AFCRL- 68_- ELECTROMAGNETIC SCATTERING FROM CERTAIN RADIALLY INHOMOGENEOUS DIELECTRICS by Nicolaos Georgiou\Alexopoulos The University of Michigan Radiation Laboratory 201 Catherine Street Ann Arbor, Michigan 48108 November 1968 Scientific Report No. 3 Contract No. F19628-68-C-0071 Project 5635 Task 563502 Work Unit No. 56350201 Contract Monitor: Philipp Blacksmith Microwave Physics Laboratory Prepared for Air Force Cambridge Research Laboratories Office of Aerospace Research L.G. Hanscom Field Bedford, Massachusetts Distribution of this document is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general public. Submitted in partial fulfillment for a Doctorate in Electrical Engineering at The University of Michigan, Ann Arbor, Michigan, 48108.

FOREWORD This report, 1363-3-T, was sponsored in part by Air Force Cambridge Research Laboratory Contract No. F19628-68-C-0071. It represents research work performed by Dr. Nicolaos Alexopoulos in partial fulfillment of his Ph. D. degree at The University of Michigan. ii

ELECTROMAGNETIC SCATTERING FROM CERTAIN RADIALLY INHOMOGENEOUS DIELECTRICS by Nicolaos Georgiou Alexopoulos ABSTRACT In this research, the phenomenon of electromagnetic wave propagation through, and scattering from, radially inhomogeneous dielectrics is studied for very high frequencies. The dielectrics are considered lossless, radially inhomogeneous in the spherical coordinates system, and of the converging or of the diverging type. The lens problem is studied by the geometrical optics technique and the radar cross-section of perfectly conducting spheres coated with radially inhomogeneous dielectrics is investigated. By assuming a plane wave as the incident electromagnetic field, the contribution in the backscattering direction due to the reflected field and the creeping waves is determined by applying asymptotic theory. This necessitates the use of the WKB and/or Langer's method for the solution of the pertinent differential equations, depending on whether there exist transition points in the range for which the solutions are required. Also, the integrals of Scott (1949) are needed in order to determine the reflected portion of the field. Such a study is interesting not only from the theoretical but also from the practical point of view, in that it lends itself useful to the understanding of radio wave propagation in radially inhomogeneous dielectrics and of the effect of coating perfectly conducting spheres with radially inhomogeneous media. It also has applications to problems of wave propagation in the ionosphere and around the earth. To begin with, a general outline of the problem and the methods of solution is given. Then, a new class of radially inhomogeneous dielectrics is introduced and it is studied by the ray tracing technique. This new class of iii

radially inhomogeneous dielectrics is also treated as the coating of a perfectly conducting sphere and the monostatic cross-section is examined when the dielectric is of the converging or diverging kind. Finally another class of radially inhomogeneous media, previously discussed by Nomura and Takaku, is considered and its effect in reducing or enhancing the radar cross-section of a perfectly conducting sphere is determined. iv

TABLE OF CONTENTS Page LIST OF TABLES vi LIST OF ILLUSTRATIONS vii CHAPTER ONE: GENERAL CONSIDERATIONS 1 1.1 Introduction 1 1.2 Scattering from Radially Inhomogeneous Media 5 1.3 Outline of Research 8 CHAPTER TWO: A NEW CLASS OF RADIALLY INHOMOGENEOUS MEDIA 22 2.1 Introduction 22 2.2 Solution for the Eigenfunctions when E(2)=(1-) /(-) 24 2.3 Geometrical Optics Approach for the New Class of Lenses 29 CHAPTER THREE: HIGH FREQUENCY BACKSCATTERING FROM A PERFECTLY CONDUCTING SPHERE COATED WITH THE NEW CLASS OF RADIALLY INHOMOGENEOUS DIELECTRICS 36 3.1 Introduction 36 3.2 The Asymptotic Solutions of S 1 () and T () 37 3.3 The Reflected Electric Field -/2 V//2 41 3.4 Derivation of the Reflected Electric Field by Geometrical Optics 50 3.5 Scattering Cross-Section Computations 54 3.6 An Outline for the Creeping Wave Contribution in the Backscattering Direction 55 CHAPTER FOUR: HIGH FREQUENCY BACKSCATTERING FROM A PERFECTLY CONDUCTING SPHERE COATED WITH A DIELECTRIC WHOSE INDEX OF REFRACTION IS N(g) = gP 69 4.1 Introduction 69 4.2 The Radial Eigenfunctions in their Asymptotic Form 70 4.3 The Reflected Electric Field 73 4.4 The Geometrical Optics Approach 80 4.5 Numerical Computations 83 4.6 The Creeping Wave Contribution 86 CHAPTER FIVE: CONCLUSIONS 94 BIBLIOGRAPHY 96 v

LIST OF TABLES Table Title Page 4-1 Computations of aN for Various, and p 85 vi

LIST OF ILLUSTRATIONS Figure Title Page 1-1 Geometry of the Problem 4 1-2 Contour C in the Complex v-plane 8 1-3 The Deformed Path in the Complex v-plane 9 2-1 Case 1: 0< y<1 26 2-2 Case 2: y> 1 27 A 2-3 Case 3: < 0, y =-h, h> 0 28 2-4 Ray Path through the Inhomogeneous Dielectric 30 2-5 Deviation Angle vs. Angle of Incidence 33 9-6 Deviation Angle vs. Angle of Incidence 34 2-7 Maximum Deviation Angle vs. h 35 3-1 The Geometry of the Problem (c = a)4 40 3-2 Contours of Integration and Different Regions Considered in the Complex v-plane 44 3-3 Paths of Incident and Reflected Rays when 0 < y < 1 52 3-4 <aN vs. 3, for 0< <1 56 N1 3-5 aN vs. 3, for 1.1 < 2.0 57 3-6 D vs. ka for y= 3/4 58 3-7 Dvs. ka for = 3/4 59 3-8 Dvs. ka for 7= 3/2 60 3-9 D vs. ka for y= 3/2 61 3-10 D vs. ka for 7 = 3/4 62 3-11 Dvs. ka for 7=33/4 63 3-12 Dvs. ka for 7= 0.993 64 3-13 D vs. ka for 7= 0.993 65 4-1 Paths of Incident and Reflected Rays 81 vii

CHAPTER I GENERAL CONSIDERATIONS 1. 1 Introduction The problem of electromagnetic scattering from radially inhomogeneous media has been considered in the past by many authors. On the subject there exist some books such as Brekhovskikh's (1960) and Wait's (1962) and numerous articles published in technical journals. The problem in its most general form was considered by Gutman (1965). Gutman assumed the electromagnetic properties of the medium to be inhomogeneous in the angular as well as the radial direction. He applied a modified form of the Hansen-Stratton vector wave-function method due to Kisun'ko in order to solve the vector wave equation and thus to determine a representation of the electromagnetic field in the medium. The solution which he obtained, however, is of a purely formal nature since it involves an infinite set of first order linear ordinary differential equatl jis. Explicit results can be obtained if the inhomogeneity is only in the radial direction and it is with this case that this research is concerned, Marcuvitz (1951) gave a rather systematic treatment of the electromagnetic field representation in a medium whose index of refraction depends on the radius in the spherical coordinate system. Nomura and Takaku (1955) studied the radio wave propagation around the earth. They considered both the earth and the atmosphere radially stratified with the permittivity being r r 2pK given by (r) = (r), p < -1, K E index pertaining to the. th layer of stratification. Tai (1958a) applied the vector wave-function method of Hansen and Stratton to obtain a complete representation of the electromagnetic field by superposing electric and magnetic types of waves each of which he expressed in terms of two vector wave functions. He then applied these general results to the particular case of a sphere whose index of refraction N(r)= r2 1/2 a 2\ 1 ~(=9-(-?

2 is that of the Luneburg lens and obtained the complete representation of the electromagnetic field inside the sphere, as well as the scattered and total fields, when the excitation source is a dipole of moment px in the x-direction and located at (r, 0, 0) = (b, wr, 0 ) in the spherical coordinates system. Flammer (1958) gave asymptotic solutions for the case of the conical Luneburg lens. His approach is not complete in the sense of comparison with the method developed by Tai and also the solutions he obtained are not exact, but asymptotic. Other radially inhomogeneous media which have been studied are the Maxwell fish-eye by Tai (1958b) and a Gaussian type of inhomogeneity by Yeh and Kaprielian (1960). Arnush (1964) studied the case of scattering when the dielectric constant vanishes on a spherical surface by using a phase-shift analysis method. Fikioris (1965a) examined the behavior of a bi-conical antenna immersed in radially stratified media and performed detailed calculations for small-angle and wide-angle biconical antennas. Farone (1965) used the Rayleigh-Gans approximation to determine the scattering by a radially inhomogeneous sphere whose index of refraction is close to unity. Finally, Uslenghi (1967) extended Tai's method for media whose permeability is also radially inhomogeneous. He established general results for the presense of resonances and dips in the low-frequency backscattered cross section. Uslenghi (1968) also studied the high frequency backscattering problem from the inverse square power lens by applying asymptotic theory. In this dissertation, the case of high frequency electromagnetic scattering from radially inhomogeneous media in the spherical coordinates system is considered. The relative magnetic permeability is taken to be unity and the excitation field is assumed to be a plane wave. Particular emphasis is placed upon the study of backscattering from perfectly conducting spheres coated with a radially inhomogeneous medium and on the computation of the monostatic scattering cross section. This study is of practical importance in that it lends itself useful to the understanding of electromagnetic wave propagation in dielectric lenses at microwave and optical frequencies, the propagation of radio waves around the earth, and the effect of coating perfectly conducting spheres with radially inhomogeneous dielectrics.

3 In assuming the incident field to be a plane wave, it is implied that the more general case of an arbitrary incident electromagnetic field can be simplified by decomposing it into the sum of plane monochromatic waves by Fourier analysis, and therefore the simplest case only is considered. In what follows, the rationalized MKSA system of units is used and the time dependence e-iwt is omitted. The following symbols are listed for convenience. o = angular frequency, k = A = tW so~O~ = wave number in vacuo, E = electric permittivity (dielectric constant) in vacuo - = magnetic permeability in vacuo, Z = y-L =,Oe = intrinsic impedance of free space (= 1207r ohm), E, p = relative permittivity, permeability inside the inhomogeneous medium (functions of r ), i = -I' = imaginary unit, E and H = electric and magnetic field vectors, x, y, z = rectangular Cartesian coordinates, r, 0,, = spherical polar coordinates. Vectors will be underlined and unit vectors will be denoted by carets. Maxwell's equations are recalled and in the notation considered they are Vx H -ikY E, (1.1) Vx E = ikZpH, with the constitutive relations V. ~E =0 ViH = 0, (1.2) and e = - = 1 in the case of vacuum. The general geometry of the problem is shown in Fig. 1-1 with region I representing the scatterer and II the free space. The superscript i indicates the incident field while, later on, s indicates the scattered field.

