 2 U21F F + 22, k2 [F -F = O k i (a) y T —+ - /I T, JF' --- "MO X I i 1 I U 1't2 21 22 2 2 2 2 U11 12 U U21 u22 (b) Figure 4.1: (a) Distributed sheet. (b) Distributed sheet discontinuity.

~u!o[ papunoJ3 () u(o Isleu-ql I-uiof up(q) aueId-Jaw q!:xaleYv (e) -laaqs painq!ls!p snonu!luo3s!p p Xq palapow sa!.laoar '.'t ani! julaw)~j,r (3) f d) A (q) I -.-..............................! x 1 A A l,. O.. O..... O..0.,,, I_,,,,,,,,,,,,,,,,,,,,,,o~,,,,,,,,........... ~r.:....:,.: -:..:.......:...:.: I. -..,.' K 4 / / / / / / 7 / ~ X /X / / / /. BIu\ S \\ \ ^ \\ \ \' SI_ _j I II I I __ __ __ __ ___-_-__-_-_ _-_-__- - -- -- dL- '- 'I - - - t~~~ '1..;. t......................................................... r( r r r r r r -.............::' ~~~~''' ': ~ ~~............... ~~i~i'ii~::~ii:~~~:::~:::::::~iiii~I~~::i::. (1) / X ^!^!^ir;i r^i~riIrir~4444 x X' X X s,. s., ',, r rA t. ' t.A_ - - - -— r-rI d) I A- XT. A?-A 1.4?.A A;r I Ct

76 metrical properties of the discontinuity and their specific value is identified here several dis-cntinuities by comparison with a few known diffraction solutions T,e orc- *'rates t he validity of the presented solution. but in general. the determirnatc of thle constants requires the enforcement of additional constraints demanding fitl continuity accross the layer discontinuity. The development of these conditions ay their use in solving for the constants are presented in chapter five. 4.1 Dual Integral Equation Formulation Consider a distributed sheet of thickness T illuminated by the plane wave {E, in Ez polarization. Fin - k(cos~o+ysino~) - (4.1. ZoHzinc, Hz polarization, as shown in Figure 4.1(a). The excitation (4.1) induces reflected and transmitted fields which are explicitly given by the properties of the distributed sheet. If this sheet models a symmetric slab, then an appropriate GSTC representation is formally given by (see chapter 2) )11 {F -F }+ x (- ) [F+ (4.2) 1 (-2) {F+ + F-}+ 22 (- ) {y [F+ -F} = 0 -c < x < x. - k2 {F++F-+22 --- (4.3) in which F is the total field, F- =F (x, y = ~/2), axF* = F (x,y = ~r/2), and dyF~ = F F(x,y) ly=4r/2. Also, l1U (-v9.)2 are differential operators which operate on the field quantity in the curly brackets, and are finite polynomials in - 2 whose coefficients depend on the slab modeled by the distributed sheet. To maintain the generality of the solution. the U1. operators are left in symbolic form and the reader is referred to chapter two for their explicit representation in terms of the material

77 constants and thickness of the layers comprising the modeled slab. In general. the order of Z l i.e. the highest derivative present) is usually the same or one more thlan tlat ofi l. and similarly the order of U21 is the same or one more than the order of /_.. T!':s. we may define the orders of the GSTCs in (4.3) to be V - = maximum { order of 14 (A'), 1 + order of i (A2) } \.e = maximum { order of I21 (A2) 1A order of U U\ (A2) } (4.4) The reflected and transmitted fields may now be easily determined by employing (4.3) to find Fre,, = Rlejk(x cos~O -sin o~ ) (4.5) Ftran = Tiejk( cos o+ysin o) (4.6) in which R1 and T1 are the reflection and transmis -n coefficients, respectively, and are given as ejkr sin o R1 2 [R en + Rodd] (4.7),jkr.sin ^o T1 = [I -. (4.8) 2 [R,,n_ Rtd. with Rven sin oU212 (cos2 0) - U1 (cos2 o) R even 21 (4.9) sin U21(cos 20)+ 1i(os2pI ) (4.9) odd sin o12 (cos2 o)U+ (cos2 ) -Da _ sin 1(oU.2 (cos2 0o) +- 14 (cos2 0) ( 1 fusinos 12 1 W\e remark that in (4. ') and (4.10), Ui (cos2 o) now represent simple polynomial functions in cos. since -&x2/k2 = cos2 o in view of the field expressions (4.5'i and (4.6).

78 Consider now the case where the right half of the distributed sheet in Figure 4 is reoiaced y-v another sheet of the same thickness. but of different properties illustrated in Figure 4.1(b). The GSTC representation of this modified sheet is f, (-0){ ){{F+ F} ~ TZl2 ( 2) {d F. (!} for -x < x < O0 and >} (_ ) - } + [F (t F [-, + (F-j} = 0 1 (- ) {F +F-} + (- ) {ay [F+ — ]} = 0. (4. 712C~2~k2 ~ ( - k. for O < I < xC. here the superscripts 1 and 2 distinguish the left- and right-hand sheets, respectively. Referring to our previous discussion, the orders of the right hand side GSTCs are given as Odd = max { order of UZ1 (A2) in A, 1 + order of U~1 (A2') in A} 2,'n = max {order of, (A2) inA,1+ order of x (A2) in A}. (4.13) The modified right hand side sheet induces a scattered field F, in the presence of the excitation (4.1), and the total field can now be represented as n + FreFl + F y > r/2 F= (4.14) Ftran + F, y < T/2 where F, is the unknown scattered field in the region lyl > i/2 and can be expressed as either [10] [52] FS (,y) - C [i Pod (coS ) + Pevn (co c)] e-ikSi(YI-r/e-ikcosada. d = my

79 t Im(a) 0 I I I I I In2 I I I I I Mb. /2 A n-<p, 3n/2 --------- - -— l —U C I I I I I I I I (a) Im(X) 1 w- I u i I - Ah -1 x Re(X) (b) Figure 4.3: (a) Illustration of C contour in the complex a plane and (b) complex A plane.

80 or F f\ {\ P (\)1 c-^^^V dA f X' _ d [ y (\) + Peen ( e kVe i(yL-/2)e - rj\ 4._ Y V/ I V1- A2 ~ uIpon invoking the transformation A = cos a. The contour C in (4.15) is the int gration path in the complex a plane shown in Figure 4.3(a), and its counterpart for the A plane is illustrated in Figure 4.3(b). Also, the spectral functions P,od (A) an, Pen \) are directly related to the Fourier transforms of the unknown ectulvaleni currents Jodd = F -F (4.1 Jeven = FS + F-, (4.Is1 via the relations Jodd () = 2 Podd (A)e k-j A (4.19) Jeven () = 2 P,, ()e-jk (4.20 Assuming that Jodd(X) x 0dd as x - 0 Jeven(x) - x Sn as x -- 0 (4.21) with 0 < Sodd < 1 and 0 < seven ~ 1, from (4.19) - (4.20) and the Abelian theorem we have Podd (A) 'odd as |A| I -. Peven () X-sevn as |A| - o (4.22) From the asymptotic behavior of Podd(A) and Pven (A), it may be easily seen that the integral representation (4.16) is well-behaved and convergent for all x and;

81 Substituting (4.1), (4.3), (4.6), (4.14) and (4.16) into the transition conditions (4.11) anj, 4.12. we obtain r ilk dA J(i (<, Podd (\) e-k \ = 0 (4.23) 1 -,\ 2 J,A) Pe iXen -(A\) e:, = 0. (4.241 for x < 0 and -C, kxA A ( osin n b eJ k - l'O /2sii 0 r 2 7 2 ) fr 2_" ( \ ) P;d (\) e-k 7s = <J /; dd^.2) T-x -52VC ~ ("\ P (~A) eJk d:\ _ 2sinoeJkxAoejkr/2 iin <oZ, / J-.^-^2 (A J eLven-j_= (A- ( -v.en (4-.26 for x > O. where A, = cos oo and gdd (A \2) = 1XA (\ V ) (T 1 A2 4.2 7) 5euen (A2) = 1 (A2) + v/T722 (,\2) (4.28 "_ A ( A) (2) + () (4.29) gven (\2) = 21 (A2) + IV 2U (A> (4-30) Zodd (A) = 211 (A) (12 (A2) - 12 ( 2) Ui (A2)1] (4.31) Zeven [1 ) 22 22 () 21 ()] (4.32) Note that strictly speaking, the integrals in (2.26)-(2.31) are not convergent because of the polynomial order of the integrands. This difficulty is common in analytical GIBC solutions and may be remedied by working with integrated field quantities as discussed in [39] and [44]. It has been shown, though, that the final solution is the same regardless of this remedy and for the sake of simplicity we will proceed with the solution of the dual integral equations as if all Fourier inverse and forward transforms existed in the classical sense. Equations (4.23) with (4.25) and (4.24) with (4.26) form two uncoupled sets of integral equations which are sufficient to yield a solution for the unknown spectra

Pdd () and Pt,,, (A). Clearly, because of the similarity between the two sets c equations. once a solution for Pd-, (A) is found. the corresponding one for P,,,A(\ follows by inspection. 4.2 Solution of the Dual Integral Equations In proceeding with the solution of Podd (A) we first rewrite (4.25) in a more suitable manner, viz. J f 0d^ d (,\2) Podd (A) sin oeZJkr/2noZdd (Al) e kA = 0.. d-.4d0 (Ajdd (A,\)( + A,) for x > 0. Closing now the integration paths in (4.23) and (4.33) by semi-infinite contours in the upper- and lower-half A planes, respectively, leads to the deduction that odd!\2) Podd (A) dd (A2) Podd () sin ooejkZ/sin=oZodd (2) Eodd (\) +.= ~ (A) (-4.3.5) rT-T? ' r \2 Eodd (-AO) v-A2 27rjid (A) (A + A) Eodd(-A) where IU (A) and L, (A) are unknown functions regular in the upper- or lower-half of the A plane. Also, Eodd (A) is an unknown entire function to be determined along with Podd (A), Uo (A) and Lo (A). To solve for these, it is necessary to exploit the analyticity properties of (4.34) and (4.35) in the different regions of the complex,\ plane. An important part of this process is the factorization of the 9 functions in (4.34) and (4.35) into a product of upper and lower functions (that is, functions free of poles, zeros, and branch cuts in the upper- or lower-half A plane, respectively). This task is described in the appendix. We have godd (A2) = odd (A) Odd (A) even+ (A2) -(A) (A)

83 5 f, (\X2) = g;. en(A) g (A) (: oi "(A\2 ) = G o end4.-36 where thie superscipts ~ denote upper or lower functions. respectively. Combining (4.34i with 4.$35 in conjunction with (4.36) and rearranging terms we obtain Add (\ (A) (5 (A) odd () _ (A\)dd (A) + =. (4.-3 g1d (A) + (A+A)dd g (A) -{)d (A) wlhere sin O,eJk/2sioZodd (Ao) Eodd (A) S (A) = '2, -3jdd(,(. 2 jdd (A2) Eodd (-Ao) The second term of (4.37) may be easily split into a sum of upper and lower functions. and when this is done. (4.37) can be rewritten as g^(A)U(A) SL'A) (A) (A) L L0(A)9d(A) gdd(Am) (, (+) L o ( 51 (o ) S (A+A) L W(A) - (Aj(o)439 (X + AO) godd (X) -od (Ao.3 The left hand side of (4.39) is now regular in the upper half of the A plane while its right hand side is regular in the lower half of the A plane. By Liouville's theorem, both sides must then be equal to a polynomial, and to determine the order of this polynomial it is necessary to examine the asymptotic behavior of the individual terms in (4-39). From (4.4) and (4.13) dd (A () ANdd/2, a (A) AN dd/2, implying that the left hand side of (4.39) behaves as lAl(Nodd+No2dd)/21'-Sodd when JAl -+ oc, provided that Eodd (A) behaves no worse than Al({vodd+Ndd)/2Sodd as Al - oo. The right hand side of (4.39) will then behave as the greater of lAl -dd1-Sodd and lAl(Ndd+iVdd)/2-1-lodd when IA -+ oc. In accordance with Liouville's theorem, both sides of (4.39) will then be equal to a polynomial of order int 2{ N 2- -_ 1 - Sd}. In terms of this unknown polynomial. we may solve first for Uo (A) and subsequently for Podd (A) t o

