Abstract Interest in applying radar remote sensing for the study of forested areas led to the development of a model for scattering from corrugated stratified dielectric cylinders. The model is employed to investigate the effect of bark and its roughness on scattering from tree trunks and branches. The outer layer of the cylinder (bark) is assumed to be a low-loss dielectric material and to have a regular (periodic) corrugation pattern. It is further assumed that the corrugation exists in only the angular direction (two-dimensional problem). The inner layers are treated as lossy dielectrics with smooth boundaries. Under the mentioned conditions, a hybrid solution based on the moment method and the physical optics approximation is obtained. In the solution the corrugations are replaced with polarization currents that are identical to those of the local tangential periodic corrugated surface, and the stratified cylinder is replaced with equivalent surface currents. New expressions for the equivalent physical-optics currents are employed which are more convenient than the standard ones. It is shown that the bark layer and its roughness both reduce the radar cross section. At frequencies where the bark thickness and its roughness are considerable fractions of the wavelength the radar cross section reduction becomes very significant. It is also demonstrated that the corrugations can be replaced by an anisotropic layer and expressions for the elements of its permittivity tensor in terms of the corrugation parameters are derived In Appendix A. Application of the equivalent dielectric layer simplifies the problem extremely. i

Contents 1 Introduction 1 2 Scattering from Periodic Corrugated Planar Dielectric Surface 2 2.1 Two-dimensional Green's Function for a Stratified Dielectric Half Space............................................. 3 2.2 Far Field Evaluation.......................... 6 2.3 Scattering from Inhomogeneous Periodic Dielectric Layer above a Half-Space Layered Medium...................... 10 2.4 Numerical Implementation....................... 14 3 High Frequency Scattering from Stratified Cylinders 19 4 Scattering from Corrugated Cylinder 26 5 Numerical Results 28 S Conclusions 29 A APPENDIX A A-1 Al Introduction A-1 A2 Theoretical Analysis A-2 A3 Low Frequency Approximation A-6 11

A4 Reflection Coefficient of Uniaxial Layered Medium A-7 A5 Numerical Examples A-12 iii

List of Figures 1 Geometry of a periodic inhomogeneous dielectric layer over a stratified dielectric half-space........................ 3 2 Contour of integration and steepest descent path in y-plane.... 7 3 Geometry of the line source and its image.............. 9 4 Geometry of scattering problem of a stratified cylinder...... 21 5 Normalized backscattering cross section ( -) of a two-layer dielectric cylinder with a = 10.5cm, al = 10cm, e1 = 15 + i7, 62 = 4 + il versus koa for TM case..................... 24 6 Normalized backscattering cross section ( a) of a two-layer dielectric cylinder with a = 10.5cm, al = 10cm, el = 15 + i7, 62 = 4 + il versus koa for TE case......................... 25 7 A corrugated cylinder geometry.................... 26 8 Geometry of the corrugated surface................. 33 9 Amplitude of the total induced current in the two-layer periodic corrugated surface versus incidence angle............... 34 10 Phase of the total induced current in the two-layer periodic corrugated surface versus incidence angle.................. 35 11 Amplitude of the reflected field from the two-layer periodic corrugated surface versus incidence angle for E polarization....... 36 12 Phase of the reflected field from the two-layer periodic corrugated surface versus incidence angle for E polarization........... 37 iv

13 Amplitude of the reflected field from the two-layer periodic corrugated surface versus incidence angle for H polarization...... 38 14 Phase of the reflected field from the two-layer periodic corrugated surface versus incidence angle for H polarization........... 39 15 The radar cross section of a corrugated cylinder for TM case with a = 10.5Ao, ai = 10Ao, L = A0/4, e1 = 4 + ii, 62 = 15 + i7, and t d = Ao/8.............................. 40 16 The radar cross section of a corrugated cylinder for TE case with = 10.5Ao, al = 10Ao, L = Ao/4, el = 4 + ii, 62 = 15 + i7, and t- d= Ao/8.............................. 41 17 The radar cross section of the corrugated cylinder for TE and TM cases using the numerical and the equivalent dielectric methods.. 42 A-1 An array of infinite dielectric slabs................... A-2 A-2 Plane wave reflection from a stratified uniaxial dielectric half-space. A-7 A-3 Real part of the equivalent dielectric tensor elements for periodic slab medium with L = A0/4, e = 4 + il, and d/L = 0.5 versus incidence angle; e, (H polarization), ey = eZ (E polarization).... A-15 A-4 Imaginary part of the equivalent dielectric tensor elements for periodic slab medium with L = Ao/4, e = 4 + ii, and d/L = 0.5 versus incidence angle; e, (H polarization). =- e (E polarization).... -16 inc Y E, E plarzaion. A 1

A-5 Real part of the equivalent dielectric tensor elements for a periodic slab medium with L = Ao/4, e = 4 + il, and fo = 45~ versus d/L; ex (H polarization), cy = 6e (E polarization)............. A-17 A-6 Imaginary part of the equivalent dielectric tensor elements for periodic slab medium with L = Ao/4, e = 4 + il, and 0o = 45~ versus d/L; e, (H polarization), Ey = e (E polarization)............ A-18 A-7 Location of zeros of (A-l0)in the kJ'-plane for for the periodic slab medium with e = 4 + il, 4o = 45~, d/L = 0.5, and (a) L = 0.2Ao, (b) L = 0.5Ao, (c) L = 0.8Ao, (d) L = 1.4Ao............. A-19 A-8 Real part of the equivalent dielectric tensor elements for periodic slab medium with e = 4 + il, and o0 = 45~, and d/L = 0, 5 versus L/Ao; e, (H polarization), cy = e_ (E polarization).......... A-20 A-9 Imaginary part of the equivalent dielectric tensor elements for periodic slab medium with e = 4 + il, and o0 = 45~, and d/L =0,5 versus L/Ao; e. (H polarization), ey = eZ (E polarization)...... A-21 A-10 Amplitude of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.25Ao...... A-22 A-11 Phase of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.25Ao........... A-23 A-12 Amplitude of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.4Ao....... A-24 vi

A-13 Phase of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.4A0.......... A-25 A-14 Geometry of a wedge-shape microwave absorber and its staircase approxim ation.............................. A-26 A-15 Amplitude of reflection coefficient of a wedge-shape microwave absorber for both E and H polarizations versus incidence angle; L = 0.4Ao, H = 1.5o, D = 1Ao, and e = 2.5 + i0.5...............A-2 vii

1 Introduction The literature concerning the problem of scattering from cylinders with rough surfaces is relatively scarce. To our knowledge the first treatment of a problem of this sort was given by Clemmow [1959] where a perturbation solution to an eigen function-expansion was obtained for a perfectly conducting cylinder with almost circular cross section, and only the E polarization case was considered. This technique is restricted to very smooth and small roughness functions. Other perturbation techniques for perfectly conducting cylinders with very small roughness have also been developed [Cabayan and Murphy, 1973; Tong 1974]. None of the existing techniques can handle dielectric rough cylinders, particularly when the roughness height is on the order of the wavelength. Study of this problem is motivated by the fact that a tree trunk can be viewed as a multi-layer dielectric cylinder with a rough outer layer. The outer layer has almost a periodic pattern and the roughness height is proportional to the diameter of the cylinder. This layer consists of dead cells with almost no water content: hence, its dielectric constant is low and slightly lossy. The inner layers that carry high dielectric fluids have very high and lossy dielectric constants. In modeling a tree, the branches and trunk usually are considered to be homogeneous smooth cylinders [Durden et al, 1988; Karam and Fung,1988]. In this report the effect of bark and its roughness on scattering is studied. Under the assumption that the bark roughness is a regular corrugation in only the angular direction (i.e., ignoring variations in the axial direction) and the radius 1

of curvature of the cylinder is much larger than the wavelength and the period of corrugation, an approximate solution to the scattering problem is obtained. In this solution, each point on the surface of the cylinder is approximated by its tangential plane. Then the polarization current in the periodic tangential surface is obtained numerically. Once the polarization current in the corrugations is found. the scattered field due to the corrugations together with the scattered field from the smooth cylinder (when the corrugation is removed) give rise to the total scattered field. The scattered field for a smooth cylinder is obtained using new physical optics surface currents that are more convenient than the traditional ones. It is shown that the corrugation on the surface can be replaced with an anisotropic layer which would extremely simplifies the problem. In Appendix A the equivalent dielectric tensor of the corrugated layer in terms of the corrugation parameters is derived. 2 Scattering from Periodic Corrugated Planar Dielectric Surface In this section we seek a numerical solution for the total field (or polarization current) inside a periodic inhomogeneous layer lying over a stratified dielectric half-space illuminated by a plane wave. The geometry of the scattering problem is shown in Fig. 1. First the two-dimensional Green's function for a stratified dielectric medium is found. Using Floquet's theorem these results are extended to the periodic case. Then the problem will be formulated as an integral equation *j)

that can be solved numerically by the method of moments. ~C E(x, y) - Figure 1: Geometry of a periodic inhomogeneous dielectric layer over a stratified dielectric half-space. 2.1 Two-dimensional Green's Function for a Stratified Dielectric Half Space For a volume distribution of electric current (Je) occupying region V in free space, the corresponding Hertz vector is given by iZo f_ _ ki-l Ii 4rk- J -, ldv', Z7 ro ____ _ _ | where Zo(= *) is the free-space characteristic impedance and the resulting fields are E= &2 nI + k (i + )82 ), (1) 2E= l cn,