4 I I A H1 -a Hi y FIG. 1-1: GEOMETRY OF THE PROBLEM The electric field of the incident plane wave is i A ikz E =xe (1.3) The scattered field is required to satisfy Sommerfeld's radiation condition at infinity throughout the free space region, specifically the condition lim r x (Vx)+ikr{ s= rl ooL (Vx)x k -=0 (1.4) A must hold uniformly in r. This is known as the Silver-Muller condition. Also, at the interface of regions I and II the appropriate boundary conditions, i. e. the continuity of the total tangential electric and magnetic fields, are applied for the determination of any constants pertinent to the solution of the problem. In order also are the following definitions in regard to the scattering cross-section of the body. The differential scattering crosssection or bistatic radar cross-section a(0, 0) is given by w 2 a(0 )=lim 47rr2 (1.5) -- oD 1Ei2

5 The total scattering cross-section is defined by the ratio of the time averaged total scattered power to the time averaged incident Poynting vector, and is related to the bistatic cross-section by 7r 27r total J41J si, o)sinOdOda. (1.6) e=o p=o Since the research herein is concerned primarily with backscattering, the definition of the monostatic radar cross-section is given for reference as V2 jEs \Q^ r00 rio 2 0 0)l | l rN= 4rr 2. (1.7) E 6=7t It is also mentioned here that the methods of solution to be applied are the geometrical optics method, exact solutions and asymptotic determination of formal solutions. The approach employed in each of these methods is well known (e.g. Uslenghi, 1967) and therefore detailed description is omitted. 1. 2 Scattering From Radially Inhomogeneous Media Assume a plane wave incident upon a radially inhomogeneous sphere of outer radius a whose index of refraction is N(~) where $= r/a (see Fig. 1-1). Applying the pertinent boundary conditions at r=a and at r=oo the far zone (r-* o) bistatic scattered electric field produced by the incident field of Eq. (1.3) is given by the well known expression ikr oo r P(cos) dP' 1 s e ir 2n+l s n +bs n n(n+l) [ P o J kr i n(n+l) an sinO n dO cos n=l n1 dP1 P(cosO) - a +bs sin] i 0 (1.8) n dO n sinO which in the backscattered direction becomes

6 ikr OD -' - (1)n(+l/2) Las b (1.9) n=l These expressions are the Mie series for the radially inhomogeneous scatterer. S S The scattering coefficients a and b are given in their general form by,(ka)-M (ka) p(ka) s n(ka)-'M (ka) (ka) s n n n S n n n. a I ( -- bn -- --- (1. 10) n C(1)(ka)-M (ka) (ka) n (ka)-M (ka) n (ka) n n n n n n where 10(na'-W (1) n(ka) - Jn+ 1/ (ka) v( - (ka)- Hn+ (ka) and the primes indicate differentiation with respect to ka. The constants M (ka) and M (ka) are determined from the boundary conditions at r=a and r=0, if the n scatterer is a radially inhomogeneous sphere throughout the range Or:a, or by the boundary conditions at r=a and r=b if the scatterer is a perfectly conducting sphere of radius r=b coated with a radially inhomogeneous medium of outer radius a. For these two cases, these constants are given respectively by n ka I= n ka u n n' and by 1 a M (ka)=- nC (,) n ka n = 2 A ^(1.12) a C (?,P) 1 aaS M (ka)- n ka aC( a Cn and they are true only if e (1)=1 which is the case in this research. The parameters involved in the previous expressions are defined as -a - ba' (1.13) ta a (1) (2) (1) (2) C (5,f3)=S (C)s ()-S (3S (), (1.14) n n n n n

7 and ~%'.(1) (2) (1) (2) C (g,)=T () T) () - T( (3) T(). (1.15) The functions S(J)(?), T()(); j=l, 2, are any two linearly independent solutions n n of d2S (M) 2 + (ka)2 (- n(1)2 } () = 0 (1.16) d92 [ (ka) 22 n and d2T () f dT (9) ) 2 d {In d _ + (ka2 n(n+ n() (g)=0 (1.17) 2 d( 1 (ka)2(2 n where e(N)=N (e). The functions S(1)(g) and T(1)(e) which are used to n n determine M (ka) and M (ka) in (1.11) are required to be finite at = 0. n n The differential equations (1.16) and (1.17) arise as follows. Consider the vector wave equation V2+ k2()] F= 0 (1.18) with k (=)=W 2E(E), F = - inside the radially inhomogeneous medium. The vector wave equation is reduced to the scalar wave equation (m) [2+ k2()] (e) = (1.19) by defining, after Tai (1958a), vector wave-functions M(m)= Vx(r/(m)) proportional to the electric field for magnetic type or transverse electric modes, and M(e)= Vx(r( e)) proportional to the magnetic field for electric type or transverse magnetic modes. Separation of variables in the spherical polar coordinates system yields (1. 16) and (1. 17) for the magnetic and electric type of waves correspondingly. Superposition of the two types of waves gives the complete representation of the electromagnetic field in the medium.

8 1.3 Outline of Research The backscattered field given by (1. 9) is amenable to numerical calculation for ka not too large. When ka > > 1, Eq. (1.9) is extremely slowly convergent and it is necessary therefore, since this research is in regard with high frequency backscattering, to subject expression (1.9) to a Watson transformation. Thus the summation is firstly transformed to a line integral by applying Cauchy's residue theorem. The backscattered field is then given by E xikr { (a - bs) f - c | dv [ a bs u1} (1.20) kr o o 121 CosT v/2v where = n+ -and and ka are assumed complex with 0 < Ik 1. The path C in the complex v-plane is shown in Fig. 1-2. Imv T C I 3/2 7/2 11/2 E * * *.*Rev 91/2 5/2 9/2 C FIG. 1-2' CONTOUR C IN THE COMPLEX y-PLANE

9 By observing that -M2 (ka)9-M _,(ka) v -, /2V 2 ((V722 (ka) -M-l (ka) (v-l (ka) (v1/ (ka) -MV-' (ka) (ka) v/21/ = - 1/2 ~v ^- 1/2 2 ^' e y ^^V /2 the path C is deformed in such a manner that it accounts for the contribution of any poles of the integrand in the first quadrant of the complex v-plane. These poles occur at the zeros of 1 ka)-M /(ka) (ka) = 0 (1.21) and (1)' ~ (1) 7 i (ka) -M 1/, (ka) _i/2 (ka) 0. (1.22) The new path is shown in Fig. 1-3. Imv x^ r R_ _\ P Rev a~ —- S-C ----- FIG. 1-3: THE DEFORMED PATH IN THE COMPLEX v-PLANE

10 Expression (1.20) can now be rewritten as bs eikr 1 Eb.s. e is S 1 v E - 2 -(a- b )- COS 7 [ dv - x kr 2 o o 2 Coswv V-If2 2 Jr cOSV [va b-yl dv+27ri (residues /2 in 1st quadrant). (1.23) Further computation of (1. 23) is achieved through asymptotic analysis. The integral along P together with the term i/2 [as-bs] gives the major contribution to the backscattered field. It physically corresponds to the reflected portion of the field. The integration over r is performed by the saddle point method over the range v=O(ka) with 6>0 but 6 < < 1 and the major contribution arising near v = 0. In performing the integration one needs the appropriate Debye expansions for the spherical Bessel and Hankel functions in the proper regions of the complex v-plane. The integrations are carried out with the aid of the formulas of Scott (1949) to O [(ka)2].This implies that the radial eigenfunctions S(), (j ) and (j) -2-/ TV_1/2(g) need to be computed to O[(ka)2]. This can be achieved by either solving (1. 16) and (1. 17) exactly and then developing the asymptotic expansions valid in the regions of interest or by obtaining the asymptotic solutions directly from the differential equations. The latter is achieved by operating directly on the differential equations by the WKB method provided that the Stokes phenomenon is not present in the regions of interest. Otherwise, Langer's theory of transition points is to be used. To obtain the pertinent asymptotic expansions for S() (), (j) v-1/2 Tv1/2(~), j = 1, 2, equation (1. 17) is put first in the normal form

11 d2U (2) J 2() v / d2() v-1/2 +)-(ka2 - / + 1 d d2 2(ka) 2e(e)(ka)2 d 2 3 1 rdI I12l [ (_ _ [ de 2 }U (t)-0 (1.24) 4 L J J2(ka) k- d /2 by setting T-l1/2()= E( U -1/2(e) (1.25) By defining now ~(()_ v-1/4 le(e2) - -; i 0 =(?) - (ka)2 E( _ V- 1/4 ) 1 d2e(g) 3 1 (Q) 2_2 + 2 g2 4 (,g d 1 /4 2 (ka)2 2 e()(ka)2 d2 [e(r)]2 1 (d( )) i=2 (1.26) the asymptotic solutions for (1. 16) and (1. 24) are found directly by applying the WKB method provided that the Q.i)(() have no zeros and that the following conditions hold 5 Q( i)(W) dQ(i)(9) 1__ 4k dg - d2 Q) 1 (i () d () << 1 E<1 8(ka)2 4ka [Q(1i)]3 /2 (1.27) throughout 13 <_ < 1. The solutions are

12 V( )[Qi)] /4 exp { ika f 7 11 + 4 dg d2 i 2 + 2 —----- -ds ------- d + 0 [(ka)-2 r (1.28) 8(ka) Q(i)(e) I with s( /2(); if i=1 v-1/2 V(j ) = (1.29) W 0) LUV) (?) if i=2 ul/2() ()= V(). (1. 30) It remains now to develop,with the aid of (1.28),the asymptotic expansions to O[(ka)2] of M 1/2(ka) and M 1/2(ka). From the definitions of C l/2(ka) and C 1/2(ka) and from Eq. (1.28) it is found that v-1/2 v-1/2 1C /2(, ): V(1) (2) (1) (2)31) C (1/2 3=)(1) (1)j3 (1)V (1) and asymptotically CV-1/2(3)-2i [Q(1)()Q(1)(] 1/ sin [ika F(1)(,D3)] [l+O [(ka)2 (1.32) Likewise Cv /2( ) = (go ()) ( [V ( 2()V( -V)(0(2)() (1.33) and asymptot2) (i) ) and asymptotically

13 -C 1/2(v~21"^ ^ 0 [Q(2)()Q(2)]/Sin[ikaF(2)(inka Q, )] x x [1 + O [(ka)2]] (1.34) with Fi 13)=;, f Q 7d[+fQ(.()] d( (1.35) and 5dQ (i) d2Q( )(S) 4 d? d2 Wf(i)) = 3-. (1.36) 8(ka) Q(i(~] From the definitions of M 1/2(ka) and M 1/2(ka) in terms of C1 /2(ka) and C 1/2(ka) one obtains v-~ ~ ~ 1 -1/d Mv-1/2(ka) ka d n [Q1)()]1/4 + i(i X -cot [ika F (,3) + 0 1(ka) 2 ] (1.37) and -/2(ka) N \n ( [E(9 1 ) + + ^f$ In E(-^ [Q W se]-c i ~, ]ka () [Q(2)()] / sec ika(2)( ] - i I( tan [ika F(2)(3)] [1+ O(ka) ]]. (1.38) From expressions (1.37) and (1. 38),the difference of the scattering coefficients follows:

14 S s I [Q(')(6)c~t(ika F( c)+otQ)(F tan(ika F(2)(g, ))] + [("-/2-v-1/2 (ka) i cot ika))+ ka a n Q Q (3) 13 a r(F) i [ Q) a 2(Q() ) -1/4) sec2 - ~aa In Q() [ -1/2XV _1/2 (ka V _1/2 (k a 2)Q2()-{k 9(( 2)) ag -iPc a:'n[ ()(~) / ) sec (ika F(2)(, l x Q) ( 1)]4 )sec ka F( )- i ) tan ika F) (, ) (ka)] i [l+o[(ka) ]]. (1.39) and (a -b ) is obtained when v = 1/2. By substituting in (1. 39) the appro0 0 proper regions of the complex v-plane and by carrying out the algebra to O [(ka) -2],the reflected portion of the field is immediately obtained after the integration is carried out along rf with the aid of the integrals of Scott (1949). The advantage of this last result is that the final form of the integrand in (1. 23) is determined for arbitrary e(g) and one can, with direct substitution of the functional form of e(g) in expression (1. 39), carry out the algebra asymptotically and perform the integration without solving for the V) (Q) eigenfunctions, in order to determine the reflected field. The contribution of the integral along P has been shown to be zero for the general case (Goodrich and Kazarinoff, 1963) as R -+ oo, and the verification for the particular cases considered here is therefore omitted.