84 find that _ j sin o_/1 _-.\2.gJ^k -/2 sin. O E,od (A) odd odd - n- ( ^odd A +2 arn n A\ (AA + A) J (AA) 4 d ()Eodd (-A ) m=1 n=O In this, -Vdd = t {1/2 (-lodd + o2dd + 1)}, and amn are arbitrary constants as et undetermined. and correspond to the coefficients of the polynomial resulting fror the application of Liouville's theorem. The chosen symmetric form of this polynorria: is not unique but will be found most useful later in constructing a reciprocal forr, for Podd (A). Following a similar procedure we also obtain P,,,, (A) as pt _ i sin o /-, 2ejkr/2sino Peven (A) 2 A + A v,ev ( AA) n(A even (A) ven (A,) L'even (-A ) m=1 n=O with Eeven (A), -even and bmn being the counterparts of Eodd (A), Nodd and ann respectively. We note that (4.40) and (4.41) imply that the powers Sodd and s:V, governing the behavior of the equivalent currents Jodd(X) and Jeven(X), respectively, are given as { N1 odd + Nodd is even Sodd = (4.42) 1/2 Nldd + N2d isodd S 1 leen,, + lNC is even Se.s.een = (4.43) 1/2 NI + N2vn is odd To determine the unknown entire functions Eodd (A) and Eeen (A), we observe that the spectra Podd (\) and Peen (A) must exhibit a reciprocal form, which may be

85 achieved by setting Eodd (A) and Eeven (A) equal to any of the following functions: Zodd (-AXAo) or Eodd () = Zdd (A) or (4.44) Zodd (A) Zeven (-AAo) or Eeven(A) = ) or (4.4.5),,en (A) where Z en (A) and Zjd,even (A) are upper and lower functions satisfying the relation Zodd,even (A2) = Zodd.ven (A) Zodd,even () (4.46) Taking into account the choices (4.44) and (4.45), we may substitute (4.40) and (4.41) into (4.1.5) and subsequently perform a steepest descent path evaluation to obtain for p - c (all surface wave contributions are neglected in this evaluation) pkp27r where (p, 0) are the usual cylindrical coordinates and Dodd (<, o) + Deven (O, Oo) is the far zone diffraction coefficient symmetric with respect to d and 6o. We have,-jir/4 sin o0 sin Dodd (0, Oo) 27r cos 2 + cos o ejk'r/2(sin o+I sin |1) g _d (cos C),dd (coS <0) 2~dd (coS o) idd (COS d o) VNodd- 1 Nodd- -m Zodd (COS 0, COS Oo) + E E amn (COS +COS o)m' (COS q COS c o)s (4.48) m=1 n=0 e-j'/4 sin >ol sin l1 Deven ([, >o) = - -- 27r cos < + cos p0 ej k/2(sin o+I sinm 1) ' ev (cos ) 9e1e (c (cos 0) eTen (cos ~) 5cvn (cos 0o) N even - Neven- 1-m Zeven (COS, COS O ) +,, (s +0 0) +cos O)m (COS 4 COS pO) n Lm=l n=O (4.49)

86 in which the functions Zodd.en (Cos 0. cos O) are given by (see (4.44) and (4.4., Zodd (- cos cos &o) or Z, (coso. los ) = Zdd (cos ) Zdd(cos o) or 1 Zodd (coS ) Zod (cos Oo) f Zeven (- COs O os Oo) or Ze,.e (COS O,COSOo) = Z,, (COS O) Z,,, (cos o) or ( Zeven (cos ) Zen (cOS O) Because the above three choices for Zodd and Zeven differ only by terms of the fort (cos Q + cos Oo)m (cs cos s &o)', it is immaterial which of them we choose, a!t lou: one of the choices may likely lead to a more compact representation. Nevertheless regardless of the choice of Zodd and Zeven, one is still faced with the determination of the unknown constants amn and bmn in (448S) and (4.49), respectively. These area. manifestation of the non-uniqueness of the finite-order GSTC sheet model employed he;.n, and their explicit determination requires the introduction of additional con-. straints pertaining to the physics of the problem as discussed in chapter five. In some cases, however, these constants can be determined by comparison with alternative diffraction solutions, and this is considered next. 4.3 Discussion of the Solution and Some Applications 4.3.1 Diffraction by thin single layer discontinuous slabs The diffraction coefficient given by (4.48) and (4.49) is very general and can model a wide variety of geometries. To check its validity, display its versatility, and assess the relative importance of the unknown constants, we consider several simple configurations which can be modelled with the proposed GSTCs. Their geometries are shown in Figures 4.4(a)-4.4(c) and include the single layer join, the material-metal join. and the material half plane, all of thickness 2w. Herewith, these are mno>

s7 y 1 /A 2w TP 2w 2wi 2w, LZ-1 x, x (a) (b >) PEC or PMC ' X......... wo X x:1 C (c) 2(w-w, 4.4 Figure 4.4: s 4w::Jm -.................,40:16 I I 1 1 U i,UU2A1,u22 111 Y rl n 2 2 2 2 11 12' 21 22 N~- X (d) Thin slab structures and their distributed sheet representation (a) Material-material junction. (b) Material-metallic join. (c) Material halfplane. (d) Equivalent distributed sheet.

by a distributed sheet discontinuity of thickness 2(w - w,) (see Figure 4.4(d)) a although. in generai. it is not necessary to employ a sheet thickness different fr that of the slab. this is useful here for comparison with previous results. If the left hand side of the slab is assumed to be thin and having a low index 0 refraction. it may be modeled by a low contrast GSTC sheet. Thus, an O(zc1 a,) approximation with terms of O(ww) neglected is sufficient for the representation of the operators or polynomials Ug1. In particular, we have: (-Ox2/k2) = 1 U2 (-ax2/k2) = jk(uw - w,) iU21 (-axik) = jk (l + ) - ( -- ) W as U1. ul 2 (-ax2/k2) = 1 (4.523 where E1 and y1 are the relative permittivity and permeability of the left hand slab, respectively, and ( i, E, polarization ui = (4.53) e1 HZ polarization Also, when w, = w, these are simply the transition conditions derived first by Weinstein [59] and later by Senior and Volakis [39]. The corresponding polynomials to be employed in (4.27) - (4.32) are given by Ul1 (-cos 0 cos o) = 1 U1 (- cos cos 0o) = jk (Ulw-Ws) 21 (-COS COS o) = jk -Ws + jk w — ws) cos ( cos o L.2 (-cos cos0) = 1 (4.54) Incorporating these into (4.48) and (4.49) and setting Zodd (COS 6, COS o0) = Zodd (-COS 0 COS 4.o) I 4.5.5'

89 Zeen (COS, COS o,) = Zeen (- Cos ~ cos 6) (4.56) viei.l. (o.o) e-'"/4 sin sin ejk1/2(sino+lsin) 2 cos + cos o0, (f- cos o cos d) - jk (ulwW - w, ) (- cos 6 cos op) 0 0 ' II odd Ij~' i ) d Od~ AL. V cos o:,odd.1) A-L (cos,; 70 dd(1) odd (cos) (co ) (o) = De-i/ sin o l sin O1 ejkr/2(sin 0+lsin I) COS 0 + COS 00 f _____[QI + [ 2 COS C COS d)] (2 (-COS COS 0)- 121 (-cos QCOS 0.) 3i [n.r (coS;; even' ) A,- (cos; ^ve )] Qeen (cosp) 52en (cos 0) blo (cos 6 + cos ~o) esven'1 'e_ CO/en' \ ev'en 'C'e,_n (cos 92) a2 =.2. where the split function._ (cos c;.,) is give n in the appendix. C2 = Jk( -— 'J c3 = -(eC - 1) (4.59) U' and odd,l - - ven,.l = u 1: u2 + 4ck2w (1/ - 1) (w w) (4.60),)?1,2 (4.60) of (4.35) and (4.58), the functions associated with the right hand side properties of the slab (i.e. those functions with the superscript 2) must be specified. Referring to the configurations in Figure 4.4(a)-4.4(c), Table 4.1 and Table 4.2 provide explicit expressions for the functions t; (-CO cos COS ) (COS Oc ) 2 d (~dd (COS O, I <0.

Table 4.1: Odd symmetry parameters for listrilltlted sheet sinlilatioll of righlt hiand side material x > 0. See Appendix I for definition of Al+ split functions.

~ ( do 2f - f Geometry, x >O UJ (-coskcos), 2(-cos cos o) en (I v (c 2) Low Contrast, O(ww,) |U = jk = - w.) +jkcosucos k ( - W.) | ' (2/2- 1) li -Mf (:os;;y e l2) A1, (jS +o; y,c "Y2) = c il Cven,2 u_ U2 /u T4k2w((<2 2-1)(w-u2w,) u: = 1Iwith 7t.2 = 2jk(w-w.,,2) Free Space Limit 2 = jl (w - w,) (1 + o cos - cos o4)1 - cos 4A+ (cos (; 7u'"'2) AI+ (cos o; 7;".12) U22 wi th~ll cVcn,2 - 222 =- k(w- w.) PEC(Ez-pol) (blo = 0) U21 =1 or PMC(Hz-pol) (bo = 0) U2, = 0 PMC(Ez-pol) or U = 0 - cos - cos PEC(Hz-pol) U = 1 _ Table 4.2: Even symmetry parameters for distributed sheet similation of right hand side material x > 0. See Appedix I for d(efinition of Al+ split functions.

92 G+ en (cos o);en (cos,) terms. By edge condition considerations. all of th stants ar,, and b6n have been set to zero except blo appearing in the definition De,^. which is non-zero unless the right hand side slab is a PEC/PNIC Under EJ/HZ excitation (see Table 4.2). 4.3.2 Diffraction by a resistive-resistive junction When w = wzs, and the material parameters of the slab geometries in Fi. ures 4.4(a) and 4.4(c) take the limits 61,2 00.1,2 0 1 O 1 jkw(61,2 - 1) I —, 1, jkw (1,2 - 1) - m (4.61) r/t,2 the resulting configuration corresponds to coincident resistive and conductive sheet junctions [37], where 71o2 and 77 2 denote the resistivities and conductivities of the respective sheets. In this case, the constant blo is forced to zero by the edge condition, and (4.57) and (4.58) simplify to Dodd (cos,, cos 0o)= sin b d (cos 0, cos 0o) (1/t1 -1/y,), 22 M_ (cos ~; l/7r1) M_ (cos; 1/7l ) M+ (cos; 1/7 /) rM+ (cos o; 1/y ' Deven (COS O, COS o) = Isin bI d (cos X, cos 0o) (7q - qe) (4.63) KlM, (cos '; 1/i) M_ (cos 0o; 1/7i ) M+ (COS 0; 1/n7) M+ (C 1 (co.62) for E, polarization and Dodd'(cos 6, COS o) =

93 sin o d(cos. cos 6o) (1//77 - 1/r/).'_ i cos o; 1/rl).I_ (cos Oo; 1/f,).1+ (coS 0; Il/r) M1.+ cos o0: 1/r7) D_-e icos &.,0o 0) -- sin o d (cos o. cos o.) (rI - r?)! L._ (cos o; l/r7)._ (cos Po; l/r ),+ (cos o; 1/r ) lM+ (cos o(; 1/I r'2) for H, polarization, where e-jr/4 sin do d (cos p, cos 0) = - 2 cos +s o (4.66),. cos a + cos as Note that if both 71. 77 -- o (i.e., the resistive sheets disappear), then Ddd (cos o. cos 0o) and De,,,,en (cos o, cos &o) tend to zero for Hz and E, polarizations. respectively. On the other hand, if both rji, r7j - oo (i.e., the conductive sheets disappear),then Dodd. (cos os s ) and (cos, cos ) tend to zero for E, and H; polarizations. respectively. This is, of course, because resistive and conductive sheets scatter independently of each other. Consequently, the field diffracted by a resistive to conductive sheet junction is the superposition of the individual sheet contributions [39]. 4.3.3 Diffraction by grounded metal-dielectric junctions Of the geometries shown in Figure 4.4, the diffracted field associated with the metal-dielectric junction (Figure 4.4(b)) is given in chapter three and can therefore be used to partially validate the derived solution. However, in order to study only the effect of the constant blo, we need to exclude the odd-symmetry portion of the metal-dielectric join diffraction coefficient. To this end, we focus on the recessed slab geometry of Figure 4.5(a), whose Hz polarization diffraction coefficient is related (throuLgh image theory) to that of the metal-dielectric join by Drs (coS b, cos >o) = 2Deen (cos 4, cos o0) (4.67