The Hertz vector potential associated with an infinite current filament located at point (x', y') in free space with amplitude Ip and orientation pj is of the form Ip(X, y) = -ZH(o(ko(x - x') + (y - y p =, y or z. (2) 4 ko The corresponding field components can be obtained by inserting (2) into (1) and then by employing the identity y1 +oo eikjy y-y'-ik,(x-x') H)(k0 /(x_ x- X)2 + (y -_ )2) — = ky dk (3) the resulting fields can be expressed in terms of continuous spectrum of plane waves. In (3) ky = /k2 - k2 and the branch of the square root is chosen such that In the presence of the dielectric half-space, when the current filament is in the upper half-space, each plane wave, is reflected at the air-dielectric interface according to Fresnel's law. It should be noted that the incidence angle of each plane wave, in general, is complex and is given by 7= arctan('). Ky The net effect of the dielectric half-space on the radiated field can be obtained by superimposing all of the reflected plane waves that are of the following form Rq ( ) eik(Y+Y')-ikz (x-), q = E or H. where Rq(n) is the Fresnel reflection coefficient. The total reflected field can now be obtained by noting that E: = -RH(v)EI

E; = RH( )E; E = RE(})E[ and since the direction of propagation along the y axis is reversed for the reflected waves, the operator a- for the x and y components of the reflected field must be replaced by -3. Thus, r o 1a[-I. - I 1 a2 + e, (y+y,)-,kx(X-x') r x [ + )a2 + Iy aa] f+O RH() kyI7d,r) _ ~Zo I — Ix 1 9l 1 2 + iky(y+y')-'-.(X-x')(4) Y= -4n [- aa + Y(1 + fy RH^ d (4) Er - IZ rf+OOy RE(Y) (Y+')-'kx(.-)dk. In matrix notation the total field in the upper half-space can be represented by GXX Gxy 0 I E= G Gyy 0 Iy, (5) 0 0 Gzz Iz where G= = -z(1 + o H (ko /x _ xI)2 + (y y,)2) -_ f' o RH(y) k( dkx], - = ao [H ()ko x- 9x)2 + (y (6 y))2 1 r+O RH( ) e"*'kY(y+y')- kx( x-) _!f+'~RH.))e~'+')'"('")d5]

Gyy = kz(1 + ) a[H21(k0 x-x/)2 + (y - + 1 fj RH(-^) elky(MY+Y)-ik(-x) dkG = -k[[Hl(ko (x - x)2 + (y - y/)2) 41 +- e-kY(y+y ) — 9 ~x+ f _0 RE(7) e' ky dkx] are the elements of the dyadic Green's function for two-dimensional layered dielectric half-space problems. If an electric current distribution Je occupies region S in the upper half-space, the radiated electric field at any point in the upper half-space can be obtained from: E(x, y) = fs[Go(x. y; x', y')Jo(x', y') + Gy(x,y; x y x y, y')dx'dy' ~E(x, y) = fs[Gy(x, y; x', y')J.(x', y') + Gyy(z, y;', y')Jy(z', y')]ddy' (7) Z3(x, y) = fs G.z(x, y; x', y')Jz(x', y')dx'dy' 2.2 Far Field Evaluation In scattering problems the quantity of interest usually is the far field expression. Here we derive the approximate form of the Green's function in the far zone using the saddle-point technique. All the elements of the dyadic Green's function have an integral of the form 1 +eikY(y+y)-ikr(2-~') I= - Rq(7) -dkx (8) T7 J 7-' ky

Using the standard change of variable kx = ko sin the integration contour is changed from the real axis in the complex k -plane to contour F in 7-plane as shown in Fig. 2. Also by defining =,__}~y 2'C S.D.P Figure 2: Contour of integration and steepest descent path in 7-plane. x - x' = 2 sin O2, Y - y P2 cos 02 integral (8) in 7-plane becomes: I = 1 Rq(7)e'^P c(Y+^2)dy The saddle point is the solution of d cos ( + 2) = 0 7

implies 7y = -~2. When kop2 > 1 the approximate value of I can be obtained by deforming the contour of integration from F to the steepest descent path (S.D.P.) given by Im[i cos(7 + 02)] = 1. There are some poles associated with the reflection coefficient function (Rq(7)) of a layered dielectric medium that are captured when the contour is deformed. the contribution of these poles gives rise to surface waves, but their effect can be ignored if the dielectric materials are lossy and observation point is away from the interface. Under these conditions we get I = Rq(-02) fS.D.P. ei'koP2cos(-y+2)dd ij~e'(ko 2 4Rko~ -~/4) where we have used the fact that Rq is an even function. Also the large argument expansion of the Hankel function can be used for distant approximations, i.e. HI1i(k0 (X - X')2 + (y - y')2) iei(koPplr/4) where pi = /(x - )2 + (y - y')2. From Fig. 3 it is seen that in the far zone the following approximations can be used also 41 = 42 = 0, Pi = p - x' sin d - y' cos 0, P2 = p -' sin 0 + y'cos o. These approximations can be inserted into expressions for the dyadic Green's function given by (6). The far field approximation of derivatives of the Hankel function 8

and integral I can be obtained by retaining the terms up to the order p-1/2 and discarding the rest. Thus the expressions for the Green's function elements in the far zone become: G,, = - 4 e i(kp-'r/4) cos2 e-iko siniX'[e-iko cosm y' - RH()eio cosy'] G~_ k~~-~, ei(~-~/~ ~-,~.,~'-,lro. ~c-'b'" b, i,-i$'".(U' + R 0oco i, Gus = /knglp/ei(kop-r/4) sin 4 cos qe- iko sin [k co + RH()e'ik CO'] G,= Zn. lgei(kop-/1r4) sin2 ce-k "i"^l' i[e-iko o"v' + RH(4)eio cos O'] Gv = -k4 -,/ ei(Ip-r/4)e in-2 e- o l- + RE(c)e',CO)e ~ ]/'] (X, = - 4 re~'~(kop-~/4)e-Si -. co.^i + RC()e cv']. (9) Y ~/ (x,y) (x',y')= Source Point (x',-y').lmage Point P (x,y)=Observation Point 2P (x',y') / Figure 3: Geometry of the line source and its image. It can easily be shown that for an electric current distribution J, the radiated far field does not have a p component and the far field amplitude defined by E =- ei(oP- r/4)S IV kop 9

has components S= ko0Z {fs os J:(xX' y)eiko sin*kx'[e-iko cos~ky' _ Ru(q)eiko cosy']dx'dy' - in Jy(x y)e-i i:[e-ikoo' + RH(p)eik cosY']dx'dy'}, (10) So = -kZ0 fs Jz(X, y')e-iko sin,x [e-iko c + RE()eiko cos Y']dxldy, 2.3 Scattering from Inhomogeneous Periodic Dielectric Layer above a Half-Space Layered Medium Consider an inhomogeneous dielectric layer of thickness t on top of a stratified halfspace dielectric medium as shown in Fig. 1. The permittivity of the inhomogeneous layer is represented by e(x, y) which is a periodic function of x with period L. Suppose this structure is illuminated by a plane wave whose angle of incidence and polarization respectively are ~o and p. For an E-polarized wave p = i and for an H-polarized wave p = -cos - sin oy, thus the incident wave may be represented by Ei = peiko(sinox-cosy) (11) A polarization current distribution is induced in the inhomogeneous layer. This current gives rise to a scattered field that can be obtained from (7). The polarization current is proportional to the total field within the inhomogeneous layer. The total field is comprised of the incident field, the reflected field which would have existed in the absence of the inhomogeneous layer, and the scattered field, i.e. Et = Ei + E + E3. 10