15 The summation over the residues in (1. 23) gives the contribution in the backscattering direction due to creeping waves. In order to determine this contribution, asymnptotic expansions are needed for the radial eigenfunctions which are valid for v near ka. These asymptotic expansions are derived directly by the WKB method if Q i( ) has no zeros in the interval 13 t 1. In case there exists a single turning point at 13 e < 1, then Langer's method is used to solve the differential equations. By writing v = mt+ka with m = 2 1/3 Langer's scheme gives, with the following definitions 0(i)() =Q i (1.40) (i)() (i) () d (1.41) and e u)() = i ), (1.44) 0 the solutions ) i)() k i1{\(i)() (1.45a) or ( i) (i ) I (j)(i)[ X~~(e)] vdEJ i gi OJB~i)(e) 1/3 V((j) 1a/4 Q ia ) d X Xu(.).3 (ka J Ai dP ) (1.45b) with i, j withi, j = 1, 2s

16 If one now defines: 3j ~'__ / 2/3 W = v d) (1.46) and (1) = 2 )]-3/2 (1.47) (i)(t) 3 x(i)((1.47) it follows that X(i) () = (ka) 2/3 W(i ) (1.48) and the solutions can finally be written in the form V(J) () w ka) )2/3S Q j= 1,2 (1.49) iQ ) i w(j) ka)2(i( = 12 The w() {(ka) 2/3 (i)() are Airy functions in Fock's notation and they are related to the Airy functions of Miller (1946) by w(1) t) ='F[Bi(t) ~ iA i(t)]. (1.50) (2) The creeping wave contribution to the backscattering direction now becomes. eikr ( (1) a 1) E c^ ^ oof fe (ka) [ (ka) -,ve ~~~~v x cr.w. kr cosM rv V V 1 _ _1 -M /(ka) 1(ka ) -(v (ka) v/ (ka) - MV-1h V -1/2 a C 0 S 7rV V -1h av V -1/ 2 -M1A -(ka) V() (ka) ) (1.51) with vi and ve being respectively the roots of

17 (ka) -/2=M - -(1) - = M (ka) (1.52) (ka) and of (-/ (ka) 7(1) - -M, (ka) (1.53) v1/2 (ka) with Imvi > O, Imv~ > 0. With the asymptotic forms _(1),/ (ka)N -im 1/2 w(l)(t)54) Vl 1/2 1 I) 1ka) im /w 1) (t) (1.55) and the substitution v = mt+ ka, the creeping wave contribution gives ikr tv lE x lre WI [ ( [ + x J cr.w. kr; IO (1)(s 2m _ aM(t) _ [M(t)] ]- v [w- VI ]-2[ m at cOsLt=t - cos L(1)2 w+, M(tI ) ]2 ^^1 aM(t) t=t where the index I scans the zeros of = -mM(t ),and w ) = mM(t ) (1.57)

18 It remains to evaluate M 1/ (ka), M V/ (ka) and then put them in the form.v-1/2' -i/2 M(t ) and M(t ) for the solution of equations (1.57) for the zeros t and t. The following explicit forms can be written down for M _ (ka) and Mv_ (ka) for the case where v is near ka: M (ka) - (1 1 1 + v-1/2 4ka {r(1)() a- Q(1)?Q) + a =1 1 a8(1)() fw() [(ka) 2/3 (1) )]w2) [(ka) 2/ 13 )] -w(l(ka)2/3 1)()] (ka)1/3 nWd()k^ W(1) [(ka 2/ 3 >)(9] ]w(l)[(ka) 2/ 3>() (] w'2) [(ka)2/3 (1)()] a(2) 2/^(1.58) X w(2) (ka)2/3(1) =1 and

19 a(ka)~ + 1 T (2) () 1 aQ ) V 1/ ka ( a +a 4ka V(2) a aQ(2)( a + 1 a Jw (2 )[ w(k[a /3() (]w2[(/ (2)(]-w(l)[(ka)2/3 ()()]x (ka) 1/3 w [(ka) / (2)(w(2)[(ka) 2/ (()]-w()[(ka)2/3 )()] 9/3 ^ 1/3 ^(2)[w12 9(2) x WI() [(ka) Q2/3 (2)] W4( ka)/ (2)( 1)(I32 X (2))[(ka) 2/3 3 (2) 3~(2))ag x (2) [(ka)2/3(2)(1]J n +(k )2/3 (2)() 2x W(1) [ a (2) ( ( [+(ka / 2)( (ka) / (2) [1) (2) 2)(2)( ] ~w (~) [(k/) (l(e] )w( (a) /2(~] -w1) Qn/(2)2(~)]| - 2/3 2 2 (1.59) x`N~w (ka 2/32)(W ()J 2 W2 x (1) (2) - (2) (2) ( ) (2) where to arrive at the forms (1. 58) and (1. 59) use has been made of the Wronskian WIl)(t) W(2)(t) -W()(t) w2)(t) = 2i * (1.60) Primes above indicate differentiation with respect to the argument and it is assumed that

20 W(1)[(ka) / (i)()]W(2) [(ka) 2/3 (1(3)] -w(1)[(ka) 2/3(i)(] w(2)[(ka) 2/ 3()()] 0. (1.61) By substituting v = mt+ka in (1.58) and (1.59),M(t ) and M(t ) can be simplified asymptotically and then equations (1. 57) are solved numerically for the first few roots when t lies in the first quadrant. Finally for particular ka, expression (1. 56) will give a numerical value for the creeping wave contribution in the backscattering direction. From the theoretical expressions which will be obtained for the reflected electric field, the monostatic cross-section based on this reflected field will be derived and it will be computed for various thicknesses of the radially inhomogeneous coating as a function of ka, for two types of radially inhomogeneous dielectrics. In Chapter Two, a new class of radially inhomogeneous dielectrics is discussed. The exact solutions for the radial eigenfunctions are derived and the geometrical optics technique is applied in order to determine the ray path of an incident ray through the medium in the optical limit (ka -+ oo). Detailed numerical computations of the deviation angle as a function of the angle of incidence and other pertinent parameters, are given. In Chapter Three, this new class of radially inhomogeneous dielectrics is considered as a coating of a perfectly conducting sphere. The detailed asymptotic computations for the derivation of the reflected electric field are presented, beginning with the application of the WKB method for the asymptotic determination of the radial eigenfunctions. The expression for the reflected electric field obtained by this method, is carried out to O[(ka) ]. As a means of comparison, the reflected field is also obtained by application of geometrical optics to O[(ka). The results of the two methods are compared and the monostatic cross-section is computed for both cases with ka varying from ka = 50 to ka = 1000. Then the percentage error in considering only the geometrical optics solution is examined. Finally, in this chapter, the creeping wave study is outlined.

21 In Chapter Four, the Nomura-Takaku (1955) radial inhomogeneity is considered, as in Chapter Three with the exception that a more detailed study is carried out for the creeping wave contribution in the backscattering direction. This type of radial inhomogeneity has an index of refraction N(~) = ~P (with p > -1, for reasons which will become apparent from the discussion of the exact solutions of the radial differential equations).

CHAPTER II A NEW CLASS OF RADIALLY INHOMOGENEOUS MEDIA 2.1 Introduction In the study of electromagnetic wave propagation in, as well as scattering from, radially inhomogeneous media, one of the difficulties is the determination of exact solutions for the eigenfunctions S ()() and n T(J)( ), and especially for the latter. Generally speaking, these eigenn functions are expressed in their exact form in terms of hypergeometric and/or confluent hypergeometric functions, which are chosen to be finite at the origin. Although asymptotic solutions for the differential equations in the radial direction can always be found, by either applying the WKB method or Langer's uniform asymptotic theory under certain restrictions on the coefficients Q(i)(C), it is with the exact solutions that one has most of the difficulties. In this chapter, a particular technique is presented, which simplifies the problem of finding exact solutions for (1. 16) and (1. 17) considerably and which at the same time gives rise to a new class of radially inhomogeneous dielectrics. If the differential equation (1.16) and the normal form of (1.17) are considered, it is seen that they reduce to one and the same differential equation if Q() = ) Q(2)( )' (2.1) which implies that one needs to solve only one equation, since in this case T(J)(=) = v) S(J)( = i( U(J)(). Relation (2.1) is satisfied if and only n n n if 1 d2(E) 3 1 (d() 2.2 2e(() ds2 4 [e()i]2 \ d = 9 9) which is rewritten as 22

23 - [d-en(e]2n - -2 (de) =0. (2.3) d? Lrd? J 2[e(F]2 d? (2.3) By substituting w(-) = d/d [ne(g)] in (2.3), the following Riccati differential equation is obtained in w(Q): d w(2) ^[w(]2= 0 (2.4) If variables are separated in (2.4), then upon integration one obtains:dg = 2 d or =- + (2.5) JJw where y is an arbitrary constant. From the substitution w(Q) = d/d5 [n c(g)] and (2.5), it follows that w= 2 1 de() (2.6)'y-~ E() dS which finally gives a solution for c(g). This solution is e(g) = A- (2.7) (e- a) where A is another arbitrary constant. For the cases of interest in this research, i.e. for radially inhomogeneous media, the constant A is determined by choosing a continuous transition from free space to the inhomogeneous dielectric, i.e. e() = 1, which yields A = (1- y), and ~=1 E( ) = I) (2.8) This type of functional dependence for the permittivity encompasses a large family of inhomogeneous media. Depending on the choice of 7, it lends itself to both converging and diverging types of dielectrics. Its most

24 valuable importance rests however in that it facilitates the theoretical study of the problem by reducing the two differential equations essentially to one. 2.2 Solution for the Eigenfunctions when c(O) = (1- Y7)2/- 7) In this section, the exact solutions for the radial eigenfunctions are determined when c(g) is given by (2.8). The various possible applications with such a permittivity function are also discussed briefly. By investigating the differential equation for the S(j)(~) eigenfunctions, when e(g) is given by (2.8), it is found that it has three regular singular points at ~ = 0, ~ = 7 and 0 = oo. This differential equation is easily reduced to a hypergeometric type with its Rieman P-symbol given by (o 7 T \ Sj)(g) = P a' b' c', (2.9) a" b" c" / with a', a"; b', b"; and c', c" being the exponents or solutions of the indicial equation at the singularity points 0, y and oo, respectively. In particular, these exponents are a 1 + n, a" =-n, (at =o) (2.10) b,= 1 + 1 -4(ka)( 1 -y b" 1- ka2(1- ( (211) by b,2, at: l,( (2.11) 2 2 and -1 +1- 4 [(ka) ( 1-,y) 2n(n+l)] _ -1-_ 1-4[(ka)2(_-)2_n(n+l)] c 2 2 (at g =00). (2.12) By reducing (2.9) to its canonical form, the result is

25 1+1-4(ka)2(1-)2 0 1 s(j)() = In+l(_-) 2 P 0 0 a / (2.13) ~n I~~~~1 2 / 1-or3 a3-1-f 2 a2 where a n+l (ka) (1-) + (n+1/2) -(ka) (1- y) (2.14) 1 4n+land a3 = 2(n+l).(2.15) The functions represented by the canonical P-symbol are well known, and a solution is chosen which is finite at the origin. Then the radial eigenfunctions are given by 1+ l-4(ka)2(1-)2 S(1)( n+l_, 2 n 2 1 (2.16) S()(e) = t+l(e-7) 2 F1 (aC' "2; 2(n+l); ~/y) (2.16) and T ()() S(1)( (2.17) n -7 n A second solution S (2), linearly independent from S (c), is obtained n n by replacing 2F1 in (2.16) with any other solution of the hypergeometric differential equation satisfied by 2F1 which is linearly independent from 2F1 itself. Now c(e) is investigated for various choices of y. Case 1 The constant 7 is chosen so that 0 7y < 1. The dependence of c(g) is plotted vs. ~ as shown in Fig. 2-1. It is seen that as e tends to 7, e(e) approaches infinity (i.e. lim. e(t) -- o ). This implies that the dielectric ->7

26 sphere acts as a penetrable barrier at ~ =7 and therefore allows energy penetration for e < 7. When 7= 0 then e() = 1/. This case corresponds to the inverse square power lens which has been studied by Uslenghi (1968). E(e) (1_,)2 7 1 1 FIG. 2-1: Case 1: 0 7y< 1 Case 2 In this case 7 > 1 (see Fig. 2-2) and E(e) < 1 for 0 < 1 <1. This case may be useful in studying diffraction of waves by plasma coated spheres or scattering from plasma clouds of spherical nature surrounded by an external medium withe(g)> 1.