94 y PEC w T' W T (a) fb x PEC PMC i r, YC x (c) u1, U 1 U1 12 U 21 22 _ _M.... b y t, 2 2 2 2 U =U =0, U =U =2 11 21 12 22 -__ A- _ 0_ _ _ _ _ I PEC (d) Figure 4.5: (a) Recessed slab (PEC stub) (b) Grounded slab with truncated upper plate (c) Recessed slab (PMC stub) (d) GIBC sheet

95 with Dete. given by (4.58) along with the PEC(H,-pol.) entries of Table 4.2. As noted above. for a right hand side PEC slab with E polarization. the edge condition demands b10 = 0 in this case. For the recessed slab geometry illuminated with an HZ polarized plane wave, the GSTC (4.11) and (4.12) become the GIBC _1 (-2) {F} + I22 ( 2) {F} =, (46) for -c < x < 0 and (- 02){ F+}++ k4 (-F2) {ayF+} = 0 (4.69) for 0 < x < xc, with the Ui. operators given in (4.52). Clearly, these GIBC can represent any of the configurations displayed in Figure 4..3(a)-(c) without regard to whether a stub (PEC or PMC) or not is placed at the junction. This information can only be carried by the constant blo as the term distinguishing the diffraction coefficients among the geometries of Figure 4.5(a)-(c). Thus, the determination of blo must somehow involve the properties of the junction across its thickness and this is discussed in chapter five. However, since the solution of the configuration in Figure 4.5(a)-(c) are already available in chapter three, blo can be identified for each geometry by comparing (4.58) with the appropriate solutions given in chapter three. Upon setting w, = 0, we find bno stub =Ijkw 10 v,pec stub jkwf _ 1b0c s (fyiw 1) [M. (5/ree'n) M. ( a.,>en1)]2 + 1/2,pmc stub _kw.. stu -k (v w-1) [M_ (e/7F.evenl) M (a el ven, / (4.70)

96 corresponding to the constants associated with the diffraction coefficients for t geometr es in Figure 4.5tb), Figure 4.5(a), and Figure 4.5(c), respectively. We n, that these constants are based on the choice of Zeven (cos, cos <o) as given in (4 6 Had this function been chosen as Zevn (cos 0, cos to) = ZC,, (cos ~) Zv,, (c o) ( the resulting constants would have been no stub = pec stub. / bo = -jkwV 'ikw + 2 jk *(v^w-i) (Mev (v/ 7enl) M (h/ 1evenI1))2+ 1/2 pec stub ' b0lo =-jkw - jkw~ 4~ m VE-11 -M- (v I 7 1 2 (4.72) and the more compact representation of the no-stub diffraction coefficient is at once evident. To assess the importance of the constant bo0 with respect to Zeven (cos A, cos 0) as given in (4.56), we plotted in Figure 4.6 the backscatter echo width patterns associated with the three configurations in Figure 4.5(a)-(c) and have compared these patterns with that computed by setting blo = 0. The chosen relative constitutive parameters for the left hand side slab are e = 2 and p = 1.2, and the entire slab is of thickness w =.04A (where here A denotes the free space wavelength). We observe that the backscatter patterns are, in general, substantially different, underscoring the importance of the constant. Although bC' sttb is nearly zero in this case. it Ad "

97 20.0 10.0 0.0 ~ ~;....i.......................................................................c -10.0 - \. -20.0 - No stub (blO=0-i.1947) -30.0 - PEC stub(blO=-0.045-1+jO.0213 \ PMC stub (bO=0.6875+.3242. 1 -Constant b10=O. __ -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees ~o PECJ Figure 4.6: H, polarization backscatter echo width for a material insert having w =.04A, e = 2 - j.0001, s = 1.2 modeled with O(w) low contrast GIBCs

98.0 1< b 20. 0 10.0 0.0 -10.0 -20.0 -30.0 _ -40.0 - 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees Figure 4.7: H, polarization backscatter echo width for a material insert having w =.10A, e = 2 - j.0001, / = 1.2 modeled with O(w3) low contrast GIBCs; blo = -.4578 + j.2593, b1l =-.0408- j.0111, b20 = -.0401 -j.0388.

99 not be so when w becomes larger. For example, if w is increased to 0.1A, a 3rd order low contrast GIBC with terms up to 0(w3) is required for an accurate simulation of the dielectric. In this case the constants blo, bll and b20 are non-zero and as shown in Figure 4.7 they play a major role in providing the correct diffracted field by the recessed slab of Figure 4.5(a). 4.3.4 Diffraction by a thin dielectic/ferrite half plane Another configuration whose diffraction has been examined in the past is the thin dielectric half plane shown in Figure 4.4(c). If a 2nd order GSTC with terms up to 0(uw) is employed for the simulation of the layer, the resulting diffracted field is given by (4.57) and (4.58) in conjunction with the "free space limit" entries in Table 4.1 and Table 4.2 for the right hand slab. The constant blo must again be specified for a complete determination of the diffraction coefficient. We remark, however, that if b10 is arbitrarily set to zero, then for w, = w the sum of (4.57) and (4.58) reduces to the diffraction coefficient already derived in [54]. Similar assumptions about the value of the constant have also been made in [32] and although this may be acceptable in some cases (i.e. for extremely thin layers or layers of certain composition), it was already demonstrated above that the constant(s) play an important role and must be accurately determined. This is the subject of chapter five. 4.4 Conclusion In conclusion, we have derived a general solution for the diffraction by a discontinuous distributed sheet representing a multilayered slab discontinuity. The solution can be specialized to a wide variety of material junctions and discontinuities by an appropriate choice for the polynomial operators Ut and U2. Unfortunately, un

100 known constants arise in the final solution due to the non-uniqueness of the finrit GSTC/'GIBC sheet representations. In comparing the derived diffraction coefficiet to a previous result for the material half plane, it was shown that the single unknown constant obtained herein was implicitly assumed zero in [54] and [32]. The impor. tance of the constant was examined for the case of three grounded slab geometries whose GIBC modeling differed only by the value of a single constant b0o. This corn. parison demonstrated that for very thin metal-dielectric junctions, the constant bo was approximately zero, whereas for thicker junctions the constant(s) played a more crucial role.

CHAPTER V RESOLUTION OF NON-UNIQUENESS ASSOCIATED WITH THE GIBC/GSTC SOLUTION In chapter four, a dual integral equation solution was presented for the diffraction by a multilayer material-to-material junction using a GSTC simulation of the multilayer slab. As expected, the solution was in terms of unknown constants and it was shown that these are dependent on the physical properties of the junction (see Figure 5.1). Consequently, an approach for determining the solution constants is to enforce tangential field continuity across the junction. This, of course, demands a knowledge of the fields internal to the discontinuous slab, which, however, are not readily available when a GSTC simulation is employed. The Weiner-Hopf solution in conjunction with the GSTC provides only the field external to the slab, and the majority of this chapter deals with the determination of the internal field from the external one given in chapter four. In the following section, a modal representation of the internal field is proposed comprised of discrete and continuous spectral components. This representation is compatible with that given by Shevchenko [45] whose eigenfunctions are chosen to satisfy the continuity boundary conditions across all layer interfaces including the air-dielectric interface. Consequently the representation is valid inside and outside 101

102 _, - - i*. %%, %i %%1, 2 %.......... I, 2%-% -I x ts X X 8 1,,,1, i s s X X X.... - q..... 1 1.;.-.................................... I,, o,,,,,....................................................... the dielectric once the coefficients of the modal representation are determined. This is accomplished by recasting the Weiner-Hopf or dual integral equation solution of the same problem from chapter four in a form compatible with the proposed modal representation, thus permitting the identification of the modal or eigenfunction coefficients. These are, of course, in terms of the unknown constants appearing in the Weiner-Hopf solution and the enforcement of field continuity across the junction leads to a linear system of equations to be solved for the constants. In the final section of the paper, several scattering calculations are presented for a few materialto-material junctions which demonstrate the importance of the constants and the accuracy of the solution.

103,~,,,,,,,,, I, '~',,,,,,,,Iowoot /,L,'IL o,/,,w,,,,,,,/,,o,,/,,, \.____..._ _________.....______ -i --- ~-2i,,,,,-,,,,,,,,,,,,1,i'i;( 2 ":: sss s,'s, It' 42 2 XI \^^:^^^.\j^^\^^\\^\\^\\^^^\J~# le_ lle0\d\I\I\^v lAlee0 O 2: Figure 5.2: Symmetric multilayer slab with irregular termination illuminated by a polarized field excitation 5.1 Modal Decomposition for the Symmetric Slab Problem Consider a symmetric slab of total thickness r with an irregular termination to its left, as illustrated in Figure 5.2. The slab is herewith assumed to consist of L homogeneous layers with the mth layer being of thickness rm and having relative permittivity and permeability ~m and I, respectively. When this truncated slab is subjected to some polarized field excitation, the field to the right of the junction at

104 the point x can be written as a sum of odd- and even-symmetry fields. That is F(x y)) = F~dd(, y) + ( F' ( y) (. wlhere Ez, Ez polarization, ~~~~F= <^~~~~~~~~~~ (5.2) ZoH, Hz polarization, Fodd (x, y) = -F~dd (x,-y) and Fe'e (x. y) = F ven (x, -y). Following [45], the odd and even fields everywhere interior and exterior to the slab may be decomposed into discrete and continuous eigenmodes as F~dd (, y) = Z A~ddodd ((Am)2, y) e-Jx1\ + B~ (y) e-JkA m=l m=1 + j codd (3) todd (A2, y) e-1kxAd3 (5.3) Y0 NYeven Feven (Z y) = E 4eveneven ((A9~)2,y) e-kA0 + 'eneven () e —'k m=l m=l + / Cove (3) Aeven (\2, y) e-i'Add (5.4) where Im{,\dd,,een} < 0 and A = /1, with the branch of the square root chosen so that Im{v/T1- 3} < 0. In (5.3) and (5.4), oodd.,een are referred to as the cross section functions corresponding to the continuous modal fields whereas oddeven are the corresponding cross section functions for the discrete modal fields associated with the surface waves. The cross section function associated with the geometrical optics fields is also 4oddeven evaluated at A = Am, where A-"is a parameter to be determined later. As can be observed from (5.3) and (5.4), the cross section functions specify the field behavior in the plane normal to the slab, and hence all information pertaining to the fields interior to the slab are embedded in these functions. They will be chosen to satisfy the orthogonality relations (where u(y) is 4(y) or E(y) for E. or H. polarization, respectively) 0 (A2,' 0 (forA#y / ^ v 'y ) v " 'dy = 0 fo r A 5 A *7-co (y)

10.5 P -m-(Y) - y d = 0 (5.6) and thus each discrete eigenmode m (y) e-Jkxx and each continuous eigenmode P (.\ y) eA - must satisfy the wave equation. Additional details pertaining to the cross section functions are givenin [45]. 5.1.1 Exterior Cross Section Functions To compute the cross section functions in the exterior slab region iYI > T/2, wee recall that in accordance with the slab simulation based on the generalized sheet transition conditions (GSTCs), the external fields satisfy the conditions (see chapter two) (- ) {2 } + Ui2 (- ) { ayF +} = 0, > 0 (5.7) U21 ( 2) {Feven+} + U22 (-2) {yFeen+} = 0, > 0. ( 5.S) where F+= F (x. y = r/2), axF+= F (x, y = r/2), and yF= F (x, y) \ 2 The operators Uij (- 32) are polynomials in -dx2/k2 and their explicit forms for the multilayer slab are derived in chapter two as a product of the functions qij (Umr, Km, ~,, -d2/k2) which are completely dependent on the properties of the mth layer. In these, the parameter u, is /-m for E, polarization and Em for H, polarization, and Km = V/mEm is the refractive index of the mth layer comprising the slab. Because of the orthogonality conditions (5.5) and (5.6), each of the cross section functions T (A2, y) and yym (y) must satisfy the odd or even GSTC (5.7) or (.5.8). In view of this odd (\2 (y) II U 2 jsin [k(jy - r/2) v/1 2] 12 (2) [ ( /2) +U,2 A 2 COS k (jyj - 7/2) v7A2

