025921-30-T TECHNICAL REPORT for NASA Grant NAG-2-541 NASA Technical Monitor: Alex Woo Grant Title: Development of 3D Electromagnetic Modeling Tools for Airborne Vehicles Report Title; Electromagnetic Scattering by Coated Convex Surfaces and Wedges Simulated by Approximate Boundary Conditions Institution: Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor MI 48109-2122 Period Covered: Feb. 1992 - Sept. 1992 ReDort Authors' H.H. Syed and J.L. Volakis Principal Investigator: John L. Volakis Telephone: (313) 764-0500

Electromagnetic scattering by coated convex surfaces and wedges simulated by approximate boundary conditions H.H. Syed and J.L. Volakis* Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor MI 48109-2122 *The work reported here was also used as a dissertation submitted to the University of Michigan Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the Doctor of Philosophy degree.

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ABSTRACT ELECTROMAGNETIC SCATTERING BY COATED CONVEX SURFACES AND WEDGES SIMULATED BY APPROXIMATE BOUNDARY CONDITIONS Asymptotic/high-frequency solutions are developed for analyzing the non- specular scattering mechanisms associated with coated convex surfaces and edges simulated by approximate boundary conditions. In particular, the standard impedance boundary conditions (SIBCs) and the second order generalized impedance boundary conditions (GIBCs) are employed for a characterization of the edge diffraction, creeping wave and surface diffracted wave contributions. To study the creeping wave and surface diffracted wave mechanisms, rigorous UTD (uniform geometrical theory of diffraction) diffraction coefficients are developed for a convex coated cylinder simulated with SIBCs and GIBCs. The ray solutions obtained remain valid in the transition region and reduce uniformly to those in the deep lit and shadow regions. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder. The diffraction coefficients for a convex cylinder are obtained via a generalization of the corresponding ones for the circular cylinder. To validate the asymptotic/high

frequency solution integral equations are derived for both E and H-polarization and solved numerically using the method of moments. Results are presented for a single and three layered coated convex cylinder. Some insights are also provided on the accuracy of the employed GIBCs versus SIBCs for application to curved surfaces. To characterize the scattering by impedance wedges illuminated at skew incidence, diffraction coefficients are derived from an approximate solution of the governing functional difference equations. This solution exactly recovers the known ones for an impedance half plane or an arbitrary wedge at normal incidence and to validate it for other wedge angles a moment method code was used. Finally, to test the usefulness of the approximate skew incidence impedance wedge diffraction coefficient for three dimensional structures, equivalent currents are derived in the context of PTD for a finite length impedance wedge of arbitrary internal angle. These are incorporated in a standard general purpose physical theory of diffraction (PTD) code and results are presented for a number of different impedance structures.

TABLE OF CONTENTS ACKNOWLEDGEMENTS.......................... ii LIST OF FIGURES............ v CHAPTER I. INTRODUCTION........................... 1 II. SIMULATION OF COATED STRUCTURES BY APPROXIMATE BOUNDARY CONDITIONS.............. 7.2.1 Standard Impedance Boundary Conditions (SIBCs)...... 8 2.2 Generalized Impedance Boundary Conditions (GIBCs).... 10 2.2.1 Single Layer Coating................. 10 2.2.2 Three Layer Coating................. 14 III. NUMERICAL IMPLEMENTATION OF GIBC........ 22 3.1 Integral Equation Formulation................. 23 3.1.1 H,-incidence...................... 23 3.1.2 E,-incidence...................... 28 3.2 Numerical Results........................ 29 3.3 Summary............................. 32 IV. HIGH FREQUENCY SCATTERING BY A SMOOTH COATED CONVEX CYLINDER SIMULATED BY APPROXIMATE BOUNDARY CONDITIONS.................... 42 4.1 Eigenfunction and Integral Representation........... 43 4.2 Field in the Deep Lit and Shadow Regions........... 47 4.3 Field in the Transition Region................. 51 4.4 Field in the Close Vicinity of the Convex Cylinder...... 53 4.5 Numerical Results...................... 55 4.6 Summary............................. 57 111

V. AN APPROXIMATE SOLUTION FOR SCATTERING BY AN IMPEDANCE WEDGE AT SKEW INCIDENCE.... 61 5.1 Modified Form of the SIBCs................... 62 5.2 Derivation of the Functional Difference Equations....... 5 5.3 Approximate Solution of the Difference Equations...... 68 5.3.1 Separation Method I................. 68 5.3.2 Separation Method II............... 75 5.3.3 Separation Method III................ 82 5.3.4 Test Results..................... 97 5.3.5 Summary........................ 101 VI. SKEW INCIDENCE SCATTERING FROM A COATED CYLINDER SIMULATED BY SIBC............... 107 6.1 Integral Equation Formulation................. 107 6.2 First Order High-Frequency Far Zone Field........ 112 6.3 Test Results........................... 113 6.4 Summary............................. 114 VII. PTD FORMULATION FOR SCATTERING BY IMPEDANCE STRUCTURES............................ 126 7.1 Derivation of Equivalent Currents.............. 126 7.2 Radiation from Equivalent Currents on a Finite Wedge... 129 7.2.1 Straight Wedge.......... 129 7.2.2 Curved Wedge....... 132 7.3 Test Results........................... 135 7.4 Summary............................. 136 VIII. CONCLUSIONS............................ 142 APPENDICES.................................147 BIBLIOGRAPHY...............................155 iv

LIST OF FIGURES Figure 2.1 Metal-backed dielectric layer...................... 9 2.2 Illustration of a three-layer metal-backed coating.......... 15 3.1 Illustration of a three-layer coated cylinder......... 23 3.2 Equivalent current model and illustration of the discretization parameters.................................. 24 3.3 H-polarization bistatic echowidth (4' = 00) for a coated cylinder having R = 1A (includes coating). (a) 61 = 0, 62 = 0.03A, 63 = 0, e = 4, 1 = 1. (b) 61 = 0, 62 = 0.07A, 63 = 0, e = 4, IL = 1. (c) 61 = 0, 2 = 0.07A, 3 = 0, = 4, = 1................. 33 3.4 H-polarization bistatic echowidth (/' = 0~) for a coated cylinder having R = 1A (includes coating). (a) 61 = 0, 62 = 0.06A, 63 = 0, E=4,l=1. (b) 61=0, 62=0.06A, 63=0, = 8, =1...... 34 3.5 E-polarization bistatic echowidth (4' = 0~) for a coated cylinder having R = 1A (includes coating). (a) 61 = 0, 62 = 0.08A, 53 = 0, E=4, I=1. (b) 61 =0, 2=0.12A, 63=0, e=4, =l. 35 3.6 Maximum allowable coating thickness for applying the high contrast second order GIBC as a function of refractive index (e-1/2) with an acceptable error of ~ 2 dB in the field magnitude........... 36 3.7 H-polarization bistatic echowidth (4' = 0~) for a three- layer coated cylinder having R = 1A (included coating). (a) 61 = 0.01A, 62 - 0.02A, 63 = 0.03A, l = 3 - jO.1, e2 = 4 - jO.3, E3 = 7 - jl.5, /1 = -/2 = -3 = 1.2. (b) 65 = 0.01A, 62 = 0.02A, 63 = 0.03A, 1 =3-jO.1, E2=4 - j0.3, 3 = 7 - jl.5, p1 /2 =/= 1..... 37 3.8 H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.32A (includes coating) and tip angle a = 90~. (a) 61 = 0, 62 =0.06A, 563 = 0, e = 3. (b) 61 = 0, 62 = 0.06A, 63 = 0, e =8. 38

3.9 H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.26A (includes coating) and tip angle a = 360. (a) 61 = 0, 62 = 0.03A, 63 = 0, c = 4. (b) 61 = 0, 62 = 0.03A, 63 = 0, = 3-jl, 2-j0.5.......................... 39 3.10 H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.26A (includes coating) and tip angle a = 30~. (a) 61 = 0, 62 = 0.03A, 63 = 0, e = 4. (b) 61 = 0, 62 = 0.03A, 63 = 0, e = 3-jl,/p = 2-jO.5.................. 40 3.11 H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.26A (includes coating) and tip angle a = 360. (a) Variable right hand side coating using a linear taper; E = 4 to 5-j 1.5 from center to tip, 61 = 0, 62 = 0.03, 63 = 0 (/ = 1). (b) Variable right hand side coating using a linear taper; / = 2 to 3 - jl.5 from center to tip, 61 = 0, 62 = 0.03, 63= 0, (e = 1)............. 41 4.1 (a) Coated circular cylinder (b) equivalent coated cylinder simulated with GIBCs............................... 45 4.2 Different regions associated with the plane wave scattering from a smooth convex cylinder....................... 48 4.3 Reflected ray path.......................... 49 4.4 Propagation ray paths in the shadow region.............. 50 4.5 Projection of the field point P in the direction normal to the surface at PN......................... 54 4.6 Elliptical cylinder configuration.................. 56 4.7 Bistatic scattering pattern for an elliptical cylinder having a = 2A, b = 1A, p = 5A, = O (a) E polarization, E, = 4, L, = 1, 6 = 0.07A (b) H- polarization, e, = 8, pr = 1, 6 = 0.2A2............. 58 4.8 Bistatic scattering pattern of a circular cylinder having b = 3A, p = 3.05A, Xi = 0 (a) E-polarization, e, = 4,,r = 1, 6 = 0.07A (b) H-polarization, e, = 8, it, = 1, 6 = 0.2A................. 59 4.9 Bistatic scattering pattern of a three-layer coated circular cylinder having b= 3A, = 0,,1 = 3- j0.1,,2 = 4 j0.3, E, - 7- j1.5, Ptl = /zr2 =,r3 = 1, 61 = 0.01A, 62 = 0.02A, 63 = 0.03A (a) H-polarization, p = 5A (b) E- polarization, p = 3.05........ 60

5.1 Impedance wedge geometry........................ 63 5.2 Sommerfeld contour y in the complex a plane............ 66 5.3 Far zone backscatter pattern for an impedance wedge having 7+ = 1.0 - j.0, rl. = 0.5 -jO.1, /3 = 30~, Epo.po, (a) internal wedge angle = 0~, (b) internal wedge angle = 100.............. 76 5.4 Far zone backscatter pattern for an impedance wedge having 7+ = 1.0- jl.0, 77_ = 0.5 -jO.1, po = 300, Epo,p, (a) internal wedge angle = 200, (b) internal wedge angle = 30.................. 77 5.5 Far zone backscatter pattern for an impedance wedge having r+ = 1.0 - jl.0, n_ = 0.5 - jO.1, internal wedge angle = 30~, Epp, (a) /o = 900, (b) lo = 70~........................ 78 5.6 Far zone backscatter pattern for an impedance wedge having 7+ = 1.0 - jl.0,?7_ = 0.5 - j0.1, internal wedge angle = 30~, Eg,Opi, (a) Po = 50o, (b)o = 400......................... 79 5.7 Far zone backscatter pattern for an impedance wedge having 7+ = 1.0 j l.0, /_- = 0.5 -j0.1, /o, = 300, Epo,o,, (a) internal wedge angle = 00, (b) internal wedge angle = 100. 83 5.8 Far zone backscatter pattern for an impedance wedge having + = 1.0 - jl.0, r_ = 0.5- j0.1, fo = 300, Epopo, (a) internal wedge angle = 200, (b) internal wedge angle = 300............... 84 5.9 Far zone backscatter pattern for an impedance wedge having r7+ = 1.0 - jl.0, 7_ = 0.5 - jO.1, internal wedge angle = 300, Eop,6, (a) 0 = 90~, (b) go = 70....................... 85 5.10 Far zone backscatter pattern for an impedance wedge having =7+ 1.0 - jl.0, 7_ = 0.5 - j0.1, internal wedge angle = 30~, Epo,p, (a) Po = 50~, (b) po 40......................... 86 5.11 A plot of (a) sin (b) cos ".89 5.12 Far zone backscatter pattern for an impedance wedge having 77+ = 1.0 -jl.0, r, = 0.5 -jO.1, 1 o = 300, E,opo, (a) internal wedge angle = 0~, (b) internal wedge angle 30............ 92 Vli

5.13 Far zone backscatter pattern for an impedance wedge having 71+ = 1.0 - j1.0, r7_ = 0.5 - j0.1, o = 30~, Eo.,, (a) internal wedge angle = 00, (b) internal wedge angle =30................ 93 5.14 Far zone backscatter pattern for an impedance wedge having 77+ - 1.0- jl.0, _ = 0.5 - j0.1, Po = 300, internal wedge angle = 30~, (a) EpopO, (b) E,P..................... 96 5.15 Different regions for the applicability of the three solutions. (a) backscatter case (b) bistatic case.................... 98 5.16 Far zone backscatter pattern for an impedance wedge having 7r+ 1.0- jl.0, r7_ = 0.5- j0.1, Pi = 300, Ep,Op, (a) internal wedge angle = 200, (b) internal wedge angle = 300.................. 99 5.17 Far zone backscatter pattern for an impedance wedge having 77+ 1.0 - j1.0, r_ = 0.5 - jO.1, internal wedge angle = 30~, E,3oo, (a),o = 50o, (b) Po = 400........................ 100 5.18 Far zone backscatter pattern for a right-angled wedge having 71+ 1.0 -jl.0, r_ = 0, fo = 600, (a) Ep,pi, (b) E+,*........... 103 5.19 Far zone backscatter pattern for a right-angled wedge having 77+ 1.0 -jl.0, t_ = 0, o0 = 300, (a) E,O3g, (b) Et4,............ 104 5.20 Far zone backscatter pattern for a right-angled wedge having r7+ 1.0 - jl1.0, 7 = 0, f,: = 600, (a) EO,p,, (b) E.,9oo........... 105 5.21 Far zone backscatter pattern for a right-angled wedge having rl+ 1.0 -jl.0, T7_ = 0, /3,= 300, (a) Emo., (b) EPol60........... 106 6.1 Geometry of a cylinder of arbitrary cross section........... 108 6.2 Geometry of a triangular impedance cylinder............. 112 6.3 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 300, o: = 900, q' = 700, and its top face having 7 = 0.25, and the other having: = 0.75 + j0.75, (a) Ep,op, (b) E...................................... 115 6.4 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 300, Po = 600, q' = 700, and its top face having 7 = 0.25, and the other having r7 = 0.75 + j0.75, (a) ERoto, (b) E,4.................................. 116

6.5 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 300, 3o = 60~, I' = 70~, and its top face having r7 = 0.25, and the other having 77 = 0.75 + j0.75, (a) Eg, (b) Eo,-,............. 117 6.6 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 300, 3o = 30~, s' = 70~, and its top face having 7r = 0.25, and the other having 7r = 0.75 + j0.75, (a) EpopO, (b) E0,01.......................................... 118 6.7 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 300, Po = 30~, 4' = 70~, and its top face having 77 = 0.25, and the other having r7 = 0.75 + jO.75, (a) EOpp, (b) Epo*,....... *....... 119 6.8 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 40~, P3 = 90~, q' = 70~, and its top face having 7r = 0.5, and the other having 77 = 1.0 + j1.0, (a) Epopm, (b) E,............ 120 6.9 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 400, io = 30~, 4' = 70~, and its top face having r/ = 0.5, and the other having 7 = 1.0 + jl.0, (a) Eo,6 (b).......... 121 6.10 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 400, Po = 300, 4' = 70~, and its top face having 77 = 0.5, and the other having r7 = 1.0 + jl.0, (a) Eo,p, (b) Ep#o,................... 122 6.11 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 500, /0 = 900, 4' = 800, and its top face having r 7= 0.5, and the other having 77 = 1.0 + jl.0, (a) Eo,0x (b) E~,.................................... 123 6.12 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, c = 500, fo = 300, 4' = 800, and its top face having r7 = 0.5, and the other having r7 = 1.0 + jl.0, (a) Epo.p, (b) E,..................................... 124 6.13 Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 500, ~3= 300,,' = 800, and its top face having:= 0.5, and the other having r7 = 1.0 + jl.0, (a) Ewto, (b) Ep.................. 125

7.1 Edge-fixed coordinate system................... 128 7.2 Finite impedance wedge geometry.................... 130 7.3 A curved three dimensional wedge illuminated by an incident plane wave............................... 134 7.4 RCS of a flat base cone due to incident E,i (HH) polarization with a = 150~, D = 2A, 9i = 900, (a) perfectly conducting cone, (b) coated cone: E, = 7-j1.5, p,. = 2-jl, coating thickness = 0.04A....... 137 7.5 RCS of a flat base cone due to incident Eki (HH) polarization with a = 15~, D = 3A, 0 = 900, (a) perfectly conducting cone, (b) coated cone:,r = 7-jl.5, p, = 2-jl, coating thickness = 0.04A........ 138 7.6 RCS of a partially coated flat base cone having perfectly conducting base with a = 15~, D = 3A, er = 7- jl.5, r = 2-jl, coating thickness = 0.04A, O' = 900, (a) E~, (HH) polarization, (b) EB, (VV) polarization. 139 7.7 RCS of a finite circular cylinder due to incident Ei, (HH) polarization with D = 3A, L = 4A, i' = 900, (a) perfectly conducting cylinder, (b) coated cylinder: Er, = 4-jl.5, p, = 2-jl, coating thickness = 0.04A............................... 140 7.8 RCS of a square plate due to incident E,, (VV) polarization with W = 10A, 8' = 900, (a) perfectly conducting plate, (b) coated plate: F, = 12.1-jO.46, p, = 2- jl.26, coating thickness = 0.08 inches, frequency = 8.004 GHZ......................... 141 A.1 Edge geometry and angle definitions.................. 149

CHAPTER I INTRODUCTION The subject of this dissertation deals with the characterization of the non- specular electromagnetic scattering from coated and impedance structures in the context of asymptotic/high-frequency techniques. Nowadays, the study of both specular and non-specular electromagnetic scattering mechanisms is crucial in the design of all high performance airborne vehicles with low radar cross section (RCS). Shaping and radar absorbing materials are commonly employed to reduce the radar reflectivity of these vehicles. However, in most cases, only the specular contributions from electrically large structures can be effectively reduced by shaping. This leaves non-specular or diffraction contributions primarily caused by edges, creeping waves and surface diffracted waves, as the dominant terms. It is, therefore, important to examine the effect of radar absorbing materials in controlling non-specular scattering. A common technique for RCS control is to use radar absorbing coatings and over the past few years much attention has been given to the electromagnetic characterization of such coatings. Methods have been developed to simulate the material coatings by approximate boundary conditions which offer several advantages in both asymptotic and numerical analyses of electromagnetic problems. For example, in the case of asymptotic/high-frequency analysis, the coating can be replaced by a single