27 c(e) y=00oo 1y> (X)! y= 1=1 FIG. 2-2: Case 2: y > 1 Case 3 Under this case (y < 0, see Fig. 2-3) the lens has been studied from the point of view of geometrical optics in section 2.3.

28 E(g) 2 h=0 1 2 h=1 h=2 h=oo FIG. 2-3: Case 3: y O, y -h, h O For a spherical lens of radius r=a made of a radially inhomogeneous dielectric with l+h N() = - (O < <, (2.18) and h 0, the exact backscattered field when the incident field is a plane wave is still given by expression (1.9) with

29 M (ka) -= 1 a nS(M)] M (2.19) n ka a n and M (ka) M(ka)- kal+h) (2.20) n n ka(1+h) where S (1) is given by (2.16) with y=-h. It is observed here that M (ka) - M (ka) is a known quantity and that it is independent of n. n n This is of great advantage in the determination of the high-frequency backscattered field, because whenever E(1)=1 (as is usually the case for dielectric lenses), the leading terms in the high-frequency expansions of M and M are equal, and since (M -M ) appears in the numerator of n n n n b all terms of the infinite series representing Eb s; two terms are generally needed in the expansions of M and M to obtain the leading term in the expansion of E *bs 2.3 Geometrical Optics Approach for the New Class of Lenses In this section it is assumed that the wavelength is infinitesimally small,i.e. ka -- oo for finite a. Under this condition, the electromagnetic wave propagation properties through the dielectric sphere are examined with the aid of optical ray theory. By considering Fig. 2-4, one traces any incident ray making an angle of incidence a at the surface of the dielectric sphere. The following parameters are also defined in Fig. 2-4: 6 = 6(a, h) = deviation angle / =M(O); O(1) -a min p = P(e) and 0 = 2p(m). min

30 tangent at P incident ray / =1 0 6 outgoing ray OA= g FIG. 24: RAY PATH THROUGH THE INHOMOGENEOUS DIELECTRIC. FIG. 2-4: RAY PATH THROUGH THE INHOMOGENEOUS DIE LECTRIC.

31 The generalized Snell law for the index of refraction is given by gN(~) sing = constant. When ~=1, N(1)=l and b=a. Then it follows that N() = +h sin a (2.21) ~+h sinV and when =min _* l +h sin a sin (2.22 min - - +h sin 7r/2 mmi min min or h sina (223) min l+h-sin a* Therefore, mn 0 unless h = 0. If h = 0, then agreement results with the inverse square power lens, where min= 0. In order to investigate how the refracted rays leave the lens, one considers the differential equation for the ray trajectory,which is -= - coto. (2.24) dp Upon integrating (2.24) one obtains the following expression PMc1 -dj- (2.25) ~ ( cot2 1 a With the aid of (2.25) and the relations 6 = 0+ 2a- 7r, 0 = 2p(min) and 2 2 2 h 2 coto = g (+h)csa - h) the deviation angle is given by 6+h,min JT O2( ~ l+h)2 csc2- (+h)2

32 which upon completion of the integration yields |lh2 i2 h sin a,,. 2 sina, h + cos a + cos a 7(l+h) - sin a 6(0, h) = -- -- --- log h-sin (r (1+ -sin a From the latter expression, the quantitative behavior of 6(a, h) has been computed for different values of a and h. In Figs. 2-5 and 2-6, 6(a, h) has been plotted vs. a. In these figures it is seen that 6 increases from zero at a= with a slope () + ) to a maximum value 6 (wit a soeda/ / max' then it decreases toward zero, which is reached at a = 7r/2 with a slope (da h. Also the maximum deviation angle is shown as a function =r/2 of h in Fig. 2-7. As h diminishes, 6 increases to infinity which max indicates that the ray trajectory inside the lens follows a logarithmic spiral toward the origin in agreement with the inverse square power lens. On the other hand, as h increases indefinitely, the maximum deviation approaches zero in agreement with the fact that lim N(, h) = 1 h — oo i.e. the lens assimilates free space.

D3Na[IDNI 10 3IIONV'SA a[IONV NOILVIAAa[:9g-'*DI o 0 0 0 0?0 -- 10 00 0.I, 0 (o O - 0j 0o ~~~~~~ I'

34 0 0 to 0 -I 01 I oI I0I 0 s0' 0 0 IION A V. A I FIG. 2-6: DEVIATION ANGLE VS. ANGLE OF INCIDENCE

300 0 2000 ~max FIG. 2-7: MAXIMUM DEVIATION ANGLE VS. h 100 0 2 6 8 10

CHAPTER III HIGH FREQUENCY BACKSCATTERING FROM A PERFECTLY CONDUCTING SPHERE COATED WITH THE NEW CLASS OF RADIALLY INHOMOGENEOUS DIELECTRICS 3.1 Introduction In this chapter, a theoretical study is carried out in order to determine the electric field in the backscattering direction, for high frequencies, when a plane wave is incident upon a perfectly conducting sphere, coated with the new class of radially inhomogeneous dielectrics. By high frequencies it is implied that ka > 1 or << a, where X is the wavelength of the incident field. The analysis follows the discussion of section 1.3 in chapter one. The perfectly conducting sphere is of radius b and the outer radius of the coating is a. The electric field of the incident plane wave is given by (1.3) and the geometry of the problem is shown in Fig. 3-1. Following the development in chapter one, the Mie series (1.9) is transformed into a contour integral in the complex v-plane, where v =n+ 1/2. Then, the reflected portion of the electric field is determined asymptotically to Oii~ ac S(j) / (J) O [ka)2]. This is accomplished by solving for SY(j) () and Tvi) with the WKB method for v = O (ka) /2+, then computing aS 1 -bv to OL(ka) 2] with the aid of the Debye expansions for (v /(ka) and (1)' Sv1/(ka) in the proper regions of the complex v-plane,and finally by per/2 forming a saddle point integration using the integrals of Scott (1949). It is recalled that the main contribution results near v=0 on the A1 path of integration. The expression thus obtained for the electric field is then used to find the monostatic cross section, which is normalized to the monostatic cross section of a perfectly conducting sphere of radius b. This normalized relation is then used for numerical computations for 0.2 < j3 < 0.99, 0.25/3< - 7 0.9913, 1.1 <7y <2 and 50 < ka < 1000. The ray tracing 36

37 technique is also applied to determine the reflected electric field to O (ka) l]. This is accomplished by considering the conservation of energy between incident and scattered fields in order to find the amplitude of the reflected field,and the eikonal relation in order to determine the phase. The result is then compared to the first term of the electric field obtained by the use of asymptotic theory. Finally, using this geometrical optics expression, the monostatic cross section normalized to that of a perfectly conducting sphere of radius b is computed, for the corresponding values of /3 and y considered previously. From these numerical data, the monostatic cross section as determined by geometrical optics is plotted vs. /3. Also,the percent error in using geometrical optics instead of the rigorous asymptotic theory to O |(ka)2] to determine the cross section is plotted vs. ka for 50 < ka < 1000. The last section of this study is devoted to outlining the creeping wave contribution in the backscattering direction. The differential equations (1.13) and (1. 24) are solved for v near ka by applying Langer's uniform asymptotic theory, since in this case it is found that Q (i() has a zero o in 13 < 1. 3.2 The Asymptotic Solutions of S 1/(e) and T v/ (). The asymptotic solutions to O [(ka)2] for the radial eigenfunctions S-(j) ) and T () () are obtained in this section, by applying the WKB method directly to Eqs. (1. 16) and (1. 24). Firstly, it is recalled that for this class of radially inhomogeneous dielectrics S(j)() = U(j)1 () and yz/~ /2 v" /2 therefore Q =Q)v -2 1/4 Q = ) Q(i) - ) (3.1) (1) u"7/ (ka)2 2

38 for i = 1, 2. From (3.1) it is seen that Q(Q) has a zero at _ y(1-7) ka -1/4 -y (v -1/4) (3.2) ~0 (1-y) 2(ka) - (v -1/4) which is almost zero. It follows, therefore, that if 3 < _< 1 such that 1 > g the WKB method can be used. By restricting 3 to values greater than t it is seen that conditions (1. 27) are also satisfied, and this further justifies the use of the WKB method. The solutions which are obtained here are valid for v O [ka) /2+ 6 with 6 >0 but 6<< 1. These solutions are given by S() () = U() ) = V()() (3.3a) from (1.28) with Q(9) as in (3.1) and T(-l()= (') S (~) (3.3b) To obtain these solutions in their final form, one has to develop the asymptotic expansions. To this end, it is found with the aid of the binomial series expansions that [Q()] - 1/4 f-y exp 2 2 1-'< 4(ka) 22} ka)-2]+ oo(ka)] + 0 (3.4) and that

39 r d 2 Q(g)Q(g) 1 exp + ikar 1 + (- d2 3) - ( 8(ka) Q (j) r*~ *~2 -exp i (ka(l-7Y) In ( —y) - 2ka( + } X + 8ka(l-y) (n ) + 4(ka)3(1-)3 X 1 4ka 3 ( — r I1 -r 3 3 2 3 + 2 3 3)] ( [(ka) ] L(ka)4 + 0 --- + 0 — V (3.5) (ka)5 [(ka)6 (3 By combining (3.4) and (3.5) the solutions in their final form are S ( /) n Y exp + i ka(l-'y) In( —y) - 2ka(1 —y) (n+)]} ( 8ka(l-) en ( 2 2 4' + v' ( 7 ) - v[ + 2 _2 I + 2- + ^31]) 1+ [(ka)2 + [ 3 + 2g 3r- (ka) + 0 4 — + 0 --- + 0 6 — (3.6) [(ka)] -(ka)1 [(ka)] (3and T (j) () is given by (3.3b). These solutions are valid provided that 0o, - 72y and |2ka(l-Y) ~ 1.