106 {112 (A2) 142(1 1. I - T-/2. 2) - / x~ {= f = / ( l, i i 7-r/2 17) * </2 (X ) 422( 1.1Y I - T/'2 y)}. Ai',=(A2, ) = {j1(A2)1 A --- — ^ — -- +U22 (A2) cos [k(lyl - r/2) 1}-A]} = {U21 (A) 12(1, 1 lYI - T/2, A2) +U22 (A2) q22(l, 1.!yI - r/2, A2)} where U.j (A2) are the same polynomials appearing in (5.7) and(5.8), and qij represent the infinite order form of the qij layer operators given in chapter two. Once each of the modes comprising (5.3) or (5.4) is substituted into (5.7) or (5.8), respectively, the differentiation implied by _-x2/k2 reduces to a multiplication by A2 and the above Todd and ""een are then readily shown to satisfy the associated GSTC. The cross section functions Oddd (A2, y) and Tev"" (A2, y) may also be rewritten in the form jEodd (A2 y) ceven (A 2 y):, Iyodd (A2) e-jk(vl|-r/2)v/l-T lY I eodd (2) ijk(l*-,/2)Vr)i even 2 (XZ) e-jI(lyl-r/ 2)VI; = even (A2) ejk(-lyl-/2)'J1 (5.11) (5.12) for |yI > r/2, where it can be easily shown that (5.11) and (5.12) satisfy the orthogonality relations (5.5) and (5.6). This representation is customarily employed for the surface wave cross section functions. In particular we set odd(y) = l-jk(yl-r/2) I > r/2 (5.13) e m(y) = ek(l/2)- 2; jy > 7/2;

107 where Xe\enJ'3 must now be chosen so that (5.13) and (5.14) satisfy their associated GSTC. By substituting (5.13) and (.5.14) into (5.7) and (5.8). respectively, we find that,\l. e i mnust satisfy the polynomial equati.. s V/l ( )22 ([md]) +il ([ ]2) = 0 (5.1 ) /1 (Aen) 22 ([ en]2) +21 ([en]2) = 0. (5. 16 and can be also identified as the poles of the slab plane wave reflection coefficient. It is also interesting to note that Podd (Aodd y) odd (y-) = >/2 (517 M12 ([Aodd] ' 'D even ([A even"]2,) rjn (y) = ( 12) y > T/2 (.5 18) implying that for the multilayer slab the cross section function associated with the discrete and continuous eigenmodes are of the same generic form given by (3.9) and (5.10). 5.1.2 Interior Cross Section Functions We consider now the determination of the cross section functions for the region interior to the slab (i.e. in the region jyj < T/2). For simplicity let us first assume a single layer slab of thickness r = 2r1, whose upper face is located at y = r1. In accordance with the preceeding, the cross section functions associated with the external fields are given by (odd(Ay) = {q11 (U1i, ir, 7 A2) q12(1, 1, I - ri, A2) y +ql2 (u1, ni, T,,, ),22(1,1, Y - T, 2)}; IYI > T' (5.19) en (v.e ) = {q21 (ul, K, T1, A2) q12(1,, IYI - r, A2) +q22 (Ul, 7 l. T1r A2) q22(1, 1 II - 71, A2)}; yI > 1 (5.20

108 obtained by setting Ulij (A2) = q^j (ull, ll r, A,2) in (5.7), (5.9) and (5.10). These orthogonal functions and each must, therefore, satisfy the continuity conditions o.dd (A^ ) = odd (\2 1+) ( Dyodd (A2, ro) = 2yl(dd (, l x) (3.25) ven (A2, ) = q (ueven (,2, T1) (, 23) Cy1peven (A2, re) = yay"ven (A2 AT+) (5.24) (due to symmetry it is not necessary to enforce similar conditions at y = -r1). It is now straightforward to deduce that possible cross section functions satisfying (5.21) - (5.24) are of the form (A2. A ) = -q12(U1, 1i, Y,,A2) (l5.25) eten (A2 y) = q22(Ul, 1i, Iy, A2) (5.26) for Iyl < T1. Also, in view of (5.21) - (5.24), the cross section functions for the surface wave modes remain as given in (5.17) - (5.18), provided (5.25) and (5.26) are used in place of jodd.even As a specific example, let us consider a low contrast 0(r1) representation of a single layer. From chapter four, the truncated GSTC operators Uij (-ax2/k2) are given to 0(T1) by aX2 - (1 U (- k2 U= - ( - )k = (5.27) Ud =2 c j ku) l 1 (5.28) U21 k2) ku (5.29) and the corresponding qij (A2) polynomials become qI1 (LUl, i,Ti,A) = q22 (Ul7,1,TlA) = 1

109 q12 (ul,. Kt,:. A) qol (21 1 T1*:) = jkull71 i (? - A 2 = l v (5.31) (5.32) and q12 (ui. 1i, JY,A2) = j'k ilyl q22 (UKiyl, A2) = 1. (5.33) (5.34) When these are substituted into (5.19), (5.20), (5.25) and (5.26) we obtain.sinfk(fylf-i ) — " \ odd( ~ -_, A 2 odd (\2 ) = +jku1 cos [kCOS (ly\- T1) vT; \Iy > 7r jkvu1lyl; IyI < T1..... ),,2 y)\ \ si\csmIk(l|r|-rT )v l-2; _s e..n / \ i ~ \~- A)) (5.35) eve (MAz, Y) = + cos (k ( rYi - T1) v'-T j; |yj > 7 (.5.36) tJ~"""' (XL, y I 1; lyIYI < 71 as the cross section functions for a single layer slab modelled with an O(T1) GSTC. For the general case of a multilayer slab, it is necessary that each of the internal cross sections functions satisfy the continuity conditions at all layer interfaces comprising the slab. From chapter two, we obtain that the boundary conditions at the interface between the L- 1th and Lth layer are P-1 ( k2) {ULF (, YL-1 +kP1 (-2 ) {OyYF (x,y =.L-1)} =:.37) 2+ (- k2 { c '~ YL- ) + p 2L-1 (-2) {@y C (Z.y = Y+ 1)} =(,3S8) k 22 k2ly= I where YL is the y coordinate associated with the top surface of the Lth layer and the

110 operators P7'- (2 ) are given by ) 1 (- ) L-l ll(m- k2) l2- ) - J1l - 3:, iK Lt _1 t,) >2 (- k^2 ) m=l q21(Um, K -M Tm,, 2 ) q22(Um,? m,-7-r(5.39) Possible interior cross section functions satisfying these conditions are odd (A2, y) peven (A2 y) I= D (Y\2) Q2(U, yI - yL- XA2) = DiL \2) q12(UL, KL YI - YL-1, XA2) +D L ( 2) q22(UL, <L, YI - YL-1, A2)} YL-1 < IYI < AJ5.40 = {D"L (A2) q12(UL, KL, IYI - YL-1, A) + eve (A2) q22(UL, L, IYI - YL-i, A2) - < Iy| < l.41 in which Deen Odd(,\2) are to be determined by demanding that the interior anid exterior cross section functions are continuous at y = r/2(= YL)- Setting y = r/2 in the expressions for the external cross section functions (5.9) and (5.10), we obtain odd (A2, = T/2) p even (A2,y = T/2) = 12 (A2) - U22 (A2) (.5.42' (5.43) and when these are equated to (5.40) and (5.41) we find podd (A2, y) even (A2, y) IY {PL-1 (A2) q2(UL, L, lYI - YL-1, A2) + P-1' (A2) q22(UL, PL, IYI - YL-1, A2)} P-1 1 (A2) q12(UL, KL, IY - YL-1, A2) +PL2~1 (A2) q22(UL, KL, IYI- yL-1, A2) (5.44) (5.45) for YL > IYI > YL-1i To derive the cross section functions for the other layers, the above procedure may be repeated in a recursive manner until all layers are accounted for. Doing so.

111 we find that a complete representation of the cross section function everywhere is t12 (,\) q22(.11 ly - /2 A2); I > T/2 '2 (A2/) = i_ b pl((2) ql2(uall y - Y-2,2\2) (5.46) W1 (\2) q(1.2(1. IgyI - /72.,\2) +ul2. (XX3) Q2=(1, 1, lyl - T/2,,2); IYI > T|9 \peten (\2. y) = 1l ('\2) ql2(UI?, IYI - y-l, I\2) (5.4 +P121 (A2) q22(Ul, l, Iy- yl-, A2); y I > IYI > ye-l k 22( Ul Ki. yl A2); I\Y < yi vwhen these are used in (.5.3) and (5.4) in conjunction with (3.17) and (5.18) we have a complete field representation for x > 0. 5.2 Recasting of the Dual Integral Equation Solution for a Material Discontinuity The expansions (5.3) and (5.4) can be used to represent the fields interior and exterior to the discontinuous slab shown in Figure 5.1. For x > 0, the material parameters used in the definition of the cross section functions (5.46) - (5.47) must then be associated with the right hand portion of the slab. Similarly, for x < 0, the material parameters in (5.46) - (5.47) must be those of the left hand portion of the slab. The diffraction by the slab discontinuity shown in Figure 5.1 was the subject of chapter four where a complete expression for the scattered field was given by employing a GSTC simulation of the slab. However, owing to the non-uniqueness of the GSTC, the resulting diffraction coefficient was in terms of unknown constants whose

112 determination was shown to require additional constraints. One such constraint i. the enforcement of a boundary condition demanding field continuity at the materi junction. but this requires a knowledge of the slab interior fields and the solution given in chapter four pertains only to external fields. However, as shownn in preceeding section. the expansion (5.3) and (5.4) is valid everywhere when used in conjunction with the cross section functions given in (5.46)-(5.47). Moreover. since the expansion coefficients remain unchanged for the exterior and interior fields. once determined.. he representation (5.3) and (5.4) can be used to find the field every. where. Since the exterior field associated with the slab discontinuity in Figure 5.1 has already been given in chapter four, it can be used to identify the expansion coefficients. This requires that the solution in chapter four is first recast into a formn compatible with that in (5.3) and (5.4), making possible the identification of the unknown constants which can then be determined by enforcing field continuity across the junction. In the following, upon stating the exterior solution we then proceed with the identification of the expansion coefficients. In chapter four, the discontinuous multilayer symmetric slab shown in Figure 5.1 was simulated by the GSTCs k1 ( 2 {F } + 12 (- ) {,OyF = 0 -o < < x <5.4S) -1 - ){F een +} + Ul2 ( } = 0, -o0 < x < 5.49) k1 2 / + 22 k / U ( ) {Fodd,+} + 12 (- ) {oyFd} =0 o 0 < x < c (5.50) (- { '} U22 (-2 ) en+} = 0, 0 < x <o. (5.51) with the superscripts 1 and 2 again denoting the material to the left and right of the junction, respectively. The total field is the sum of the even and odd components