2 boundary condition suitable for the application of the Wiener-Hopf technique [1,2] or some other function theoretic approach [3,11]. In numerical analysis, the replacement of the profile of the coating with a single boundary condition eliminates the need to introduce unknown polarization currents inside the coating. This, significantly, reduces the total number of unknowns and results in a more efficient solution. The standard impedance boundary condition (SIBC) [4] has been frequently employed to simulate composite material coatings on metallic structures. However, it is well-known that the SIBC simulation is accurate only if the coating is very thin and has a high index of refraction. One of the major reasons for this is because the SIBC cannot model the polarization current components which are normal to the coating. To improve the accuracy of the SIBC simulation, higher order impedance boundary conditions were recently developed [5] and found to be capable of simulating thicker coatings with greater accuracy. These conditions involve higher order derivatives of the fields beyond the first and can be thought as a generalization to the SIBC. They are,therefore, referred to as the generalized impedance boundary conditions (GIBC) and one of their advantages is again a simplification in the analysis. In our study, the SIBC and second order GIBC will be used for a characterization of the edge diffraction, creeping wave and surface diffracted wave contributions. The characterizations will be carried out via asymptotic/function theoretic methods which have been traditionally used in high-frequency studies (i.e. where the scatterer is many wavelengths in size). An important advantage of the asymptotic/highfrequency techniques over numerical methods is that the scattered field from a complex structure is a priori subdivided into contributions from its different components. This gives an insight into the importance of each scattering contributor which is particularly useful for designing vehicles with given radar cross section. Among the most

popular asymptotic/high-frequency methods are the geometrical theory of diffraction (GTD) and physical theory of diffraction (PTD). The GTD/PTD formulations can permit characterizations of non- metallic geometries but require the appropriate diffraction coefficients associated with a specific discontinuity in surface curvature or material composition. These diffraction coefficients play a role similar to the reflection and transmission coefficients of the GO reflected and transmitted rays and can be determined from an exact analytical solution of a corresponding canonical geometry. Unfortunately, this has only been rigorously accomplished for a handful of geometries which primarily include metallic edges and wedges at normal and skew incidences [6]-[8], second order surface discontinuity [9,10], impedance wedges at normal incidence [11], thin dielectric and resistive edges at normal and skew incidences [12,13], impedance or material discontinuities in a plane [14,15] and the right-angled wedge at skew incidence with one of its faces perfectly conducting [16]. Solutions may be also found in the literature for creeping wave characterizations and for some non-generic configurations. The main goal of this dissertation is to develop asymptotic/high-frequency solutions for analyzing the non-specular scattering mechanisms produced by coated convex surfaces and edges simulated by the approximate boundary conditions. Also, to validate the accuracy of these asymptotic solutions, integral equations are derived on the basis of the employed SIBCs and GIBCs. The integral equation solutions are then used for benchmarking the proposed asymptotic expressions. Some insights are also provided on the accuracy of the employed GIBCs for the subject geometries. The organization of this dissertation is as follows. In chapter two, both the SIBCs and the GIBCs are presented for simulating a material coating on a metallic substrate. The GIBCs involve higher order derivatives having coefficients which are

4 determined using the exact reflection coefficient of the metal-backed coating. As an illustration, the simulation of metal-baked uniform and three layer dielectric coatings is given. Chapter three deals with numerical solutions on the basis of a second order GIBC. The GIBCs have so far been presented for simple planar surfaces such as a uniform coating on a ground plane [5,17] and a resistive sheet [17,18]. Of practical interest, however, is the application of these conditions to multilayer and inhomogeneous coatings and layers forming arbitrarily curved surfaces. In this case, the derived GIBC for a planar coating is extended to the curved surface by postulation. The accuracy of such an extension as a function of curvature is, of course, in question and its examination for surfaces other than circular, spherical or elliptical requires a numerical solution. In chapter three, a numerical implementation of a second order GIBC, simulating the coating on a metallic cylinder of arbitrary shape, is considered. Integral equations, for both E and H- polarization, are derived and solved numerically using the method of moments. Results are presented for a single and three layered coated circular and ogival cylinder and the accuracy of the given GIBCs verses SIBCs is discussed. This numerical implementation is, also, needed to validate the asymptotic/high-frequency solutions, for a convex coated cylinder which is presented in chapter four. In chapter four, the non-specular scattering mechanisms like creeping waves and surface diffracted waves are considered as applied to a coated convex cylinder. Rigorous UTD (uniform geometrical theory of diffraction) diffraction coefficients are presented for this geometry simulated with SIBCs and GIBCs. In particular, ray solutions are obtained which remain valid in the transition region and reduce uniformly to those in the deep lit and shadow regions. These involve new transition functions

5 in place of the usual Fock-type integrals, characteristic to the perfectly conducting cylinder. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder. The diffraction coefficients for a convex cylinder are obtained via a generalization of the corresponding ones for the circular cylinder. Results are presented which validate the accuracy of the ray solutions by comparing them with those obtained from the eigenfunction solution and the numerical solution of chapter three. Chapter five deals with a most crucial source of non-specular electromagnetic scattering, that of diffraction by an impedance wedge. A normal incidence solution for the diffraction by an impedance (SIBC) wedge having arbitrary included angle is already available [11]. On the contrary, for skew incidence, solutions have only been obtained for impedance wedges having included angles of 0 (half plane) [12,13,14,15,19], r/2 (with one face perfectly conducting) [14,15,16,20], r and 37r/2 (with one face perfectly conducting) [14,15,19]. This is because enforcement of the SIBCs on the wedge faces leads to a set of four coupled functional difference equations which can only be decoupled for the special cases mentioned above. In chapter five, we present an approximate solution of the coupled difference equations by again employing Maliuzhinets' method [11]. The derived solution recovers those for an impedance wedge of arbitrary included angle at normal incidence and an impedance half plane at skew incidence. The reduction of the approximate solution to that for the right- angled wedge (the only other known exact solution) is done numerically. In chapter six, we examine the accuracy of the approximate skew incidence dyadic diffraction coefficient (developed in chapter five) for a variety of different wedge angles. This is accomplished by comparison with numerical data obtained from a new moment method solution of the coupled set of integral equations for skew incidence

on an impedance polygonal cylinder. The solution of this coupled set of integral equations provides the surface currents which are integrated to yield the scattered field on the diffraction cone. The corresponding (first order) high-frequency solution for the far zone scattered field for the polygonal cylinder is then compared to the results from the moment method code. The skew incidence diffraction coefficient for an impedance wedge is essential for computing the electromagnetic scattering of practical three dimensional impedance structures. To test the usefulness of the approximate skew incidence solution for such 3-D structures, equivalent currents are presented in the context of PTD for a finite length impedance wedge of arbitrary internal angle. These are derived in chapter seven and are incorporated in a standard general purpose physical theory of diffraction (PTD) code [21]. Many patterns are computed on the basis of this code which demonstrate the accuracy of the formulation for a number of different impedance structures. These include typical shapes such as plates, finite length cones and cylinders which have been partially or fully coated. The PTD implementation required a dyadic physical optics (PO) diffraction coefficient which is derived in appendix A.

CHAPTER II SIMULATION OF COATED STRUCTURES BY APPROXIMATE BOUNDARY CONDITIONS The use of composite materials, in the form of a uniform or non-uniform coating applied to a metallic substrate, has led to the development of methods for simulating material effects in electromagnetic scattering. A possible approach is to employ approximate boundary conditions which offer several advantages in both asymptotic and numerical analyses of electromagnetic problems. For example, in the case of asymptotic/high frequency analysis, it allows an accurate replacement of a coating or a layer with a single boundary condition amenable to a Wiener-Hopf analysis [1,2] or some other function theoretic approach [3]. In numerical analysis, the profile of a coating can be replaced by a single boundary condition on the surface of the coating. This eliminates a need for introducing unknown polarization currents within the coating and thus leading to a more efficient solution. Traditionally, the standard impedance boundary condition (SIBC) [4] has been employed to simulate dielectric coatings on perfectly conducting objects. However, as is well known, the SIBC provides limited accuracy and is only applicable to lossy and/or high contrast coatings. One of the major reasons for this is because it cannot model the polarization current components that are normal to the dielectric layer.

The SIBC has, therefore, been found to be best suited for near normal incidence unless the coating's material properties are such that penetration within the coating is limited. The SIBC is a first order condition in that its definition involves a single normal derivative of the component of the field normal to the modeled surface. Recently [5], however, a class of boundary conditions were proposed whose major characteristic is the inclusion of higher order derivatives (along the direction of the surface normal) of the normal field components. These were originally introduced by Kane and Karp [22], Karp and Karal [23], and Wienstein [17] to simulate surface wave effects, but have been found to be rather general in nature. In fact, they can be employed to simulate any material profile with a suitable choice of the (constant) derivative coefficients. Appropriately, they are referred to as generalized impedance boundary conditions (GIBCs) and can be written either in terms of tangential or normal derivatives provided a duality condition is satisfied [5]. Unlike the SIBCs they offer several degrees of freedom and allow an accurate prediction of the surface reflected fields at oblique incidences. This was demonstrated in [5] for the infinite planar surface formed by a uniform dielectric layer on a ground plane. It was found that the maximum coating thickness accurately simulated by a given GIBC is related to the highest order derivative included in the condition. 2.1 Standard Impedance Boundary Conditions (SIBCs) The most common approximate boundary condition is the standard impedance boundary condition (SIBC). It allows us to solve a two media problem while solving explicitly for the fields only in one. This is achieved by relating the tangential fields along the surface which takes into account the effective material composition of the

second medium. The idea of impedance boundary conditions was first introduced by Leontovich (see [24]) and, later, Senior [25]-[27] reviewed and expounded upon the use and limitations of this principle. For the metal-backed dielectric layer shown in figure 2.1, the impedance boundary condition can be mathematically written as y ( x metal Figure 2.1: Metal-backed dielectric layer. E-(n * E)-n 77Z(n x H) (2.1) where i is the normal to the surface, 77 is the surface impedance, Z is the free space impedance and E and H are the total electric and magnetic fields respectively. For the metal-backed layer of thickness 6 in figure 2.1, 77 is given by (an ejwt time dependance has been assumed and suppressed throughout) 7 = j-N tan(NkS) (2.2) where k = 27r/ is the free space wave number, N = V/eri, is the index of refraction whose magnitude is assumed to be large, Er and tr are the relative permittivity and permeability of the layer, respectively. Senior [26,27] has shown that a dual relationship exists for (2.1) where ZH -- -E/Z, E -, H and 7r -- giving H- (n H)n = _ (n x E) (2.3) 77Z

The vector form of the SIBCs given in (2.1) and (2.3) can also be applied to curved surfaces. However, it is required that the penetration depth of the electromagnetic field must be small compared with the minimum thickness and minimum radius of curvature pg of the body, and the local wavelength in the material must also be small compared with pg-. These conditions can be summarized as [27] tImNlkpg > 1 (2.4) It is to be noted that SIBCs support both electric and magnetic surface currents J and M, respectively, where 7J= n x H, M = - x (2.5) and the use of (2.1) yields the relationship between the two currents as M = — rlZ x J (2.6) 2.2 Generalized Impedance Boundary Conditions (GIBCs) 2.2.1 Single Layer Coating Consider the planar single-layer coating shown in figure 2.1. An appropriate boundary condition that simulates the coated structure takes the form [5,23] m (-jk)m = m (2.7a) M a, am m (= 0 (2.7b) Z=(_ )m j ymk l where the constants am and a, are specific to the material and geometrical properties of the simulated structure and must be derived to allow an accurate reproduction of the field in the region y > O.

11 Expressions given in (2.7) are scalar forms of a class of GIBCs. Because, however, they involve normal derivatives, they are not convenient to implement numerically neither do they provide a physical meaning of what they represent. It is, therefore, important to express (2.7) in terms of tangential derivatives. As shown in [5], this can be accomplished provided am and a' are not chosen independently. For example, when M = 1, (2.7a) and (2.7b) can be reduced to (2.1) and (2.3), respectively, where we identify 77 = ao/al = aL/a' as the surface impedance. For M = 2, (2.7) becomes c2EY al aEy ao E Oy2 jka2 ay k2a2 (2.8) 2 Hy al c9H a, ~ ~92f4, ~ &II~,_ -~~-H 0 0y2 jka' y k2aH' = which can also be written in terms of tangential derivatives and, thus, exposing their non-local character. From [5] we have that (2.8) are equivalent to ( 6~[E. ~a a61 a2+ ao [ 2 (x^x [E a2 V(*-a.)]) =- a ZY x -[H V(.H) ] (2.9) jkal al a, jka', provided the duality condition a/ a2 + ao a1 (2.10) a' + a' al is satisfied. The vector form of the boundary conditions (2.9) allows us to write it in a coordinate-free form, thus, making it applicable to non-planar surfaces. Referring to figure 2.1, (2.9) may be rewritten as AK jF al A a2+ ao? aX fh\ [x E a[ V(nE-)) - a- Znx[H jkaEV. ZH) (2.11) where denotes the unit normal to the surface at the point of at the point of application.

There are a number of methods that can be employed for deriving the constant coefficients am and a'. One approach is to expand the plane wave reflection coefficient in a suitable form leading to the identification of the constants. A second method is a generalization of that employed in [1], where the boundary condition at y = 0O is derived by transferring the fields from the ground plane to y = O via a Taylor series expansion. Below we derive a second order GIBC by using the first of the methods mentioned above. Consider the plane wave Hz = ejk(=o~''+"i') (2.12) incident on the surface y = 0, shown in figure 2.1, satisfying the GIBC given in (2.7). The implied reflection coefficient is then E (- 1)na, sinm' MR(') = _,,,_Mo(2.13) E am sinm' m=O with an analogous expression for the reflection coefficient associated with E.-incidence. We are interested in finding the constants am when the GIBC simulates the single layer coating shown in figure 2.1. This can be accomplished by expanding the exact reflection coefficient corresponding to the given coating in a form comparable to (2.13) and thus allowing identification of the constants am. For an H-polarized plane wave incidence, the exact reflection coefficient can be written as Rk(qY)=- s.Wtan (ksl 2 c') + jc:, sin'( R(2') - cos2 q' tan (k/N2 - cos2 4') - je, sin' (2.14 ) The corresponding reflection coefficient for E-polarization is Re(q!') = - cs2 qV' - j/,sin 4'tan (k6N2 - cos2') (2.15) R/NE - oes' + jp,sin' tan (k P/N i-os2 1') where Rh(4') and Re(I') are both referred to y = 0+.

13 2.2.1.1 Low Contrast GIBCs The low contrast conditions can be derived by introducing the approximation tan x; x in (2.14) for INI small [28], giving ( k6 sin2 b' + jCr sin2 o' + k6(N2 - 1) Rh(q') - k6sin2'-+ej sin2' + kS(N2 -1) (2.16) k6 sin2 q' - je, sin2 q' + kS(N2 - 1) from which the constants a, are easily found to be (N2 - 1)k6 a. = (2.17a) fr al = -j (2.17b) k5F a2 = -k (2.17c) Er Following a similar procedure, we may also derive the constants a' corresponding to Es-incidence. We have a' = 1 (2.18a) a1 =jklpr6 (2.18b) 2a2: 0 (2.18c) 2.2.1.2 High Contrast GIBCs The high contrast conditions assume (NI to be large and, therefore, using the approximations VN2 - cos2 N 1 sin2' (2.19) 2N 2N

14 tan (2- sin2 2N- sin2 (2.20) in (2.14) leads to the constants a = (N- [tan(kN) - tan (2.21a) a = -jfr [1 +tan(kSN) tan 2N (2.21b) a2 = -N tan(k6N) -tan (N + kb N2N -2NI (2.21c) L {1 + tan(k6N) tan (2N) }] a' = (N-2N) [1 + tan(k6N) tan (2)] (2.22a) a, = js [tan(k6N) - tan ( )] (2.22b) a2 = [ + tan(kN) tan ) - kS N-2N (2.22c) * {tan(kSN) - tan ( 2A) }] In the above, the boundary conditions implied by (2.17)-(2.22) are referred to the surface y = 0+. 2.2.2 Three Layer Coating

15 arbitrary constitutive parameters. The geometry of the coating is illustrated in figure 2.2. For H-polarization the exact reflection coefficient referenced to y = 0 can be written as x 62 22/////////// 63 3 3 metal Figure 2.2: Illustration of a three-layer metal-backed coating. FN(S') Rh(') - FDN(C') (2.23) where FN,D(q') C= El23kyokyl ky2 cos(ky363) cos(ky262) cos(k yll) -el263kyok2 cos(ky363) sin(ky2S2) sin(kyl 1) - fie2kyokylky3 sin(ky363) sin(k.262) cos(kl 51) - eCe2kyOky2ky3 sin(ky383) cos(ky252) sin(kyl1 )

: je2e3ky1ky2 cos(ky363) cos(ky262) sin(kylSi) j jEl 63kyl k2 cos(ky363) sin(k2y52) cos(kylSi) ~ j3%2k2 ky3 sin(ky363) sin(ky262) sin(kll6i) jelE2kyl ky12ky3 sin(ky363) cos(k~y22) cos(kyl S) (2.24) with kyn = k/Nn2 - cos2 4' (2.25) N, = V/i and kyo = k sin q'. 2.2.2.1 Low Contrast GIBCs To put (2.23) in a form compatible with (2.13) we must now expand the terms sin(ky,nn) and cos(ky6nn) in powers of sin 4'. The simplest case is to assume that kynEn is sufficiently small allowing us to set sin(kSn,n) t kyn,, (2.26) and cos(ky,n6); 1 (2.27) Clearly (2.26)-(2.27) also implies that N,nn must be small and thus the resulting boundary conditions will be valid for low contrast and thin coatings. Substituting (2.26)-(2.27) into (2.23) and retaining only terms up to and including O(6n) we obtain (N - 1) k + (N22 - 1) k2 + N32- 1) k3 (2.28a) E1 E2 C3

17 a = -j (2.28b) a2= k ( - + - + ) (2.28c) XVhen these are subsequently introduced into (2.9) or (2.11) we have a second order (low contrast) GIBC for simulating a thin three-layer coating. The generalization of (2.28) to any arbitrary number of layers is obvious provided the total coating thickness remains small. Following a similar procedure we may also derive the constants a' corresponding to E.-incidence. We have a'= 1 (2.29a) at = jk(151 + p2S2 + p363) (2.29b) a2 = 0 (2.29c) These imply a first order condition, but by retaining higher order terms in the expansion we find that; - = P12- (-Cl 12- )- (2- N 1) (2.30a) al = jk,uLP2 (i5161 +,2S52 + /363) (2.30b) a2 = -(a1 + a2) (2.30c) in which c1 = k2p1p362 [-jpkl6163 + 63]

and C2 = k2P261 [/1262 + 1363] 2.2.2.2 High Contrast GIBCs For large N, the approximations (2.26)-(2.27) are not valid. In this case it is more appropriate to set kn = k k Nn + sin2') (2.31) and expand the resulting sine and cosine terms appearing in (2.24). This, however, leads to a 12th order GIBC that is obviously impractical to employ analytically and numerically. Instead, a more reasonable approach is to assume that the three layers comprising the coating have varying refractive indices. In practice, the top layer usually has a small refractive index whereas the bottom layer has a larger refractive index. With this assumption we set sin(kIy1l1) ~ ky,161 (2.32) cos(k,,il);:s 1 (2.33) ky2 = kN -2 cos2q' k N2 sin2') (2.34) sin i P, 2N) sin2' (2.35) cos ( k62 sin2 " q'): 1 (2.36)

19 and ky3 - k7N32 - cos2q $' kN3 (2.37) Employing (2.32)-(2.37) into (2.23), after much tedious -algebra we find that a sixth order GIBC is required [28] to recover the resulting reflection coefficient. This can be truncated to a second order GIBC with the pertinent constants for H-polarization given by ao -je2e3 (N2-1) (N2 -2N ) kl - jl63 (N22-1) tan [(N2 -2N2) k62] + je (N12- 1) N3k61 tan [(N2-2N ) k62] tan(N3k63) - je1e2 N2 2N) N3tan(N3k63) (2.38a) al - -E1le23 (N2 -2N2 + 12e3 (N22-1) k61 tan [(N2- 2N) k6] + ee2N3tan [(N2-2N ) tan( ) + e12e2 (N2 -2N ) N3kS1 tan(N3kS3) (2.38b)

20 a2 = je2e3(N2 -1) (N2 - 2N)k22 tan [(N2 - 2) k2] 2N2 N 2N -N -je2e3 (N -2N )k61 -je2e3(N 2-1) 2N21 -jfe3 (N22-1) tan [(N-2N ) k62] + jr (N: - 1) N3k6 1i2 tan(N3k63) + je2N3ak6ltan[(N,-2 k62] ta(N3k63) + jee2 (N2 - 2N) N32N2tan (N3k63) +N2"2N2 2NN - je1e22N N3tan(N3k63) (2.38c) 2N2 Similarly, for E-polarization, we find ao = 2a3 (N: - 1) 2N2 ke tan(N3k63) + 13 (N2 -)tan [(N) tan [(N2 1 ) tan(N3k63) + p2(N: - 1)N3k6 tan[(N - 2N:) kE2] - I1/2 [(N2 - - ) N3] (2.39a)

21 al -j/l1p23 N2- 2N tan(N3k53) + jii2i (N2 - i) kS1 tan [(N 2N ) k62] tan(N3k63) - j1I N22N3 tan [(N2 - -) k62] - 1/1.2 N2- 2N2 N3k6j1 (2.39b) a2 =12L3 (N1 - 1) kei tan(N3k6 3) 2N1 + 2/13 (N2 2N2 ) 6 tan(N3k63) 1 kNk62] - 82X3 (N — 1) ( N2-2 ) kEl 2 tan [(N2- 2NJ )'k]tan(N3k63) + P1p3 tan [(N2 - 2N2) k tan(N3k63) + 1Y3(N22 -1)22 tan(N3kS3) + 12N3kl tan [(N2 - 2N ) k2] + (2 N - 1)N3kSl 2N2 1 N3 + 11 ( 2N )N32N tan N - kj(N22-32c)2 2- 2N1 (239c)

CHAPTER III NUMERICAL IMPLEMENTATION OF GIBC In this chapter, we pursue a numerical implementation of the boundary conditions (2.11) in a manner similar to that employed in conjunction with the standard impedance boundary conditions (2.1)-(2.3) [29,30,31]. Of interest in this implementation is an examination of the accuracy of the second order GIBCs over that of the SIBCs in simulating a dielectric coating over a metallic structure. So far, GIBCs have been generated for simple planar surfaces such as a uniform coating on a ground plane [5,17] and a resistive sheet [17,18]. Of practical interest, however, is the application of these conditions to multilayer and inhomogeneous coatings and layers forming arbitrarily curved surfaces. In this case, the GIBC as derived from the corresponding planar structure/surface is extended to the curved surface by postulation. The accuracy of such an extension as a function of curvature is, of course, in question and its examination for surfaces other than circular or elliptical requires a numerical solution. For a closed surface, the GIBCs equivalently replace the entire effect of the coating, thus, eliminating a need to introduce polarization currents within the dielectric. This is an important advantage of the GIBCs leading to a simpler formulation and a reduction of unknowns in the numerical implementation.