40 A ~A~ ~ ~ ~~ AA k z; — A 2c FTEHM2bIEP 2a FIG. 3-1: THE GEOMETRY OF THE PROBLEM. (c=ay)

41 3.3 The Reflected Electric Field With the asymptotic forms of S (j) ( and T() ) one easily v 1/2 v 1/2 proceeds to determine the reflected electric field to 0 ka)2]. It is pertinent, that first of all the asymptotic expression for a _/ 2 bV / be derived. To this end, a step by step procedure is employed in determining these coefficients. Firstly, the definitions for M I (ka) and v 1/2 M 1/(ka) are recalled in terms of C _/(, 3) and C /(, 3). In this v"1/2 z/2 v 12 case Mv _/(ka) is simplified in the following form. /2 lowI I Ta F ac (I M) M- (ka) =ka)+ - j u C, )- ((-<>) v-/'2 -/2ka(l-{[yC) (,aV L- I 2=l (3.7) From (3. 6): c ( ka 1 2r -)( y) exp 4 (( 1 - 2 + (ka) + (^7 )2 sing( ) 1+[ (ka) 2+ v 0 +0- 1 Vk + + e[ Jsn ([( kJ a)4 j+0 [6(ka) (a ka) (k) (3.8) From this derivation for C -1/ (t,3) one obtains: + O[(ka) 2] + oR^ L v4 ]or6 5]+O[_ 1 ) (3.9) L(ka)a n(ka)] L(k (ka) and

42 My1 /(ka) - 2ka(1-y) at ) + V 12 a~2k _ _ _ _ _ (1-,y) csc2g (9, g) ag(eZ, a g(,3) I + apI a 1 + ka 1 2_ ka \1 _1 v - 2l _f_ ((03-y)(cot g (9 )) aJ \2 24 (ka)2 (2 l-'y I } r Cka -2^ " 2 r4 V6 r v6+ r8 8 1 + oka [( + 0 ka) 3 + o +ka)4 + Oka) +k)6 (3.10) where r 1 1'c-\ 1 2 g(,3) = ka(1 ) - 8ka(l —y) f- -y 2 ka( -y) X n +2 1 +-y ( 1 ) i (ka)33(l_r)3 3)1 I 22 21+ (3.11) _2 -2 3 (3 3 and henceforth g(Q, 3) = g (1,) =g. (3.12) 1=1 By observing that 1 ag 1- )+ ka) + 0ka)4 (3.13) ka2 2~ + =10L kaj (ka)2 (ka)4 and by simplifying (3.10) after the term (3-7y)(cot g (g,3)) g is factored out in the denominator, the following expressions are derived: i i 12 2 M 1(ka) ka(l,-y) + 2 (ka)2 cotk +O(ka)(2] + + ]+)3] [(ka)4J ([(ka)5 ka and

43 J +21 se 2 1- tan M-/(ka)- - 2ka ) 2ka(l) sg1- 2 2)tan g L ka-12)(ka) + ka + [ +~[( + ~[( ] + ~[( )6 (3.15) It remains now to obtain the asymptotic expansions for the spherical Hankel function of the first kind and its derivative in the proper regions of the complex v-plane. The regions in the complex v-plane are shown in Fig. 3-2 (Watson, 1952). For the case of the reflected electric field the Debye asymptotic expansions are needed in region one. These expansions are ( i __ 72-iT [ F 2 (1) (ka) s i/ e 4iii 1- +O'kaf)] + ~L )3 (3.16) _,/2 k) sin h q 4ka 3 and ~f777 e {^i + -- } [ + O(ka)-[ 2] + gi )lj(ka)' / h t ika } [ 1 2 i [ (:)] (3.17) -(ka)2 with the following relations being recalled v= kacoshr, = v (tan h r-r7), (3.18) the restriction - < arg (-i sin h r7) < - (3.19) 2 2 and the requirement that | is sufficiently far from v = ka in the fourth quadrant while it runs close to the imaginary v-axis in the second quadrant. It is noted here that the notation has been changed somewhat from that of Watson (e.g. Watson uses y instead of r7 ) for convenience. If now the following asymptotic expressions are taken into account:

44 Imv I 6b 6a /r2 4 2 // /^ /I/ \ / | \ I 1 Re \___C Rev -ka ^i -ka C 5I 3: CS OF IN AD DT CONSIDERED IN THE COMPLEX v-PLANE. FIG. 3-2: CONTOURS OF INTEGRATION AND DIFFERENT REGIONS CONSIDERED IN THE COMPLEX v-PLANE.

45 -20t v e exp {i v - i - - i2ka 1 - i + 0 J (3.20) 12(ka) (ka) 1 2 r 4 _ _6 sinh r vi 1- 12 2 1 + 0 4 + 0 6 (3.21) 2 (ka) L(ka)-4 (ka) M^(ka) - M ka(- + - 2 ) (2 csc 2 2 v-112 V- 1/2 ka 1 -,Y 2 2 2k\ (ka) X sec )2g2+ J})[1 i+ 0 [ka + 0 +0[v + 0 + o ] (3.22) ka).ka)(ka) (ka) _ then together with (3.14, 15, 16 and 17) the difference of the scattering coefficients is found to be 2 4 as bs ( e {i v i } {I + - 1y_ a g - 2, iexp i T v - i - i2ka + i2 1 + i2ka i - / ^/2 \ i^ ka g^ 4ka tang expi2g V sin exp in- -i2ka+i2 1+ [ka(l-a) ] [2 ka(1-y) eka + O[ka)2]+ 0 L 3]+0 v4v235 0[ +0,4. (3.23) L(ka) L(ka)J, (ka) L(ka) By substituting in (3.23): g= el+ E2 (3.24) where =+2 1 2 E ka(1l-ky) I)n 11 n/n + - (3.25) ka(1-Y\en — 2 ka( 1 — y)

46 e2 = 8ka(1-y) 1 8 (ka) 3(-1 )3 + -+72 1) - 1)] (3.26) such that e2 << el, then (3.23) is given in the following simpler form: a_1/2- b1-1/2 i ka lj 4ka 12(ka)3 + 9-, + 4 + 3 2V1 n (k ik + I 3] + 4ka(l-y) 1- + 4 (ka)3(-)3 Ln 3 2 / (ka( (1-)Y 3 2 1 71 ) (tan l)[exp(i2E1)] ^3 3 J - ka (l —y) (sin 2e exp [iv v- i i2ka + i2e 2 e. ~i v- - ka1- + sin2eseCe 2 exp ir v - 2 ka(l-y) 4ka(l-y) 1 1 2 + o[(k +0] - ik - i2ka + i2e} 1 + 0[(ka2] [ +0 [ +4] ka + 0 5 + O[ 6] (3.27) Lka (ka -J If v = 1/2 in (3.27),then as bS J e-i2ka [-(l-'y)-n l — Y -)1 + oka)2 + 0 [(ka)-2 0 0 4k L-J- +0 —- ++0 4 - +0 ] results. (3.28) (ka)L3 +(ka Li(ka).5 0ka)[

47 If now one writes ikr,. I'\ -i7TV 1 Eb.s. A e s ss ye s -refl. kr i2 o a-b -J 2e [a _ b 2 dV (3.29) then by substituting (3.27) and (3.28) in (3.29) and by performing the integration along r1 with the aid of the integrals of Scott (1949), the reflected electric field is obtained in an explicit but asymptotic form. It is recalled that the integration along r1 is a saddle point integration over the range v = O [(ka) 1/] with the major contribution arising for v near zero. Scott (1949) evaluated such a class of integrals which in their general form are ly ^ oo ey 2 E = e — e w dw (3.30) oq J iy 1+ w - oo ey with q = 0, 1, 2, 3,..., 0 < y < 7r/2. By performing the saddle point integration for v < ka Scott found that the major contribution arises for v near zero. Some of the integrals which he computed and which are of importance here are 1 2 1 E N - - - + O(e) (3.31) co o 2e 6 E 1 -- + O(~o) (3.32) ~ 1 2e and E 1 - + O(e~) (3.33) o,2 3 Scott used these integrals in order to determine the backscattered electric field from a perfectly conducting sphere, when the incident field is a plane

48 wave, for ka >> 1. Throughout expressions (3.30) - (3.33), e and w are given by i 47r ka and w = i2wv. In the problem considered here, expression (3.29) is reduced to several integrals of the type (3.30) by substituting the asymptotic expression for a /i- b 1 in (3.29) and by letting w = i2rv. In this V-112 V-112 case, in each occurring integral e is a function of the 3 and y parameters as well as of ka. A typical integral e.g. results if the first term of (3.27) is consideredi.e. the term -i exp (irv - i(v2/ka) - i2ka + i2c. The integral along l in this case becomes: -ve v a s b ] dv = - e X -2rve1 2 r -cw 2 we 1or dw e(E dwn -e -) -w 2 6.. l+e 47r _27rve (3.34) where for this particular integral, E - - ~ -n' -and 47r-ka 1 v = (ka) /2+. By proceeding in a similar manner, the following expression is finally obtained for the reflected electric field:

49 b. s. eikr ( ) -i2ka [1-(l-y)ln 9 ] -refl. kr 2(1-In i-) X 1{ 4ka(1 -/) [n ~(ly) - 1I -'3 - + 4ka(l-) 6 (1-3(- Y) 13 +2 22 3 + 4'y - - _ 6 Y.3 - 2iy -2n/ + 1 3 + (1 - jnti - g +(1-in13-)) {exp [i2ka (1 -) In 137] +l-Ing —13 1 — + / [ l+7-2( n,+ Y) exp -i2ka(l-y)n 1 (3.35), --- - - -07 ] } + o [(ka)2 ] (3. 35) l-q( The correctness of this expression is checked with the known result for the perfectly conducting sphere. It is thus observed that if 13=1, i.e. if the thickness of the radially inhomogeneous coating is zero, then (3.35) reduces to b. s. A (a2 ikr-i2ka i -refi. 2rb=a 2 bs x (_ ) eikri2ka {1- 2ka+ ~ L) ]} (3 36) which is the well known result for the reflected field by a perfectly conducting sphere. It is also observed in (3.35) that since Imk < 1 but positive, in the limit 13=y the expression (3.35) reduces to b.s. A i ikr-i2ka (3 Erefl. - 8kr(-y) e which indicates that if P=7y the reflected field contribution is very small.

50 3.4 Derivation of the Reflected Electric Field by Geometrical Optics. Although the geometrical optics contribution can be determined from the first term of expression (3. 35), it is of interest to derive this contribution by the ray tracing technique. Such a derivation gives not only a means of comparison with the results obtained by rigorous asymptotic theory, but also a physical insight into the problem. In considering the geometrical optics solution for the reflected electric field, it is implied that ka - oo. Furthermore, the reflected electric field b.s. A b.s. is polarized as the incident one, i.e. E = x E, and it satisfies the x vector wave equation. This vector wave equation easily reduces to [V2 +(ka) N2(E)] Eb. s =O (3.38) c ~X where 5V = *.(3.39) b s A solution is assumed for E in the form x o w b.s, Eb. s. ik() +i7r (3. 40)'E e(3. 40) X=0 (ik) whose leading term is E.s Eb.. ei + (3.41) X 0 and which is the geometrical optics solution for the reflected electric field. b. s The amplitude E' is easily determined from the principle of con0 servation of electromagnetic energy between the incident and scattered electromagnetic fields. This of course implies that the inhomogeneous dielectric is assumed to be lossless. The phase 8(0) is determined from the eikonal equation [1 V 0e()]2 = N(E) (3.42)

51 which follows from substituting (3. 40) into (3. 38), performing the differentiations and then equating terms in powers 1/k to zero. b.s. In order to determine E explicitly, a tube of rays of cross-sec2 0 tional area 7rd is assumed incident upon the coated sphere in the +zdirection. By considering the amplitude of the incident plane wave to be unity, the electromagnetic energy carried in the incident tube of rays is given by gincident rd ( C =-^. (3.43) 2Z On the other hand, the energy of the scattered field is 2scattered ^~ b I 2 scattered - [E 27 dS (3. 44) r-2 [o+p] where 6 = electromagnetic energy. From the last two relations, it follows that b. s. m _ gd E =lim 0 d 2-0 r r dS (3. 45) l J ^r-2 Cp] where dS is the element area in the spherical polar coordinates system. The limits of integration can be understood from Fig. 3-3. Upon completion of the integration, the denominator inside the radical of (3. 45) is 27r 7t dS = 27rr2[1-cost2(a+p)]] (3.46) T-2 [Op] and therefore the magnitude Eb. s. becomes E. = lim 2 n(3.47) o 2rsin(c+p)

A x - z a cu~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~, I a ar I I /~~~~~~~~~~~~~~ Fd 3 PNCIE A FIG. 3-3: PATHS OF INCIDENT AND REFLECTED RAYS WHEN 0 < y< 1.