The polarization current in terms of the total field is given by J(x, y) = -ikoYo((x, y) - 1)Et. (12) Upon substitution of expression (7) for ES into (12) a set of integral equations for polarization current can be obtained; for E polarization we have J(x, y) = -ikoYo((x, y) - 1)(eikosinxl[e-ikcosy+ RE(o)eikcos1y] + fo' f+ Jz(x', y')Gzz(x, y; x', y')dx'dy'}, and for H polarization J,(x,y) = -ikoYo((x, y) - 1){cO kos sit[_e-ko coo + RH(O)eiko c y] + fo f+I[J.(x', y')G.(x, y; x', y') + Jy(x', y')Gy(x, y; x', y')]dx'dy'}, Jy(x, y) = -ikoYo(e(x,y) - 1){- sin oeik in[e-iko csy + RH( o)eikO COSOy] + Jo f+ [J (x', y')Gy,(x, y; x', y') + Jy(x', y')Gyy(x, y; x', y')]dx'dy'}. (14) Since there is no closed-form representation for the kernel of these integral equations, finding the solution, even numerically, seems impossible. But by employing Floquet's theorem the integral equations can be reduced to a form which is amenable to numerical solution. The fact that the permittivity of the inhomogeneous layer is periodic in x,excluding a phase factor, all the field quantities are required to be periodic in x. Therefore the polarization current must satisfy Jc(x + nL, y) = J,(, y)eiksin"OnL (15) 11

Now by using (15) the integration with respect to x can be simplified significantly by breaking the integral into multiples of a period, that is +00 I, = f G (x, y; x',y')J(zx',y')dx'dy' ~~~-00oo^~~~~~~~ (16) n =-00 At this stage, if the variable x' is changed to x' + nL and property (15) is used, Iz, becomes xo+L I = GPz(x, y; x', y')Jz(x', y')dx'dy' where +00 GP(x, y; Zx, y') = E Gz(x, y; x' + nL, y')ek sin"nL n=-oo rl, —00 If the expression for G,z as given by (6) is inserted in the above equation and the order of summation and integration is interchanged, and then the identity -+0 +o00 Z einL(kosinno+k.) = 2r [ b[(k. + kosinq o)L- 2n] n=-oo n=-oo is employed, the periodic Green's function simplifies to 7 +00 knx (x-X') GPx(x, Y: x, Y) L Z [e ikyle Y + RE(7n)eikn9(Y+Y)] (17) L n=-cc ny where 27rn kn L -ko sin 0o, k,= k\2 - k and knz'n = arctan( ). 12

Other elements of the periodic Green's function can also be obtained in the same manner -00 fly GP(, Y; x', y ) = (1j+ a 2 [eik l -y'l - RH(y)eikn(y+y')] eik (X; aYZ k xy n=-oo G(, y; Y', y') =- Io +- V [eikn yl-y' -+ RH(yn)e kny(Y+Y')] e-kfl(kn ) xy, (Z, y; x' y') Y= -Y- 2o o~ 4 )e knniz(' G PZ(x, y;, y )= =- (1 + (10 " ~ [eiknlY IY' + RH(yn)eikf(Y+Y')] e-t'k (18) The integral equations (13) and (14) now take the following form Jz(x, y) = -ikoYo(e(x, y) - i){ei'o sinO[eikcO cS Oy + RE(o)e i c y] t +L/2 +j fJ Jz(x', y')Gr(x, y; x', y')dx'dy'}, o -L/2 J(zx, y) = -ikoYo(E(x, y) - l){~cos oeik n[_e-k o cos~oy + RH(R(O))iko cos Y] t +L/2 + f [J(x', y')GP(x, y; x', y')+ J(x',')G (,;', y)ddy'}, 0 -L/2 Jx(x, y) = -ikoYo(e(x, y) - 1){- sin Ooeiko sin o0 [e-iko cos oy + RH(ko)e'ko CS o Y] t +L/2 + f f [Jx(x', y')GP (x, y; x', y') + Jy(x', y')GPy(x, y; x', y')]dx'dy'}. o -L/2 (19) Far away from the surface (y > o0), contribution of only a few terms of the summations in (18) are observable. These terms correspond to values of n such that kny is real and they are known as the Bragg modes. Among all the Bragg modes the mode corresponding to n = O carries most of the scattered energy and this is specifically true when L < oA. The scattered field due to this mode is a 13

plane wave and for E and H Polarization, respectively, we have EE - Z t +L/2 [ikosin EE - 2Lcoso f f J(zx',y')[e ikco' + REeik~c~ y']e-ikosinox'dx'dy' ~ -L/2 " *eik (cos 0o y+sin 00oc)^ (20) L co TZSO St +~ L/2 J(, ko cos 0o _ RHeiko cos O e- iko sin o d xdy cEH - +2L cos o f f J(x',y')[e-ik O'- RH -kY']eik dx'dy' 0 -L/2 t +L/2 -sin0 f f Jy(x', y)[e-ik cosyY' + RHek cosoY']e-ik sinO3'dx/dy' 0 -L/2 ) (cos oO: - sin oyl)eiko(cosOY++sin<Ox) (21) 2.4 Numerical Implementation It is very unlikely to find an analytical solution to the equations as given by (19) even for the simplest form of (x, y). However an approximate numerical solution can be obtained using the standard moment method with point matching technique. In this method the cross section of the inhomogeneous layer over one period is discretized into small rectangular segments over which the dielectric constant and polarization current can be assumed to be constant. Now in equation (19) the integrals over one period of the inhomogeneous layer can be broken up into summation of integrals over each segment where the polarization current is constant. Let pq designate a cell whose center coordinate is (xp. yq) = (pAzx. qAy), where p and q are some integers and Ax and Ay are dimensions of the rectangular segments. If the polarization currents as given by (19) are evaluated at the center of uv-cell (point matching) the integral equations can be cast into matrix equations. 14

The matrices formed by this technique are known as the impedance matrices. The solution to this matrix equation gives the polarization current at the center of each segment. After a simple integration of the periodic Green's functions over the area of pqcell, it can be shown that the entries of the impedance matrix for E polarization (TM case) are of the form 2ik? (6(uv) 1)+oo L ( e((u, v)- 1) E [e ikny y-Yql + RE (n)e ikny(Y +Yq)]sin(knAy/2) L U,, e=- kn, xk" sin(knAx/2)e-ikat(xu~ -) v # q /"'(e(U, V)- 1),Z [_ieikv"yAy/2 + i +RE(n)ei2k" Z(u,v;p,q) n=-oo * sin(knyAy/2)] kik2 sin(knxAx/2)e-ikn"x(xu -p) v = q u $ p n=-oo -1 + 2/' (e(u v) - 1) Z [-ieik'"uw/2 + i + RE("yn)ei2k'nuY sin(knyAy/2)] knk2 sin(knA/2) v = q u = p. (22) For H polarization (TE case) the integral equations for JI and Jy are coupled, which result in coupled matrix equations that can be combined into a single matrix equation. The resultant impedance matrix consists of four sub-matrices of the following form ~1 Z2 2= zz2 Z3 z4 15

whose entries are given by n=-oo 2ik2+00 RHQkn2Yi2knyyv Z(uv;,q) = sin(,knyy/2) sin(-kn2/)e (kn px/2)e Vikn(u) v = q u $ p -1 + L -(e(u, v) - 1) Z [-ieiknYy/2 + i - RH(n)e Yv sin(kAy/2)]^ 2.sin(^,Aal/2) v = q u = p= (23) -(e(u, v) - 1) E [sgn(y, - yl)e'knIYv -yql + RH(Y)eikny (Yv +Y)] n=-oo Z2(u, v; p, q) = (k sin(k-Ay/2) sin(knAz/2)e -kn("u ") V q (e(u) - 1) E RH(7n)ei2knyv sin(kA Ay/2)e - * sin (Aknzz/2 )e-'ik1" ("u-P ) v = Q, (e(u, )-l) E) [sgn(y, - yq)e&kUnlY-Yel -RH (^)eiknY(Yv+Y-)] Z(; sin(kAy/2) sin(knl*/2)e -ikn(xu -p) U $ q Z3(U L'; P, q) = + (e(U)- 1) Z -RH(yn)et2knyY9 sin(k,ay/2)fn=-oo0 *sin(knaA/2)e-'kn"('u-rp) =, (25) 16