23 3.1 Integral Equation Formulation 3.1.1 Hz-incidence Consider the plane wave H' = ejk(zc~08'+ysin') (3.1) incident upon a three-layer coated cylinder of arbitrary cross section as shown in figure 3.1. For a two dimensional curved surface having large radius of curvature, the GIBC given in (2.11) simplifies to nr P.E.C ~23' 2 2 1 2L Figure 3.1: Illustration of a three-layer coated cylinder. E,= ao+ a2ZH + Z- 2H (3.2) a1f 2k2a1 as2

24 where we have also employed the relation jZ aOH En Os (3.3) WVe note that (3.2) is identical to that implied by (2.9) if we let E. -+ E,, Ey -+ n and 0/ax -- d/Os. To compute the scattered field by the configuration in figure 3.1, a traditional approach is to introduce the electric and magnetic surface equivalent currents (see figure 3.2) A Al Jg~~" = (3.4) t=S ~mth element ~~~~~ters~~.~th J n x H = SH =.J (3.4)

25 M E x i = E, z= M (3.5) on the outer surface of the coating where E and H denote the total fields on C. From (3.2) and (3.4)-(3.5) we then find M() =[ao + a2 + a2] Z ]ZJ () E C (3.6) Enforcement of the GIBC, thus, eliminates one of the unknown current components. In the case of a first order GIBC (SIBC) the relationship between M, and J, is linear as implied by (2.6). The appearance of the second derivative in (3.6) is therefore attributed to the higher order condition and provides an added accuracy in the simulation of the coating. To construct an integral equation for the surface current J,, we now enforce the condition Hz = J, = Hz + H" (3.7) on C where Yk d~,(~)no(~'(kp)d'+ J Hz- 4 Mcz(J)H2(kp)ds'+ 1 )d(kp)d (3.8) is the scattered field. In (3.8), H(2) represents the zeroth order Hankel function of the second kind and d/dn' denotes differentiation along the direction normal to C at the integration point. Also, p = I- T1 (3.9) where r and T' are the two dimensional position vectors defining the observation and integration points, respectively. The differential element ds' denotes arc length along the integration contour C.

26 Substituting (3.8) into (3.7) and making use of (3.6) leads to the integral equation ao + a2 d a2 Hz()= J,(F) + J ) k -jn - a,2]H)(kp)ds', F E C(3.10) 4 k al dn' kal asI2 To solve this numerically we may discretize C into N straight segments, as shown in figure 3.2, and assume a constant current distribution on each segment. By enforcing (3.10) at the center of each of the N segments, we then obtain the discrete system [Vm]= [Zme][Jat] (3.11) for a solution of the current distribution J.. In (3.11), [J.e] is a column matrix with J,, e = 1, 2,..., N denoting the current on the eth element of C. [Vm] is the excitation column matrix whose elements are given by Vm = Hz(,m), m = 1,2,..., N (3.12) in which rm denotes the location of the observation/test point (see figure 3.2). Finally, [Zmt] is a square impedance matrix whose elements can be expressed as ZZ,mt = + Z + Zm (3.13) with Z~m =k ao + a+ | H()(kp)ds) = r-m (3.14) Zmt - -1J~ d H2)(kp)dskpd', (3.15) - 4' 4or =a [ H? ~)(kp)] s (3.16) 4 ~k~a1A 1 xk,_p)ds', -r-m

27 in which r'F = SeSt + ntnt denotes the center of the eth integration cell of width A. The vectors et and fi represent the unit tangent and unit normal, respectively, to the surface at et. It is necessary to evaluate the integrals in (3.14) and (3.15) analytically for small r and this can easily be accomplished after Z t and Z2t are written in terms of local coordinates. To do so, we define the observation vector m, as r-m = smsm + nmnm = smtse + nm,tne (3.17) = s' S + n'nl (3.18) and by substituting these into (3.14) we obtain k a +k a 2o *m+H2)(kVf2 + a H ( + 2)d, (3.19) zlmt a=A 0n 4 al Jt-B'nt- 2 where = n=m - nt. Similarly, for Z2 we have =2 = d |, H o(2) (k/~2 + n2) ds (3.20) t 4 dfi Jn,t-,,, The analytical expressions for the integrals in (3.19) and (3.20) have been developed in [32] by using a small argument expansion of the Hankel function to O(p4, p4 ln p). Once the current distribution has been determined from (3.11), the far zone fields can be easily evaluated using (3.8). By introducing the large argument approximation of the Hankel function into (3.8) we find sHl n-kr ok+"ea 2 jk(n ta+2stt3) sl,...too =l aO r a]

28 where Vt = (int' x) cos + (nft' ) sin 2&e = (scA* ) cos b + (s.t y) sin (3.22) and (r, q) denote the usual cylindrical coordinates of the far-zone observation point. The echowidth of the structure is defined by 2rr IHz (3.23) r —oo I zi12 and can be computed via (3.21). 3.1.2 Ez-incidence When the coated cylinder is illuminated by the plane wave E- = ek(oso+y'n#) (3.24) the equation (2.11) simplifies to ao - a2 a 82 Ez a+ ZH - Ez (3.25) z al' k2(a2 + a' ) as2 and we observe that this is the dual of (3.2) upon letting am a' and invoking (2.10). The equivalent currents are given by J = x H = - = Jz (3.26) M = E x n= —AEz = M (3.27) From (3.25) we also have Jz()= ( a + a 2) Yn 6!M,(r), + E TC (3.28) - aj k2al 8Js2/YM()

29 which is dual to (3.6). To construct an integral equation for the magnetic current Ms we follow the same procedure as that employed in the case of He-incidence. On C, Ez = -Ms = E'z + Ez (3.29) where Ez is the scattered field given by E=- Jz J=(f')H 2)(kp)ds' - J Md (/)dH(2) (kp)ds' (3.30) Introducing (3.28) into (3.30) and substituting into (3.29) leads to the integral equation ra' +a' d a2 02 - Ez'() = M,(F ) + 2) M(') k a2 -j + a- 2 ] H(2(kp)ds', (3.31) _EC This is dual to (3.10) and its numerical solution can be accomplished in a manner parallel to that employed for (3.10). Also the far zone field is given by the dual of (3.21). 3.2 Numerical Results This section addresses the numerical accuracy of the high and low contrast second order GIBCs in comparison with the SIBCs and other standard formulations. A determination of the GIBC's range of validity by comparison with simulations based on exact solutions or other validated numerical techniques is also presented. Figure 3.3 presents a numerical comparison between the GIBC, SIBC and the exact solution with TEincidence on a uniformly coated circular cylinder. As seen, the SIBC is quite accurate in simulating thin coatings. However, as the coating

thickness increases, it no longer remains accurate and this is demonstrated in figures 3.3(b) and 3.3(c). We note that the low and high contrast GIBCs employed to generate the curves in figure 3.3 were obtained by setting 61 = 63 = 0. Figures 3.4 and 3.5 show a comparison of results based on the low and high GIBCs for different permitivities and coating thicknesses. In particular, figure 3.4 includes data corresponding to constant thickness, single coatings. As expected, the low contrast GIBC shows a better agreement with the exact solution in the case of low contrast material and the high contrast GIBC is more accurate for higher permitivities. The curves in figure 3.5 correspond to constant permittivity coatings and it is now seen that the high contrast GIBC is more accurate over the low contrast GIBC as the thickness of the coating is increased. Data obtained from numerous tests suggest that the agreement between the low contrast GIBC and exact solutions is acceptable for f 6 < 0.15A with H-polarization and for W 6 < 0.20A with Epolarization. The range of validity of the high contrast GIBC is unfortunately a function of the refractive index and thickness as depicted by the plot in figure 3.6. The aforementioned conclusions on the range of validity of the high and low contrast GIBCs were based on data for a single uniform coating on a circular cylinder of radius 1A. Of course, other parameters such as the radius of the cylinder and homogeneity of the coating are of importance. Although these were not accounted for in this preliminary study, some general suggestions can be noted based on our experience. In particular, when the relative permeability is not unity, it is reasonable to assume that the range of validity of the conditions will be between those stated for the E and H polarizations. Also, when the conditions are applied to model multiple layer coatings, an average permittivity can be computed before making use of the criteria given above. Representative scattering patterns from a multilayer cylinder

are given in figure 3.7 and as seen, the agreement with corresponding numerical data [32] is good. Since the presented GIBCs were derived on the assumption of a planar surface, it is of interest to examine their accuracy in simulating surfaces having small radii of curvature and/or discontinuities in the first derivative. With regard to the first issue, the conditions have been employed to model smooth coated cylinders down to at least A/4 in radius without any appreciable deterioration of their accuracy. The second is addressed by using a second order GIBC to simulate a uniformly coated ogival cylinder. Scattering data based on the low and high contrast GIBC are shown in figure 3.8 and compared with data based on a finite element boundary integral [33] method. We note that the accuracy of the GIBC formulation tends to decrease with increasing eccentricity of the ogive. In general, though, the agreement between the exact and GIBC-based solutions remains good for tip angles (a) greater than 60 degrees. However, the accuracy of the GIBC simulation can be greatly improved by replacing the sharp tip of the ogive with a rounded one (having radius of about 0.01A) and increasing the sampling around the tip. This is demonstrated in figures 3.9 and 3.10 for tip angles of 360 and 300, respectively. The agreement between the FEM and the GIBC solutions tends to deteriorate as the tip angle goes below 30 degrees. The results presented up to this point correspond to uniform coatings. As noted earlier, though, the GIBCs can be employed to simulate coatings of varying thickness and layered material composition. Figure 3.11 presents data for an ogive coated with a constant thickness coating having non- uniform e and/or tp.

32 3.3 Summary Second order generalized impedance boundary conditions (GIBCs) were used to derive integral equations for scattering by two-dimensional coated structures-of arbitrary cross-section and implemented using a moment method procedure. Results based on this numerical implementation were then compared with corresponding data based on an exact solution or from other validated computer codes. It was found that the proposed second order boundary conditions provide an improved simulation of the coating in comparison with the traditional standard impedance boundary condition and guidelines were given for their region of validity. The primary reason for the improved simulation is because the second order GIBC includes the effect of the polarization current components normal to the coating in addition to the tangential ones. In general, the presence of edges deteriorates the accuracy of the simulation particularly for thicker coatings. This is due to the inherent non-uniqueness of these boundary conditions at abrupt terminations, a situation which can only be remedied by introducing additional field constraints at the terminations [51,52]. The second order GIBCs were found to be equally effective in simulating non- planar coatings having laterally non-uniform but smoothly varying material properties or thickness.

33 20 IS Exact - - - -Low Con. 1 0....... SIBC (a) 1 -5 o ~-10 -20 j 0 30 80 90 120 150 180 Angle (degree) 20 1 5 Sr 10 (b) 1: -10 5 0 30 60 90 120 150 180 Angle (degrees) 20 Exact 1 ~.. H i-g Con. A~ 10 0~~,','. /~ ~ c 5 i 0 0 (c) -5 ~J o 10 -20 -20 I. I 0 30 80 90 120 150 180 Angle (degrees) Figure 3.3: H-polarization bistatic echowidth (qY = 00) for a coated cylinder having R = IA (includes coating). (a) 61 0, 62 = 0.03A, 63 = 0, 4, p=1 (b) 6i = 0,7 62= O.07A, 63 = 0, 4, A = 1.- (c) 6i = 0, 62 = 0.07A,1 63 = 0, e = 4, p = t;r'i~~ -... szcon,0....,I..i, -20' 0 30 6) 0 g O120 150 180 _-u~. (d,,~e,) 20...... 15~~~'1 aJ __gh Con. c~~~~~~.. ~e; t T M ~~~z t.~ _.~.. y!ji C ~~~~~~~~~~~ ~~~~~~~~ = r r~~~~~~~~~~~~~~~~~~. t~~~~~~~~~~~".,-,. z ~r ~ vV V,.' -1, O: -15 -0 ~r I',, 20 3 0 9 2 8 Anir] (degrees)

34 20 h 1 5 _ __ Exact. High Con........ Low Con. 10 10 o -10 -20.' 0 30 60 90 120 150 180 Angle (degrees) 20 (a) _e 15: 10 t- o S _hCon. C: 0 -10 ______ Exact -2 -..,Low Con.. -20 0 30 60 90 120 150 180 Angle (degrees) (b) Figure 3.4: H-polarization bistatic echowidth (q' = 0~) for a coated cylinder having R = 1A (includes coating). (a) 6, = 0, 62 = 0.06A, 63 = 0, e = 4, p = 1. (b) 6 =0, 62 =0.06A, 63=0, e=8, #= 1.

35 20 1 - Exact X- - - High Con. 10.1.~.. Low Con. 5 0 o -10 -15 -20 0 30 60 90 120 150 180 Angle (degrees) (a) 20 15 1 0 -0....... _.........ow con..2L 4~...... ~ 5(b) o- Exact 0 -- 1 0' High Con... Low Con. -20 0 3 0 60 90 120 150 180 Angle (degrees) (b) Figure 3.5: E-polarization bistatic echowidth (q' = 00) for a coated cylinder having R = 1X (includes coating). (a) 61 = 0, 62 = 0.08A, 63 = 0, c = 4, p = 1. (b) 61 = O, 62=-0.12A, 63 - 0, e - 4, p = 1.

36 0.3 0 0.4 4) magnitudg ot the retractive index able error of ~ 2 dB in the field magnitude.

37 25 Moment Method, 15 ____ Low con. -~ 5 (, a)'I (a) -5 y -25 I I 0 30 60 90 120 150 180 Angle (degrees) 25 ZS, 15 at 0 5 (b) 0 c -15 Moment Method a,_ _ __High con. -25 0 30 60 90 120 150 180 Angle (degrees) Figure3.7: H-polarization bistatic echowidth (q' = 00) for a three- layer coated cylinder having R = 1A (included coating). (a) 1 = 0.01A, 62 = 0.02A, 63 = 0.03A, c1 = 3-jO.1, e2 = 4-jO.3, e3 = 7-jl.5, pIl = 2 -= 3 = 1.2. (b) 61 = 0.01A, 62 = 0.02A, 63 = 0.03A, e = 3 - jO.1, 2 = 4 - jO.3, e3 = 7-jl.5, p1 =2 = p3 = 1.

38 25 FEM 93 15 -_ _ _ Low con..~ 15...... High con. 0 -15 - -25 - J 0 30 60 90 Angle (degrees) (a) Low con. length L5 1.32. High con. -62.06A, 62 =, 3 = 0, 8. 0 30 60 90 (b) ~2 =O.06, 63=, ~=.~ ( b,=O 2.6,6 -',C=8

39 15 cD -5 (a) -25GIBC with rounded 0........ GIBC with sharp tip -35 -45 0 30 60 90 Angle (degrees) 20 t 0 (b) -10 X -20 _F -EM O C~..,.~ " GIBC with rounded...... ~....... GIBC with sharp tip -30 -40.. I.., 0 30 60 90 Angle (degrees) Figure 3.9: H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.26A (includes coating) and tip angle a = 360. (a) 61 = 0, 2 = O.03A, 63 = 0, = 4. (b) 61 = O, 62 = 0.03A, 63 =-, E - 3-jl, = 2 - j0.5.

15,, -25 H GIBC with rounded tip o'....._.7.7.7 GIBC with sharp tip -35 0 30 80 90 Angle (degrees) r5 SP 5 _-~,, o,,, (b) ~ -15 - -25 _ FEM., ____ GIBC with rounded tip -.-....GIBC with sharp tip r -35.. -45 I I 0 30 60 90 Angle (degrees) Figure 3.10: H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.26A (includes coating) and tip angle a = 30~. (a) =1 0, ~2 = 0.03A, 63 = 0, e = 4. (b) Si = 0, S2 = 0.03A, 63 = 0, e = 3 jl,/ = 2 -jO.05.

41 FEM 15.d 5 -5 0 30 60 90 120 150 180 Angle (degrees) (b) 15 5 -25 0 30 60 90 120 150 180 Angle (degrees) (a) Figure 3.11: H-polarization backscatter echowidth from an ogival cylinder having total length L = 1.26A (includes coating) and tip angle a = 360. (a) Variable right hand side coating using a linear taper; e = 4 to 5 - 31.5 tip, 6, 62 0.03, 63 0, (E-15 ). 0~~~~~~~nl dges Fiue.1 -oaiainbcsatrehwdhfo noia yidrhvn

CHAPTER IV HIGH FREQUENCY SCATTERING BY A SMOOTH COATED CONVEX CYLINDER SIMULATED BY APPROXIMATE BOUNDARY CONDITIONS This chapter deals with the subject of non-specular scattering mechanisms arising due to creeping waves and curved surface diffraction. A rigorous UTD solution is developed here for the diffraction by a coated convex cylinder simulated with the approximate boundary conditions (SIBCs and GIBCs). In addition, a uniform asymptotic solution is obtained which remains valid when the observation point is in the close vicinity of the cylinder. One of the objectives of this exercise is to demonstrate the use of the approximate boundary conditions, presented in chapter two, for the development of analytical solutions for curved surfaces. These asymptotic solutions are then validated using the eigenfunction solution for a coated circular cylinder and using the numerical data based on the integral equation formulation developed in chapter three for a general convex cylinder. The problem of scattering by a smooth convex impedance cylinder has received much attention. Wang [34,35] presented ray-optical solutions for the impedance and coated cylinders. His results are valid only in the deep lit and shadow regions and do not apply to the case where the observation point is in the transition region. Wait 42 ~ ~ ~ ~ ~~~add

and Conda [36,37] developed a solution which is valid in the transition region and for observation points on and off the surface. However, as pointed out by Pathak [38] it does not uniformly reduce to the ray solution [39,40] exterior to the transition regions. Also, it is not valid on the portion of the surface in the transition region and these limitations were the primary motivation in Pathak's work [38] for the perfectly conducting convex cylinder. Recently, Kim and Wang [41] presented a solution applicable to a coated cylinder that remained valid in the transition region. They employed a heuristic approach to obtain the numerical values of the resulting transition integral applicable to a coated cylinder. Their solution is uniform but is not applicable in the close vicinity of the cylinder. The UTD solution to be presented here parallels that given by Pathak [38] for the circular perfectly conducting cylinder. However, in the case of the coated cylinder the resulting UTD expressions are in terms of Fock-type integrals whose efficient evaluation is of primary interest. In the following, we first present the eigenfunction solution based on the second order GIBC simulation of a circular coated cylinder. By employing Watson's transformation this is written in integral form which is then cast in a ray representation. The ray solution is subsequently generalized to the case of a general convex cylinder. Finally, the evaluation of the Fock-type integrals is discussed and some results are presented which validate the accuracy of the GIBC ray solutions. 4.1 Eigenfunction and Integral Representation Before considering the problem of scattering by a convex impedance suface, it is instructive to first develop a solution for the diffraction by the canonical geometry of a circular impedance cylinder. Generalizations to any convex surface can then be

44 made on the basis of the expressions obtained for the circular cylinder. Consider the plane wave Ui Uoeikx -= Uoe3kPos (4:1) incident on the coated circular cylinder shown in figure 4.la, where u0 is the amplitude of the wave and u' denotes either the E, or Hz component of the incident field. We propose to simulate the circular cylinder with an equivalent one (see figure 4.lb) satisfying the GIBCs. This implies that (2.11) takes the form E = (a2 + ao) _ a2Z iO2HZ E,O= - ZHa (4.2) a, k2b2a1.0$2 for Hz-incidence or =(a2 + ao) a'2 Z E, ZHO - k2b2(Ia )2 (4+3) in the case of E_-incidence. In the above, am and a' were derived in chapter two for the low and high contrast GIBCs. An eigenfunction representation of the total field in the presence of the cylinder can be generally written as 00 u= J [Jn.(kp) + A.H(2)(kp)] e-jn (4.4) n=-oo where Jn(.) is the Bessel function of order n and H(2)(.) denotes the nth order Hankel function of the second kind. To find the constants An we enforce the GIBC given by (4.2) or (4.3) at p = b. This yields A J.(kb) + Q(n)Jn(kb) (4.5) H(2)'(kb) + Q(n)Hn(2)(kb) in which the prime denotes differentiation with respect to the Bessel or Hankel func

45 Y P(PI,) x (a) Y P(P'O) (b) Figure 4.1: (a) Coated circulaz cylinder (b) equivalent coated cylinder simulated with GIBCs.