53 From the definition of p(~) as given by (2. 25), one obtains P= f dp() = -sina - (-y)d.. ( (3.48) 1J~ J^~~~ ^ (1-7) -(E Q- sin a By observing that d = a sin a, it easily follows that b.=s. asin a E =lim a-0-O 2r sin -sin(a 2 (3.49) and finally by carrying out the limit = { 2r l —y/ (3. 50) results. Ihrdetermining the phase 8(g), it is first mentioned that the factor of Tr in (3. 40) is added in order that the 1800 phase shift, from the total reflection of the incident rays at r=b, be taken into account. The explicit form of 8(g) is evaluated from (3. 42). By taking the origin (Q=0) as the zero phase reference point, the solution of the first order linear differential equation =de(D) a a~ 1/3 (3.51) is 8() = aa-2a [-(l-^7)n(1)]. (3.52) It follows then from (3. 41) together with (3. 50) and (3. 52) that the reflected electric field is given by ikr k1 ) -i2ka 1-(1-T-)ln ( b. s ^ e _ ka(1-Ty) -i2ka [^ —x kr x (3.[53) X kr 2 1-in/- nO]

54 It is observed that this latter expression is in agreement with the first order term of (3.35). 3.5 Scattering Cross-Section Computations It is of interest to perform numerical computations, so that the effect of the radially inhomogeneous coating in reducing or enhancing the monostatic cross-section of the perfectly conducting sphere of radius b, is determined. These computations are carried out for different values of 3, y, and ka. The cross-section of the perfectly conducting sphere of radius b is denoted by ab and it is normalized to a, where a is the cross section of this same perfectly conducting sphere coated with the new class of radially inhomogeneous ab dielectrics with outer radius a. The normalized cross-section aN = is N co determined for two cases. Under case one, aN is derived by considering the N1 reflected electric field given by geometric optics. Under case two, oN is N2 derived by using expression (3. 35). The expressions for a and a are N! N2 given respectively by 2 -2 1-a.n-?/2 yaN1 =.. (3.54) and N- =132 { 1 ( 1-en3- / 3\2 n1 l1aN2 k ~-1 4 ] ( 1- 1+ 4ka(1- sin [ka(1-y) In(-)] x 2 4/ (ka) - (i'y 1+7y-2 [ n3+ -/])] +16(ka)2(1-2 ( ) - 1- ( 1-n 3 3 3 13 + if3(1 —n — 7/-) 2 + (1-f n3- q/) cos [2ka(1 - yn()3- ] +- 1+^y-2(Qn13+ W,))] ) j -1 }(3. 55) x,\ + l+T-2(,n 7 r/3)J )]2

55 uN - N Along with aN and -N the relation D 2 x 100 is computed for 1 N2 NN2 0.2: 13 < 0. 99, 50, ka < 1000 and various values of 7. The parameter D gives the percent error in using the geometrical optics approximation, in order to determine the cross-section, instead of (3. 35). The most interesting numerical results are shown in Fig. 3-4 through Fig. 3-13. In Fig. 3-4 the parameter oN is plotted vs. 3 for 0 < < 1. For this range of, - the inhomogeneous dielectric is of the converging type and the coating enhances the cross-section of the perfectly conducting sphere as one would expect based on physical reasoning. On the other hand when y> 1 the inhomogeneous dielectric is of the diverging type and as it would be expected the coating reduces the crosssection of the perfectly conducting sphere. In Fig. 3-5 one observes that this is the case and that for smaller 13 and 7 very close to unity, the reduction of the cross-section is considerable. The percent error D is presented in the remaining figures vs. ka for various values of 3 and 7. It is deduced from these figures that the percent error is insignificant for ka as low as 50. The conclusion then is that the geometrical optics technique is indeed a powerful, very accurate and very simple tool in studying the cross sections of perfectly conducting spheres coated with radially inhomogeneous dielectrics. An exception to the above conclusion is the case y = 0. 99 3. It is seen in the last two graphs that for this case, the error is as high as 74 percent when ka = 50. However, this should be expected if it is recalled that the radial eigenfunctions S(J) ) and / ( )) as obtained by the WKB method are valid provided that V- /2 V_ 1/2 *0 and 2ka(1-y)|> > 1. When y= 0. 993 and 13= 0.98 it clearly follows that this latter condition is violated and therefore the error for this case is explainable. 3. 6 An Outline for the Creeping Wave Contribution in the Backscattering Direction In this section, part of the analysis required in order to obtain the creeping wave contribution in the backscattering direction, in an explicit form, is presented.

56 "N1 1.00. 8 0.6 0. 4 7=3/4 / = 13/2 0. 2 y=3 3/4 =00. 99. FIG. 3-4: UNI VS. 3, FOR 0 < y< 1 NI2.4.. Ij 0.2 0.4 0.6 0.8 1.0

- I'O O0 CO hI. 0 0 0 0'1 o Q II II II II II 1S3 t-, CJ1 CO l33 t-* 11 1 11 1 0.CO LG Too to ~ ~ ~ ~ 6

58 D FIG. 3-6: DVS. ka FOR y= 3/4 2 0/o 3= 0.2 -2 0/o 2 ~/o / = 0.4 -2 0/o 1 ~/o 1 =0/o0.6 /o 1 /o - 50 100 150 200 250 300 400 500 600 700 800 900 1000 ka 50 100 150 200 250 300 400 500 600 700 800 900 1000

59 D FIG. 3-7: DVS. ka FOR y= 1/4 1/2 O/o 13 = 0.90 -1/2 O/o 1/2 O/o 3 = 0. 92 -1/2 O/o 0.01 ~/o -0.01 0/o: 0.94 0.1 o/o.-/< 13 = 0. 96 -0.1 ~/o 0.2 O/o 3 = 0.98 -0.2 0/o 0. 05 0/0 = 0.99 -0.05 ~/o I 5 I' I' I' I' I' I 0 8ka 50 100 150 200 250 300 400 500 600 700 800 900 1000

60 D FIG. 3-8: DVS. ka FOR 7y= 3/2 1 0/0,= 0.2 1 0/0 1 0/0 l^b 13=0.4 -l~/o 1 o/o 0 -1 0/o 1/2 % - = 0.90 -1/2 O/o 1/2 O/o _1/2\ /~~_~~ J ~3 = 0.92 -1/2 %/o 50 100 150 200 250 300 400 500 600 700 800 900 1000 ka

61 D FIG. 3-9: D VS. ka FOR -y = 3/2 0.01 ~/o - 1 = 0.94 -0.0lo/o 0.2 /c 3 0. 96 -0. 2 /o 0. 2 0/o \3 = 0.98 -0.2 0/o. 05 /o 1 3= 0.99 - 0. 05 0/0 - I,, -i - -I' I'- I,- I, - I J - I ka 50 100 150 200 250 300 400 500 600 700 800 900 1000

62 D FIG. 3-10: D VS. ka FOR y 3j3/4 2 0/o~~~~~~~~~~~~=2%_ -2/o =. 1 0/o Iy00,B = o. 4 -1/2 o/o - \ _3=0.6 -1 ~/o ~~/~^~ ~ ~~, = 0. -1/2 /o -V2 % _ 5I I' I 5I I 3 I 5 I -0 ka 50 100 150 200 250 300 400 500 600 700 800 900 1000

63 D FIG. 3-11: DVS. kaFOR y=31/4 1 O/o1/ > ~ ~ ~ ~ ~ ~ ~ ~~~1 = 0.92 -1 o/o 0. 2 o 3 0. 94 -0.20/0 0.2%/o 13 = 0.96 -0.2 0/o!/2~%/~ \Z3 = 0.98 -1/2 O/o 0. 1 0/0 13= 0.99 -0.1 ~/I i I 5 I' I 0 I' I, I 7 ka 50 100 150 200 250 300 400 500 600 700 800 900 1000

64 D FIG. 3-12: DVS. ka FOR = 0.9913 1/2 ~/o ~20/1 _ 3=0.2 -1/2 /o 2%0/ 13 = 0. 4 /2 O/o \ 3=0.4 -20/o 2lo - 3 = 0.6 -0/ - ___^ __ ____\__ 1,/3 = 0.9 -i2 /o -10 % 50 100 150 200 250 300 400 500 600 700 800 900 1000

65 D FIG. 3-13: DVS. ka FO{ = 0. 993 10 o __ = 0. 92 -10 ~/0 10 ~/o f3 = 0.94 -10 ~/o 200/0 1 = 0.96 -20 ~/o 100 /o 3 = 0.98 -100 0/o 50 ~/o -......_ ^^~~13 /= 0. 99 -50 /o 5 I I, I i I, I, I, I 8 ka 50 100 150 200 250 300 400 500 600 700 800 900 1000

66 The asymptotic expansions for the radial eigenfunctions S( ( (e) and T (/ (), V-1/2 V /2 which are valid for v near ka, are given in terms of the Airy functions in Fock's notation. These expansions must, more accurately, be valid for v = ka+ mt, where m= (ka/2) / and t is of the order of unity. Since, in this research, interest is confined to the contribution of the first few creeping waves in the backscattering direction, particular attention is paid to those poles in the complex-v plane which are located nearest the Re v-axis. A parameter T is, therefore, defined such that T t — (3.56) 2 m which implies that 11< < 1, and this latter condition corresponds to considering only the first few creeping waves in the 0 = Xr direction. With definition (3. 56), the coefficient Q(~) of the differential equation (1.16) becomes 2 - 1 + T — (1- )2 1 4 4(ka) (3 which are at -2 2 in order to examine whether the zeros of Q(Q) lie within /< 1 one first finds the zeros of 0 (3. 58) which are at 5 = 1 (3.59) ~01( =0 and = __(3.60) ~02 2-y It follows that since y < 3, 02 is outside, E< 1 and therefore 01 is the only simple turning point in that range. By defining a parameter 2 T= 7 — ---- 4 2 4(ka)

67 then since = O[(ka) 2/3] and similarly T = O[(ka)2/3], by a perturbation technique the simple turning point is found more accurately to lie at = 1- (1-) T+o0(T2) (3.61) 2 2 ) Since T = O(T ), then to the same order of approximation T7T and to = 1- (1-3)T+O(T2) (3. 62) This turning point is within,,< o ( 1 and therefore Langer's technique is now used to solve the differential equation. By writing the coefficient as 1- T\2 _ +T Qo() = (1- ) - the differential equation to be solved is a2 (j) (j) = 2 S-1/ (Q) + (ka) Qo(0) Sv_1/2 () = 3.63) d v-1/2 The solutions of (3. 63) are given by S) o 1/4 _ ()QK ) ow [(ka)2/3 (/)] (3.64) and /i (Oto S(j) v-() j = 1, 2. (3. 65) In the above relations 2/3 (=) ( 1Qo( d)) d 2 (3.66) and therefore one calculates:

68 f d(1 Y) +2T d y- -T In -2n{ T T 2 ( 2y- T) 2+ 2y(1+T)-(l+T) +( 2y- T) + y(1+T) - i T X 0o x in - (+T) 2'y - 7T +2 y(l+T) -Y(1+T) -(1+7) +(1+T)| - ~~~1 J ^~~~~~~~~~0 {(r2 -2y- ) 2 2 - (1-y) n (1 -) (E-)'Y2-2Y- T) +21+T) -'Y2(l+T) + +1 (3.67) In order to proceed in computing numerically (1.56) which gives the contribution of the creeping waves in the backscattering direction, equations (1. 57) must be solved for the zeros t and t. This in turn necessitates simplification of M /2 (ka) and M 1/ (ka) asymptotically in terms of T or 2 t/m. From relation(3. 67), the asymptotic expansions to O(T /2) of JI 1! 7 d~ and IJ Td d~ are obtained first. Then ~(1), 0o (3), a/y,, etc., are computed, to give finally the asymptotic forms of M(t) and M(t). This work has not been included here due to the cumbersome expansions. The same technique, however, is carried out in Chapter Four for a simpler type of radial inhomogeneity, and the constants M(t) and M(t) are there given explicitly.