n=-oo 0 L (e(u, v) -l) I2 {(1 _ >)eikil - k~)[~'~l~-rl + Ra(%)e~'* ~-o+ 4>] knk2 sin(knyAy/2) sin(kAnzA/2)e- kn(-uP)} v # q (e(, v) -) [ —i(1 - )e'iknAy/2 +i + RH(n)(l - l=-00 0 0 Z4 (U, v; p q) = sin(knyAy/2)ei2knYYU ] knk sin(knxAZx/2)e-iknx(x-:p) Z4(u, v; p, q) - v = q u 7 p v=qufp -1+ L (e(u,) - 1) [-i(l - )eikA/2 + i + RH(- ) n=-oo - k0 *(1 - ) sin(knAy/2)e2k ] sin(knA/2) v = q u = p. (26) In (24 ) and (25) sgn(x) is the sign function defined by +1 if x>0 sgn(x) = (27) -1 if < O. All series in (24), (25), and (26) are exponentially convergent and thus the series can be truncated for relatively small n. This is also the case for (23) except for one term in the summand of the expressions corresponding to v = q. The convergence rate of S= E s c k/k2) Sin(knAx/2)e-tkn'(x -p) (28) n=-oo n, ny is very poor and in order to improve the convergence rate the standard trick is to add and subtract a series whose summand is asymptotic to the original series. For large n 27rn.27rn nm s L'ny L L7 I ~

and the summand is approximated by i L -- 2n sin(k - nxx/2)e-ikn"(x-x) ko2 27rn where the asymptotic forms for knz and kny are only used in the amplitude factor. It can be shown that + ~oo eina j -= = isgn(ce)(7r- a c ) (29) n=-oo,n:AO where c is a real number. By employing (29), a closed form for the asymptotic series (Sapp) can be obtained and is given by -L ikn(xu"-P) [sgn(xp -_ xu)( - 2xpL I) sin(ko sin oAx/2) Sapp = k -- cos(ko sin q-x/2)j zp xu 2o(1 - ) cos(ko sin OoAx/2) u = X Now (2S) can be written as _ - 2 k- L i (k2k i sin(ko sin soAx/2) S i nk 2 k k - sin(ksl~x,)Z)ee + ki ksin O n= — oo' sin/ ny 0 nOQ +Sapp in which case the series converges very fast. The right-hand-side of the matrix equations may be represented by an excitation vector whose elements, for E polarization, are b( u. v ) = ikoY'o((Z(,,y) - 1)eko Sin O[e-'k coso0 + RE( o)ek COS Ov] The excitation vector for H polarization is made up of two sub-vectors with entries bl(u, v) = ikoYO(e(xu, yv) - 1) cos koeik"e sin0u' [-e-ikO Co SOy + RH(o )ei<o COSo0y, ] b(u. v) = -ikoo(6(x,. yJ) - 1)sinoin Ooe' inou[e-ik oco O0Yv + RH (o)e "k Co800 V1 18

We point out that the inhomogeneous layer may have an arbitrary thickness profile with a maximum height t. In such cases we may assume that the layer has a constant thickness t and the permittivity corresponding to air-filled points is 1. 3 High Frequency Scattering from Stratified Cylinders For dielectric cylinders with large radii of curvature, physical optics may be used to obtain the scattered field provided the dielectric has sufficient loss to prevent significant penetration through the cylinder. The dielectric loss also suppresses the effects of creeping waves which enhances the physical optics results. If the dielectric cylinder is stratified, the physical optics approximation could still be used if the radius of curvature of all the interface contours are much larger than the wavelength. Two types of physical optics approximations can be applied: 1) surface integral and 2) volume integral approximation. In surface integral physical optics the equivalent surface currents are approximated by electric and magnetic surface currents of the infinite tangential plane. In the latter method the volumetric polarization current is estimated by finding the internal field using geometrical optics ray tracing. Of the two techniques, the surface integral physical optics is much easier to employ. New physical optics surface currents [Sarabandi et al 1990] that are more convenient to use than the standard ones [Beckmann 1968] are examined. These currents can be obtained by noting that the reflected plane wave from 19

a dielectric interface can be generated by equivalent electric and magnetic current sheets. These currents are normal to the plane of incidence and their density is proportional to the incident field amplitude, polarization, and associated Fresnel reflection coefficient. Suppose the incident field is given by Ei = Eoeikokr, H e = Hoe r and the normal to the cylinder surface is represented by the unit vector n. The unit vector normal to the plane of incidence is _ n x k, [ n x ki i in terms of which the new physical optics electric and magnetic currents are given by Je = -2Yo(Eo i) cos RiRE(Oi)eikk t (30) Jm = -2Zo(Ho t) cos kiRH(k)eikir t (31) Here, RE and RH are Fresnel reflection coefficients and,i is the local angle of incidence given by Xi = arccos(-2 k i). In shadow regions on the surface (>i > ir/2) the currents are zero. Suppose a stratified cylinder with arbitrary cross section is illuminated by a plane wave travelling in -x direction (~i = -x) as shown in Fig. 4. The outer 20

surface of the cylinder is described by a smooth function p(O). For E-polarized wave (Eo = z) only electric current and for H-polarized wave (Ho = Yoz) only magnetic current is induced on the cylinder surface as given by (30) and (31). It can easily be shown that in the far zone of the cylinder in a direction denoted by s3, the far field amplitudes for E and H polarization respectively are given by SE = ^ I cos iRE( i)e-ikP()(cos+cos(~'-'))Tp2(~,) + p, 2(e)dq' (32) S= If J cos piR(i)e- 0(')(co^'+o' -))(, p 2 ) + p'R2()d' (33) where the integral is taken over the lit region and p' is the derivative of p with respect to q. If the surface of the cylinder is convex the integral in (32) and (33)....::.. I Figure 4: Geometry of scattering problem of a stratified cylinder. can be evaluated using the stationary phase technique. The stationary point (OsSP) 21

is the root of the equation d-p(s') (cos o' + cos(<' - <)) = 0. By noting that at the stationary point Oi = 0s/2 and,o20(/) -,(O/ P(OsP) Pcos(sp - /12)' and also by defining d2 g = d (2 O) (cos q' + cos($' - S))]'=(p equations (32) and (33) become <S XsSP) k F7 SE = CS(- )RE() cos(s- /2),-,:0p(S,(co2 ) co ((s, p - - I ) (g), e,-ikop(5sp)(cos qSp+cos(qSp -3b,)) ( e-isgn() 34) = f^ D s P(XSP) k 0 2" 2 cos(Osp - /2)V 2 I -ikO P( sp)(cos5sP +cos(k4sp — fs)) }.-isgn(g) (:35) For a circular cylinder of radius a these expressions simplify to SE = kora cos(0,/2)RE( )e-'2ko~os(/2)e"7/4 (36) s1 = V/,o raos s/2)RH(6 )e-2koacos(~/2)ir (37/4 9 93~ ~ /)~j") -(37) To verify the validity of the physical optics expressions with new set of physical optics currents we compare expressions (37) and (3S) for a layered circular cylinder 2)2

with the exact series solution [Ruck et al, 1970, pp. 259]. Let us consider a twolayer cylinder with inner and outer radii of al = 10cm and a = 10.5cm respectively. The dielectric constant of the inner and outer layers respectively are 15+i7 and 4+ il. These values are so chosen to simulate a tree with smooth bark. Figures 5 and 6 compare the normalized backscattering cross section (o/7ra) of the cylinder for E and H polarizations using physical optics expressions and exact series solution. In these figures the cross section of the cylinder in absence of the outer layer (bark) is also plotted to demonstrate the effect of the bark on reducing the cross section of the cylinder. For frequencies above 2 GHz (koa = 4.2) the agreement between the two solution is excellent. The bark layer plays the role of an impedance transformer which reduces the cross section of the cylinder by 14 dB around koa = 16. 23