46 tion argument, a-j (a+ao - n2 a2) for Hz-incidence al k2b2al Q(n) = (4.6) - -n 2 2) for E -incidence and when a2 a2 = 0, Q(n) reduces to j o = -j77 for Hz-incidence Q(n) = Q = (4.7) -j = -j- for E.-incidence Since (4.4) is a slowly convergent series, especially for large kb, our objective is to obtain asymptotic representations for ut suitable for practical applications. We, therefore, seek uniform expressions for the total field in the lit, shadow and transition regions. Such expressions should also recover the well known geometrical optics field, where applicable. To obtain a ray representation of (4.4) in conjunction with (4.6) we must first recast ut in integral form. Employing the usual Watson's transformation [6] and noting that Q(n) = Q(-n), we find [38] Ut U1 + U2 where U1 = U0 f [J*(k) - J.'(kb) + Q(v)J,,(kb) H(2) e_,dv (4.8a) co [o]:o (p HP),(kb) + Q(v)H?2(kb) or =00 Hl/(1)(k - )'(kb) + Q(v)Hv(1)(kb) Ul Hv(1)-(kp) - H(2)(kp) e""'"dv (4.8b) 0- Hv2'(kb) + Q(v)Hv2)(kb) and 0U2 = Uof [J(k- J(kb) + Q(v)J,(kb) H(2)(kp)] eV'/2 -o fo (p)- I/HA2'(kb)+ Q(v)H/) (kb) oo[0

47 (4.9) e-jv(21r+) + e-jv(2X7r —) dv i-e-j2vr in which (4.10) and Q(v) is given by (4.6) upon letting n - v. The field component ul, defined above, includes the geometrical optics and dominant surface diffraction contributions whereas u2 denotes the creeping waves which circulate around the cylinder more than once. We are interested in an asymptotic evaluation of ul and to do this we must separately consider each of the space regions illustrated in figure 4.2. 4.2 Field in the Deep Lit and Shadow Regions In the deep lit region, the geometrical optics incident and reflected rays represent an accurate first-order high-frequency approximation of the total field. The geometrical optics field can be extracted from u1 upon performing an asymptotic evaluation of (4.8) yielding [42] UGO(PL) M Ui(PL) + Ui(QR)R,h P e (4.11) P+ ke (4.11) in which Ra a/a cos O'+a Cos2 O R a, - a+ + 2 0 (E, - incidence) (4.12a) ao - a cos + a2 os2 i(4.12b) Rh (H,,- Oincidence) (4.12b) a +- a, cos - + a2 Cos2 e' are the reflection coefficients associated with the coating [5]. The parameter Oi is illustrated in figure 4.3 and e is the distance (must be large) from the surface reflection

48 Lit Region PL point QR to.the.observ o. F y sReflected Incident (:Lit) SB Ra Rays ~........................:-....... i.i:...... Shadow Region Figure 4.2: Different regions associated with the plane wave scattering from a smooth convex cylinder. point QR to the observation point PL. Finally, p is the reflected ray caustic distance and for a convex cylinder it is given by pB(QR) cos e0 p= Pg(QR)COSOi (4.13) Pg(QR) is the radius of curvature of the surface at QR and is equal to b for the circular cylinder. For observations in the deep shadow region, a residue series representation of ul is more appropriate. From (4.8), we obtain = H()(kp)em(-) (4.14) U1 = - m= H/2) ( kb)o( [Hv2)'(kb)+ Q(v)H2)k)l (kb4.14),

49 PL, Ray Path ly Re ec,ed ui x Figure 4.3: Reflected ray path. where v,r are the zeros of the transcendental equation H,2)' (kb) + Q(Vm)H(2)(kb) = (4.15) Obviously, (4.14) does not provide a ray-picture interpretation of the creeping-wave diffraction which is desirable for generalizations to non- circular cylinders. To recast (4.14) in a form compatible with the Keller type GTD format, the Debye approximation must be employed for the Hankel function for kp > Iv, 1. This yields ul,) - ui(Q/)Ts,h - (4.16) where Ts,h = Z Dm(Qi) * Dnm(Q2) (4.17) mil

50 is the corresponding diffraction coefficient and for the circular cylinder Dm(QI) = Dm(Q2) (4.18) ei - H 4k2b m2(kb)~ {Hv2)(kb) + Q(,)Hv2)(kb)} ] with 0 as defined in figure 4.4. Y Q Figure 4.4: Propagation ray paths in the shadow region. A generalization of (4.16)-(4.18) for convex cylinders is now possible by making the replacements b pg(Q1,2) (4.19)

e-jY, exp -[j ft'(Q2) vp(t')dt] (4.20) Consequently, the attachment coefficient D,(Ql) is no longer equal to the launching coefficient Dm(Q2)4.3 Field in the Transition Region The geometrical optics and creeping wave solutions presented above are not valid in the transition region close to the shadow boundaries as illustrated in figure 4.2. New uniform expressions are, therefore, required to overcome this limitation. These can be derived following the procedure adopted by Pathak [38] for a perfectly conducting convex cylinder. For the lit region we find U1(PL) (PL) X U'(PL)+ ui(QR)Rs,h ek (4.21) where R,,h is now given by -4 1{.(x')~1 Fei 1i aR8,h=- — ~exp — j }12 [ 2z'F {1 -F(2k cos2')} + G(', q)] (4.22) F(x) = 2jV/xe-. i e e-j2dx (4.23) x'= -2m(QR) cos i' (4.24) m(QR) 2= {kP(QR) } (4.25) and Pg (QR) cos O' p = 2 (4.26) F(x) is the UTD transition function [43] and G(x', q) is defined by G(x',q) = /-.oo W(r) - q(r)W2(r) (4.27) ~~ji: J-00 "2\' 1 - Y\~' I (4.27)

52 in which q(r) = mQ(v) (4.28) whereas V(r) and W1,2(r) are the Fock-type Airy functions [44] 2jV(r) = W1(T)- W2(r) (4.29) W1,2(T) = 1 e' -t3/3dt (4.30) The contour rl runs from ooeAi 73 to oo -je. and r2 is the complex conjugate of rl. We remark that in the case of an SIBC simulation, q(r) becomes a constant. For the shadow region, we have e-jkcs ul(P,) t ui(Qo)Th,, (4.31) where the diffraction coefficient To,h is now redefined as =T -m(Q)m(Q2)/ek-jkt [2/. { - F(ksa)} + G(x q)] (4.32) in which jt(Q2) m(t') dt Jt'(0l) ps(t') t (Q2) dt' (4.34) and a (4.35) 2m(Q)m(Q2) (35) As is usually the case, (4.21)-(4.35) were first derived for the circular cylinder and were subsequently generalized for non-circular convex cylinders.

53 4.4 Field in the Close Vicinity of the Convex Cylinder In all of the above derivations we have assumed that e is large. Consequently, the given expressions are not adequate for field computations very close to the cylinder. In this case, it is possible to obtain a suitable approximation of the integral (4.8) by replacing the Hankel and Bessel functions in terms of Airy integrals and then employing a Taylor series expansions for these integrals. Following a procedure similar to the one given in [38], the resulting expressions for an arbitrary smooth convex cylinder are u1(P) Z U'(PN) [e-hz - ( 1 (jhz')n + e 3 {A(z') - A2(z') (4.36) when PN is in the lit region, and u,(P) ui(Q,)ei-kt [P9(PQ)] {AI(z)- A2(Z)} (4.37) when PN is in the shadow region. In (4.36) and (4.37) h 2h3 h4 h5 h3 h4 h5 Al(D) = gi(D) +j= g(D) - j. 91(D) - - 1'(D) -j4-!g0 (D) (4.38) A2(D) = hg2(D) + j! g2(D)- 2T! 92(D)- 92'(D) (4.39) 1 ~~ e-jDr ~91 (D)= 1 dr (4.40) V/ir JOO W2(Tr) - q( r)W2(,r ( 92(D) = - | W(-)- qdr (4.41) f2O r =o W2(r) - q(r)W2(r) z= -m(PN) cos Oi (4.42) for PN in the lit region and = t'(PN) m(t') (4.43) zt,(o,) pg(t')

54 for PN in the shadow region t (PN) t = f dt' (4.44) Jt'(Q1) kd(PN) h = k(PN) (4.45) m(PN) d(PN) = p-pg(PN) (4.46) The points P and PN in the lit and shadow regions are illustrated in figure 4.5. We remark that when the cylinder's surface obeys the SIBC, g1,2(D) simplify to Q_ (Field Point) f (Field Point) P'..'P l.I d ZP P N N (In Shadow Zone) (In Lit Zone) Figure 4.5: Projection of the field point P in the direction normal to the surface at PN.

1 d jD. gl(D) g(D) = oo W(r) - qW2(r)(4.47) g2(D) -. mQg(D) (4.48) In the case of slowly attenuating creeping and/or surface waves (4.36) is not adequate and an improved result may be obtained by adding (4.36) and (4.37) with t > irb (in case of a circular cylinder of radius b). Clearly, the addition of (4.37) corresponds to the contribution of the creeping wave that has travelled the minimum distance on the cylinder's surface to reach PN. The contribution of those creeping waves that travelled more than once around the cylinder is given by u2 and could be added to ul if greater accuracy is required. The functions gl(D), g2(D) and G(x, q) are Fock-type integrals that are formally the Fourier transform of a slowly varying factor consisting of a quotient of terms containing Airy functions and their derivatives. A computationally efficient scheme was proposed by Pearson [45] for the evaluation of these integrals. The scheme is an extension of the Fourier trapezoidal rule devised by Tuck [46] to treat the rotated-ray exponential behavior occuring in the integrals. The sampling frequency used in the computation is dictated by the slowly varying Airy-function factor in the integrands. Sufficiently accurate results have been obtained for both lossy and non-lossy coatings on a perfectly conducting cylinder using this scheme. 4.5 Numerical Results The UTD expressions derived in the previous sections provide a complete set of equations for the computation of the total field in all regions of interest. In this section, we present some calculated data which validate the accuracy of the derived expressions by comparison with data based on the moment method and eigenfunction

solutions. P(P, ) y 0 \ x Figure 4.6: Elliptical cylinder configuration. To show the validity of the UTD solution in the case of a convex cylinder, a special case of an elliptical cylinder (see figure 4.6) is considered in figure 4.7. Data based on the moment method solution are compared with the one obtained from the UTD solution in conjunction with the second order low and high contrast boundary conditions. Figure 4.8 verifies the asymptotic solution, developed for the field point in the close vicinity of a convex cylinder. We remark, however, that the approximations used for the Hankel functions in the derivation of (4.36) and (4.37) become less accurate for some values of c, and,. associated with lossless coatings, and this can be avoided by using more accurate approximations for the Hankel functions. Finally,

figure 4.9 demonstrates the use of GIBC in simulating multilayer coatings by simply redefining the material constants a, and a' as discussed in [47,48]. A difficulty in implementing the expressions derived in this work was the evaluation of the Fock-type integrals G(x, q), gl(D) and g2(D) as well as the determination of the zeros corresponding to (4.15). The Fock- type integrals were evaluated by employing the method described in [45] and the zeros of (4.15) were determined using the routine given in [49]. 4.6 Summary Rigorous ray solutions of the scattered fields were presented for a coated convex cylinder. These were developed in the context of the uniform geometrical theory of diffraction and specific expressions were given for the scattered fields in the lit, shadow and transition regions as well as for observations in the near vicinity of the cylinder. That is, UTD expressions were derived for all regions exterior to the coated cylinder. These are suited for engineering computations and are given in terms of the generalized Pekeris or Fock-type functions whose evaluation was efficiently performed via the Fourier Trapezoidal rule suggested by Pearson [45]. In comparison to the solution given by Kim and Wang [41], the ray representations given here are based on a second order generalized impedance boundary conditions which permits the simulation of thin multilayered coating as demonstrated in the included examples. Also, in our implementation of the transition fields we employed a rigorous rather than a heuristic evaluation of the Fock-type integrals. Further, we have presented accurate field representations for observations on or near the vicinity of the coated cylinder and these can also be used for computing the radiated fields by a source or an aperture on the surface of the convex cylinder.

58 0.0 P -10.0;s -20.0 -30.0 Lr Moment Method (GIBC) -......... UTD (GIBC) -40.0,. I 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle ~ (deg) (a) 10.0,.,..,..,.-.,.., 0.0 X~ -10.0 I -20.0 -30.0 -Momet Mrod (GoeC)......... UTD (GIBC) -40.0...'' I.. (.. I.. I -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle * (deg) (b) Figure 4.7: Bistatic scattering pattern for an elliptical cylinder having a = 2A, b = 1A, p = 5A, qS = O (a) E- polarization,, = 4, p, = 1, 6 = 0.07A (b) Hpolarization, e, = 8, / = 1, 6 = 0.2A.

59 10.0 0.0 -10,0 tL -30.0 -40.0 rfummw (GnIO -60.0 AsynmpodcE mnm (GIBOC) -70.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle * (deg) (a) 10.0 I 0.0 -10.0 2 1 -20.0 I'- -30.0 -50.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observation Angle (deg) (b) Figure4.8: Bistatic scattering pattern of a circular cylinder having b = 3A, p 3.05A,, = 0 (a) E-polarization, e, = 4, /, = 1, 6 = 0.07A (b) Hpolarization,, = 8, lr = 1, 6 = 0.2,A.

60 10.0 0.0 - -10.0 -20.0' -30.0 { ----- UITD (GIBC) -40.0.. 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Observaion Angle (deg) (a) bOD0....!' =, =3..., = 4 j.3, =.., 0.0.... 0.0 -10.0. -20.0 -3 0.0t 40.0 I -5.0. 0.0 30.0 6M0 90. 1200 150.0 180.0 Obsrtion Angle t (deg) (b) Figure 4.9: Bistatic scattering pattern of a three-layer coated circular cylinder having b = 3A, di = 0, e,1 = 3- j0.1, e,2 = 4- j0.3, er3 = 7- jl.5, ~,, = 2 = 1,3- = i, 6 = 0.01), 12 = 0.02A, 63 = 0.03) (a) H-polarization, p = 5A (b) E polarization, p- 3.05A.

CHAPTER V AN APPROXIMATE SOLUTION FOR SCATTERING BY AN IlMPEDANCE WEDGE AT SKEW INCIDENCE So far, in the previous chapters we developed and validated diffraction coefficients for smooth coated convex cylinders simulated with the approximate boundary conditions (SIBCs and GIBCs). These diffraction coefficients can be utilized to study the non-specular scattering mechanisms caused by creeping waves and surface diffracted waves. In this chapter we will consider other sources of non-specular electromagnetic scattering, namely those associated with diffraction from first derivative discontinuities as in the case of edges and wedges coated with lossy material. A normal incidence solution for a wedge with arbitrary included angle and having SIBCs imposed on its faces is already available [11]. However, the diffraction coefficient obtained from this solution is only applicable to two dimensional structures and for practical applications it is necessary to derive coefficients applicable to three dimensional structures. This can be accomplished by solving the impedance wedge problem with a plane wave excitation at skew angles. Unfortunately, the exact solution to this problem has only been obtained for a few wedge angles. In particular, solutions are possible only for wedges having included angles of 0 (half- plane) [12,13,14,15,19], ir/2 (with one face perfectly conducting) [14,15,16,20], xr and 37r/2 61~~~~~~~~~src

(with one face perfectly conducting) [14,15,19]. The main difficulty in obtaining the skew incidence solution for the diffraction by an impedance (SIBC) wedge is the lack of techniques to solve the resulting four coupled functional difference equations. For the specific wedges mentioned above the four difference equations can be decoupled yielding the solution given in the references but their decoupling for other wedge angles has so far eluded researchers. In this chapter, an approximate solution is developed for an impedance wedge at skew incidence using Maliuzhinets' method [11]. This solution exactly recovers the known ones for an impedance half plane and the normal incidence wedge solution. A major effort is devoted to the validation of the approximate solution and for this purpose a moment method code was developed for computing the scattering by an impedance polygonal cylinder excited at skew incidence. The formulation forming the basis of this code is deferred until chapter six. 5.1 Modified Form of the SIBCs Consider an impedance wedge geometry shown in figure 5.1 having an external angle of 2b = ni with surface impedances 7l+Z and q7_Z at 0 = + and q = -, respectively. This impedance (SIBC) wedge is illuminated by a plane wave impinging from the direction = -x sin P, cos' - sin sin O' + z cos p (5.1) where 1 4' < and ft represents the skewness angle and is equal to 7r/2 when the wave is incident normal to the wedge. The z components of the incident field are of the form Ez = ee-j't- ZH = hze.-k'' (5.2)

0=-1 Figure 5.1: Impedance wedge geometry. with Z being the intrinsic free space impedance. We observe that the total field must have the same z dependence as the incident field because the wedge is infinite along the z direction and the SIBCs are independent of z. Therefore, from Maxwell's equations E = jksin2 p cos p + 1p(ZH)} (5.3) 1 1 aE2 a(54) ZH, - jk sin2 ro jcos f i -pp(ZHz) 5p' (5.5) ZH,= jksin2 {Ezp + cos opa (ZHz)} (5.6) and these can be used to rewrite the SIBCs in a more convenient form not involving the z components of the electric and magnetic field. Beginning with the usual forms

of the SIBCs given in terms of the tangential fields, we have Ep = T:Fr+ZHz at b = ~:I (5.7) Ez = ~rk:ZHp at 0 = ~t: (5.8) Differentiating these with respect to z we obtain (tangential differentiation of the boundary conditions is allowed) a3Ep'k zH z (5.9) a fEz ~+ ZHo (5.10) oz ~ z Alternatively on multiplying (5.7) with p and then differentiating (5.7) and (5.8) with respect to p yields a a (Ep) =:rl+Za (pHz) (5.11) Op op 0E5 _ 01H (5.12) and these are of course equivalent to the previous two sets. Other alternative boundary conditions can be obtained by combining the sets (5.9)-(5.10) and (5.11)-(5.12). In particular, by dividing (5.11) with p and adding the resulting equation to (5.10) we get P1 aE- i 7 ( aH H a T ) ~ ~Z Hz (5.13) - o~ = ~ z \-p p where we have also used the divergence condition V.*+ E =0 (5.14) P O-~(E~)+ + —-=z