113 and can be represented as = F: d(x.y) +Feen (x.y) F F(x.y)+ Fs (z,y) + F,fr(xy) y > /2 Ft,,, (z, y) + F, (, y) y < -r/2 Finc (x y) = ejk(oso+sin-o) (.5.52) where (.5.3) is the incident plane wave field, Frefl (x,y) denotes the reflected field. which from (4.5) and (4.7) is given by kr sin o Fr.jl (r? y) = ) [Rven + Rldd] ejk(zcosO-ysin ) (5.54) with R-d and R en given by (4.9) and (4.10), respectively. Similarly, Ftrns (x. y) is the transmitted field. which from (4.6) and (4.8) is given as jkr sin m o Ftra (x,) = e-k [RetYn - Rodd ejk(zcos'o+Yimo). (55) Finally, F-(x. y) is the field scattered by the discontinuity and upon employing the dual integral equation method in conjunction with the GSTC (5.48) - (5.51), we find (from chapter four) F (x, y) =Fdd (, y)+ Fven (, y) (5.56) where F odd r00 I- 1 0 v '1 _ Fdd (xy) = I 1~A~v'T~2 odd odd/ odd+ odd s y ) YI J| y -oo 2 7 X + Ao 1 (A) odd (A) (A) Qodd (A,) Nodd-l VNodd-1l-m e -jk(yAlvl +e-jkAxd> Zodd (-AAX) + amn ( + Ao)m (AA) - =(5.57) L m=l n=O _i =_,oJ V...... ejkr/2((.+ Fs"' ( 'Y) - 27r +, even (A) ieven (A,) ()even (A) (o. n% (AO) ooucn. I - 2+ 2x (Xo) z.n (-AAo) + E n(A + Ao)m (AAo)n ' -.5) m=l n=o V -A' 8

114 In these expressions, A\ = cos o. and (; - A "a ) god. ven (\) ICe 52+ ( ) 2 -ge ven Z(A) go - Zodd (Zeven (11 vd,, = it(1/2 (dd + -;0d + 1)} Vven =nt {1/2 (en + n + I) (d (A) od ( = o (\2) + -,'1T (A2) n (A) = even (2) = 11 (A2) + / 122 (,2) 2 11 2 d _)= 52 ((A) o = 4 (A2) + v (A2) n (A)= e (A2) = z4n (A2) + A-22 (A2) AAO) = [z1q (-AA0) Z12 (-AA) -12 (-AA,) 11 (-AAO)1 AA0) = [U,(-X,)U2 (-AA,)-U2 (-AAX)U (-A) Aodd = 0(A) of Add (2) odd,2 = 0(A) of 2 dd (A2) vet en =O 0(A) of even ( 2) een 0(A) of even (A2) (.5.60 (5.61) (5.62) (5.63: (5.64i (5.6,5' (5.66' (5.67,. (5.69) (5.70) As seen, the scattered field expressions are in terms of unknown constants amn and bmn and to determine these via the procedure outlined above, we must first rewrite F (x, y) in a form compatible with (5.3) and (5.4). To do this we need to identify from (5.52) to (5.58) the discrete and continuous spectral components. The discrete portion of the spectrum is, of course, comprised of the geometrical optics and the surface wave fields. These can be identified by detouring the integration path in (5.57) and (5.58) as shown in Figure 5.3. In particular, for x < 0 the integration path may be deformed to one over the branch cut in the upper half of the A plane. capturing any surface wave poles attribute d to the zeros of ) and g:e n Similarly, for x > 0. the integration path may be deformed to one over the

11.5 Im(X) 4 1 Re(X) I< I (a) SW pole Im(X) Im(X)..C.__, Re(X) 1!I IL I 4 1C I I L. IV I 0 I (b) Re(X) -1 Ir,,, ~,rb m -.-,- I -x Cl1 Ef_ -hl — 4_ l = ~l ll d-P- - %%TIVY IVI I & Lff GO pole GO pole p _ v~~~ I = I I SW pole (c) Figure 5.3: (a) C contour in the complex A plane. (b) Deformation for region 1 integrals. (c) Deformation for region 2 integrals.

116 cut in the lower half of the A plane causing the capture of the geometrical optics Pol at A = -Ao in addition to any surface wave poles attributed to the zeros of 5god(\ 2+ (,\ and g'n (A). Through the above deformation of the integration paths in (5.57) and (5.58) ve obtain Fod (,y) = Fwhere for yi > r where for lyJ > T/'2 FJodd (, y) + Fodd (, y) + Fodd (x, y) < 0 2F0dd(, Y Fdd (x, y) + Fdff (x, y) x > d { Feven (X, y) + Flevn (Xy) + Fleven (y) < F2even (Xy) + F2Cven (x, y) + F'even (X, y) x > 0 - 90? 2U - $ t d*i * (5.71) (.5. 2) j lvi ZSY) = g1 {ejklylisinc o + Ro dd &-jk(/yl-r)sinmo} (574) jkxcos4>o Fp,odd (x, y) = e {ejklysin'o + R ev e-jk(lyl-r)sino} ( 5 r 0 /y- i r j.lyl jkx cos 6o F2,even (,y ) -= - eik {e jklyls ino + R edde-jk(ll-r)sin o} (.746) 2 Rodd -sin +OTU2 (cos2 0) -U121 (cos2 0) (5.74) sjkn cos2 (os2 ) +21 (c2 F-~even (X e ejg |sI Iosin oo _ Reven(5.76 3j kxcos 6o goee (X(y) = + R 2 Ree _ sin 4>OU22 (cos2 0o) - -U2 (cos2 0) 2 sin ~OU/2 (cos2 40) + U122 (cos2 0) ( sin i s/12 7 RFvn, _ 3sin o 22 (cos2 Y) - U2x (cos2 o)dd IYI ev eni ~O d 2 l d d 21$ d A odd d(5.7 8) ioU (c (o)odd) + S2o ) 21 2 (5.7(co) )' odd j. (sindo+v 'T~(A d ~_k\2od e2 jkly]VI_ ( A, _?_,,,o,odd) m( Fy Nddw - sin d0 eJ2 (sQot ) ejkVI- - SW) = - Z dd [a dd(A)] w 1_ A' odd0 + XA godd (A,2,odd) godd (0) 5o (A,0) [ Ao, Nodd-l Nodd-l-m o m 2dd Zodd (-AAodA ) + amn (Ad + A) 1 o) (5.79) m=l n=O s,,odd. ~(sin ~o+/1-( ') ~ 2, odd /I~2odd2 2,_dd y - sin do e e- I' e-3l l/ I-( F? "(y'i') = =,Al + AO god 2 odd 9f (A godd (AO) [2^+ ^, A 2,odd N~odd -1 Nodd -1 - n-,7 / ^ 2,odd>~ A nO V^ f 2,odd. ^ ^ 2,odd\ (F rOn Zodd -AoA' + mn +AO A A 7m=l n=O

117 F1 (,) (ve -sin- e (( 1 ' ) ejkXA e-ikIY 1/(A ven)2 =U A1) 'E l *et A0 geven (Aieven) geen ( A) Qr ( (A0) [ I \1 evn v e n - 1 vn -1-ml1 Lm=l n=O F.even (X.y) " - sin o e e-krA1' ev - ke-kIVI -(A" ) FLen=1 + Aeven (< 1 even) ven () (); [Zeren (- A0Aeven) + n Z bmn ((A A) + ]oS A. Ao (.2) dm=l n=O |y| Icbc 2A' A.+ Ao (dd (A) gddd (AO) d f\) (o F-~'-Vodd- 1Vdd-1-m ( -A / e-ik| Y I| -— ekAdA L*dd (->>AA0) +[ a+Am' (Ao + A_)+ O J (5.83) Fdf ( ) |9 7 A + A 51- (A) govdd (A0) v- (Ak) 50dd (A0) No(dd - 1 Noadd- m -jk. Zo (-Ao) + E amn ( + Ao)m ( eJkIY'/ekzd (5.S)i m=l n= o Fs2,en y)= JCl ejk-2(+) dlf ( \ Y) I r T A + A Q51 d (A) 51- (Aod2+(A) 52+ (A o),evenl N n-l-m -j, 2e-jikrAjd [zoen ^(-AA,) + Z Zb ab. m (A + A)m (AAl)n] en-Al (5.86)A Lm=l n=0o, / "- A 5.s4) 'In these, the components Fgo F dff denote the geometrical optics surface wave, and branch cut (or diffraction) contributions to the total fields. Also N1'2 iS the number of captured surface waves (i.e, (A)those with normaized (A propagatio)n constants =such that Im {A nor Im {} <). *k-r/2 r'TA V/iA27 F d2'e IJk '/'41 -A-2 e 1 =) blff v 'yl Jc^ 2r A + Al (A)~, (Ao). (A) ~god (Ao) N~de -lI~odd-l-m I y 'kxA z*od (-AAo) + aS b.(A + A)(AAio )n e-(e-o)dA —"(I'D -— 4) m=l n=o v 2 and branch cut (or diffraction) contributions to the total fields. Also N? is the

118 To identify the expansion coefficients A,, B,, and C (A) appearing in (5.3) and (.4). (5.73) - (5.86) must first be rearranged. By making use of (4.9) - (4.10), (5.61 - (5.64), and (5.577) - (5.78), the geometrical optics terms can be rewritten as y sin ~,ejkcs0~ejkr/2sin~ [ 4 (2 ) j sin (k sin o [|y - r/2]) F9d (xy) - godd (A 2) L'1 Jsin PO +12 (A2) cos (kcsin, [Iy - r/2])] (58) 1 2 (,odd y / sin oej ej/ (>2) sin (k sin 6o [I - T/2]) go (X. y) A y Y) YI ~ ~ "go (A2) 11 sin ~o +U2 (A2) cos (k sin 0o [yl - r/2])] (5.89) F2 (-2 levyen (sie'kCO 'ejk/2sin Ln (X2 j sin (k sin, [lYj - r/2) g0 21g o (X\v- en (A2o) ' sin 1 +122 (A2) cos (ksin 40 [Iy - r/2])] (5.90 The bracketed terms in (5.87) - (5.90) are now readily recognized as the cross section (5.4) the geometrical optics fields may be expressed as F, o (y) = Add (A) od (2 y) jkc 0 (5.91) FTb k tm (x, y) = Add (A0) add (A2 y) e3kxcoso (5.92) Flevy (x,y) = A4en (A0) Aisves (t, y) eko 0s (5.93) Feen (xy) = Ax, (A0) en (, y) ei 0 (5.94) where the A expansion coefficients are identified as AFd (A) = sin Ae- /__ (5.96) A2a (A>O) 5- (A2o) A()5.9 4even (A) = sin oej k/2i (59 l 0~gOvd, (A-2)

sin - s,eJk'/2 "ino A4,2' ( o) = (en.92) (98 2 (Xo Also. ]dd,et en are the same as those given in (5.46)-(5.47) but are associated with material parameters to the left of the slab and likewise oddeven are associated with the material parameters t tthe right of the slab. To identify the B expansion coefficients we observe that the surface wave terms are already of the form appearing in (5.3) and (5.4) and can be readily rewritten as rl,odd SW Fo~dd (X, y) = Z B' (Xo) [Zodd (-XoAl ) + n dd m (Lrodd odd (\iodd] 2rA N+ - 2dd X mannA + AO) (A i ddA)L]fiodd [,y e y ) 99) m=l n=0 I, 2, odd F2,odd (z, y) = 2,odd (Ao) [Zodd (-A 2,odd) 1=1 o~dd-1 odd-1-m + E E ma (+\2odd +,A)m (2 )odd )A]odd ([2odd]2,Y) e oi\i.100) er e L~even (Aven /,evenleven Z B1' ( 1.Z dd ( \ \AOAc) o (X y) = E B (Ao) [Zodd (-AoAl ) /=1 Jievcn -l.V ve n- l-m 2 Z Z bm (leven+Ax )m (A,evenAIvneenen(Aen even (eeven e ) m=l n=o0 J2,,even r2,even,, 2,even / [ ^ \ 2,even, ==1 - - A+Cte' >'vn: ~2e i evn\ee k v+ E bmn (A1 +An)m (Al A rO T 22 Al61'62y e-'a. 102) m=l n=O with Bl Aodd - sin ejkr/2sin eo Bo (Xo) = + _] 5 (o) (o) 5 '* ^ *'. -A?.dd + A 0 [2 odd ( ( 1\,odd g (A ) L J j 1,odd 0- '+ 2+ 0. =A (5.103) 2,odd _ -sin0 e k1/2sin Oo B1~ (AO) A - A o (A d) d(0 a ]A odgd ) X ~,odI + xI ~~_~ (X?,o ) A (X2) od 2, (Ao) - O A ~=~'a (5.104)