O. --.' ~ ~ I-'............ i.... i.....|......... -...... -5. -10. -15...... P.O. 2-Layer...v.|-.. P.O. I-layer 0 Exact 2-layer -20. 0. 5. 10. 15. 20. 25. 30. 35. k0a Figure 5: Normalized backscattering cross section (i) of a two-layer dielectric cylinder with a = 10.5cm, al = 10cm, e1 = 15 + i7, e2 = 4 + ii versus k0a for TMI case. 24

0..- -------------—..........'.....-...........-.................... -5.- % o -10. -15. 15, F $ P I 0 --- P.O. 2-Layer 9 5 F-.........-P.O. 1-layer 0 Exact 2-layer -20.... 0. 5. 10. 15. 20. 25. 30. 35. k0a Figure 6: Normalized backscattering cross section ( -) of a two-layer dielectric cylinder with a = 10.5cm, al = 10cm, l = 15 + i7, 2 = 4 + il versus koa for TE case. 25

4 Scattering from Corrugated Cylinder Consider a corrugated dielectric cylinder with arbitrary cross section as shown in Fig. 7. Assume the corrugation geometry is such that the humps are identical and of equal distance L from each other. Further assume that if the corrugation is removed the surface of the cylinder would be denoted as before by p(p) and the radius of curvature at each point is much larger than the wavelength and L. Under A6Y Out l~cm X Figure 7: A corrugated cylinder geometry. these conditions each point on the cylinder surface can be replaced, approximately, by a periodic corrugated surface. The accuracy of this approximation is in the order of physical optics approximation for smooth cylinders. Suppose the cylinder is illuminated by a plane wave travelling in -x direction and let us denote the tangential coordinate at the center of each hump by (x', y') where y1 coincides with the outward normal unit vector (n{()). If the origin of tihe 26

prime coordinate system corresponding to the mth hump is located at (pmr Qm) we have m T Pm = Ae~ - (38) g=1 e+i=. -— L (39) p 2(0- )+ p2(0) where Ali is a known quantity. The local incidence angle at the mth hump can be obtained from PM = arccos(n(km) x) and the induced current in the mth hump can be approximated by that of the periodic corrugated surface when the incidence angle is Sl. The scattering direction is denoted by qs as before and the scattering direction for the mth local coordinate is given by pm = ~s - <d^m The far field due to the mh hump (Sm), depending on the polarization. can be obtained from (10) and we note that those humps with I 0s l> wr/2 do not contribute to the far field. The total contribution of the cylinder corrugation to the far field is the vector sum of the fields due to each hump modified by a phase factor to correct for the relative positions of the humps. Therefore SZ = E Sme- k~"e- -iko(r cosos+ym sin,.) (40) rn 2 7

where Xm = Pm COS 0", Ym = Pm sin &. The total scattered field may now be obtained from S = SC + s,, where S, is the far field amplitude of the smooth cylinder. 5 Numerical Results To examine the effect of surface corrugation on scattering from corrugated cylinders, we consider a two-layer circular cylinder with uniform corrugation. The pertinent parameters are chosen as follows: each hump is a Ao/8 x 0o/S square with dielectric constant el = 4 + ii, the distance between humps is L = Ao/4, the thickness and dielectric constant of outer layer are Ao/2 and 4 + ii respectively, and the radius and the dielectric constant of inner laver are lOAo and 13 + i7T respectively. For the corresponding periodic surface (see Fig. 8) all the components of the induced current in each hump are obtained by the moment method and the amplitude and phase of the total current (f, J(x', y')dx'dy') are plotted in Figs. 9 and 10. Figures 11-14 show the amplitude and the phase of the zeroth Bragg mode as given in (20) and (21), the reflected wave in the absence of the corrugations, and the sum of the two waves (total reflected wave) as a function of incidence angle. For E polarization (Fig. 11) the total reflected wave is less than the reflected \wave 28

in the absence of corrugation (reduction in the scattered field). In this case, as far as the total reflected field is concerned, the corrugation can be replaced by a homogeneous dielectric layer with thickness \o/S and e = 2.6 + iO.58. For H polarization (Fig. 13) the total reflected wave is weaker than the reflected wave in the absence of the corrugation for angles less than the Brewster angle and vice versa for angles greater than the Brewster angle. Once the induced current versus angle is obtained the bistatic scattered field can be computed from (41). Figures 14 and 15 show the radar cross section due to the corrugation (a,) and smooth cylinder (a,) and the total radar cross section (ac + ao). To examine the role of the outer layer, the radar cross section of the cylinder (1-layer) when the outer layer is removed is also plotted. It is seen that the smooth bark reduces the scattered field by 3 dB and the corrugation on the bark further reduces the scattered field by another 8 dB. In Fig. 16 the radar cross section of the corrugated cylinder, for both polarizations, are compared with a smooth cylinder when corrugation is replaced by an equivalent uniaxial dielectric layer as explained in Appendix A. The thickness and dielectric tensor elements respectively are t = Ao/8, y = = 2.6 + i0.58, and 6e = 1.81 + i0.78. Excellent agreement is obtained. 6 Conclusions A hybrid solution based on the moment method and physical optics approximation is obtained for corrugated layered cylinders. The only restriction on the physical 29

dimensions is the radius of curvature (r) of the cylinder where we require r > A0. Also new physical optics expressions for the equivalent surface current on the dielectric structure is introduced. Also it is shown that when period of the corrugation is smaller than the half a wavelength the corrugation can be modeled by an uniaxial dielectric layer which extremely simplifies the problem. This method is employed to investigate the effect of bark and its roughness on the scattering from tree trunks and branches. It is shown that the bark and its roughness both reduce the radar cross section. The low contrast dielectric bark layer manifest its effect more significantly at higher frequencies where the bark thickness and its roughness are a considerable fraction of the wavelength. Acknowledgement This work was supported by the NASA under contract NAGW-1101.:30

References [1] Beckmann, P., The Depolarization of Electromagnetic Waves, Boulder, Co: The Golem Press, 1968. [2] Bodnar, D.G, and H.L. Bassett, "Analysis of an anisotropic dielectric Radome," IEEE Trans. Antennas Propag., pp. 841-846, Nov. 1975. [3] Cabayan, H.S., and R.C. Murphy, "Scattering of electromagnetic waves by rough perfectly conducting circular cylinders," IEEE Trans. Antennas Propag., 21, pp. 893-895, 1973. [4] Clemmow, P.C., and V.H. Weston, "Studies in radar cross section XXXVIDiffraction of a plane wave by an almost circular cylinder," Radiation Laboratory Report No. 2871-3-T, The University of Michigan, Sept. 1959. [5] Karam, M.A., and A.K. Fung, "Electromagnetic scattering from a layer of finite length, randomly oriented, dielectric, circular cylinders over a rough interface with application to vegetation," International Journal of Remote Sensing, 9, pp. 1109-1134, 1988. [6] Kong,J.A., Electromagnetic Wave Theory, New York: John Wiley & Sons, 1985. [7] Morita, T., and S.B. Cohen, "Microwave lense matching by simulated quarterwave transformers," IRE Trans. Antennas Propag., pp.33-39, Jan. 1956. 31

[8] Padman, R. "Reflection and cross-polarization properties of grooved dielectric panels," IEEE Trans. Antennas Propag., 26, pp. 741-743, 1978. [9] Sarabandi, K., F.T. Ulaby, and T.B.A. Senior, "Millimeter wave scattering model for a leaf', Radio Sci., accepted for publication. [10] Tong, T.C., "Scattering by a slightly rough cylinder and a cylinder with impedance boundary condition," Int. J. Electron., 36, pp. 767-772. 1974.:32

Y d _L - ------- Figure 8: Geometry of the corrugated surface 33

1.0 i 0.9 ___ z-component (E Pol.) 0.8......... |x-component (H Pol.) 0.7 _ ----- y-component (H Pol.) 0.6 "0 0.5 0.4 0.3.... 0.2..... 0.1 0. 10. 20. 30. 40. 50. 60. 70. 80. 9. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure 9: Amplitude of the total induced current in the two-layer periodic corrugated surface versus incidence angle. 34

180.0 -'.....''. —......|... |'."...|....|............. 180.0 I 140.0 100.0 -........ —- --- " — -. A'-'''' 60.0 - 20.0 Se~ [ --- z-component (E Pol.) -20.0 -........ x-component (H Pol.) ----- ycomponent (H Pol.) -60.0 - -. -100.0 -140.0 -180.0 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure 10: Phase of the total induced current in the two-layer periodic corrugated surface versus incidence angle. 35