However, (5.13) can be simplified further upon noting that OH,z ap = jkYE (5.15) when this is introduced in (5.13) along with (5.7) we obtain the condition OE, -jrlkpE, + Ep = 0 (5.16) which is just another form of the SIBC not involving any of the z field components. Proceeding in a similar manner, we can combine (5.9) and (5.12) to obtain OH, 1 _apH_ + Hp = O (5.17) which is obviously the dual of (5.16). As we shall see in the next section, equation (5.16) and (5.17) are the most convenient forms of the SIBCs necessary for the application of Maliuzhinets' method in the case of skew incidence. 5.2 Derivation of the Functional Difference Equations To apply Maliuzhinets' method [11] for the solution of the impedance wedge diffracted fields we begin by first representing the total fields in the presence of the wedge as Ez = 2 l'j eJkPsi"ocS9(a + O4)da (5.18) jkz os (5.19) ZHz = e +2 jejkp"~c~o h(a+ b)da (5.19) These are complete spectral representations where -y is the Sommerfeld contour shown in figure 5.2 and Se,h are unknown spectral functions. The goal in the subsequent steps is the explicit determination of Se,h on the basis of the governing Maxwell's equations and the boundary conditions (5.16)-(5.17). On

66 -~r \ o 7r \ Figure 5.2: Sommerfeld contour y in the complex ca plane. enforcing the boundary conditions (5.16)- (5.17) at ~ 4 along with (5.3)-(5.6) we obtain lejkp sin p. cos asin ct +in cos sin aeSh(a, ~ I) r7. sin 8o +Cos as,(a~ + 4)} = o (5.20) ejkp sin a ~- sin co {cos rio sin aS, (ac~ 4- I) cos rSh(a ~ i )} = 0 (5.21) These are four coupled integral equations and we observe that Se,h are decoupled when cos oo, = O corresponding to the case of normal incidence. Note that the top and bottom face impedances 77~ appear in the dual factor multiplying the integrands

67 integrands be even functions of a. A mathematical statement of this is A (a) {cos aS,C(a ) 4-) + coso sinaSh(a )} = cos aS,(-Ya -~ ) - cos Io sin aSh(-a t i ) (5.22) A (a) {cos 0o sinaS,(a 4) )- cos aSh(a )} = - cos p, sin aSe(-a ~ ~) - cos aSh(-a ~ 4) (5.23) where C,h a) -sin a ~ sin Oh (5.24) with ne 77 - sin - si (5.25) Equations (5.22) and (5.23) are functional difference equations which are coupled and cannot in general be solved for Se,h(c) in a closed form. In the case of normal incidence (cos 30 = 0), the difference equations take the form A"(a)S(a ~+ f) = Se(- a ~t ) (5.26) A (a)S(a 4 (c ) = Sa(-a ~ 4) (5.27) and each of these difference equations can then be solved independently via Maliuzhinets' method. Equations (5.22) and (5.23) can also be decoupled for some special cases by employing certain suitable linear combinations of the spectral functions S, and Sh [15]. In particular, separation can be achieved for (a) a half plane ( = ir) (b) a two part plane (4 = r/2) (c) a right angled wedge with one face perfectly conducting (4 = 3ir/4 and = 7r/4).

5.3 Approximate Solution of the Difference Equations In this section, we shall consider approximate solutions of (5.22)-(5.23) for arbitrary impedance wedge angles and surface impedances 7rn. As mentioned earlier, these are coupled and cannot be separated for an arbitrary wedge angle b (4 < ir) and 3o, (0 ~< /o < 7r/2). However, approximate solutions can be obtained which will be shown to be of acceptable accuracy over a certain angular sector exterior to the wedge. In the following, three approximate solutions to the difference equations (5.22)-(5.23) are presented whose combined validity range encloses the entire region of interest. Since most wedges of practical interest have 7r/2 < ~4 < 7r, we shall restrict our study to this region. 5.3.1 Separation Method I For this separation method we shall assume that ~4 is near ir (P = ir being the half plane case) and at the same time f,3 is close to r/2 (,6 = ir/2 being the normal incidence case). The resulting solution for Se,h is expected to be most accurate for wedges having 7r and/or 8,,;r/2. We also expect that the approximation will recover the known half plane solution and that at normal incidence with arbitrary wedge angle. On our way to obtain such an approximate solution, we first divide (5.22) and (5.23) by cos a giving A(a) {cos,0 tan aSh (a ~ 4-) + S,(a f )} = - cos /o tan aSh(-a -~?) + S,(-a ~ I) (5.28) and Ah(t) {cos f0 tan aSe(a ~ ~) - Sh(a ~)} = - cos io tan aS,(-a ~ i) - Sh(-Ca i ~) (5.29)

69 We now have two options in proceeding with the decoupling of (5.28) and (5.29). IVe may, for example, opt to satisfy the boundary conditions on the upper face (q = +D) exactly and only approximately on the lower face (q = -D). Alternatively, we may instead choose to satisfy the boundary conditions on the lower face of the wedge exactly and only approximately on the upper face. Proceeding with the first option, we introduce t,(, + 4) = cos I/a tan ctSh( + b) + Se(ce + 4) (5.30) t2(a + 4i) = cos,o tan aS.(a + ) - Sh((a + 4D) (5.31) Then tl(a - 4) = cos o, tan(a - 24)Sh(a - 4) + S,(, - ( ) (5.32) t2(a - 4) = cos,o tan(a - 2)S.(a - f) - Sh(a -') (5.33) and on noting that cos p, tan(a - 2f) = cos,o tan a - cos #,I tan 2D {tan(a - 2D) tan ac + 1} (5.34) we can rewrite (5.32) and (5.33) as tl(a - ) = cos,, tan aSh(ca - ) + S,(a - 4) + bl(ca) (5.35) t2(a - $) = cos p, tan aS,(a - ) - Sh(a - 4) + b2(a) (5.36) where bl,2(ca) = - cos,3o tan 21 {tan(ca - 25) tan a + 1} Sh,e(CY - I) (5.37)

Substituting now (5.30), (5.31), (5.35) and (5.36) into (5.28) and (5.29) yields the new difference equations A+Ch(C)tl,2(a + l) = t1,2(-Ca + Q) (5.38) for the upper face and ACh(a)t1,2( - ) = t1,2(-a - ) + P1,2(a) (5.39) for the lower face of the wedge with p1,2(a) = A, h(a)bl,2(a) - b1,2(-ca) (5.40) To decouple tl and t2 on the basis of (5.39) and (5.40) it is necessary that we set p1,2(a) = 0 and this approximation affects primarily the boundary conditions imposed on the lower impedance wedge face. To observe this let us replace a with a+ in (5.38) and with a- 4 in (5.39). When the resulting equations are subtracted, we obtain Ae,h(a + )t1,2(ca + 25) - A,h(a - )tl,2(a- 2b) = fi,2(a) (5.41) where 2 cos /,sin 2~ fi,2(Ca) = -P1,2(a -2- )2 A cos 25 + cos44 cos 2a + sin 44 sin 2a e,h(a - (I) ~Sh,(a - 24) - 2 cos sin 2 Sh, (-a) (5.42) cos 2( + cos 2a For fi,2(a) = 0, the two difference equations (5.41) become linear first order homogeneous functional difference equations and can be solved via Maliuzhinets' method for t1 and t2. It is interesting to note that fi,2(a) is identically zero when cos /e sin 2D = O. Thus on setting fi,2(a) = 0, the resulting solution for tl,2(Y) is the exact one for the cases where

(a) po = ir/2 (normal incidence with arbitrary wedge angle) (b) 4 = 7r (half plane with arbitrary 1,o) (c) 4 = ir/2 (two part plane with arbitrary,3). Therefore, any solution of (5.41), based on the assumption that cos, sin 2 - 0, will exactly recover all three cases mentioned above. Once tl,2(Ca) are found from a solution of (5.41), the corresponding Se,h(C) spectra are obtained by inverting (5.30) and (5.31). First, we rewrite these as tl(a) = cos, ( Sh(oa) + Se(a-) (5.43) cos (a - ~) sin(a- ~) t2(a) = cos,O s(a _ )Se (a) - Sh(c) (5.44) cos(a -) and from these we readily find that cos (a - 4P) Se,(C)=1-sin2 (a - {cos(a - )tl(a) + cos f, sin(a -,)t2(a)} (5.45) 1 - sin 8,o sin:(a - $b) Sh (a) = 1s2 o sin2(a - ) {cos fo sin(a - ) )tl(a) - cos(a - 4)t2(a)} (5.46) ( - sin2 /o sin2(a - $) From Maliuzhinets [11], a solution of (5.41), with fi,2(a): 0, which is free of poles and zeros in the strip IRe(a)l < ( is the auxiliary Maliuzhinets' function I(a, +h, 0,h) defined by the product e(u,9i9,h) = (C + 2c,h ( 2 + ~,h -Tr 0 + (P 2 +,hpt,a) (5.47) where T1t(a) is the Maliuzhinets' function for which simple analytic expressions are given in [50]. For large jIm(a)l, this function has the property Imlir)ao I'(a)=O {expIIm)l} (5.48)

72 where 0(-) is the Landau symbol. We may thus express t1,2(a) as t1,2((a) = al,,2(a) (a, O+ e,) (5.49) e,h, e,h where al,2(a) satisfies the equation.1,2(c ~ i ) = al2(- a ) (5.50) and from (5.38) and (5.39) its presence is allowable since Ae,h(a)(a -~ =) = T(-a:: >) (5.51) In accordance with (5.50) the most general solution for U1,2(a) is al,2(a) = sinm' (5.52) where m is an integer. However, before proposing any form of al,2(a), we note certain characteristics that must be satisfied as dictated by the physics of the problem. They are: (a) al,2(ca) must have a first order pole singularity in the strip IRe(ca)l < i at a = q' to allow recovery of the incident field (b) Se,h(a) = O(constant) for large IIm(a)l (since the fields at the edge take a constant nonzero value) (c) al,2(a) should contain the pole singularities associated with the term 1/cos(a - (I) which lie in the strip IRe(a)l < 7r. This is a consequence of the solution procedure. In view of these requirements, a1,2(a) should be of the form Alsi cos A Ce,h + De,h 5,sin- sin ( sin +- sin) (sin -sin ) ( n n n n n n The coefficients A,2, Ce,h and D,h are constants to be determined and al = -!= - 3 represent the zeros of cos(a - ~) in the strip IRe(a)l < r.

The constants A1,2 are obtained by requiring that the final solution recovers the incident field. For this, it is necessary that cos( q(' - IA) e = 1 - sin2 - sin2(S' 4 ) {cos(4' - )A1 + cos f3 sin(k - ()A2} (5.54) cos(' - 4D) hz = sin2 2 lsin(( -) {cos o, sin(k' - -)Aj - cos(k' - ) A2} (5.55) h 1 - sin2/fo sin2(' - (I) giving A1 = cos i, tan(q' - 4)h, + ez (5.56) A2 = cos 3o tan(q' - -)ez - h, (5.57) To determine the constants Ce,h and De,h, we note that the expressions for Se,h given in (5.45) and (5.46) have four undesirable poles, all in the strip IRe(ca)l < r. These are associated with the multiplying factor {1 - sin2 A/ sin2(a - (I)}-l and are given by 61,2,= (- ) + o a, =o (2r/2 _< < r) (5.58) with a0 =jln (tan ) (5.59) so that cos a0 = csc,o and sin a0 = -j cot,3o. Each one of these poles corresponds to an inhomogeneous plane wave which grows exponentially with increasing kp in some region of space and it is thus necessary to remove these poles by choosing the constants Ce,h and De,h such that the numerator of Se,h goes to zero at these poles. From (5.45), the conditions, which must be satisfied to eliminate these undesired poles are jrIt ( —2 00) +.t2 7(1'a ) = 0 (5.60)

jt (j - 2+ aou) + t2 3(q- 2~a=)0o = (5.61) which are four equations sufficient for determining the constants Ce,h and De,h. A set of conditions, identical to (5.60)-(5.61), can also be obtained by similar arguments imposed on (5.46). Thus cancellation of the undesired poles for S(ca) is carried over to Sh (C). To obtain the z components of the far zone diffracted field, we first deform the Sommerfeld contour shown in figure 5.2 into a steepest descent path. A non-uniform evaluation of the integrals via the steepest descent method then yields -jkp ainpo e-j E e2rkpsine {Se(q - 7r) - Se(q + ir)} (5.62) V2'kp sin $o zHZ.2'kp sin Bo, {Sh(S - 7r) - Sh(q + 7r)} (5.63) V12irkp sin Sh5 We note that in the far zone, the above z components of the diffracted field are proportional to the corresponding lo, and q components via the relations E - sin loE,o ZH = - sin 13oE 1 (5.64) Similarly, for the incident fields E= sin flOEo ZHZ = sin l,oEO (5.65) The solution of the difference equations (5.28) and (5.29) as presented above recovers the exact solutions for the three special cases, namely those corresponding to, normal incidence (o = 7r/2) with arbitrary wedge angle, a half plane (arbitrary,, 4D = 7r) and a two part plane (arbitrary,, 4 = 7r/2). An important feature of this solution, lacked by other available solutions [13,14,15,19], is that a single computer code can be employed to compute the scattered field for all three special

cases mentioned above. However, our primary interest is to examine the accuracy of the solution for other wedge angles. We expect, of course, the solution to deteriorate as the wedge angle and/or the skewness angle /3 move away from the values corresponding to the special cases. This is illustrated in figures 5.3-5.6 where far zone backscatter (q = Y') patterns are given as a function of the observation angle X and for different wedge angles. Figures 5.3 and 5.4 show the patterns for internal wedge angles of 00, 100, 200 and 300 with 3, kept at 300. In figures 5.5 and 5.6 i3 is varied from 900 to 400 while the internal wedge angle is kept constant at 300. From the plots in figures 5.4(b) and 5.6(b) we observe that the pattern deteriorates in the region beyond the reflection boundary of the upper face or in other words when the lower face becomes visible. This is concluded from the appearance of a false pole and a false null in the pattern not corresponding to any physical characteristic. Such a behavior is not unexpected because as noted earlier our solution emphasized the upper face boundary condition whereas the boundary condition for the lower face was only approximately satisfied since we arbitrarily had set fi,2(a) = 0. Next we introduce new linear combinations t1 and t2 to separate the difference equations so that the lower face boundary conditions are satisfied exactly whereas the upper face boundary conditions are only approximately satisfied. The derived solution should be more accurate for observations near the lower face of the wedge. 5.3.2 Separation Method II To separate the equations for the lower face (+ = -4) of the impedance wedge we introduce the linear combinations t1(a -') = cos PotanaCSh(C - b) + SC(a-r 4) (5.66)

76 20.0............... 10.0 0.0 -10.0 -20.0 (a) >c -30.0 -40.0 -50.0 -60.0 -70.0............... -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0. 10.0 0.0 -10.0 -20.0 (b). -30.0. -40.0 -50.0 -60.0 -70.0............. -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.3: Far zone backscatter pattern for an impedance wedge having + = 1.0jl.0, 71- = 0.5- j0.1, /o = 30~, Eaoap, (a) internal wedge angle = 0~, (b) internal wedge angle = 10~.

77 20.0..................... 10.0 0.0 -10.0 2~4 -20.0 (a) o -30.0 -40.0 -50.0 - -60.0 - -70.0,........ -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 10.0 0.0 -10.0 2 -20.0 (b) -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 (deg) Figure 5.4: Far zone backscatter pattern for an impedance wedge having r+ = 1.0 - jl.0, r- = 0.5 - j0.1, /3 = 300, Epoyo, (a) internal wedge angle = 200, (b) internal wedge angle = 300.

78 20.0...........J......._.... 10.0 0.0 -10.0 -20.0 -30.0 -40.0 -50.0 -60.0 -70.0..... -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0.............. 10.0 0.0 -10.0 -20.0 (b) cc -30.0.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.5: Far zone backscatter pattern for an impedance wedge having + = 1.0 - jl.0, r/_ = 0.5 - j0.1, internal wedge angle = 300, Eo,30, (a) /30 = 900, (b) Io = 700.

79 20.0 10.0 0.0 -10.0 -20.0 (a) m -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 0 (deg) 20.0. 10.0 0.0. -10.0 -20.0 (b) c -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.6: Far zone backscatter pattern for an impedance wedge having rt+ = 1.0 - /1.0, r/_ = 0.5 - j0.1, internal wedge angle = 300, Eo,3,, (a) /3o = 500, (b) /o = 400.

t2(a- ~) = cos /o tan aS,(a - I,) - Sh(a -) (5.67) This implies that tl(ca + ~) = cos/,o tan(ca + 2(D)Sh(ac + b) + S,(ca + () (5.68) t2(c + 4) = cos /, tan(ca + 2M)S,(c + D) - Sh(a + () (5.69) and on following the procedure outlined in the previous section we get the difference equations A+,h(a + O)tl,2(a + 2) - A,(a! )tl,-(a- 2) = f,2(a) (570) where fl,2(a) is now given by ________2 cos rosin 2~ + fl,2(a) = PI,2 (a + 2's) =A+.f(a) = p2(a +=cos 2- + cos 40 cos 2a + sin 40 sin 2a h( + ~Sh,e(a + 24D) - cos 2 Sh,,e(-a) (5.71) cos 20 + cos 2a Again we note that fi,2(Ca) is identically zero when cos /, sin 2- = O. This implies that on setting fi,2(a) = 0 the resulting solution for tl,2(a) is the exact one for the cases where (a) /p = ir/2 (Normal incidence with arbitrary wedge angle) (b) - = 7r (Half plane with arbitrary,o,) (c) = ir/2 (two part plane with arbitrary,/o). The new expressions for Se(a) and Sh(a) are Se(a) = cos(a + ) {cos(C + ()tl(a) + cos Po sin(a + b)t2(a)} (5.72) S1 -) = 1 2, - sin sin2(a + q) cosi(a + (I)) {cos fo sin(ca + ~)tl(ca) - cos(a + (I)t2(ca)) (5.73) Sha = - sin2/~o sin2(a~ + (~)

and the solutions for t1,2(a) are again of the same form as in (5.49). The expressions for o1,2(a) will also be the same as given in (5.53) except for the replacements a1 - -ca and Ca2 -- -a2. On imposing the requirement for the incident field we get A1 = cos /30 tan(' + ())h + e, (5.74) A2 = cos o tan(+' + 4)e, - h, (5.75) and the constants Ce,h and De,h can be found by cancelling the undesired poles of S,,h. These are now associated with the term (1- sin2 o, sin2(a + -)}' and for IRe(a)l < ir are given by 1,2 =.(- ) 4- ( o 63,4 = - -) r to (7/2 < < 71r) (5.76) where ca, is the same as before. From (5.72), the conditions which must be satisfied to eliminate the undesired poles are jtl (- a0)o +t2 -( -ao) =0 (5.77) ijt, (3 - a o) + t2 3(3 _ 2 — to ) =0 (5.78) leading to a 4 x 4 matrix for the solution of the constants Ce,h and De,h. A nonuniform steepest descent path evaluation of (5.18) and (5.19) will, then, yield the far zone diffracted field as given in (5.62)-(5.63) with the new expressions for Se,h as given in (5.72)- (5.73). Figures 5.7 and 5.8 show the far zone backscatter (q = q') patterns of an impedance wedge for internal wedge angles of 00, 100, 200 and 300 with Po kept at 30o. In figures 5.9 and 5.10, i is varied from 90~ to 40, while the internal wedge angle is kept constant at 30~. Again, we note that the accuracy of the patterns

worsens as we move away from the special cases of a half plane (internal wedge angle=00) or away from normal incidence wedge (Pfl = r/2). When the internal wedge angle is not zero, close inspection of figures 5.8(b) and 5.10(b) reveals that the pattern deteriorates in the region beyond the reflection boundary of the lower face or in other words when the upper face becomes visible. This is concluded from the appearance of a false pole and a false null in the pattern not corresponding to any physical characteristics. Such behavior is not unexpected because our solution emphasized the lower face boundary condition whereas the boundary condition for the upper face was only approximately satisfied. The pattern is reasonable as long as the observation point remains near the lower face. 5.3.3 Separation Method III We next turn our attention to the case when both faces of the wedge are visible. In order to obtain a solution of acceptable accuracy in this region, the difference equations associated with the upper and lower faces of the wedge have to be decoupled without introducing approximations to the implied boundary conditions. This can only be done if all trignometric terms in Se,h(a) have a period of 2C. However, because of the presence of sin a and cos a this is not possible unless they are replaced by an expansion having that property. Suitable candidates for functions of this expansion are sin -, cos -, sinm -i-, cosm, (5.79) where m is an integer. From equations (5.62) and (5.63), we observe that in the final evaluation of the far zone diffracted field, a eventually takes on the real values of k -r. Also, to determine the constants Al,2 we substitute -' (a real valued variable) in place of ca. This suggests that we may treat a as a real variable and represent

83 20.0...... I... 10.0 0.0 -10.0 -20.0 -30.0 (a) -40.0 -50.0 -60.0 -70.0...... -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0......... l 10.0 0.0 -10.0 -20.0 (b) = -30.0. -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 (deg) Figure 5.7: Far zone backscatter pattern for an impedance wedge having t+ = 1.0 - jl.0, TL_ = 0.5 - j0.1, /3 = 300, Ep0of, (a) internal wedge angle = 0~, (b) internal wedge angle = 100.