120 -- sin o0 ejkr/2sin o BQ ve \ ve \ o) =+_n ec _ _ een 5 p2.cven, \ \_ - sin o. ejkr/2 sin B2.,.,. (:,\o+ A0 (Ae) _en (A0) ('5.106, It remains to identify the C expansion coefficients and to do this, it is necessary to employ the transformation,3 = /l-T2. Doing so in (5.83) permits us to rewri;t F" as diff as l.odd (, y =Y J -3 + -A o) Fdff (XIy) l - jIf 3e -..3.ikr. jk'r/2( V/1i-A2,,+ii+^ ^ dd (v/~2 + 6) Qd (Ao) od (/ ) g (Ao) [ N --- od d-1 Nodd- -m m n Zodd AO 1-;31 +J]l ( (n + Ao 32 A.10,) m=l n=O where the branch of the square root is chosen so that Im(V/-is) > 0 and 6 is a vanishingly small positive number. The integral may now be split into its positive and negative portions and then recombined to obtain l(,odd Y L j kVI-i2/iZodd (V, ~A) diff (X) y 27r vT (I-r + Ao) odd (A.) gd (/ ) g2o (A0) { 5dd (2 g Vd) }~ (A- ) JO where Zodd( 1 )I Zodd (-Ao /3-) Nodd-l Nodd-1-\ m / \ + E E a(mn Ao + -AI ) - A) (5.109) m=l n=O

121 By invoking (3.61), we can show that, odd -2 39, 1 + V,/'l --—,-3 2 g, (VI-3 - 1i ' V - )2 [lll (l - 3:)]2- (2 [ (1 - 32)12 ink3 -) + cos [/ (1 _ 0) si n()[l j - (I - /) + cos (k3 [Iy - r /2])U (i - I I 1 1 1 where the term in square brackets is identified as -dd (1 32, y) as given in (5.46) for |y| > r/2. Substituting (5.110) into (5.108) finally yields Flodd/ _ (, y) odd (3) - odd Tro(1 _ 2 y k) ej /7-dj (5.111) with the expansion coefficient Cdd (03) given by Codd(3) =. l P.j,..32 1v-A2,;r3 (/I- + +Ao) e, jk-r/ (/)2 "____ 5o2d (\VTT ) 52d (o) - ('o) {[U (1 -- 2)]2 [- 2 [ (1 - 32)]}) 1-3 (AIn a parallel manner we obtain In a parallel manner we obtain F2 odd ( 00 Ediff (x,y) = i f/ Y -- C2 (/3) Zodd ( /3, A0) odd (1 - 32, y) e-jikV/l-2d3 (5.113) (ven/) Zeven ( >0) (1 - /32 y) e-ikenvd cl Z vn / 7A pi_, -kx ', o00 Fl.even (X,y) Fdiff J (5.114) F reven (X y) = j ceven (/3) _cn ( A0) iven (i - /32 y) e2ik/-d (5.115) where ee(n ( 2 X. o) Zeven- NA 1-l- 2) N;even-l Neven.-l-Im + E E bmn(Ao+ m=l n=O 1- /2) (A 1/2 5.)116)

122 and the spectral expansion coefficients are given by C" idj 7 -j 32 1-.o 3 2 1 - 2 -A(I ejkr/2V'c-odd; (,/dCi ) 52+ (vTI ) gdd (A) yodd (A) {[UA (1 - /32)]2 - 3 [,22 (1 - 3)l} -,-krl2 (-)?32' + Xo odd (v/F_,) g1id (A~) 2 dd (Ao) {[ul ( - /32)] - _ 2 [u 2 (1 _ 32) ]'l C evenj /32 0 c2 (,3 = (T +A: v'l-,~2' (vVT-,x + X o) odd V/~i~,,o d02-()2 - (k/'V) dd (Ao>) ((o){ L1 (1 - / 32)]2- - 2 [U22 (1 - 32}19) From (5.3) and (5.4), the modal terms (5.91) - (5.94), (5.99)-(5.102), (5.111), and (5.113) - (5.115) provide a field representation which is valid everywhere. Since the odd and even fields are decoupled, two independent representations are obtained for each of these fields. 5.3 Determination of the Constants To determine the constants amn and bmn, we may now enforce the tangential field continuity conditions F(z = O-,y) = F ( = O+,y); yI < r/2 (5.120) 1 axF(x,Y)3=o- = 1 xZF(x,y)=o+; IyI < r/2 (5.121) uziy) u2(y) with f U1,2 (y) Ez-pol u,2, (y) = (5.122) 1 e1,2 (y) HA-pol

123 and the subscripts 1 and 2 denoting quantities attributed to the left and right side of the slab. Substituting (5.71) - (5.72) into (5.120) and (5.121), we obtain FOdd (X =O, y) + Fodd ( = 0-, y) Fd (d = y) Fo x = o y) + oFs2d ( = 0+, y) + Fodd( = 0+y) (f.123) 1 9aX [F1 odd (X, y) + Fl,odd (, y) + Fl,odd (x y )]0 2d [2,odd (,y) + F2odd (, + odd (y)]= (512) -~-d' d'ff.2 gFoe ( =Oa y) + FvLen (x = 0-,y) + Flevn ( = Fu (x = O y) + F2eeI (x = O, y) + F, ( 0 y) (5.12.5 - 1 (a [F,even (,y) + p2Feen ( xy) + Fi eve ( y)j _= _ 1 x(x y) d+ F2 (x Y) (5.124) U2- 2( geo wo to be solved for all amn and,bm. In particular, for an odd GSTC of x(-dd) to tlne left and of QO(12dd) to the dright of the discontinuity, the number of amn to be determined is equal to N odd (.Odd - 1) = Ndd + N2dd is even U2 - go (xI )- Ndd + Ndd is odd To determine all a constants, (5.123) and/or (5.124) must then be enforced or sampled at a minimum of Na points across IyI < r/2 and 0 <,0 < r. Similarly for an even GSTC of O(NFvCn) to the left and of O(N(Vfn) to the right of the discontinuity, determi ned is equal to ) even ven v:oV (~Vo - 1) (NN vcn+la(cn (Jlven+Nvena-2); N 1 + No2a is even =,2 { )8 N'l + No 2 is odd 8 even ~ even (5.128) and thus, the b constants can be determined by enforcing (5.125) and/or (5.126) at a minimum of T Nb points. Substituting for the fields in (5.123) and (5.124) as given in the previous section.

124 a //a a a20 a21 22 a /a,a36 a31 '/ Aa40 Figure 5.4: Indexing scheme for constants. we obtain the equations Na Vdd (\o, y) = apQdd (m (p), n (p), A, ) (5.19). p=l NVa V (0, Y) = Od~F (m (p), n(p) Ao y) (0 130) p=1 where ap = am(p)n(p) with (n + m - i)(m + n) + M (n+m-1)(m+n)+m (5.131). (p) 1 {/1 +8(p-1)-1 } {1 Jl+8(p-1l)+l )1 m(p) = p- Int{ (2 ) Int ~-{ 1 (-1)+}(5.132) 2 2 2 n (p) = nt {- m (p) (5.133) which are in accordance with the ordering of the amn constants as the order of the GSTC is increased (see Figure 5.4). Also, F ( 11 () Al 0 - 2 (A~) 2 ( jl,odd + Bl,odd( o () Zodd) lo odd ([ 1,odd] ) /=1 2 odd - 2,odd () Zdd (- 2,odd) todd ([ 2,odd]2 ) 1=1 (A /== v

Fodd (A y) = u - 125 + j Codd 3) Zo (-\) dd (i - 32, Y) d - od C (3) Zod 1- A32o) dd d ( 32y).134) A,, odd( ) d (,y>) -, AA ododd (.oddy 2 N, odd 2,odd d (odd Al Bl ( S o ) -7 \ \ 12,odd lp odd (\ 2,od0d 2, ul (Ti) N2 odd 2,odd 2,odd() (.A ) 4 dd + U2l (Z) oI. ),odd + 2, cdd ( 2A dd ( y d3 Jo U 2(y) Jo s2F (, (o — Z(5odd (1 -,.13 y) d5 (5..135) and Qd(m, n, A,, y) \Nl,odd.2, odd S B2,odd (A) (, + Add) (A,odd) odd ( [A2,odd' ) j coddc (1) + Am )) ( ) d + j Cmdd (G) (1/ A + A-) (lo2 ) J dd (1 - d 32, y) d (5.136) Qa, (Aio,y) ) yl.Iodd,odd 1 1 odd \ E, B.'1 (tA) (A/ + i odd m 1odd odd dd 2 L "' ul(y) dA2,oddB.dd (oA0) (O + A 2,odd)m (AA 2,odd) odd ([ 2,odd] 2, ) 1=1 u2(y) 2' y j00O oddT(fl) ( 1 Ao)m (( no) 2 ) J~ i'2 () (3A) (/i (2+ Ao) (/-o) (2 yi- ) d( (5.137)

126 Provided 'odd are even, the integrals in (5.135) and (5.137) converge (see Table 5 i and (5.129) - (5.130) can be solved for the constants ap by matching (5.129) at; points and (5.130) at N,2 points such that VN\ + Na2 > Ala To solve-for the b constants we substitute for the fields in (5.125) and/or (5.126 gi ving a n gl 0n Nb VVen ((oy) = EbpQe (m(p), n(p) A, y) p=l iNb ev (A y) = pQa (m (p), n (p), A, y) p=l (5.138) (.3.139) with Feve (Ao, y) 'T/even (A ) =,xF [ (o =y) Aeven (Ao) lven (A)ty) - Av"en (Ao) "ven (A2, y) Al (O) 1 o,y -A2 (AO) /2 AO -1,even + e BI een (AX) Zeven -AAl leven) 2ven ([A,even], ) 1=1 + j C1 (3) Zeven (.-4A,),ven (i - 32 y) d: - j cr71 (/?) Zven (-J 1 A0) r'! ( 1- 32,y) d/ (5.140) eA eAven (A), A 0ve ( \ 2 - N"e lven A even n even ((_A)teven) ([t \even] + even, Beven (Ao) AZevn (A I _,even) even \2 even ) A.euv ( y) \Zi1 e) 2even (1 /32 ) d 1 VT^ 0^,even A2 2,ke even A 2 -N 'v' (Po> fz,^ (1 - ve), I( ) Al - Ao l v/2 o,,~ (~o)~,^ ~ 2 02(?) fn o, +J j/A2Te7n Cven()) (A o ve (1 - ev y) d3 + Zeen1 A u(y2 (. ) (5.141),5y41 and Qeve (Ao) =

1 27.Vid or -'-odd or Junction Branch cut integral Order of branch \ e en -2t'en Field (odd or even) cut integrand even even E, or Hz Fdf (x= ) -3 L __Fdiff (Z = Ot y) 1/1-31 odd odd E, or H, Fjff (x = 0-, ) I 3/-2 diff (X = ly) iff (x = 0+, y) odd odd |H or 4 F (.ff(x =|Oy) | | odd even E, or Hz Fd( = 0-,y)|-3/2 _____ F2 +x=Q\y}|/?|-5/2 F2ff (x = 0+,Y) |l-1 even odd Hy or Ey FJ ( = 0,y) | -3/2 LII__________^ Il2________F/(x = die e od d o 0 -, y )|- 2 odd even H or Ey Fjif(x= 0-,y) I-1/2 - - dF(=,) I Table 5.1: Asymptotic behavior of integrand for the functions Fdff (x =, y).