1.0 0.9 -__ //_ 0.8 ---- Total / 0.68......... Zeroth Bragg Mode / 0.7. ---- No Corrugation /,/ 0.5 0.4 0.3 0.2 0.1 - - 0.0 _= 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. incidence Angle (Degrees) Figure 11: Amplitude of the reflected field from the two-layer periodic corrugated surface versus incidence angle for E polarization. 36

180.0 140.0 100.0 60.0.......... —20.0 -20.0 i -60.0 Total -100.0 -......... Zeroth Bragg Mode —...- No Corrugation -140.0 -180.0 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure 12: Phase of the reflected field from the two-layer periodic corrugated surface versus incidence angle for E polarization. 37

1.0 0.9 ____, 0.8 Total....... Zeroth Bragg Mode 0.7 ---- No Corrgation I 0.6 c3 /1 X 0.5 - 0.4 / / [.3 """ A- -/'' 0.3 F 0.2......... - - *"-*3~~~~~~~...., / 0.1 - \! 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure 13: Amplitude of the reflected field from the two-layer periodic corrugated surface versus incidence angle for H polarization. 38

180.0 140.0, 100.0- - ---- - _ 60.0- / 20.0 2 0. 0- -- -- — _______ —________________ -20.0:, - Total -100.0 ---—......... ZerothBragg Mode --- No Corrugation -140.0 t- 5- i -180.0.....-........ |: 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure 14: Phase of the reflected field from the two-layer periodic corrugated surface versus incidence angle for H polarization. 39

15.0 l~5.0'.........,....i........,................, 10.0 - - 10.0,,-.... —--------------------------—. —------—..., 5.0;; -5.0 | -----—. Corrugation ------ Smooth (2-Layer) -0 -Smooth (1-Layer) i..... -10.0 - -150. -120. -90. -60. -30. 0. 30. 60. 90. 120. 150. Scattering Angle (Degrees) Figure 15: The radar cross section of a corrugated cylinder for TM case with a = 10.5AO, a, = 1oAgo L = Ao/4, el = 4 + ill E2 = 15 + i7, and t = d = So/8 40

15.0.... 10.0. -' t s I. Smooth (2-Layer) l I | \.Smooth(l-Layer) -5.0 - f l * \ K \r ".. " —- \ -150. -120. -90 -60. -30 0 30 60. 90 120 150 Scattering Angle (Degrees) 41' > i i'.'. I \. Figure 16: The radar cross section of a corrugaoted cylinder for TE case with 100.5o,-i / - L Ao = 4 4...~ = 15 + i7, and t = d = Ao/8. 41

10.0 [000",,,,o m'n,!' --- -Numerical Result (E Pol.) -5.0 -------- Numerical Result (H Pol.) o Equivalent Layer (E Pol.) a Equivalent Layer (H Pol.) -10.0 -150. -120. -90. -60. -30. 0. 30. 60. 90. 120. 150. Scattering Angle (Degrees) Figure 17: The radar cross section of the corrugated cylinder for TE and TM cases using the numerical and the equivalent dielectric methods. 42

APPENDIX A SIMULATION OF AN ARRAY OF DIELECTRIC SLABS WITH AN EQUIVALENT ANISOTROPIC MEDIUM Al Introduction Our purpose in this appendix is to find the equivalent dielectric constant of a medium comprised of homogeneous dielectric slabs of identical material which are equally spaced. Depending on the polarization of the fields and the boundary conditions imposed, a variety of different modes can be supported by this structure. Modeling of grooved-dielectric surfaces with anisotropic homogeneous media, to our knowledge, was first studied by Morita and Cohen. They analyzed this problem by simulating the infinite periodic array using a partially-filled waveguide, then, by solving the appropriate transcedental equation the propagation constant in the corrugation was obtained. These results are used in many problems such as matching dielectric lens surfaces [Morita and Cohen, 1956] and designing broadband radomes [Bodnar and Bassett, 1975; Padman, 1978]. Since the interest is in the modes that are excited by an incident plane wave, simulating the array by a partially-filled waveguide which forces the tangential electric field to be zero at the waveguide walls may not be appropriate. Here we impose a condition that supports the modes which would be excited by a plane wave incident on the structure. This approach will provide a formula for the elements of the equivalent dielectric tensor which is different from that reported by MIorita and Cohen. The validity of this technique is verified by numerical solution A-i

of corrugated surfaces in Section A5. In Section A4 the reflection coefficient of uniaxial layered media is derived to extend the applicabilities of the ecquivalent dielectric technique to periodic surfaces with arbitrary cross sections. A2 Theoretical Analysis To proceed with the analysis, suppose that similar dielectric slabs of an infinite array with thickness d and dielectric constant e are parallel to the y-z plane. Further assume that the period of the structure is denoted by L. The geometry of the problem is depicted in Fig. A-i. If the x-y plane is the plane of incidence, the solution is independent of z and therefore the waves can be separated into E- and H-polarized waves. Each period of the medium can be divided into two regions iY Figure A-i: An array of infinite dielectric slabs. and, depending on the polarization, the z component of the electric or magnetic A-2

field must satisfy the wave equation, i.e. + 02 + eko) J("(x,y) = o < < d (Al) ( + - + ko) a"(xy ) = d < x < L (A2) where -(x, y) = Ey(x, y) or Hy(xy). Using separation of variables and requiring the phase matching condition, the solutions of (Al) and (A2) take the following forms'i(x, y) = [Aeik: + Be-ikx] eiky (A3) "'(5x, y)) = [Ceik' + De-ik] e k (A4) where ky is the propagation constant in the periodic medium which must satisfy (kY)2 + k2 = ek (A5) (k'1)2 + k2= (A6) In an attempt to find the unknown coefficients, we notice that the tangential components of the electric and magnetic fields must be continuous at x = d which constitutes two equations. Two more equations can be obtained by applying Floquet's theorem for periodic differential equations which requires xI(x + L, y) = aOr(x, y) for some constant a. To set an appropriate value for a suppose the medium is simulated with an equivalent homogeneous dielectric. If a plane wave illuminates the half-space of the equivalent homogeneous medium at an angle po, the x dependency would be of the form ek~ SIn"Ox. This dependency suggests that we need to A-3

impose a progressive phase condition, i.e. let cr = eikosin0L. Therefore the other two equations become Etan(O + L, y) = eik~ sin"oLEtan(, y) (A7) Htan(O + L y) = eiksinOLHtan(0, y). (AS) Application of the mentioned boundary conditions for the E polarization gives the following equations Aeik^d + Be-ikCd = Ceik'd + De-ikd k1 [Aeikd - Be-6ikd] = kI [Ce i'd - De-i"d]( A + B = [Ceik' + De~i4'1] e-ikosinfoL k- [A - B] - kJ [Ceik' - De-ik"] e-ik' sin"oL Since we are interested in the nontrivial solution of the above linear equations the determinant of the coefficient matrix must be set to zero. This condition provides an equation for ki and k,' and is given by -(? + i) sin(kfd) sin (kf(L - d)) + 2 cos(kld) cos (kl(L - d)) (X10) 2 cos(ko sin OoL) Dispersion relations (A5) and (A6) give rise to another equation for k' and k"1. i.e. (k)2 - (k)2 =- ko( - 1) (Al l) The transcedental equation (A10) together with (All) can be solved simultaneouslv to find the propagation constants. It is worth noting that in a limiting case A-4

when the periodic medium approaches a homogeneous one then kI -* k7l and (A10) reduces to cos(kL) = cos(ko sin SoL) which implies kx = ko sin 40 as expected. After solving (AlO) and (All) for kI', the propagation constant in the ydirection can be obtained from k2= k2 _ (k')2 (A12) It should be pointed out that the solution for k4 is not unique and therefore this structure can support many modes corresponding to different values of k7I. The dominant mode for this structure corresponds to a value of kI' such that the imaginary part of ky is minimum. For the equivalent homogeneous medium with permittivity e6, the propagation constant in the y-direction would be k= k2 (e -sin2 40) (A13) Comparing (A12) and (A13) it can be deduced that ez =sin24 o+l- (k) (A14) From the symmetry of the problem it is obvious that if the electric field is in the y-direction the equivalent dielectric would be the same, that is 6y = e,. Using a similar procedure the following transcedental equation for H polarization can be obtained k-(k + f) sin(kd) sin (kIL ( -d)) + 2 cos((L cos (kL 2 cos(ko sin o0L) A-5