84 20.0......................., 10.0 0.0 -10.0 -20.0 (a): -30.0.~ -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 - 10.0 -20.0 (b) C -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 (deg) Figure 5.8: Far zone backscatter pattern for an impedance wedge having r+ = 1.0 - jl.O, r_ = 0.5 - jO.1, ri, = 300, E.,,,, (a) internal wedge angle = 200, (b) internal wedge angle = 300.

85 20.0..... 10.0 0.0 -10.0 - -20.0 - (a) - 30.0 - -40.0 -50.0 -60.0 -70.0... -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 10.0 0.0 j,.O, -10.0 -20.0 (b) -30.0 -40.0 -50.0 -60.0 -70.0I -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.9: Far zone backscatter pattern for an impedance wedge having r+ = 1.0 - jl.0, r/_ = 0.5 - jO.1, internal wedge angle = 30~, Eo,po, (a) /3o = 900, (b)~ o= 700.

86 20.0.,,... 10.0 0.0 -0 -10.0 -20.0 (-30.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0....... 10.0 O 0.0-10.0 -20.0 ~~((b), -30.0-40.0 -50.0 -60.0 -70.0................ -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.10: Far zone backscatter pattern for an impedance wedge having r/+ = 1.0- 0jl.0, r/ = 0.5 - j0.1, internal wedge angle = 300, Eo,01, (a) /o = 500, ~0, (b) po = 40~.

87 sin a and cos a in terms of a Fourier series given by cos a = ao + Z amcos (I (5.80) m=1 sin = b,, sin mr (5.81) m=1l where a = 2 coada, a. = cosacos m da (5.82) and 1 i mTra bm = sinasin, da (5.83) The above series representations will approach sin a and cos a, inside the real interval [-4, ~], depending on how many terms in the series are used for the approximation. Obviously, the expansion will deviate substantially from the actual values of sin ac and cos a outside this interval. Since k Ir (with -4 < q < 4) replaces a in the evaluation of the far zone field, we would like the expansions (5.80)-(5.81) to remain accurate inside the intervals [-4 - 7r, T - r] and [-4 + Ir, b + r]. From the above we conclude that this is not the case, and consequently the Fourier series expansion is not a suitable choice. Another possibility for representing sin a and cos a by a function of period 2D is to use a few terms of the expansion with am, bm other than those given by (5.82) and (5.83). In particular, we may choose the beginning terms of the series, namely sin and cos (where we have chosen b1 = 1,a, = 0 and al = 1 for the time being), to approximate sin a and cos a, respectively. To establish a measure of this approximation in representing sinca and cos c for other T, we refer to the plots of sinff- and cos 2 in figure 5.11. Clearly, for l = X these approximations precisely

88 recover sin ca and cos a. Introducing this approximation to replace sin a and cos a where they appear in Se,h(a) and in equations (5.22) and (5.23), we obtain Ae(a) {cos Sc(a ~ ) + cos po sin - Sh + ~) -- rac 7era cos T-S(-a -~ ) - cos 3o sin -Sh(-a ~ 4) (5.84) A (a) {cosf, Se (it ) - co, q Sh(a ~ ~) = ira ira - cos,o sin -- S,(-a - ) - cos — Sh(-a + S a ) (5.85) where A,h(ac) is the same as defined in (5.24). To separate the above equations, we introduce the linear combinations ira ira tl(C + <) = cos/,o sin TSh(a +') + cos Sre(a + ) (5.86) t2(a + 9) = cos,o sin - Se(a + D) - cos -C-Sh(a + (I) (5.87) which also imply that tl( -) = cos fi sin -q Sh(a - ID) + cos - Se(a - 4) (5.88) t2(ca -) = cos, sin -S(a - ob) - cos -Sh (a - ) (5.89) Introducing these expressions in (5.84)-(5.85), we obtain A+h(Ca)t1,2(a. + I) = tl,2(-a + P) (5.90) Ae,(aC)t1,2(a - C ) = tl,2(-a - ) (5.91) which can be readily decoupled for a solution of tl,2(a). In particular, on replacing a with a + 4 in (5.90) and with a - ~ in (5.91) and then subtracting (5.91) from (5.90) we get A+h(a + 9)tl,2(a + 2~) - A,h(a - )t,2(a - 2) = 0 (5.92)

89 1.5 1.0 0.5' (a) o.o -0.5 U I ~'4' I. r 1.5 1.0 0.5,. i1.!........................ E: e,r b. I ~~~~-j,, r 01.0. -i~~~~~~~~~~~~~~~~~~~.' -360.0 -270.0 -180.0 -90.0 0.0 90.0 180.0 270.0 360.0 ca (deg) 2=360 deg......... 20=340 deg.-.-.. 20=330 deg...... 242=300 deg..... 20=270 deg Figure 5.11: A plot of()si~,(b o of (a) s~in "~'r, (b osT

90 This is a difference equation identical to (5.41) with fi,2(a) = 0 and can be solved via Maliuzhinets' method for all wedge angles. We observe from (5.84) and (5.85) that the solution of (5.92) for an arbitrary wedge angle will recover the exact solutions for the cases when (a) /o = vr/2 (Normal incidence with arbitrary wedge angle) and (b) 4 = 7r (Half plane with arbitrary,o). Once tl,2(ca) are determined, the spectra Se,h(a) are found from the relations =1 -CO 2c _ jCos - (+ as, sin )2 1 - sin2 p0sin cossin t( (1 1 i ra ira A Sh(a) 2 2.ra coso sin — t (a) - cos-t2() (5.94) 1 - sin2 osinCOS (5.94) and these were derived by using the expressions t,(a) = - cos 10 sin - -Sh(C) - cos o-S-(a) (5.95) Ira ira t2(a) = -cos fo sin - -S,((a) + cos -- Sh(CI) (5.96) which follow from (5.86)-(5.89). Let us now solve (5.92) for tl,2(a). Following the procedure outlined before, tl,2(a) can be written as in (5.49) with u,2(a) changed to!cos'a n n The above expression for rl,2(a) is chosen to satisfy the conditions (a) Ca,2(a) must have a first order pole singularity in the strip IRe(a)ld < at a = q' to allow recovery of the incident field (b) S,,h(a) = O(constant) for large IIm(c)l (since the fields at the edge take

a constant nonzero value). The constants A1,2 are again computed so that the residue of the pole at a = q' yields the incident field. We have A1 =-cos/ o sin -cos (5.98) A2 =-coso #sin r ez + cos hz (5.99) From (5.93) and (5.94), the undesired poles of Se,h in the strip IRe(a)l < r are given by =1,2 = aCt, iti, 3,4 = -Ca, ~ jai (5.100) where a, =, ai= nIn tan ) (5.101) The constants Ce,h and De,h are determined in a similar manner as described earlier to cancel these poles. The specification of t1,2(a) and Se,h(a) is now complete, and the far zone diffracted field can be calculated from the non-uniform steepest descent path evaluation of (5.18) and (5.19). This far zone diffracted field, with q = k', is plotted in figures (5.12) and (5.13) for a half plane (internal wedge angle=0~) and for a wedge (internal wedge angle=30~). From figure (5.12), it is interesting to see that there is no deterioration (i.e. no false peaks or nulls) in the pattern for the region between the two reflection shadow boundaries. However, in figure (5.13), the comparison of the cross-polarization patterns of the wedge with that of the half plane reveals that in the case of the wedge there are two false poles appearing at b = I'T (reflection shadow boundaries). To resolve this problem, we note that for a half plane (f = 7r)

92 20.0..... 10.0 0.0 -10.0 -20.0 (a) -0 -30.0 -40.0 -50.0 -60.0 -70.0. -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 10.0 0.0 r -1.zo -. (b) i -30.0 -40.0 -50.0 -60.0 -70.0A -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.12: Far zone backscatter pattern for an impedance wedge having % = 1.0jl.0, rI- = 0.5 - j0.1, fo = 300, EBo~,, (a) internal wedge angle = 00, (b) internal wedge angle = 300.

93 20.0 -- i.............. 10.0 0.0 -10.0 -20.0 (a) -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0. 10.0 -10.0 -20.0 (b) 5c -30.0' -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.13: Far zone backscatter pattern for an impedance wedge having r+ = 1.0jl.0, _- = 0.5- j0.1, /o = 30~, E,m, (a) internal wedge angle = 00, (b) internal wedge angle = 30~.

94 sin =0, at a = -(-, 0, TX (5.102) and cos = 0, at a = — +, - - (5.103) 2 2 which ensures that the poles in question do cancel out. However, for an arbitrary wedge angle ~ i7r, (5.103) is no longer true. This suggests that we should replace the undesired poles just as it was done in the case of a half plane. We note that this modification still permits our solution to reduce to the exact ones as discussed above. In addition to this, the inspection of (5.84) and (5.85), yields that the normal incidence (p#, 2= ) wedge solution can still be recovered. Replacing cos by cos + cos in (5.93)-(5.96), we obtain that *{ (cos r + cos )ol(s)+cos/ sin t2(t)} (5.104) cos2,o sin2 + (cos + cos { cos o sin tl( (a)- (cos + cos tsin 2(a) (5.105) These expressions imply that A, = -cos,fsin - hz - + cos - e, (5.106) A2 =- cos/o, sin e+ os + cos -hz (5.107) and we observe that the incident field is recovered as required.

95 The poles of Se,h to be cancelled are now associated with the multiplying factor {cos2 0sin2 zr + (Cos X + cos ) } and for IRe(a)I < r, these are =1,2 = r ~ jcri, 3,4 = - ia~ jai (5.108) where a = cos-i i = n In tan (5.109) r2 sinfl o 2' provided i, < 1. When j|. > 1 the poles are given by =1,2 = jai,, 3,4 = ~j ai2 (5.110) with ai,2 = 2 ln (x1,2 + /,2 1) (5.111) where - _:_ cosfo I Xl,2 n= - c: 2..os2 - - sin2 i3 (5.112) sin /2 o sin 2 jo n The constants Ce,h and De,,h can now be determined in a similar fashion as described earlier to cancel these poles. Figure 5.14 shows the far zone backscatter (4 = 4') patterns of an impedance wedge having an included angle of 300. It is clear from figure 5.14 (b) that the replacement of cos, by cos!" + cos n takes care of the false poles. So far, we have developed three different solutions of the difference equations represented by (5.22) and (5.23). The first and the second solutions were obtained by separating the difference equations associated with the upper (4 = +P) and the lower (q = -A) faces of the impedance wedge, respectively. The third solution was constructed by approximating the trignometric functions sin a and cos a terms with

96 20.0............. 10.0 0.0'~ -10.0 -20.0 (a) -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 10.0 - 0.0 -10.0 -20.0 (b) co -30.0.~ -40.0 -50.0 -60.0 -70.0..... -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.14: Far zone backscatter pattern for an impedance wedge having Lr+ = 1.0jl.0, r7_ = 0.5- jO.1, /o = 30~, internal wedge angle = 30~, (a) EOo0p, (b) E;.

functions which were periodic in 2(I. This led to the separation of the difference equations associated with both faces of the impedance wedge. We note that all three solutions recover the half plane (D = ir, with arbitrary P3,) and the normal incidence (/3 = ir/2, with arbitrary wedge angle) cases exactly. These three solutions can now be employed to compute the far zone diffracted field with each solution to be used in the angular region where it is expected to do better than the others. In particular, the first solution should be employed when the observation angle q is in the region starting from the upper face q = +D and ending at the shadow boundary (incident or reflection) reached first from the upper face. The second solution should be employed when the observation angle f is in the region starting from the lower face S = -<X and ending at the shadow boundary (incident or reflection) reached first from the lower face. The third solution should be used in the region where the observation angle q is between the two shadow boundaries (both reflection shadow boundaries or one incident and one reflection shadow boundary) of the wedge. Figure 5.15 shows the regions where the three solutions should be applied. Using this criterion, the far zone backscatter (- = 4') patterns of an impedance wedge are plotted in figure 5.16 for internal wedge angles of 200 and 300 while /3 is kept constant at 300. Figure 5.17 shows similar plots for two other values of f, (namely 50~ and 400 ) with the internal wedge angle kept at 30~. 5.3.4 Test Results The presented approximate solutions for an impedance wedge are expected to be accurate when the internal wedge angle of the wedge is close to zero (a half plane) and/or when the skewness angle Po is near 900 (normal incidence). However, it is interesting to compare the results obtained on the basis of these approximate wedge

98 Reflection S.B. Method I (a) Method III Method II Reflection S.B. Method II Method 11 Incident S.B. (b) Reflection S.B. Method I Method III Method II Reflection S.B. Figure5.15: Different regions for the applicability of the three solutions. (a) backscatter case (b) bistatic case.

99 20.0........I 10.0 0.0 -10.0 -20.0 (a) C -30.0 C -40.0 -50.0 -60.0 -70.0............. -180.0 -90.0 0.0 90.0 180.0 * (deg) 20...... 10.0 0.0 -10.0 -20.0 (b) -0 -30.0 -40.0 -50.0 -60.0 -70.0............. -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure 5.16: Far zone backscatter pattern for an impedance wedge having %I+ = 1.0j1.0, r/ = 0.5 - j0.1, 3o = 300, Epo~,, (a) internal wedge angle = 200, (b) internal wedge angle = 300.

100 20.0...... 10.0 0.0 -10.0 -20.0 (a) m -30.0 -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 10.0 0.0 -10.0 E: -20.0 -.(b) c -30.0 ( b)' -40.0 -50.0 -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 (deg) Figure 5.17: Far zone backscatter pattern for an impedance wedge having + = 1.0jl.0, r/_ = 0.5- j0.1, internal wedge angle = 300, Eooo;, (a) 3o = 500, (b) Io = 400.

101 solutions with the known exact solution for a right-angled wedge (internal wedge angle = 900) having one face perfectly conducting. This is the only known solution not recovered by the given approximate ones. The accuracy of the approximate solutions is obviously expected to be low in this case because the internal wedge angle of 90~ is quite far from the special cases recovered by the approximate solutions. Figures 5.18-5.21 show backscatter patterns for both like- and cross- polarizations with 3,o having values of 600 and 30~. We observe that the approximate solution is more accurate for fo = 600 than it is for P3 = 300. This is understandable because Po = 600 is closer to 0 =3 90~, in which case the approximate solution recovers the exact one. We remark that the presented cross-polarization plots were computed everywhere by the approximate solution of method III since the other two approximate solutions become highly inaccurate for this wedge angle. 5.3.5 Summary In this chapter, we considered the skew incidence scattering from an impedance wedge having arbitrary included angle. Application of the boundary conditions (SIBCs) on the wedge faces resulted in a set of four coupled functional difference equations. Three different approximate solutions were presented to solve these difference equations. The first and the second solutions were obtained by separating the difference equations associated with the upper (q = +4) and the lower ( = -f) faces of the impedance wedge, respectively. These solutions were found to recover the exact ones for the three special cases, namely those corresponding to, normal incidence (fl, = 900) with arbitrary wedge angle, a half plane (arbitrary f,, internal wedge angle = 00) and a two part plane (arbitrary P,, internal wedge angle = 1800). An important feature of these two solutions, lacked by others [13,14,15,19], is that

102 a single computer code can be employed to compute the scattered field for all three special cases mentioned above. The third solution was constructed by approximating sin a and cos a, appearing in the difference equations, with functions having a period of 24. This solution reduces to the exact half plane (arbitrary P,, internal wedge angle = 00) solution and the one for normal incidence (o, = 900). A scheme was suggested to use all three approximate solutions in their respective regions of accuracy. Finally, the results from the approximate solutions were compared with those obtained from the exact solution of a right-angled wedge.

103 20.0 10.0 0.0 -10.0.g -20.0 (a) e, -30.0 -40.0 -50.0 Approx. Solution......... Exact Solution -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0.... 10.0 0.0 -10.0'L -20.0 (b) -40.0 -50.0' Approx. Solution......... Exact Solution -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure5.18: Far zone backscatter pattern for a right-angled wedge having,+ 1.0 -jl.0, _-0 =, po = 600, (a) E,soso (b) E+#.