128 IV,aven - Z B 1 (A\) (A + Al even) (A \1even Xeven ([A,even2 ) ~2 o; ^o + A^ J ^o~l Y -I ^[A/ j jY 2. even +- B2,even (A0) (A0 +X A2even)m (/ \2,even) n"; even ([\2.even1 /=1 -ceven3 + )m ( 3Ao) ven (l y) d:3 + j Ceven (3) (1 2 + Ao) m (13A) ven (1 - 2e) dQ (5.142) o ((y) = '\Te w \ve^levn (A 0) (A\ + iEven) m (0 lteven) n even ( [\even] 2 \ 2=x2 u32(4) N seven swS W 1 ~l~even ven, Ao AOl (y)) ) I 1 provided N e () 2, even (5.42,eoen even ge and (1 ) c b sov o t o cn (.139) at b points such that b N2 N. Results ased on the solutionof -/ v" -—,\ 2Aeven (0) ^ 7-1 -,,y dI J(5 ) - ( ) an (5.138) (19) are con ered next., (5.143) Is before bp n we a,(p)() wth p, m(p), and n(p) given in (5.131)-(5.133). Again, s evera difrtion are even, the integrals in (5.141) and (5.143) converge and (5.138) - (5.139) can be solved for the constants bp by matching (5.138) at Nbj points and (5.139) at Nb2 points such that Nbj + Nb >_ NM. Results based on the solution of (5.129) - (5.130) and (5.138) - (5.139) are considered next. 5.4 Validation of the Solution In this section we address the validation of the GSTC solution. In particular, several diffraction patterns are presented for selected material junctions and these are compared with data obtained by other means. Issues related to the numerical implementation are also discussed, including those pertaining o tthe convergence of the solution and sampling criteria. Finally, some family curves are given for selected

129 junction geometries in which the slab thickness is varied. Figures 5.5.. 5.6, and 5.7 present the echowidth of several recessed slab geometries. These results were computed with the GSNIF-GIBC solution of chapter three and the more general GIBC solution of chapters four and five. The slab having r =.04, e = 2, on = 1.2 (see Figure 5.5) may be adequately simulated by a low contrast second order GIBC, whose solution is distinguished by the presence of a single unknown constant blo. The computed value of this constant is given in Tables 5.2 and 5.3. which also conlain the values of the constants pertaining the simulations given in Figures 5.5 to 5.14. It is observed that the constants predicted by the GIBC and GIBC-GSMIF solutions are practically identical, and the corresponding diffraction curves overlay one another for the PEC stub case as well as the PMC stub case. The solution constants for the GIBC solution corresponding to the PEC stub were determined by applying the boundary condition Ey = 0 at the junction. In particular it was found that a satisfactory solution for bo10 could be obtained by enforcing this condition at a single point along the junction and for a single angle of incidence. The nearly-exact values in Table 5.2 result from enforcing the vanishing electric field at a single junction point for four distinct angles of incidence. We remark that there is no need for additional sampling points along the junction, since the interior I function is constant with respect to y for a second order low contrast GIBC. A question may arrise, however, as to why it is desirable to sample at a greater number of sampling points than the number of unknowns. This is because the GIBC solution should ideally satisfy the boundary condition over all angles of incidence and at all points on the junction. Hence by using a sampling grid which spans the junction, it is possible to obtain a solution which satisfies the boundary conditions across the junction in an average sense. Once the overdetermined system is generated, a

130 solution for the constant(s) may be obtained by standard least-squares techniques With regard to the computation of the field quantities at the edge, care rns be exercised in evaluating the branch cut integrals. As seen from Table 5.1, the convergence of these integrals is not always guaranteed, since the integrand must have an asymptotic behavior of JAIX1-6 with 6 > 0. It may then be deduced from Table 5.1 that one cannot match Ey or Hy at the junction unless the order of all GIBC/GSTC for finitely-conducting bodies is even. We remark, however, that although the branch cut integrals for the PEC and PMC cases above always converge (they behave asymptotically as JAK-3/2 and JAK-5/2 respectively), their evaluation is not trivial by virtue of the infinite limits of integration. Herewith the infinite interval is transformed to a finite one. In addition, in case of a pole near the integration path, the addition and subtraction process described in [51] is employed to regularize the integrand. In Figure 5.6, the slab thickness is increased to.1A and it is now seen that fourth order conditions are required, resulting in three constants to be determined. It is found that nearly 12 junction constraints are needed to adequately specify the constants by enforcing field continuity at three points across the junction for four angles of incidence. The agreement between the GIBC, GIBC-GSMF, and GSMFexact are excellent, and the same is, of course, true for the GIBC and GIBC-GSMF constants (see Table 5.2). Note that the error in setting the constants to zero in this case is significant. The final recessed slab geometry has r =.4, e = 5, y = 3 modeled by second order high contrast GIBCs. Again all solutions agree quite well as do the values of the constant blo as given in Table 5.2. Figures 5.8 to 5.14 present diffraction patterns for material half-planes of increasing thickness and are compared to data from a numerical model. The numerical

131 model was constructed by first generating the transient response of a finite length slab using bandlimited frequency domain data. The contribution from the half-plane edge was then extracted by time gating the transient response. This numerical model is valid except near grazing, where the surface wave and ray field excited by the back edge arrives at the front edge in concert with the incident plane wave. Figures 5.8 to 5.11 present the echowidth for low contrast simulations of a half-plane having e = 32 o = 1.2. Clearly, the agreement between the numerical data and GSTC solution is excellent. The reader should also note the ere small values obtained for the constants as the thickness tends to zero, and this is in agreement with the second order GSTC solutions proposed in [53] and [54]. However, for thicker half-planes the constant plays a more significant role as evidenced by the erroneous result predicted in Figure 5.11 when the constants are set to zero. Data based on two high contrast simulations are presented in Figures 5.12 and 5.13 for a material half plane having e = 5 and A = 3. Because of the higher sampling required, numerical results could only be furnished for a thickness of up to.05A (see Figure 5.13). We observe that the results for the.01A thick half-plane given in Figure 5.12 are in agreement and the same is generally true for the curves in Figure 5.13 despite the obvious instabilities of the numerical data. In Figure 5.14 a GSTC simulation is constructed for a two layer half-plane having 71 =.005, e1 = 5, I1i = 3 and r2 =.03, E2 = 2, I2 = 1.2. The agreement of the GSTC solution with the numerical data is quite good except at edge on incidence, and judging by the abnormal behavior of the numerical data in this region it is conjectured that these data are in error. Finally, in Figures 5.16 to 5.21, family curves are given for various half-planes and grounded junctions. These are limited in thickness by computational restric

132 tions stemming from the rapid increase in the number of constants as the order i increased. and additionally from the numerical intensiveness of the routine Which determines the unknown constants. It is, therefore, necessary that the proper order (the one providing adequate simulation and converged results) of the GSTC be determined beforehand. This can be generally found by leaving out the constants and computing the diffraction coefficient for increasing order of GSTC until convergence is reached. The constants can then be determined for the order of the GSTC ren. dering convergence. This procedure was found quite adequate and was employed to generate the data in Figures 5.15 to 5.21.

133 20.0..... 10.0 T0.O - | 20.0 2"d order l.c. GIBC(GSMF), PEC stub 2" order I.c. GIBC, PEC stub j~ 40- - ' —' ' ' ' ' ' ' ' -- — " - ' '...__... ------------- -30.0 2 ndorder l.c. GIBC(GSM F), PMC stub 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees = T.04X, ~=2-j.0001, 1 1.2 / Figure 5.5: Hz polarization backscatter echo width for a recessed slab with r =.4, = 2, = 1.2 modeled by 0(r) second order low contrast GIBC (see Table 5.2 for constants).

134 20.0 10.0 0.0 -10.0 -20.0 -30.0 -40.0: —. -. -00 ~~\~~~~~ I on (exact) (GSMF)-.o.. nss............ consts.=0 i GSMF modal soluti 4th order l.c. GIBC 4th order l.c. GIBC 4.. ord r........... GIBC,.. 4'" order I.c. GIBC, I.... I I I. I..,,I 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees Figure 5.6: H. polarization backscatter echo width for a recessed slab with r =.1, e = 2, /z = 1.2 modeled by 0(r3) fourth order low contrast GIBC (see Table 5.2 for constants).

135 20.0 _ 10.0 0.0 -10.0 -.. m. __.^,....\ -20.0 GSMF modal solution (exact) -30.0 2"d order h.c. GIBC (GSMF) 2"d order h.c. GIBC -40A * 0., I,,,,,.' ' |.... *.. |... *!, I,, 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees jT=.4A.e 5-j.0001.M>=3 M Figure 5.7: Hz polarization backscatter echo width for a recessed slab with r =.4, e = 5, u = 3 modeled by O(cK-) second order high contrast GIBC (see Table 5.2 for constants).

136 0.0 -10.0 -1'0. O _ -20.0 -30.0 _ -40.0 -50.0 -60.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees T=. 005), E= 2'.000 1 t~l. Figure 5.8: H, polarization backscatter echo width for a material half-plane with r =.005, e = 2, P = 1.2 modeled by 0(r) low contrast GSTC (see Table 5.2 for constants).

137 10. 0 0.0 -10.0 - -20.0 -30.0 \ B -40.0 0(r1) I.. GSTCs Numerical -50.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees T=..05x, ~2-j.0001, =l1.2 Figure 5.9: Hz polarization backscatter echo width for a material half-plane with r =.05, e = 2, p = 1.2 modeled by 0(r) low contrast GSTC (see Table 5.2 for constants).

138 10.0 0.0 -10.0 Po C p< b) -20.0 -30.0 -40.0 -srn _n 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees to T=.1 ox, e=2-j.0001,,U-.2 i 8''SS N''''''''''S Figure 5.10: Hz polarization backscatter echo width for a material half-plane with r =.10, e = 2, j = 1.2 modeled by 0(r) low contrast GSTC (see Table 5.2 for constants).

139 10.0 0.0 -10.0 -20.0 -40.0 O( r);. 'C;GST 0( O 3) I.c. GSTCs consts.= - 4 0..,............................................... Numerical -50.0 "" i...~~ i ~ ~ i ii I l..,l I I I -50.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees T=.20X. ~=2-j.0001. I=1.2 Figure 5.11: Hz polarization backscatter echo width for a material half-plane with r =.20, 2, =, 1.2 modeled by 0(r) and 0(r3) low contrast GSTC (see Table 5.2 for constants).

140 0.0 -10.0 -20.0 C0 -30.0 -40.0 -50.0 A -60.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees T=. olX C=-j.0 00 1 q3 Figure 5.12: Hz polarization backscatter echo width for a material half-plane with r =.01, e = 5, It = 3 modeled by 0(^~) (even symmetry) and O(-~1) (odd symmetry) high contrast GSTC. The free space side is modeled by O(r) (even symmetry) 0(r2) (odd symmetry) low contrast GSTC. (see Table 5.3 for constants).

141 10.0 0o.0 -20.0 - -30.0 _ -40.0 _ -50.0 0.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees 0 _i xt e=.5-j.001, =3 Figure 5.13: Hz polarization backscatter echo width for a material half-plane with r =.05, e = 5, _ = 3 modeled by O(c~) (even symmetry) and O(c-l) (odd symmetry) high contrast GSTC. The free space side is modeled by 0(r) (even symmetry) 0(r2) (odd symmetry) low contrast GSTC. (see Table 5.3 for constants).

142 10.0 0.0 -10.0 -20.0 -30.0 -40.0 -50. 0 L 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees.2 Figure 5.14: H, polarization backscatter echo width for a two layer.07A symmetric material half-plane with Tr =.005, E1 = 5, /l1 = 3 and r2 =.03, e = 2, = 1.2. Layer 1 is modeled by O(KO) GSTC, layer 2 and the free space layer are modeled by O(r) (even symmetry) 0(r2) (odd symmetry) low contrast GSTC (see Table 5.3 for constants).

143 10.0 0.0 -10.0 -20.0 -30.0 -40.0 -50.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees Figure 5.15: Hz polarization backscatter echo width curves for different thickness material half-planes with e = 2,, = 1.2. All cases are modeled by 0(r) low contrast GSTC (see Table 5.3 for constants).

144 20.0 10.0 0.0 "0,< b -10.0 -20.0 -30.0 -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees PEC no~r. ram-% Figure 5.16: 0 H, polarization backscatter echo width curves for different thickness grounded slabs with e = 5, p_ = 3. The number to the left of the colon denotes the 0(r) of the l.c. GIBC for the free space side, and the number to the right of the colon denotes the O(,c-1) of the h.c. slab GIBC.