The equivalent dielectric constant e6 can be obtained from (A14) where kII is the solution of (A15) and (All). Since e, 4 cy = Ez the equivalent medium is uniaxial with the optical axis being parallel to the x-axis. A3 Low Frequency Approximation An analytical solution of equations (A10) and (A15) for kI and kI' cannot be derived, in general, but using Newton's or Muller's method numerical solutions can easily be obtained. One of the cases where approximate expressions for ex and y. = eZ can be derived is the low frequency regime where L < 0.2A0. In this approximation the sine and cosine functions are replaced with their Taylor series expansion keeping up to the quadratic terms. Therefore equations (A10) and (A15), respectively, reduce to (k )2d + (k"I)2(L - d) = (ko sin 00)2L x{~~~~ x ~~~~~(A16) (k )2d + (kII)2(L - d) = (k sin 00)2 L. These equations together with (All) can be solved easily to obtain d L-d Cy =fZ +=- + (A17) L L ittt en d ie, te el, s -- t+ e ad teno d + a then pA-18) (e - ()ed)L - d) + eL2 ~iL - d) + d Note that when d - L, the elements of the dielectric tensor approach the permittivitv of region I, i.e. er, y,, e -+ E and when d -- 0, then 6 e, f, -+ 1, as expected. There are two interesting points in (Al7) and (A18) that should be A-6

mentioned. The first is that the equivalent permittivities are not functions of frequency (co). The second point is that ~y (= ez) is not a function of incidence angle and variation of e. with incidence angle is very small. This variation is higher for larger values of e. A4 Reflection Coefficient of Uniaxial Layered Medium Consider a multilayer dielectric half-space as shown in Fig. A-19. Suppose each dielectric layer is uniaxial and the optical axes of all the layers are parallel to the x axis. Further assume that a plane wave whose plane of incidence is parallel to the x-y plane is illuminating the stratified medium from above at an angle <o. The interface of the nth and the (n + l)th layers is located at y = dn. - - Y e d2 ___ —------- d, ___________________________ dN Figure A-19: Plane wave reflection from a stratified uniaxial dielectric half-space. The dielectric tensor of the nth layer is defined by D" = nE"E and is assumed A-7

to be of the following form en, 0 0 En 6= O 0 En 0 (A19) 0 0 E, and its permeability is that of free space (Pn = /o). In this situation two plane waves are generated in the dielectric slabs: one ordinary and one extraordinary. For the ordinary wave the electric field and the electric displacement are parallel and both are perpendicular to the principal plane (the plane parallel to the optical axis and the direction of propagation) [Kong, 1985; p. 68], hence the magnetic field is in the x-direction (see Fig. A-19). For the extraordinary wave, however, the electric displacement and the magnetic field lie in the principal plane and the y-z plane respectively which force the electric field to be parallel to the x-axis. Therefore the ordinary or the extraordinary waves can be generated, respectively, by a magnetic or an electric Hertz vector potential having only an x component. For ordinary waves D" = oeEnE and the magnetic Hertz potential (H7/ = Iox) must satisfy the wave equation, i.e. V2n + kc kJI, = 0 (A20) The electric and magnetic fields in terms of the magnetic Hertz potential are given by En = iktboV x i (A21) Ho= V x V x 7 (Ai2) A-S

In the case where there is no variation with respect to z (A21) and (A22) reduce to En = -ikoZo y (A23) 9y Ho =- + O y a (A24) 9y2 9xay The electric and magnetic fields associated with the extraordinary wave, as discussed earlier, can be derived from an electric Hertz vector potential, i.e. H = -ikoYoV x I," and from Maxwell's equations v x En = k v x Ine which implies E = k2 IJ + VPn (A25) where Pn is an arbitrary scalar function. Using (A19) the electric displacement in the nth layer may be represented by De= o[(n- En )Eex + CnEe] (A26) Inserting (A25) into (A26) and noting that Dn = VV. 7n - V211 A-9

we get an V2H- + enkoll + (en - n) + V (en -V. -: ) = 0 (A27) So far no condition has been imposed on the scalar function (n. To simplify the differential equation (A27) let V. Hn = 6nDn be the gauge condition. Therefore the electric Hertz vector potential must satisfy 2 - En0 rd1 V2 rI + en k2I + 6n -n e = 0 (A28)'n 0X2 In the special case where a/az = 0 the magnetic and electric fields of the extraordinary waves can be obtained from an n H = ikoZo "z (A29) ay n1 a= 2n" 1 a2 + n (A30) ~ k L, + ax2)~+ xA30y The solution of the differential equations (A20) and (A28) subject to plane wave incidence can be represented by ei(k"xkOYY) and ei(k ex key) respectively and upon substitution of these solutions in the corresponding differential equations the following dispersion relationships are obtained (ko)2 + (ko,)2 = eko (A31) 1 1 -(k)2 + (k,)2 k (A32) en e 6rn Imposing the phase matching condition, i.e. k n = k~ = ko sin (), the dispersion relations simplify to ky = ko,A-sin o A-10

key = ko n -sin2 40 Vn The Hertz vector potentials in the nth layer can be written as nI = [A'e-k e Y + A eiko'Y] eikosin0 (A33) In = [Bt-e ey + BneikY] eikosin4ox (A34) In (A33)-(A34) the subscripts i and r in the coefficients denote the propagation in the negative and positive y direction respectively. Since there is no variation with respect to z, the incident wave can be decomposed into parallel (H) and perpendicular (E) polarization which would excite extraordinary and ordinary waves respectively. For E polarization the electric and magnetic fields in each region can be obtained from (A23) and (A24) using (A33). In region 0, A? is proportional to the incident amplitude, and -A~/A~ = RE is the total reflection coefficient. In region N + 1, which is semi-infinite, AN+'1 = 0. Imposing the boundary conditions, which requires continuity of tangential electric and magnetic fields at each dielectric interface, we can relate the field amplitudes in the nth region to those of (n + 1)th region. After some algebraic manipulation, the following recursive relationship can be obtained A? (-An+1/An+') rn + e-'2k 1 dn+ Awh (-A+/A )r + e-e2k where /n -sin2 - en+l - sin2 \0tn - sin + n - sin A-11

Starting from -A+l'/Ai1+ = 0 and using (A35) repeatedly RE can be found. For H polarization incidence the extraordinary waves are excited and using (A34) in (A29) and (A30) the electric and magnetic fields in each region can be obtained. Following a similar procedure outlined for the E polarization case an identical recursive formula as given by (A35) can be derived. The only difference is that A is replaced by B, koy by ke,, and Fn becomes Fn which is given by n _v/x(n+1)en+lV/n - s2 - /n si/en+l - sin2 o /x(n+1)'n+l6n - sin2 1 + V'e','nf+l - sin2 ) A5 Numerical Examples As mentioned earlier (A10) and (A15) can be solved using numerical methods. Here we use Newton's method to find the zeroes of the functions given by (A10) and (A15) in which k4 is expressed in terms of k"I using (All). Before searching for zeroes we note that these functions are odd functions of k', that is, if K is a solution so would be -K and both would give identical solutions for the equivalent permittivity as given by (A14). To study the behavior of the equivalent dielectric tensor elements, we consider a medium with e = 4 + ii and L = Ao/4. Figure A-3 and A-4 depict the variation of the real and imaginary parts of ex (H polarization) and ey = eZ (E polarization) versus angle of incidence for d/L = 0.5. It is shown that the dependency with the incidence angle is very small and these results are in agreement with (AlT1) and (A18) within 10%. The real and imaginary parts of the equivalent permittivities as a function of d/L for incidence angle So = 45~ are shown in Figs. A-. and A-6 A-12