104 20.0...... _. 10.0 0.0 -10.0 a3 -20.0 (a) 300 -40.0 -50.0 Approx. Solution........ Exact Solution -60.0o -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0 10.0 - 0.0 - -10.0 -20.0 (b -30.0 -- -40.0 Approx. Solution -50.0 - Exact Solution -60.O -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure5.19: Far zone backscatter pattern for a right-angled wedge having r/+ = 1.0 - j1.0, r/_ = 0, /o = 300, (a) EOoO-, (b) E.~9

105 20.0......1...... 10.0 0.0 XdJ -10.0 -20.0 (a) -30.0 - -40.0 -50.0 / APApproL Solution......... Exact Solution l -60.0 -70.0........ -180.0 -90.0 0.0 90.0 180.0 0 (deg) 20.0 10.0 - 0.0 -10.0 -20.0 (b) ~ -30. - 7. -40.0 -50.0 C Approx. Solution.........Exact Solution -60.0 -70.0 -180.0 -90.0 0.0 90.0 180.0 * (deg) Figure5.20: Far zone backscatter pattern for a right-angled wedge having (+ = 1.0 - jl.0, In_ = 0, p, = 600, (a) E+pt, (b) Epo+v

106 20.0.........iWTW......._..' 10.0 0.0 3X, -10.0 -20.0 - (a) - 300 - "'C \-40.0 -50.0 C I Approx. Solution........ Exact Solution -60.0 -70.0I -180.0 -90.0 0.0 90.0 180.0 * (deg) 20.0.. 10.0 0.0 -10.0 -20.0 (b)...... -30.0'~ -40.0 -50.0 Appox. Solution Exact Solution -1870.0 -180.0 -90.0 0.0 90.0 180.0 O (deg) Figure5.21: Far zone backscatter pattern for a right-angled wedge having 3+ = 1.0 - j1.0, II_ = O, /o = 300, (a) E+o, (b) EPOgi

CHAPTER VI SKEWVV INCIDENCE SCATTERING FROM\ A COATED CYLINDER SIMULATED BY SIBC The skew incidence dyadic diffraction coefficient developed in the previous chapter is an approximate one and it is necessary to establish the extent of its accuracy. This is achieved in this chapter by comparison with numerical data. These are obtained from a moment method solution of a coupled set of integral equations for skew incidence on an impedance cylinder of arbitrary shape. The surface currents obtained from the solution of the coupled integral equations are subsequently integrated to yield the scattered field on the diffraction cone. A corresponding first order high frequency solution for the far zone scattered field is then generated using the approximate diffraction coefficient and compared with the moment method data. 6.1 Integral Equation Formulation Consider a closed cylindrical surface of infinite extent along the z-axis and having a constant cross section in the xy plane. Let C denote the contour enclosing the cylinder's cross section in the xy plane and n' to be the outward normal to C. Also, we define t = z x n, so that (t, n, z) form a right handed rectilinear system at each contour location as shown in figure 6.1. 107

108 A A/ p\l t C A Figure 6.1: Geometry of a cylinder of arbitrary cross section. Since the impedance cylinder is infinite along the z-direction, the scattered field will have the same z-dependence as the incident field. We assume the incident field to be the plane wave r — = o ejk(x cos' sin l+y sin O'sin #~-z cos l~) (6.1a) =; Zs' x i (6.1b) where St denotes the unit vector along the direction of incidence. By invoking the equivalence principle, the scattered field from the impedance cylinder can then be given as the radiation of the surface electric (J) and magnetic (M) currents J(t) = n x H_ = tJ(t) + iH(t) (6.2)

109 M(t) = x = tTi(t) + MZ(t) (6.3) in which (E, H7) denote the total electric and magnetic fields in C and t is a measure of the arc length along C. From (6.2) and (6.3), it follows that Jt = H,, J = -H (6.4) Mt = -E,, Mz = Et (6.5) and when (6.4) and (6.5) are coupled with the SIBCs (see (2.1)-(2.3)) Et = rZH,, Ez = -rIZHt (6.6) to be enforced on C, they yield Mt = -ZJ,, M = rZJt (6.7) The scattered fields can in general be written in terms of the vector potentials as E = -V x F-jkZA + -ki,VV. A (6.8) 1H = V x A-jkYF + kZ VV F (6.9) jkZ By using (6.7), we have A =- J {iJt(t') + zJ (t')} H2) (kr sin,3O)ejkzcos Idt' (6.10) C 4J {iJz(t') - 2Jt(t')} 7(t')H,, (kr sin pS)e-Jkz cs~3odt' (6.11) in which i' denotes the unit tangent to C at the integration point, H,2)(.) is the zeroth order Hankel function of the second kind and r= IP- P'1 (6.12)

110 where p and p' denote the projection of the observation and integration points in the xy plane, respectively. Expanding (6.8) we find that the z-component of the scattered electric field is given by YE 4 Jk.sin2 /I / d(t,)Ho2)(kr sin 8o)dt' c j/ (('4 ~ r)H(2)(kr sin P3o)dt' c jk sin /3 cos /o j JP,(t') {(i t)(i. * ) +(An')( i)) HO2)(kr sin Po)dt' (6.13) where H2)(.) denotes the first order Hankel function of the second kind, r= (P ) (6.14) and the factor containing the z-dependence has been suppressed. A similar expression for the scattered magnetic field is ksin2 r(t')J(t')H2)(kr sin /3o)dt' c jk sinflo j - 4 JJt,(t')("'. i)H~2)(kr sin #P)dt' c +jk sin Po cos P.o j (t')J,(t') {(i i')(t. ) 4 +(n.')(n * i)} H2)(kr sin /3)dt' (6.15) where n' = t' x Z and we have again suppressed the e-jkzCOS0o factor. A solution for Jt and J, can now be obtained by constructing a set of coupled integral equations to be solved numerically. A usual procedure in obtaining such a set is to enforce the boundary conditions (6.6) on C. Expressing the total field as a sum of the incident and scattered fields, in view of (6.4), (6.5) and (6.7) we have E' + Ez = t7ZJ, (6.16)

111 and H' + Hz = Jt (6.17) Substitution of (6.1)(without the z dependence),(6.13) and (6.15) into (6.16)- -(6.17) now yields a suitable set of integral equations, for the solution of Jz and Jt on C, as YE~ = r(t)J,(t)k sin' j E k tin 4 J (t)H' 2)(kr sin Po)dt' + jk sin / j71 (t')Jz(t')(H' *')H(2)(krsin,30)dt' C jk sin Po cos po JJt(t') {(tt')(t- ) 4 +(n n')(n r)} H(2)(kr sin lo)dt' (6.18) H = Jt)+k sitn2o J r (t')Jt(t,)Ho(2)(kr sin do)dt' c jk sin f J(tP)('. ))dt' C jk sin P0 cos o j tl(t')Jo(t') {(i.')(i. ) r 4c +(n. i')(i. r)} H(2)(kr sin #o)dt' (6.19) To discretize the integral equations (6.18) and (6.19) it is, of course, necessary to discretize and evaluate the line integrals appearing in them. By employing a pulse basis expansion for J, and Jt, a matrix system is generated whose impedance matrix elements can be evaluated numerically or analytically in terms of the functions defined in [32]. Upon evaluation of the surface currents J,(t) and Jt(t), the far zone (p - oo) fields at (p, ) can be obtained from (6.13) and (6.15) by letting t = -, n i, r p, and making use of the large argument expansion for the Hankel functions. It is also to be noted that in the far zone, the above z- components of the scattered field are proportional to the corresponding f, and q components via the

112 relations E= - sin 03oE;o, ZHS = - sin 3oE; (6.20) Ez = sin to,E~,, ZHz = sin P3,E, (6.21) 6.2 First Order High-Frequency Far Zone Field To validate the approximate skew incidence diffraction coefficient developed in chapter five, we need to compute a first order high-frequency (GTD) far zone scattered field from a triangular impedance cylinder shown in figure 6.2. For this geometry, the first order high-frequency far zone field is represented by F\ir Figure 6.2: Geometry of a triangular impedance cylinder.

113 3 e-jksp _ E C EZ EDp *E (6.22) p=l where =: =/~o01o~DPo' + 30 IDo,0, + 1/3oD+, + 6'Do, (6.23) is the skew incidence dyadic diffraction coefficient for an impedance wedge, s is the distance between the diffraction and observation points and the subscript p simply denotes that the parameters are measured with respect to the pth edge of the triangular cylinder. 6.3 Test Results To demonstrate the validity of the diffraction coefficient, we refer to figures 6.36.13 where the echowidth = lim 2rs =.1 (6.24) &-~~O Jr1 12 is plotted as a function of scattering angle. In figures 6.3-6.7, bistatic scattering patterns are plotted for,do = 900,600 and 300 while the internal wedge angle a is kept constant at 300. Clearly, the agreement between the first order GTD solution and the moment method is quite good. The small disagreement observed at around 00, 1800 and 3600 is due to the absence of multiply diffracted fields as confirmed by the results for the normal incidence case (which is exact) shown in figure 6.3. Figures 6.8-6.13 show bistatic patterns for the triangular cylinder with the wedge angle ca = 400 and 500. Again, the approximate GTD solution is found to be accurate even for /, down to 300. The normal incidence (Po = 900) patterns 6.8 and 6.11 are given as a reference to show the regions of disagreement due to the absence of multiply diffracted field. The above test cases are only a sample from a rather large

114 set of validation data. In all cases, we found a remarkable agreement between the first-order GTD solution and the corresponding moment method data for internal wedge angles up to 500~. 6.4 Summary To validate the approximate solution for an impedance wedge, a coupled set of integral equations for skew incidence on an impedance cylinder of arbitrary shape were derived and solved via moment method. The surface currents obtained from the solution of the coupled integral equations were subsequently integrated to yield the scattered field on the diffraction cone. A corresponding first order high frequency (GTD) solution for the far zone scattered field was then generated using the approximate diffraction coefficient and compared with the moment method data for a variety of different wedge angles. A remarkable agreement was found between the first order GTD solution and the corresponding moment method data for internal wedge angles upto 50~.

115 30.0.................... 20.0 10.0 0.0 (a)'10.0 20.0 - I 1st Order Approx. GTD..... Moment Method -30.0 3050.......... A......... 0.0 90.0 180.0 270.0 360.0 * (deg) 30.0....... 20.0,~ 10.0 0.0', -0't -30.0 1st Ordr Approx. GTD 40.0.........MOment od 0.0 90.0 180.0 270.0 360.0 * (deg) = 0.25, and the other having TJ 0.75 +jO.75, (a) Eapb (b) E~~y.

116 30.0.... 20.0 00 10.0 0.0', -10.0 ~~~~ —20.0 - ist Order Approx. GTD......... Moment Method -30.01........ I........... I 0.0 90.0 180.0 270.0 360.0 * (deg) 30.0. 20.0 10.0 (b) -10.0 / -20.0 - stOrd Approx. GTD -......Moment Method -30.0 0.0 90.0 180.0 270.0 360.0 * (deg) Figure 6.4: Far zone bistatic scattering pattern for a triangular cylinder with top top side width w = 4A, a = 300, /3 = 600, ~' = 700, and its top face having ing r/= 0.25, and the other having 27 = 0.75 + j0.75, (a) Eop,-, (b) E~,.

117 10.0........ _ 0.0 - Ist Order Approx. GTD s' -loo P\ A,~~~.-. Moment Method -10.0 -20.0r (a) 70 -30.0 -40.0,_......... I....... -50.0.. _ 0.0 90.0 180.0 270.0 360.0 * (deg) 10.0 0.0 - 1st Order Approx. GTD Moment Method -10.0 -20.0 (b) -30.0 -40.0 -50.0............ _...... I..... 0.0 90.0 180.0 270.0 360.0 * (deg) side width w = 4A, a = 30~, /o = 600, q' = 700, and its top face having 7 = 0.25, and the other having r = 0.75 + j0.75, (a) Eep, (b) EOo0,.

118 30.0................... 20.0 10.0 0.0 (a)'a' 0 \ I v * a 6 -10.0 -20.0 1 st Order Approx. GTD -20.0 -. Moment Method -30.0...... 0.0 90.0 180.0 270.0 360.0, (deg) 30.0 20.0 i0 0.0 (b).). (b) CI 0.0 7 -10.0,: a a -20.0 Ist Orda Approx. OTD......... Moment Method -30.0......... 0.0 90.0 180.0 270.0 360.0 (deg) side width w = 4A, a = 300, /3 = 300, ~b = 700, and its top face having 0i = 0.25, and the other having / = 0.75 + j0.75, (a) E~o~, (b) E~W.

119 10.0...,...,, I......1 0.0.10.0 1 -20.0 - -30.0 40.0 _ C~~~~~~1st Orde Appx GTD -40 ---- Moment Method -50.0..'..I I.. 0.0 90.0 180.0 270.0 360.0 (deg) 0.0........ I I 0.0 -i0.0 00 -20.0 (b) -30.0 O Ist Ordt Apprx. GTD -40.0......Moment Method -50.0......... 0.0 90.0 180.0 270.0 360.0 * (deg)

120 30.0.......... 20.0, 10.0 Co 0.0 -50.0 I I (a) 0 -20.0. -30.0 1st rdr Appro. GTD -40.0 -........ Moment Method - 0.0..I.......I........ 0.0 90.0 180.0 270.0 360.0 * (deg) 30.0 0.0, 10.0 20.0 -10.0.a -20.0,/.a -30.0 Ist Okda Apjox GTM -40.0.... Momet Method -50.0........ 0.0 90.0 180.0 270.0 360.0 (deg) Figure6.8: Far zone bistatic scattering pattern for a triangular cylinder with top side width zu = 4A, a = 40~,,o = 90~, b' = 70~, and its top face having 7 = 0.5, and the other having r = 1.0 + jl.0, (a) Eo,p,, (b) E4,,.

121 30.0 20.0 *d 10.0 0.0 -10.0 -20.0 -30.0 1 st Order Approx. GTD. 3.- Moment Method -40.0 -50.0... 0.0 90.0 180.0 270.0 360.0 (deg) 30.0....... 20.0 ~g. 10.0 0.0 - -20.0 I -30.0 1st Order Approx. GTD Moment Method -40.0 0.0 90.0 180.0 270.0 360.0 * (deg) Figure 6.9: Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, a = 400, = 300,' = 70, and its top face having r = 0.5, and the other having r = 1.0 + jl.0, (a) E,opo0, (b) E,~,.

122 30.0...... ......."-.. I......... 20.0 10.0 0.0. -10.0 0~ -20.0 ~ 1-30.0 - 1.Ist Orde Approx. GTD40.0:......... |- Moment Method -40.0 -50.0........ 0.0 90.0 180.0 270.0 360.0 * (deg) 30.0........ go 10.0 0.0 Cu -10.0.0 -20.0 side -30.030, and its top face having 0.5, and the other having 1.0 + jl.0, (a) E (b) TD -40.0[......... Mommt bktbod -50.0 0.0 90.0 180.0 270.0 360.0 (deg)'7 = 0.5, and the other having r/ = 1.0 + jl.O, (a) EbP,, (b) EPo9,.

123 30.0...... 20.0 0 10.0 0.0 -10.0 (a)'~ -20.0 0= -30.0 f -30.0 1st Order Approx. GTD -40.0 - -........ Moment Metbod -SO.O J A................ 0.0 90.0 180.0 270.0 360.0 * (deg) 30.0....... I... 20.0 0, 10.0 Co i0.0 (b) 1 0.0, 0~I ~20.0 -30.0 1st Orde Approx. GTD -40.0........ Moment Method -50.0........ 0.0 90.0 180.0 270.0 360.0 (deg) Figure 6.11: Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, ca = 50~, /3 = 900, k' = 800, and its top face having r= 0.5, and the other having r/ = 1.0 + j1.0, (a) Efjom, (b) E~,.

124 30.0.. |...... 20.0 ~ 9 10.0 0.0.) -10.0 (a) 72 -20.0. i-30.0 [~ "is,r ~, 1st Order Approx. GTD -40.0 - -------— Moment Method -50.0....... 0.0 90.0 180.0 270.0 360.0 * (deg) cc 3-0.0 +.. 30.0. 20.0 1. 10.0 Co -30.0 (b)''.~ -20.0.. = fa ~~~~30~~.0~~~~ ((~1st Order Approx. GTD 40.0.........Moment Method -50.0........ 0.0 90.0 180.0 270.0 360.0 (deg) Figure 6.12: Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4,, a = 500, 3o = 300, b' = 800, and its top face having r/= 0.5, and the other having r = 1.0 + jl.0, (a) Eo,o, (b) E~,.

125 30.0.........i.,...... 20.0'_ 10.0,.5 0.0 -10.0 (a) /-20.0 -30.0 -: y ~ -30.0: __ Ist Order Approx. GTD -40.0 -------- Moment Method -50.0.. 0.0 90.0 180.0 270.0 360.0 * (deg) 30.0....... 20.0 - - 10.0 -0.0 (b) -1. 0 - -20.0 LT~a~~ i!. 1st Order Approx. GTD 40.0......... Moment Method -50.0... 0.0 90.0 180.0 270.0 360.0 * (deg) Figure 6.13: Far zone bistatic scattering pattern for a triangular cylinder with top side width w = 4A, c = 50~, 3 = 300, q' = 800, and its top face having r = 0.5, and the other having r = 1.0 + jl.0, (a) EIP, (b) EPoS,.

CHAPTER VII PTD FORMIULATION FOR SCATTERING BY IMPEDANCE STRUCTURES The approximate solution developed in chapter five for an impedance wedge is essential for studying the scattering from practical three dimensional structures. In this chapter, equivalent currents are derived, based on the approximate dyadic diffraction coefficient, for computing the scattering by a finite length impedance wedge of arbitrary angle. The derived equivalent currents have been implemented in a standard general purpose PTD code [21] and results are presented demonstrating the accuracy of the formulation for a number of impedance and (dielectrically) coated structures. These include typical shapes such as plates, finite length cones and cylinders which have been partially or fully coated. The PTD type of implementation requires a dyadic physical optics diffraction coefficient which has been derived in appendix A. 7.1 Derivation of Equivalent Currents Consider an impedance wedge of infinite length oriented along the z-axis of a spherical coordinate system as shown in figure 7.1. The normalized face impedances on the upper and the lower faces of the wedge are 7+ = Z+/Z and - = Z-Z respectively, with Z being the free space intrinsic impedance. The incident electric 126

127 field on the wedge geometry is assumed to be a plane wave described by (Ty) = Eie-jks',.ei (7.1) where E' is the constant complex amplitude of the incident field at the origin, si is the unit vector in the direction of propagation, r is the position vector of a point in space and e" is a unit vector indicating the polarization of the incident electric field. The incident field at a point z = z' on the edge can, therefore be expressed as =(z) = Eie jkz'cosP ei (7.2) and to simulate the source of the field diffracted by the edge, we place a fictitious electric line current 7(z') = Ie(z)e-kz' cop't (7.3) at the edge of the wedge, in which t denotes the unit vector tangent to the edge. The magnetic vector potential at the field point due to this current is given by oo e-jksd A() = (Z') 4rsd dz' (7.4) where sd is the distance from a current element of incremental length dz' at z = z' to the field point. Assuming that T(z') is slowly varying and k is large, (7.4) can be evaluated by the method of stationary phase to give I' (z'0)e —/~ -jk,: ~o, ao e-'jksd ~Az(r) t \/ e- kt:cosp (7.5) 2 v'_ -k sin P, eS where z' is the stationary point corresponding to the case when 3o = / Using this result the f,-polarized component of the far-zone radiated field due to the line current It, where /o = sd X, is then found from Eo-, jkZsin/3,oAz (7.6)

128 z=t od t7o QE iA S Figure 7.1: Edge-fixed coordinate system.

129 This must be equal to the known diffracted field from the wedge given by =jksd - Eo =-EtP Dpopo + E#IDpoOk eikz' o (7.7) where Dp,pO and Dpo, are the diffraction coefficients of the wedge and will be defined later, E, and E+, are the complex amplitudes of the d,'- and q'-polarized components of the incident field at the phase origin with /'o = si x q'. Comparing (7.6) and (7.7), the electric line current IP is readily identified to be I(z')= Tj [E~,(z') Dp.O(z') + Ey,(Z')DPo,'(z')] (7.8) Following a similar procedure one can obtain the expression for the magnetic line current Im to be Im(z') = ki [E~,(z')+,,(z') + Ea,(z')DO (z')] (7.9) 7.2 Radiation from Equivalent Currents on a Finite Wedge 7.2.1 Straight Wedge Consider a straight wedge of finite length e oriented in the direction t as shown in figure 7.2. For this case, it is convenient to assume a local coordinate system (X', y', z') for the wedge so that t coincides with the z'-axis. The origin of this local system is chosen to be at the center of the wedge and is defined by the global position vector r. We again postulate the existence of filamentary electric and magnetic currents I[ and Im on the finite edge. The amplitudes of these equivalent currents are assumed to be the same as those found for the infinite wedge. The field diffracted by the finite wedge can then be computed by integrating these currents over the finite length of

130 z A:<ft~~I r/, o -C1 2 x / Figure 7.2: Finite impedance wedge geometry.