145 20.0 10.0 0.0 -10.0 -20.0 -30.0 -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees PEC ran~ Figure 5.17: Hz polarization backscatter echo width curves for different thickness grounded slabs with e = 11, / = 7. The number to the left of the colon denotes the 0(7) of the l.c. GIBC for the free space side, and the number to the right of the colon denotes the O(c-') of the h.c. GIBC for the right hand slab.

146 20.0 - 10.0 0.0 -10.0 -30.0 -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees 12 PEC Figure 5.18: Hz polarization backscatter echo width curves for different thickness gro.ided junctions, with e1 = 2, yl = 1.2 and e2 = 5, 2 = 3. The number to the left of the colon denotes the 0(r) of the l.c. GIBC for the left hand slab and the number to the right of the colon denotes the O(K-1) of the h.c. GIBC for the right hand slab.

147 20.0 - 10.0 0.0 -10.0 - -20.0 - I -30.0 _ I.I -40.0 L 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees PEC Figure 5.19: Hz polarization backscatter echo width curves for different thickness grounded junctions, with el = 5, 1t = 3 and e2 = 11, /2 = 7. The number to the left of the colon denotes lae O(tc-1) of the h.c. GIBC for the left hand slab and the number to the right of the colon denotes the O(c-l) of the h.c. GIBC for the right hand slab.

148 20.0 10.0 0.0 m.0,< Nb -10.0.,.t. '=.005, (1:1,1) / I \ K.=.025,,.:" /\ eeee,\..ee // \..",." ~ / \ % |.s |... I I...I... 1 -20.0 -30.0 -40.0 0.0 30.0 60.0 90.0 120.0 150.0 Angle in degrees 180.0 - 0 -i, E=5-j.OOO01, =-3 PEC.w JN ^J^ W ^.^^ 'lDPEC Figure 5.20: Hz polarization backscatter echo width curves a grounded junctions, with 1e = 2, /1 = 1.2, r1 =.025 and E2 = 5, #2 = 3. The first number to the left of the colon denotes the 0(r) of the l.c. GIBC for the left hand slab and the first number to the right of the colon denotes the O(rc-) of the h.c. GIBC for the right hand slab. The second number on either side of the colon denotes the 0(r) of the l.c. GIBC for the free space slab needed to give the two sides the same thickness.

149 20.0 10. 0 0.0 -10.0 -20.0 -30.0 -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Angle in degrees PEC ran~ Figure 5.21: H, polarization backscatter echo width curves a grounded junctions, with el = 5,,1 = 3, r1 =.05 and e2 = 11, /2 = 7. The first number to the left of the colon denotes the O(x-1) of the h.c. GIBC for the left hand slab and the first number to the right of the colon denotes the O(n-1) of the h.c. GIBC for the right hand slab. The second number on either side of the colon denotes the 0(r) of the l.c. GIBC for the free space slab needed to give the two sides the same thickness.

150 FIGURE 5.5;IBC (GSMF) -PEC STUB bl0 = -0.0451+j0.0212 3IBC (GSMF) -PMC STUB blO = 0.6875+jO.3242 FIGURE 5.6 IBC (GSMF) blO = -0.4578+j0.2593 b20 = -0.0401-j0.0388 3IBC blO = -0.4596+jO.2599 b20 = -0.0400-j0.0388 FIGURE 5.7 3IBC (GSMF) blO = -0.0831+j0.0489 FIGURE 5.8 GIBC-PEC STUB blO - -0.0451+jO.0213 GIBC-PMC STUB blO = 0.6875+jO.3242 bll = -0.0408-j0.0111 bll = -0.0410-j0.0110 GIBC blO = -0.0837+jO.0489 3IBC blO = -.0000+j.0000 FIGURE 5.9 3IBC blO =.0022-j.0009 FIGURE 5.10 GIBC blO =.0008-j.2323 FIGURE 5.11 3IBC 2nd order blO = 0.0152-jO.0166 3IBC 4th order alO - -0.0734+jO.0138 a20 - -0.0022+jO.0103 all- 0.0145-jO.0010 bll - 0.0146+jO.0097 bl2 - 0.0035-j0021 b21 - -0.0004+jO.0125 blO - b20 - b30 - 0.0350-j0.0368 -0.0013+j0.0565 -0.0022-j00.0015 Table 5.2: Values of solution constants for curves presented in Figures 5 to 12

151 FIGURE 5.12 alO = -0.0012-j0.0000 blO = 0.0019-j0.0005 FIGURE 5.13 alO = -0.0798-j0.0054 blO = 0.0171-jO.0239 b20 = -0.0019+jO.0011 FIGURE5.14 alO = 0.0003-jO.0000 blO = 0.0180-jO.0102 b20 - 0.0002-jO.0001 FIGURES.15 t-.005 blO - -.0000+j.0000 t=.05 blO -.0022-j.0009 t=.10 blO.0008-j.2323 t=.20 blO - 0.0152-jO.0166 bll = 0.0090+jO.0002 bll = -0.0060+jO.0002 Table 5.3: Values of solution constants for curves presented in Figures 12 to 15

152 5.5 Summary In this chapter a method was proposed and demonstrated for determining the unknown constants. Specifically, an eigenfunction expansion was presented as a representation for both the interior and exterior fields. The solution of chapter four was then recast into this form, allowing the unknown constants to be determined by imposing additional continuity conditions across the junction. Various scattering patterns were given alidating the derived diffraction solution for several material half-planes and junctions.

CHAPTER VI SUMMARY The goal of this dissertation was to develop a plane wave diffraction coefficient for thick multilayered symmetric slab junctions using simulations based on the generalized impedance boundary conditions and generalized sheet transition conditions. This task was accomplished in four chapters dealing with the derivation of the GIBC/GSTC, the formulation and formal solution of the plane wave diffraction by a class of slab junctions, and the subsequent resolution of the unknown constants which arise in these solutions. To model multilayered slab junctions, it was first necessary to develop GIBC and GSTC for multilayered planar slabs, and this was the subject of chapter two. Recurrence relations were initially developed to relate fields in the adjacent layers of a multilayered structure, and these were subsequently used to derive infinite order boundary/transition conditions, conveniently expressed in a matrix product with each matrix corresponding to a layer. Low and high contrast approximations were then introduced to approximate the individual elements of each layer-matrix leading to a finite order GIBC/GSTC for the multilayered slab. Since each individual layer in the slab was characterized by a separate matrix, a low or a high contrast approximation could be employed for each individual matrix as dictated by the refractive 153

154 index of the layer. The accuracy of the derived conditions was evaluated by corr. paring the GIBC/GSTC reflection coefficients to their exact counterparts and design curves were given for various single layer geometries based on the maximum phasor error of the approximation. In chapter three, the plane wave diffraction by a recessed slab in a ground plane was formulated and solved via the GSMF in conjunction with the dual integral equation method. The motivation for studying this geometry stemmed from the availability of exact data for uniform slabs. It thus served as a reference for evaluating the accuracy of GIBC in junction simulations. Furthermore, the employed GIBC simulation resulted in a unique solution, thus bypassing the non-uniqueness issue associated with GIBC/GSTC simulations of more arbitrary material junctions. The diffraction by a material discontinuity in a thick dielectric/ferrite slab was considered in chapter four. The slab was modeled by a distributed current sheet obeying generalized sheet transition conditions (GSTCs). This representation was then used to develop dual integral equations in terms of even and odd unknown spectral functions, which were proportional to the sheet currents. The solution for the spectra paralleled standard procedure but resulted in expressions involving unknown constants, revealing the non-uniqueness of the GSTC. It was demonstrated that the unknown constant(s) could be determined explicitly for the recessed slab discussed in chapter three by comparison with the results therein and it was also shown that the obtained solution reduces to simpler known solutions, including that for combinations of resistive and conductive sheets junctions. One way of determining the unknown solution constants discussed in chapter four is to employ field continuity across the junction. This however requires knowledge of the interior fields and in an effort to determine them, an eigenfunction decompo

155 sition was developed for arbitrary symmetric multilayered slabs valid in the exterior and interior regions. The solution given in chapter four was then recast in this form permitting the determination of the interior slab field via analytic continuation. The unknown constants were then resolved by applying field continuity accross the junction. Specifically, a point matching scheme was proposed in which the junction was sampled across its width for different angles of incidence. The resulting unique solution was then verified for a number of recessed slab geometries by comparison with the GSMIF solution ini chapter three, and also for thin material half-planes whose scattering patterns were computed numerically. It was observed that the constants are very small for thin slab structures, but become significant with increasing slab thickness. Finally, family curves were generated for a number of half-plane and junc- tion geometries, and it was found that the convergence of the GIBC/GSTC solutions can be estimated by leaving out the solution constants. Clearly, the most challenging part of this work was the determination of the unknown solution constants. The mere fact that this was possible proved that the GSTC/GIBC are useful for practical simulations. Certainly, the method used in chapter five for determining the constants could be employed or paralleled in other applications. However, as noted therein, the evaluation of the constants for higher order GSTC simulations becomes numerically intensive and it would, therefore, be desirable to find alternative means for accomplishing this. For example, instead of point matching one could explore the orthogonality of the expansion modes or perhaps use a more efficient evaluation of the integrals. In this work we explored one application of the GSTC/GIBC simulation, that of diffraction by multilayered material slab junctions. As can be expected there are numerous other applications where the GSTC/GIBC can permit analytical solu

156 tions. Examples include the extension of this work to skew incidence; possible characterization of junctions other than vertical where this solution may be employed in conjunction with a numerical one; and the diffraction by material junctions on curved surfaces at normal and skew incidences. Also, the characterization of multiple diffraction effects among material junctions is a straightforward process following the method already employed in [13].

APPENDICES 157

158 APPENDIX A MULTIPLICATIVE SPLIT FUNCTIONS In this appendix we consider the splitting of g (A2) = UA (A2) + VT tUB (,2) (A.1) as a product of two functions, one of which is free of poles, zeros and branch cuts in the upper half of the A plane and the other having the same properties in the lower half of the A plane. That is, we seek to write g (A2) in the form G ( ) = S+ (A) 5_ (A) (A.2) where the superscript + and - indicate an upper or lower function, respectively. Noting that NA A (A2) = An [1-A2]" (A.3) n=O NB UB(A2) = E n [1- ] (A.4) n=O with NA = NB or NA = NB + 1, we may rewrite g (A2) as Ns (A2) =IE: Sn (A.5) n=O where N, =-Max(2NA,2NB + 1) and Sn = An/2 if n is even and Sn = B(n,-)/2 if n is odd. However, since we seek a multiplicative splitting of (A.5), a more convenient

159 form to represent Q (\2) is (A2) =So (+ (A.6) in which 7,A denote the zeros of the polynomial "vi, S1 (-A)i. We immediatelv now identify that each of the product terms in (A.6) can be factored as 1+ - M+ (A;) 1)M_ ( ) (T) where K+ (A1/; ) =,M+(A) (AS) M+ (A;-y) is the split function characteristic to the impedance half plane having a constant surface impedance 1/y [34]. With the branch choosen so that Im(v/l-A) < 0, I+ (A' v) is explicitly given by fM+ (A; ) Im() < M+ (AX;) = M- (-A;7) = ( — ) ' Im ) (A.9) '(cs'/)="I (7-/2) l + V2 Cos 2 )] [i + v.cos (os 2 lV,+ (cos Q; 1l/177) = Lt 2-) (A.10) [' [' (3</2 - a - 0) - i) (r/2 - a + 0)12 In this, Im (/) > 0 A = cos a Im (\|1-1/72) < 0 0 = sin-'(r1) with 0 < Re(O), and k,, (a) is the Maliuzhinets function [22] whose evaluation in algebraic form has been given in [50]. We remark that in the limit as 7 -> 0, M'I+(A;7y 0) = - (A.12) J"Y7

160 and as? -> oc M+ (A;y ) = 1 (A.13) The determination of g: (A) is now rather trivial. By substituting (A.7) into (A.6) we easily obtain Ns g (A) = 5 (-A) = So II f+ (A; 7n) (A.14) n=1

BIBLIOGRAPHY 161

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