respectively. As L/Ao increases the propagation loss factors (Im[k,]) of different modes become comparable to each other. For example, if L/Ao = m/2, where m is an integer, there would be at least two modes with equal propagation loss factor. Figures A-7a through A-7d show the location of zeroes of (A10) in kl'-plane for d/L = 0.5, 40 = 450, and four values of L/Ao. The real and imaginary parts of the equivalent permittivities as a function of L/Ao for d/L = 0.5 and ho = 45~ are shown in Figs. A-8 and A-9. The discontinuity in the equivalent permittivities at integer multiples of Ao/2 are due to the abrupt changes in the location of the zeroes in the k>I-plane which correspond to the dominant modes. To check the validity and applicability of this technique we compare the reflection coefficient of a corrugated surface using the moment method as discussed in Section 2.4 with the reflection coefficient of the equivalent anisotropic medium as derived in Section A4. The geometry of the scattering problem is depicted in Fig. 7. We consider a case where L = Ao/4, d = t = Ao/8, el = 4 + il, and 62 = 15 + i7. The tensor elements for these parameters are found to be e, = 1.81 + i0.15, ey = EZ = 2.6 + i0.58. Figures A-10 and A-11, respectively, compare the amplitude and phase of the reflection coefficients for both E and H polarizations. Excellent agreement between the results based on the moment method and the equivalent layer is an indication for the validity of the model. As frequency increases, because of presence of higher order modes, the discrepancy between the moment method and the equivalent layer results becomes more eviA-13

dent. Figures A-12 and A-13 show the reflection coefficient of a corrugated layer with L/Ao = 0.4, d = t = 0.2A, e1 = 4 + il, and 62 = 15 + i7. In this case e = 1.81 + i0.2, y = e, = 2.77 + i0.78 and the agreement is still very good, but for L > Ao/2 where more than one Bragg mode exists the model fails to predict the reflection coefficient accurately. Success of the equivalent layer in modeling rectangular corrugations can be extended to arbitrary periodic geometries. By approximating the cross section of the periodic surface with staggered increments of equal height, the surface can be viewed as a stack of corrugated layers (see Fig. A-14). The height of each layer AH must be chosen such that AH < A/10 where A is the wavelength in the material. Then each corrugated layer can be modeled by an equivalent anisotropic slab and the reflection coefficient of the resultant uniaxial layered medium can be obtained. To demonstrate this method consider a wedge-shape microwave absorber with permittivity e = 2+i0.5, period L = 0.4Ao, wedge height H = 1.5Ao, and base height D = 1o. The number of layers considered here is 30 and the corresponding reflection coefficients for both E and H polarizations as a function of incidence angle is depicted in Fig. A-15. A-14

5.0 4.5 4.0 E Polarization......... H Polarization 3.5 3.0'co 2.5 2.0 -............. -----------............-" 1.5: 1.0 0.5 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure A-3: Real part of the equivalent dielectric tensor elements for periodic slab medium with L = Ao/4, e = 4 + ii, and d/L = 0.5 versus incidence angle; e: (H polarization), e, = e, (E polarization). A-15

1.0 ^ 0.9 0.8 |- - -. E Polarization -i HPolarization 0.7 0.6!" 0.5 0.4 0.3 0.2 - O'1'........................................................................ 0.0, 0. 10. 20. 30. 40: 50. 60. 70. 80. 90. Incidence Angle (Degrees) ~~~~~Figure A-4:- 4 anr Figure A-4: Imaginary part of the equivalent dielectric tensor elements for periodic slabmedium with L /4, 4 + i, and dL = 0.5 versus incidence angle; (H Polarization),,y (E polarization). A-16

5.0.... 4.5 4.0 - E Polarization........ H Polarization 3.5 3.0 0.0 I. At......I......I..I. 0.0 20.5 0.6 0.7 0.8 0.9 1.0 2.0 K'"" d/1 Figure A-5: Real part of the equivalent dielectric tensor elements for a periodic slab medium with L = Ao/4, e = 4 + il, and <o = 450 versus d/L; e. (H polarization), y = e, (E polarization). A-17

1.0:r 0.9 E Polarization 0.8 -...... H Polarization j 0.7 - y^ /0.7 _ 0.5 / 0.2 0.4 - / 0.3 - /' -.0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 d/1 Figure A-6: Imaginary part of the equivalent dielectric tensor elements for periodic slab medium with L = A0/4, E = 4 + il, and ko = 45~ versus d/L; e: (H polarization),,y = c, (E polarization). A-18 A-18

Io. 6.oo00 --------- --------- ------------- — I —- - - ---— I.000. --- -- --------.00 _ - 10,00 ~ ~ _'' 0*0 ~ - 4 - -! ~ ~ ~ - -- - -O -..M O - - - 2. 0 0 0 - 6. 0 0 -- - - I,,, o.o000 -- - - - - --- ---- - -- - O. - -----------. - - - - - _-_ - - - - - _ __ -2.000.....f-. —.-0 -.. —..-..... O.(OOO,,,.. —.-..*. —— * —** —-*^ ii^ ^..,ooo. —-.- ------ -J- ---- " —-' —— _-_-' _._.*.. aI I I~ Ii II \ 1,. o,,j, ~1 ~ I"\,,., -2.ooo'..,'-,., I 8.000'/...... t I \ I I, -10.0 0 L.. -1000 - -... - - - - - - - - - - - - - - - -10o.0o 4.o00 -O000 2.000o 68000 0 -10.00 -6.'000 -2.000 2.000.0O - 0 Figure A-7: Location of zeros of (A-lO)in the k.I-plane for for the periodic slab medium with e = 4 + il = 45~, d/L = 05, and (a) L = 0.2Ao, (b) L = O.5Ao, (c) L = 0.8Ao, (d) L = 1.4Ao. A-19:c... i i~~o \d / =, I ~ I:~~~~~~~~A1

5.0,'..... |........... 4.5 4.0 E Polarization -—. ——. H Polarization 3.5 3.0 2.5 - 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1A Figure A-8: Real part of the equivalent dielectric tensor elements for periodic slab medium with e = 4 + il, and 4o = 450, and d/L = 0, 5 versus L/Ao; ex (H polarization), e, = e, (E polarization). A-20

1.0......'.. 0.9 E Polarization ----- H Polarization 0.8 0.7 0.6 0.5 0.4 0.3 - / / * a - 0.2 - 7 0.1 0.0...... 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Figure A-9: Imaginary part of the equivalent dielectric tensor elements for periodic slab medium with e = 4 + ii, and 4o = 450, and dIL = 0,5 versus L/Ao,; c (H polarization), Y = e, (E polarization). A-21

1.0 I. I, I, I I III.. I....,.,.... i,.... I.....I,... 1.0 0.9 - Numerical Result (E Pol.) 0- -8_ -. Numerical Result (H Pol.) 0.8 O Equivalent Layer (E Pol.) / 0.6 0.5 0.4 0.3 02 r~a[.. J.... i>Io ]...'' " 0.2 0.0 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. incidence Angle (Degrees) Figure A-10: Amplitude of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.25,o. A-22

180.0 160.0 140.0 120.0 Numrical Result (E Pol.) c v.... Numrical Result (H Pol.) bo) 100.0 Equivalent Layer (E Pol.) / Equivalent Layer (H Pol.), Q 80.0 I' 0i 1~:/C 60.0 -, 40.0 -,, 20.0 - 0.0' * * * *.. I.. i.... I... 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure A-ll: Phase of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.25Ao. A-23

1.0. I. * I....I II.. - I....II* I I I....i' 0.9- --- Numerical Result (E Pol.)......... Numerical Result (H Pol.) / 0.8 0.8 E- / - 0 Equivalent Layer (E Pol.) I, 0.7 C D Equivalent Layer (H Pol.) / 0.6 /'0 0-5 - ^ /o 0 0.5 0 0.4 0.3 0.2.oo 0.1 g-: - —' -'B-''I —------- 00 I.... ICA 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure A-12: Amplitude of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.4o. A-24

180.0. i I.. I I I Numrical Result (E Pol.) 160.0 p. -..... Numrical Result (H Pol.) i 140.0 o Equivalent Layer (E Pol.) C Equivalent Layer (H Pol.) o, ______________________________0 w 120.0 - _ ou i 100.0-,0 - 80.0 al0 - "' 60.0 0 40.0 o' ee 1 20.0 0.0 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure A-13: Phase of reflection coefficient of a corrugated surface for both E and H polarizations versus incidence angle; L = 0.4Ao. A-25

-L L isY -L L H D AH Figure A-14: Geometry of a wedge-shape microwave absorber and its staircase approximation. A-26

UNIVERSIY OF MICHIGAN 3 9015 03695 6202 0.I I, -10. - -E Polarization H Polarization m -20. -30. -40. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. Incidence Angle (Degrees) Figure A-15: Amplitude of reflection coefficient of a wedge-shape microwave absorber for both E and H polarizations versus incidence angle; L = 0.4A0, H = 1.5AXo D = 1Ao, and = 2.5 + iO.5. A-27