CHAPTER IV HIGH FREQUENCY BACKSCATTERING FROM A PERFECTLY CONDUCTING SPHERE COATED WITH A DIELECTRIC WHOSE INDEX OF REFRACTION IS N(Q) = ~P 4.1 Introduction Nomura and Takaku (1955) considered an interesting class of radially inhomogeneous dielectrics in their study of radio wave propagation in an inhomogeneous atmosphere. They assumed the atmosphere to consist of stratified layers of radially inhomogeneous media. The index of refraction of the K th layer was taken to be N(M) = P K, with p> -1. This index of refraction represents a class of radially inhomogeneous dielectrics which are of the diverging type. The larger the exponent p, the greater is the divergence of the electromagnetic rays. Nomura and Takaku (1955) solved the wave equation and superposed the solutions of TE and TM modes in order to obtain a complete representation of the electromagnetic field. The following radial eigenfunctions were obtained for the corresponding TE and TM modes. T (S =ka^ H I I for TM modes sj) =2 H 0(j) ka for TE modes T(j) (') =ka' p+ ( ) v-I/2 2(2p+ 1)~ v"k p+1 with VI' = and v" = p+1 p+l From these solutions, the restriction p > -1 becomes clear if one observes the argument of the Hankel functions. By assuming a dipole excitation source and by applying the Watson transformation the authors obtained a residue series, which represents the radio waves traveling around the earth. Nomura and Takaku also applied geometrical optics to trace the ray paths in the inhomogeneous atmosphere, and performed numerical computations by assuming different values of p for various environmental conditions. 69

70 In this chapter, the above mentioned radial inhomogeneity is considered as being the coating of a perfectly conducting sphere of radius b. The outer radius being taken as r = a, the normalized index of refraction is written as N() = P, p> -1. In the same manner as in Chapter Three, a plane wave is assumed incident on the coated sphere with its electric field given by (1. 3). The backscattered electric field is put into an integral form by applying the Watson transform on the Mie series and the explicit asymptotic expression for the reflected electric field is obtained by integrating along the path rl (see Fig. 1-3) with the aid of Scott's integrals (Scott, 1949). The creeping wave contribution is given by the sum of the residue series as in (1. 23). The monostatic cross section is finally obtained from the geometrical optics reflected electric field and it is computed for different thicknesses and different values of the exponent p. 4. 2 The Radial Eigenfunctions in their Asymptotic Form In solving the differential equations (1.13) and (1.14) exactly, one may encounter difficulties in developing their asymptotic expansions to O[(ka) ]. In this case the exact solutions are Hankel functions of complicated argument and index and their asymptotic forms may be derived from the well known Debye expansions of these functions. Nevertheless, it is easier to obtain these asymptotic expansions by applying the WKB method if possible. In order to apply the WKB method the normal forms (1. 16) and (1. 24) are considered in S)/ (j ) and U (). From these differential equations, it is seen that V_~ /21 V2~-1/2 Q () -2p _v42 1/41 ad(1") (ka) 2.2 _2 2 -1/4+ p(p+l) and Q,2M) -P22 (4.2) (2)~ ~ (ka) 2

71 have zeros at * 2-2 1 2(pH1) 0= L (ka)2 3] 0an (2ka) 2 correspondingly. It follows that since v = O[(ka)/ 2+6], with 6 << 1 and 6 > 0, then 10 < < 1 and 20 < < 1. For finite p, 20 > 10 and therefore if b > 20 then the WKB method can be used to obtain solutions of S )/ (Q) and U.J 1 () which are valid throughout b,< < 1. It must be mentioned here ]-1/2 / that in this case the WKB method is applicable because conditions (1. 27) are also satisfied for b,< < 1. By considering (1. 28) valid over the range v = [(ka)/2+ 6], it remains to develop the asymptotic forms of[Q(i)() -1/4 and perform the integration and carry the asymptotic algebra in the exponential term, in order to obtain the explicit asymptotic expressions for S ( () and (.i~~~~~~j) SO(j~~~~~~~) 1/2 Uy/1 (e). In particular, for the functions S^j (L), by noting that — 2 1p2 1 2-2 (-1/2P + [Q]-/ exp n ( } - p /2)+ [1) 4 (ka)2 2(pt) (ka) 4) (4.5) and 5 dQ (1)() dQ (1) exp+ ika') 1+ 4 d (1) d d (1) ( 1) Jeik Cp+l v2-1I4v _ __ P-1 -2 1 1 51 -3p ovexp ika p+ i 2ka ( p+1)P+l +- ( 13( ) 8(ka) 3(p+1) 48ka p+1(p+ 1) 9 8(ka) 3(p+l) [(2p1)] (P+L)}+o[(ka) - 2] [ o ([ +3] [(4] [+0 (ka)] (4.6)

72 the solutions (j) -p/2 +eP + ika2 1/4 1 4_ i3 Q yexp p + i a- + V~- 1h Ptl P-2ka (p+l) 1p+I - 8(ka) 3 3(p+1) 3(p+1) 5i 1 i p(2p-1) 1 1 V2 __-2__:F48ka +3p - + 1 48ka 3p 4ka p +l + 4 (k 2 2(p+)}(1+0[(ka) ]+ O[3 + [ 4 ] [ 6 2 [ 8 ) + 0 4+ 0 V 5 +O0 6 (4 7) (ka) (ka) (ka) are derived, which are valid for p> -1 and b > 20' The superscript j denotes 1, upper sign (4 8) 2, lower sign By proceeding in a similar manner for the eigenfunctions U (j), it is found that with [Q(2)(t] -1/4 -p/2 2+ 2 (ka(+) 4 I) p-f- 2 exp < ika Q(2) (Q)d exp ika + i 2a + (2) +1 2ka (p+l) P$+1 _______ ~. Vi..p__1 i}(+o1v23]+ 8ka( 1)p+li 2ka p+ l -24(k)3 (P (3 ka) ] [(ka)k ] (ka) [(ka)a / and

73 5f 4 dQ Q(2)) () 5 4 d - Q(2) ( ) Ja) 2 Q )5/2I 48ka 4ka p- p+T1 O(ka) 2]+0 + O[-1 ) (4.11) +4kap (ka) t(ka).1/ the solutions ( ) -P/2'+1 r 2 1 ___ U (exp ika ^-i F + v (e/2 exp p+1 2ka (ptl)p+1 8ka(p+1)t + a1 2v^ 2(P^ l+ + l3] + k(ka)4+ (ka) (ka)6/ (4.12) result and they are valid for the same restrictions as S () (c). From (4.12), d-2 /2.-I V1/2 T (j)l (U) vp 1/ ( ) (4.13) is obtained. It is furthermore observed that T( = + exp[i 2 S1]s (. (4.14) 4. 3 The Reflected Electric Field In this section, the reflected portion of the backscattered electric field is derived. With the aid of the asymptotic expansions for the radial eigenthe parametrsV/2 V M/2" /2 M y/ (ka) are computed. Then the difference of the scattering coefficients ^-/2 funcin h aajtr V'2F ) C-/( P Mzka n Mvz~a) re-op/2d hntedfeec f h cteigcefcet

74 is found and finally by integrating along the path P in the same manner as in Section 3. 3 the reflected electric field is determined. From (4.7), (4.13) and the definitions of C1,,/ (, 3) and C 11/2 (, 3) one obtains: l(-, /22i () (ka)2 2 2(p+1 2(+1)]} -2 v2 1 4 61 8 \ +o[(ka)-2]+ 0 3 + 4 —I 4J lv+i 6) (4.15) + ] (ka )3 (ka) 4 [(ka)5 (ka) C/,.1( )2i( P/ exp (4 2 [2(1 + 2(p 1)]}1sin [g(2)1 (1 1 u-~(f [_2 1r/ 4 2 [68]' +oL(ka)-2J+ - 0 + O + O 61) (4.16) (ka) IL(ka) ka) (ka) where () p Sl V" p 2ka p+ l ^l.)~ 91 )(t3) = ka (Pt1 1 13Ptl)+ v-1i/4 1 1 1 )t1 4 1 _1 5 1 24(ka)3 p+1 3(p+1) 13(p1) 48ka 3p 3P + p(2p-1) (J _ -1 (4. 17) 4ka(p+l) ep1 + ) (4.17) and g(2)( (1l) 2ka p+1 pp+1) * 418) From (4.15, 16, 17, 18):

75 My/ (k) 2ka + (1- 2 -2) cot g(1)(, 1 (l+O[(ka)-2] + +oI -VI 0 vo- -- + 0I 1 2 r 4 8 (ka) L(ka) (ka) (ka) and M V-/2 (ka) 2ka - 2 - 2 k ) tan g(2) ('3 - P (+ tan g(2)(,3)) X ra 2ka1 )] 2 6 [44 6i X (+0[(k-Oa) 1 -+0 34- — 5 6 (4.20) \ J (ka) (ka) (ka)) (ka result, where the expansions 1 ag(1)(, 1 2 1 2+ 1 I 1-.2 2 4] ka d1 [I + ka)-2]+ (4.21) ka =1j 2 (ka)2 (ka)4 and 13) / -. 1-t32(p+1) +O[(ka)-2]+04] 2 (ka)2 agmm' 9) ap AP 2 (ka) 2 A 2(p 1) (ka) 4 (4.22) have been used to arrive at (4. 20) and (4. 21). By writing g(l) - g(l)(') ~=1' g(2) -g(2)()' (4.23) g(2) = g(1) + e3 = l + 2 3, (4.24) with ka 1 - p+l 1 (4.25) ^E1= p ~^ + 2ka p+1 1 p4-l (4.25)

76 2 8ka p+l 1 - (i 3(p+1) -48a 13z"24(ka) 1P 48ka p + 4ka(p1) (1 /, (4.26) e = P ( ) ) (4.27) and by using the trigonometric approximations tan g(2~ tan l + (e + 3) [1 + tan ] (4.28) and cot g(1)l cot e - e2 [l+cot2e] (4.29) the following relationships are obtained: M (ka) - M (ka) ka sin2e - 2 ct2E +0 — v 5 + Ov 6 - (4.30) (ka)nJ (ka) -i / 2-2i 2e3sin2l E Oe., t a n 2 3 1 M (ka) + Mo(ka) - 2 - i1 ~~1 v~2 2 22~~ 1 -2 j2)- 2 1+tan2e 1+0 kka)2 0[V23]+0 4 1+ "(ka)" 2kaP l (ka) 3 (ka)4 r 6 [ 81\ +0 -V +0 V ) (4.31) (ka) (ka) and

77 -2 k M (ka) M / (ka)v - 2+ o[(ka) ]+ 3+ 0 + (ka)2 (ka) (ka) r 6 81\ + ol -]+ 0- 6) (4.32),(ka), (ka)6/ With these expressions and the Debye asymptotic expansions for i/ (ka) and (1)' - 2/ (ka) the asymptotic form for the difference of the scattering coefficients is found to be: 2 / irv-i J - i2ka+i2e1 4 1 a l/- bs ie k1+ 4 -i - + 2iE + v -1/2 v -ih 4ka 12(ka)+ 2i ~2 2 \ ip v-i - +i4E1 -i2ka iv - i - i2k + 4k e ka - e ka0 (ka ) + 43Cka 4ka. \ 2 r 4 r 6 8 +0-i v 3+0V O +0 +0 o 60 ) (4.33) (ka)3 - ka)4 kka) (ka) 6 and when v = 2 -i2ka [1- 1 (-l /p as s_ bs s Pil r x aI-bi/ =a -b e l —- x V-v2 v-12 o o 4ka j P+ [p ]K~ -2i v2]+ 4V6L~]0[ %] )' 01(a)[]) La) O[(ka) +(kao ) (4.34) results. The reflected electric field is now given by