131 the wedge. In the far zone, we obtain the diffracted field.d tjk] [ZIe(')A x (S x t) + Im('t)(s x t] 4rr dz' (7.10) where T' is the position vector of a current filament on the wedge and the propagation direction of the radiated field is defined by the unit vector s = ~ sin 8 cos q+ + sin 8 sin q + + cos O (7.11) Substituting the expression for I(T') and Im(f') in (7.10) and noting that 4E~( r) = sin,B5oE'(') (7.12) and E;() =-(t x t) *E'(n) (7.13) sin oE the far-zone radiated field is more explicitly given by rd 2 -i 1 [A x (A x tA){(j e')D'0, - (i x:). D 27rr sinpo J-t/2 Do} -(s x i){(i. ei)Op). - (t x S).iDOq}] ejk( -")'rdz' (7.14) where the incident field on the edge is assumed to be Ei(') = Ei(F')ei = e-jk'e (7.15) and i is the propagation direction of the incident field given by -Z= -i sin O' cos q - y sin O' sin qi + z cos i' (7.16) If required, one can write the ui-polarized component of the above field as U. d r= (.eji-) [(F. t {( )i)Dpop- ( x z) *- 2-rr sin {o e | i - x ){i-(x) (7.17).l_ lo/

132 where ui is a unit vector transverse to s (the unit vector in the direction of propagation). Also in the above we have used the fact that Flo+ =~+z'' (7.18) Performing the integration defined in (7.17) yields u,e= u t;')F[(i * i) {(t e i)Dp,0. - (a x z) * e'D,0' 21rr sin o, -u (t x s) {(t" - e')D~-(t x z)~ eID, 1 sinc{( 2)} (7.19) In this the diffraction coefficients D can be chosen depending upon the type of high frequency theory to be employed. In particular, D = DGTD - DP (Physical Theory of Diffraction or PTD) (7.20a) D = DP~ (Physical Optics) (7.20b) D = DGTD (Geometrical Theory of Diffraction) (7.20c) where DGTD is the approximate diffraction coefficient derived in chapter five and DP0 is the physical optics diffraction coefficient for an impedance wedge derived in appendix A. 7.2.2 Curved Wedge Consider now the curved impedance wedge shown in figure 7.3. The curve describing the edge is arbitrary and may vary continuously. If a plane wave is incident on the edge, the total field diffracted from it can be calculated by first dividing the

133 edge into small straight segments and then summing the individual contributions, i.e., Td (total) = E d (segment) (7.21) Suppose the excitation field at one of these small (incremental length) segments is given by =e-jkt.' (OAiai +'iai) (7.22) and the far-zone field associated with the diffracted field from this differential length along the edge may be written as Ed e-jkr r (Obe + qb0) (7.23) From (7.19), the scattering amplitudes be and b4 of the diffracted field by each segment are then related to the incident field amplitudes through the matrix be feei fe'i aei (7.24) bLL floi fe i fai where er -jkrx fue = 27rr sin 3o ( -;)' [(u t x z) ei-ui. (i x ) {(i. i)D, -(t x z). D } ]t sinc { k( -2 ) } (7.25) where u stands for 0 or X whereas e' represents 0' or qY, and since the discretization segments are small we may set sinc { (-) ~}2 i 1 (7.26) 2 j

134 scattered (eo,~) r,2 z segment 1 incident F r 3 u e r i s wn (01en l 1)

135 Equations (7.21) in conjunction with (7.23)-(7.25) can be employed to calculate the total diffracted field from an arbitrary curved edge. The RCS due to the edge diffraction is then given by aue, = 4r If,, 12 (7.27) It should be noted that d in (7.21) is only the partial field and, therefore, the RCS in (7.27) is the partial RCS of the scatterer. To compute the total far-zone scattered field we must add a complementary field to the diffracted field. Since in this work we will pursue a PTD type of implementation, the complementary field will be a Physical Optics (PO) field and the total field will be, = Td + E o (7.28) Hence, for the total RCS, the physical optics field must be calculated separately and then superimposed with the diffracted field to give ati, 47 ift: i 2 (7.29) where fuci = fuse + fP,? (7.30) 7.3 Test Results The derived equivalent currents have been implemented in a standard general purpose PTD code called McPTD [21] which employs only first order high-frequency far zone fields. In particular, the codes McWedge (for a general curved PEC wedge geometry), McLine (for long straight PEC wedges) and McRing (for a PEC wedge forming a ring) have been modified for the case of an impedance wedge. To test the accuracy of the equivalent current formulation, we refer to figures 7.4 - 7.8 where

136 the RCS of a number of different structures is plotted as a function of incident angle Xi. The results obtained from McPTD have been compared with the measured and moment method data. The numerical data is obtained from a body of revolution code which does not employ any approximate boundary conditions and is, therefore, considered to be exact for coated structures. The RCS patterns given in figures 7.4 - 7.6 are for a flat base cone with half cone angle c = 150. Figures 7.4(a) and 7.4(b) display patterns due to incident E,j (HH) polarization for a perfectly conducting and a coated cone, respectively, with base diameter of 2.OA. A decrease in RCS in the case of the coated cone is observed because of the lossy coating. In figure 7.5, patterns for a larger cone having base diameter of 3.0A have been presented. Figures 7.6(a) and 7.6(b) show RCS patterns of a partially coated cone, having a perfectly conducting base, due to incident E,i (HH) and E,i (VV) polarization, respectively. RCS patterns for a finite circular cylinder are displayed in figure 7.7 whereas figure 7.8 shows RCS patterns for a plate. 7.4 Summary To demonstrate the usefulness of the approximate diffraction coefficient, obtained in chapter five, equivalent currents were presented in the context of PTD for a finite length impedance wedge of arbitrary internal angle. These were eventually incorporated in a standard general purpose physical theory of diffraction (PTD) code [21] and results were presented which demonstrated the accuracy of the formulation for a number of different impedance structures. These included typical shapes such as plates, finite length cones and cylinders which were partially or fully coated.

137 30.0 i',, 1 ~ 20.0 10.0 0.0 (a) j -10.0 -20.0 McPTD -30.0 MoM -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Incident angle'i (deg) 30.0 20.0 ID 0.0 (b) -20.0 -30.0. mom -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Incident angle i (deg) Figure 7.4: RCS of a flat base cone due to incident E,i (HH) polariation with = 15~, D = 2A, 0' = 900, (a) perfectly conducting cone, (b) coated cone: er=7-jl.5, /,=2- jl, coating thickness=0.04A.

138 30.0 ~ i ~ ~ ~ i ~ 20.0 130.0 -10.0 -20.0 MCPTD -300- 1.0.. mom 40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Incident angle' (deg) 30.0 20.0 10.0 -0.0 (b) V -10.0 A...... -20.0 -30.0. 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Incident angle 3i (deg) Figure 7.5: R.CS of a fiat base cone due to incident E, (HH) polarization with a 30=7-j1.5, 0 =2- ji, coating tbickness=0.04A.

139 30.0 20.0 10.0 30.0 (a) c X -10.0. I 1 -20.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Incident angle' (deg) 30.0 ~ 20.0 i0.0 0.0 wit -10.0 -20.0 o o -30.0 MOM -40.0 0.0 30.0 60.0 90.0 120.0 150.0 180.0 Incident angle 4 (deg) with cm = 15, D = 3A, e7=7-jl.5, p,=2-jl, coating thickness=0.04,, 0~= 90~, (a) E~, (HH) polarization, (b) E~, (VV) polarization.

140 30.0. 20.0 (a) -20.0. -30.0 McPTD -30.0......... -40.0. 0.0 30.0 60.0 90.0 Incident angle'i (deg) 30.0 20.0 x 10.0 ) -30.0....... mom L -40.0,~% - 0.0 60.0 90.0 Incident angle i (deg) Figure 7.7: RCS of a finite circular cylinder due to incident E, (HH) polarization with D = 3A, L = 4A, 0' = 900, (a) perfectly conducting cylinder, (b) coated cylinder: E7=4-jl.5, IZ=2-jl, coating thickness-0.04A.

141 30.0 20.0 - McPTD Measured 10.0 0.0 (a) 4-10.0. -20.0 -30.0 -40.0 0.0 30.0 60.0 90.0 Incident angle'i (deg) 30.0 20.0 McPr Musured / 0.0 30.0 60.0 90.0 Figure 7.8: CS of a square plate due to incident E (VV) polarization with W = (b)' c -10.0 -20.0 -30.0 -40.0 0.0 30.0 60.0 90.0 Incident angle ei (deg) j0.46, #= 2- jl.26, coating thickness=0.08 inches, frequency=8.004 GHZ.

CHAPTER VIII CONCLUSIONS In this study, asymptotic/high-frequency solutions were developed for analyzing some non-specular scattering mechanisms associated with coated convex surfaces and edges where the coating was simulated by approximate boundary conditions. Both the standard impedance boundary conditions (SIBCs) and the generalized impedance boundary conditions (GIBCs) were employed for a characterization of edge diffraction, creeping wave and surface diffracted wave contributions. Second order generalized impedance boundary conditions (GIBCs) were derived in chapter two for a single and a three-layer coating and these were employed in chapter three to construct integral equations for scattering by two-dimensional coated structures of arbitrary cross-section. These integral equations were implemented via a moment method procedure and the results were compared with reference data based on analytical or other validated numerical solutions. It was found that the proposed second order boundary conditions provide an improved simulation of the coating in comparison with the traditional (first order) impedance boundary condition and guidelines were given for their region of validity. The primary reason for the improved simulation is because the second order GIBC accounts for the presence of polarization current components normal to the coating in addition to the tangential 142

143 ones. In general, the presence of edges or abrupt changes in the material leads to simulation inaccuracies which become more apparent as the coating thickness increases. This is due to the inherent non-uniqueness of these boundary conditions at abrupt terminations, a situation which can only be remedied by introducing additional field constraints at the terminations [51,52]. The non-specular scattering mechanisms like creeping waves and surface diffracted waves were considered in chapter four as applied to a coated convex cylinder. Rigorous ray solutions of the scattered fields were presented for a coated convex cylinder simulated by SIBCs and GIBCs. These were developed in the context of the uniform geometrical theory of diffraction and specific expressions were given for the scattered fields in the lit, shadow and transition regions as well as for observations in the near vicinity of the cylinder. Specifically, UTD expressions were derived for all regions exterior to the coated cylinder and these were shown to be suited for engineering computations. As expected, the derived UTD expressions were given in terms of the generalized Pekeris or Fock-type functions whose evaluation was efficiently performed via the Fourier Trapezoidal rule suggested by Pearson [45]. In comparison to the solution given by Kim and Wang [41], the ray representations given here, for the surface diffracted field, are based on a second order generalized impedance boundary condition which permits the simulation of thin multilayered coating as demonstrated in the included examples. Also, in our implementation of the transition fields we employed a rigorous rather than a heuristic evaluation of the Fock-type integrals. Further, we have presented accurate field representations for observations on or near the vicinity of the coated cylinder and these can also be used for computing the radiated fields by a source or an aperture on the surface of the convex cylinder.

144 In chapter five, a most important source of non-specular electromagnetic scattering, that of diffraction by an impedance wedge at skew incidence was considered. Application of the boundary conditions (SIBCs) on the wedge faces resulted in a set of four coupled functional difference equations. Three different approximate solutions were derived on the basis of these difference equations. Two of the approximate solutions were obtained by separating the difference equations associated with the upper and lower faces of the impedance wedge, respectively. These were found to recover the known solutions for three different sets of wedge angles and skewness angles, namely those corresponding to, normal incidence (fl, = 900) with arbitrary wedge angle, a half plane (arbitrary,80, internal wedge angle = 00) and a two part plane (arbitrary f,, internal wedge angle = 1800). An important feature of our solution, lacked by the others [13,14,15,19], is that a single computer code can be employed to compute the scattered field for all three special cases mentioned above. A third approximate solution was constructed by approximating the trigonometric functions sin ct and cos aC appearing in the difference equations with functions having a period of 2C. This solution again was shown to reduce to the exact solutions of the half plane (arbitrary,80, internal wedge angle = 00) and that of the normal incidence wedge (,@, = 90~). For other 8,/ and wedge angles, the three approximate solutions were found to be of acceptable accuracy in certain regions of space and a scheme was suggested to combine all three approximate solutions for approximating the diffracted field in the entire angular section exterior to the wedge. Some comparisons with the exact solution for the right-angled wedge showed that the accuracy of the approximate solutions was still acceptable even for the internal wedge angle of 900 which was quite far from that of a half plane. To validate the approximate solution, developed in chapter five for an impedance

145 wedge, a coupled set of integral equations for skew incidence on an impedance polygonal cylinder of arbitrary shape were derived and solved via the moment method. The surface currents obtained from the solution of the coupled integral equations were subsequently integrated to yield the scattered field on the diffraction cone. A corresponding first order high frequency (GTD) solution for the far zone scattered field was then generated using the approximate diffraction coefficient and compared with the moment method data for a variety of different wedge angles. A remarkable agreement was found between the first order GTD solution and the corresponding moment method data for internal wedge angles up to 500. Finally, to test the usefulness of the approximate skew incidence wedge diffraction coefficients for practical three dimensional structures, equivalent currents were presented in the context of PTD for a finite length impedance wedge of arbitrary internal angle. These were incorporated in a standard general purpose physical theory of diffraction (PTD) code [21] and results were presented which demonstrated the accuracy of the formulation for a number of different impedance structures such as plates, finite length cones and cylinders which were partially or fully coated. The PTD implementation required a dyadic physical optics (PO) diffraction coefficient which was derived in appendix A. For the future work, a characterization of the diffraction effects by impedance discontinuities in smooth convex cylinders should be pursued. Of interest is the case where the observer and the source are both at a finite distance away from the discontinuity. This will accommodate the situation when a creeping wave strikes the discontinuity and the observation point is also near or on the surface of the impedance cylinder. To treat this case, at first a parallel analysis should be carried out for an impedance insert on an impedance plane. The results of this analysis will then be

146 extended to the case of a similar impedance discontinuity on the convex cylinder on the assumption that the cylinder radius is large. Alternatively, a direct analysis could be pursued. The pertaining analysis will certainly be much more complex and the resulting diffraction coefficients may not, therefore, be of practical use. Using the diffraction coefficients based on the forementioned analysis, it will be possible to characterize the scattering and radiation by apertures and conformal antennas on a convex impedance cylinder. An analysis of the mutual coupling among such devices is another application of interest. The proposed research in the above paragraph is concerned with the two dimensional applications. A natural extension of this work would be an analysis for three dimensional applications. In this case, the electric and magnetic fields are coupled on the cylinder surface and thus heuristic extensions of the two dimensional results to three dimensions, usually done in the case of perfectly conducting structures, are not applicable. In general, therefore, a separate analysis is required in the case of skew incidence. The solutions developed in this work, for scattering by an impedance wedge at skew incidence, were obtained by solving a first order homogeneous functional difference equation. However, these solutions are approximate and a next step should be to improve the accuracy of these approximate solutions by employing, for example, an iterative procedure. Such a procedure will require a solution of a first order inhomogeneous functional difference equation and, therefore, will make the analysis more complicated. Another future task should be to pursue a rigorous solution for an impedance wedge of arbitrary wedge angle with GIBCs imposed on its faces.

APPENDICES 147

148 APPENDIX A PHYSICAL OPTICS DIFFRACTION COEFFICIENT FOR AN IMPEDANCE WEDGE AT SKEW INCIDENCE The impedance wedge diffraction coefficient presented in chapter five predicts infinite diffracted field at the incident and reflection shadow boundaries. However, this difficulty can be overcome by subtracting a physical optics diffraction coefficient from it resulting in a finite field at the shadow boundaries. The diffracted field obtained in this manner is only a partial field and is called a fringe field. The total scattered field for a particular finite structure having edges can be computed by calculating the physical optics field for the structure, separately, and then adding it to the fringe field. In this appendix, a dyadic diffraction coefficient, for an impedance wedge, is derived using physical optics. In other words, we seek a physical optics approximation to the diffraction tensor given in chapter five. Consider a plane wave incident on the upper face of an impedance wedge (see figure A.1) and is represented by =,e-jki'*, ZiT = _e-jki-F (A.1)

149 y a+ Fe'gendngedfntos.

150 where i' = _-xsino0, cos' - sin, sin q' + A cos/3o (A.2) with q' measured from the upper face of the wedge. It is convenient to proceed with just the z-components, e, and h2, of the incident field. The rest of the components can be expressed in terms of the z- components, using Maxwell's equations, as 1 e, = (cos Io cos O'ez + sin O'hz) (A.3) ey sin (cos o sin q'e, - cos q'h,) (A.4) h = (-=sin O'e_1+ cos f, cos 0'h) (A.5) sin ft h = sin (cos b'e, + cos0 o sin q'h.) (A.6) The total field is a sum of the incident field and a reflected field. We assume the z-component of the reflected field to be of the form EZ = Re,e-jkir',r zH, = Rhhz e-jkir' (A.7) where Re and Rh are the reflection coefficients for the z- components of the electric and magnetic field, respectively, and r' denotes the propagation direction of the reflected field and is defined as gr = -- sin f/, cos q' + A sin o, sin q' + z cos /0 (A.8) The upper face of the wedge satisfies the impedance boundary conditions (at y = 0) E= rl+Z H, E=- r+Z Hz (A.9)

151 and using these, the expressions for Re and Rh can be derived to be Reez = t[{(sin f + 7+ sin fi) (sin q'- s o)- cos2 3p cos2b,}ez -2 cos /3, sin 4' cos 4'h,] (A.10) Rhh = [2 cos /o, cos q' sin S'ez + {(sin q' - 77+ sin (,io)'sin' + - /OS26COS2 /hZ] h(A.11) with 77+(A.12) = (1 + rt+ sin /, sin q')(r7+ + sin,o sin q') Now the surface electric and magnetic currents induced on the upper face (y = O) of the wedge are J~ = n x 7, M — n x P (A.13) where ni = for the upper face and H and he denote the total field which is the sum of the incident and reflected fields. It is convenient to write these currents in the form ZJ-. e-jk(zcosoxsinG cosi') (A.14) Msi = me-jk(z COS,So-x sin ocos') (A. 15) where j and mT are determined, using (A.10) - (A.15), to be = 2y sin 4' cos /, cos S'e, + (sin' + sin) hz} -{(sin4~ + risinin/,o)e-cos/ cos d'h} (A.16) r~~~~~-o/3co4/. ~

152 m =-27 sin (sin' + [7 sin + i o)e. - cos o cos q'h}:r - r+ {cos cos (b'e: + sin' + + hz} z (A.17) The far-zone scattered field due to the induced surface currents on the upper face (y = 0) of the wedge is given by k e-'ik"-I'" =J4 {'Zs X [s x:,(E) + s x M(t)), ds' (A.18) where the integration is performed over the upper face of the wedge. The s unit vector is defined as s- = sino p cos + + sin fo sin 0 + z cos /o (A.19) with 4 measured from the upper face of the wedge. Assuming now that the edge diffraction is mainly due to the surface currents concentrated on a narrow strip along the edge, the edge diffracted part of the field Z' can be written as ~J 4lo/_00 {z,X [S X 7.(x,,Z)] + s x M.(,, Z,) e-jk/(,-x,)2++V+(_-z,)2 dz'dx' (A.20) v(x - x')2 + y2 + (z- z-)2 where the absence of the upper limit, in the above integral, on the x' integration denotes that only the asymptotic endpoint contribution at x' = O is taken into account. Employing the method of stationary phase for the z' integration, one can evaluate the integral in equation (A.20) and the expression for the diffracted field is given by Enp sin p_ 4 sin _.(cos. +_ c.s ~, {s x (s x j)+ s x m} (A.21)

153 with J and mT defined in (A.16) and (A.17) respectively. The edge diffraction tensor given in (A.21) can now be transformed into the edge-fixed coordinate system [43] and the resultant diffracted field from the edge is given by E~,o() |oDP(<D, (,'; Po, r.7+) DPo (,,; 0 Ro,) 1 E() -Al(s) Ed(s) -D_o0,,;L3o,77+) Do (D,,;/o, 7) L E(QE) (A.22) where s is the distance between the diffraction and obervation points and A (s) is Al(s)= ejk e' (A.23) The 2 x 2 matrix in (A.22) represents the physical optics dyadic diffraction coefficient similar to the one presented in chapter five. The components of the diffraction matrix can be expressed as DPO~(, q'; /3o,r+)= A2(q,';qYo) E(j, c'; 3o,o, +)+cos,FoF,)] (A.24) DpO (qS, 0Y; 3o, 77+)= A2(q),,;o)[G(c;, q;/3o,7+)+cos#3oH Y,; /,, 1 with A2(q, O'; /o) = (1 - sin2 Po sin2 q')-'(1 - sin2 o sin2 q)-1 (A.25) and E7 cos q cos q' - cos P/3o sin b cos q' U" (A.26) G" cos/~o sin q cos q cos q cos qY

154 Fn cos 08 sin q sin 4' cos q sin' | U, (A.27) H,, - cos q sin q' cos p, sin q sin' L V" where the notation E(q, b'; lo, 7/+) = En, E (0, q'; fo, n1') = E,, etc., has been employed for simplicity. The U and the V functions are given by UI7~ = U~PO 07 oi77+= sin q' U"-1 UPo(- =cos 4 + Cos q' (cos2 po - sin2o cos cos ) (1 - + sin o sin') (A.28) (1 + q7+ sin Po sinqS')'V" = VP~(q, 0'; io, 77+) = -sin po cos Po sin d (1 -7+ sin o0 sin ) (77+ + sin,80 sin $') (A.29) For the case when the lower face of the impedance wedge is illuminated the above expressions will be applicable with the replacement'-+ —,'-, nr - n7r - q and,80 -,. -.

BIBLIOGRAPHY 155

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