78 ik[r-2a 1- 1 -3])]. + ka -i + 2i b. s. e ____ - __ Jl+ 4 ___12(ka)3 x kr {2 -i27rv 1+e ~~~~~~2 v~~~2 ka p+ P+1 p p+l e d -e ve 2t 1 E f 1] 2 11 2ka -ip p-I-i ve -i2 e l+ei12 2.i v' j, ka 2rve - EW ye. ~~ dv+ __ we (n J7Vdw (4. 36) 1+e i2 (27ri)2 2veY 1+e-w i/2+c i where v (ka), e = - w = i2w and 0 <y<ir/2. 47r ka It is easily seen that the integral on the right hand side of (4. 36) corresponds to E of(3.31). Then 0,O 2 -iV Je ve.....dv 2 [.- + O(E1) (4.37) 1+e-i2wv (2~ri) 2 (436

79 or 2 ka 1 e -i dvi ka 1 O-[(ka).- (4.38) -i+ e Another type of integral occurring in (4.35) is v - i ka P1- 1-i ve dv= 1 v e dlJ 31 e -i2rv 6dv 12(ka) 3 1+e2v 12(ka)3 (2ri)6 iy 2 2rveiy 5 - ew xi w e dw (4.39) -W j_ 27rveiY +e The right hand side integral of (4. 39) corresponds to the type (3. 30) with q= 2. Then from (3. 33): iy 2 J 27rveiY 5 -Ew i_ _ 1 w e dw+ 1 -- --- eh0^'.) -dw ew + 6 + O(e 2(ka) (iy w 12(ka)3 (27rii) L -- ~27g~~~~~Trve ~(4.40) In this case e = - 1 - p (- l ) (27^2 p+' \ p+1 By proceeding in a similar manner, the saddle point method integrations are completed and the final result obtained for the reflected electric field is

80 b.s. A ikr / 1 \ 1 - -i2ka 1- Pt [1 1P+]] Eb. ^ e I 1 p+l r ~A1- 1 _ refl -x 1kr [pl 1- 1 _ x li p+l [ 1 p1 i f'- 12 13p1, l,+-'__ll_ I(....__..1_ii_ r( 13P<l)] ~ p +- (l ) 8 2 1 14ka /3 ( - -'+ 3- 31- +1 1 i1( 2 (1- p l)] ]) [(a) / P+(4. 41) xo t. r t (3. ) for the pe ctly c ctig sp e w n 1. + 2 1 p+l p+l p+/ 12 3pl 2(p+l) pi ka 1 -P 2ka (p+1 -^ -/ -mkj^E'^ Jf k (4.41) 1- p + p - p2 1 3P This expression is valid for p> -1 and j3> 20 Furthermore it reduces to the result (3.36) for the perfectly conducting sphere when J3= 1. 4.4 The Geometrical Optics Approach The ray tracing technique, as it was shown in Chapter Three, is very useful not only because it is helpful in checking the results obtained by rigorous asymptotic theory to O[(ka)1] but also because it clarifies to a good extent the physical phenomena which take place. In this section, the ray tracing technique is again applied to obtain the optical ray paths in the radially inhomogeneous coating and the reflected electric field to O[(ka) -]. It is assumed that a tube of rays of diameter 2d is incident on the coated sphere. Upon incidence on the inhomogeneous medium the rays diverge away from the perfectly conducting sphere, as shown in Fig. 4-1. It is expected, therefore, that, based on physical reasoning, the coating will reduce the

1.01 A~~~~~~~~~~~ x + FIG. 4-1: PATHS OF INCIDENT AND REFLECTED RAYS

82 monostatic cross-section of the perfectly conducting sphere. Indeed, it is shown in Section 4. 5 that this is the case by computing numerically the monostatic cross-section. The reflected electric field is now determined. By assuming that it is given by b. s. Eb. s. ik ( i)+ iD (3.41) x o for ka -* oo when a is finite, the amplitude is found first by applying the principle of conservation of energy between incident and scattered fields. The factor 7r in the exponent of (3. 41) is due to the abrupt change of phase which b s occurs due to reflection of the incident ray at r=b. The relation for E o is given by (3. 45). In this case the angle p for an arbitrary incident ray is p p(g)d = - sin, (p+1) 2 (4. 42) in J^?V - sin a or, by integrating 1 1 -1 rsina 1 -1 P= - os (s [psir -cos ina]) (4.43) and it is shown in Fig. 4-1. From (3. 45) and (4.43) one obtains b. s. a sinac E lim (4.44) 0 2r 1- - 1 F sin ]l - Co-1 1 a —O 0 sin a- l (coss [sinpa] By expanding the inverse cosine terms in a series form restricted to principal values (4. 44) becomes:

83 b.s sin c E lirm o 0 2r 31 a 3 sin {- - a- 4- - - - * -n * p +l 6 63(p+1) + a sin a ~ 0 2r. f e2 2 {- ( Of [- + 6 603(+ ).) a 1 2r 2r1- p+l (1- P a 1 (445) In order to compute the phase 0(e), the eikonal equation (3. 40) is considered, which in this case gives the following differential equation: d( = ap (4.46) With the phase reference point being the origin, the solution of (4. 46) is 9(e) = a -2a 1- - 1) (4. 47) The reflected electric field is then given by ikr-i2ka [1- 1 +)] E -x ( )^ ( a ) (4. 48) and it agrees with the first order term of (4.41). and it agrees with the first order term of (4. 41). 4. 5 Numerical Computations Based on the expression for the reflected electric field derived by geometrical optics, the monostatic cross-section of the coated sphere is found to be

84 2 7ra c [ 1 l(4. 49) By proceeding as in Chapter Three,the cross-section of the perfectly conducting sphere ab is normalized to ca. Then computations are performed for 0. 1<:/, 0. 99 and p = 1, 2, 3,4,5. The computed normalized expression is 2 N -: -- o 1321 (1- 11)] (4.50) and the result is shown in a tabulated form in Table 4-1. It is seen from Table 4-1 that the monostatic cross-section of the perfectly conducting sphere of radius b is reduced considerably, as the thickness of the radially inhomogeneous coating is increased (1 decreases) and as the exponent p increases. The calculations of a and D have been omitted for N2 this case, since it is felt that the results of Section 3. 5 give a rather general idea of the error involved in using geometrical optics down to ka=50 to compute the monostatic cross-section.

TABLE 4-1: COMPUTATIONS OF uN' FOR VARIOUS 1 AND p 3 p=l p=2 p=3 p=4 p=5 0.10 25.50252 1115.562 62537.64 4000331 277781.5x 1015 0.20 6.760001 71.68446 985.9604 15665.03 271441.2 0.30 3.300279 15.23891 89.95126 621.5734 4738.555 0.40 2.102500 5.522501 17.69254 66.13759 275.8715 0.50 1,562500 2.777778 5.640625 12.96000 33.06250 0.60 1.284445 1.758080 2.583759 4.093380 6.987287 o 0.70 1.132704 1.315469 1.572172 1.940411 2.480574 0.80 1.050625 1.111267 1.184356 1.273019 1.381314 0.90 1.011142 1.023178 1.036193 1.050280 1.065543 0.91 1. 008921 1. 018474 1.028711 1.039690 1.051473 0.92 1.006969 1.014358 1.022228 1.030583 1.039470 0.93 1.005276 1.010831 1. 016684 1. 022851 1.029354 0.94 1.003833 1.007838 1.012022 1.016395 1.020967 0.95 1.002633 1.005363 1.008192 1.011126 1.014168 0.96 1.001667 1.003382 1.005147 1.006962 1.008830 0.97 1.000928 1.001876 1.002843 1.003832 1.004841 0.98 1.000408 1.000822 1.001242 1.001667 1.002098 0.99 1.000101 1.000203 1.000305 1.000408 1.000512

86 4.6 The Creeping Wave Contribution The contribution of the creeping waves to the backscattered electric field is studied in this section. In chapter one, it was mentioned that this contribution is given in terms of an infinite summation of residues in the first quadrant of the complex v-plane, where the residues closest to the real v-axis occur for v near ka. The strongest contribution in the backscattering direction comes from the residues nearest the real v-axis. It is this case which is examined here. First it is recalled that the asymptotic expansions of S( l/2() and T ( /() valid for v near ka are required, so that M v/(ka) and M i (ka) can be determined from their definitions C-/2 Z (/-1/2 in terms of C o /(ka) and C /(ka). By considering the differential 2 v'~/2 equations (1. 16) and (1. 17) with Q Mi)() and Q(2)( ) given by (1.26), (4. 1) and (4.2), it is readily seen that for 12) << 1, or p finite and (ka) 2 >> 1, one can define in approximation (ka)2 2 2 Q(e) ) Q)= Q 1( ) = Q(2 ) 2 2 (4.51) (ka) 2 By setting v = ka + mt, where m= (ka/2)1/3 and by defining a parameter = t/m2 (4.52) such that |T << 1 for the first few creeping waves, it follows from (4.51) that Q(-) = 2p (4.53) It is immediately seen that Q(Q) has a zero at

87 1 = -(l+T)p p > - 1 (4.54) op \ 2 /> which is outside the interval / < ~ < 1 but nevertheless very close to it. Langer's theory is therefore used in order to obtain the solutions. These solutions are and T-) () P (~)) W(j [ka)2/ 3()] (4.56) with j = 1,2. Also () = (2 Q4 d. (4.57) op By following the outline in Chapter One, Mv i (ka) is still given by (1.58) with ( i)() = r(),whereas M 1/(ka) in this case is: 4(ka) 1/3 a(') (k) X Ml(ka) =ka +v/ (ka) - I K V-112 2 ka +- 1 _1, g() 3 4 () a3,8 Q(s) a3, -\2 a ((4.58) X (Wl"^g) "2) [u(A2 ([W()] 2) [W() (4.58) + (ka)2/3 a() (l) ]"2) - [ ] 2 [)] 1 (a Lt)[L()] [(2)L -1( ]w()12)L()] j =1 with w(c) = (ka)2/3(), w(3) = (ka)2/3(), (4.59) and it is assumed that 3 f 1 so that

88 Wa [1l)] (2) ['~] -W( ~] 2) [l)] / 0 * The asymptotic expansions M(t) and M(t) of (1.58) and (4.58) will be developed,so that w( —) -= M(t) m and w()= -M(t) m w(1) I can be solved numerically with the aid of the asymptotic expansions and diagrams of Logan and Yee (1962), to yield the zeros t, and t*. To this end the following procedure is followed. Firstly the integral i foQY d5 is evaluated. op One finds that i.1 ----------- J 7\2 d 2(p+-l) T \2 f j2(p+p1) + T)2 d? - - op 2 12 12i.. __________ 1+ 1 ( P+1-) sin, 2 (4.60) from which

89'f1-^ Edg =-+ v c(+T sin +T)) (4.61) p+ 1 pi 2 op results. The asymptotic expansion of (4.61) in r yields 2 3/2 3 (p+ l) 4 fd1i e r (1j)2 - 40 + 0 [T2 ] (4.62) op from which - /3 7r 1 2/3 (1) |= [ Q() d (p1 ) 2/3 + T (4.63) op results. With the aid of (4.62) and (4.63) one has 1 f 1 M a), 1 aQ,12 - 1 1- 2 ([V5/2] (4.64) 4ka {(?Q) a a() e/p=1i 2ka 5 + 0 and,2 ae-' 3 [2(p]1/3 ( -2 191 2 3]. e + - 15 28800 + o Other computations pertinent to finding M(t) and M(t) are 2- 2 -1 dr =. +1. P + — - sin,- (4.66) op

90 3-i r 2 )3/2 f^ ~~e (2 p[-l2(P1) [ +1+a22 o[3] (4.67) op with o= 5 [8- 3p32(p+l)]. (4.68) 1 r8+92(P+l) 124(p+l) 1e 10- [L 1 2(pt1)J (4.69) and c'2 8 [7 _ 2(p+1)+4 24(p+1) (4.70) ^s s E_ 2(p+.w']2 Also [jp4-1J)] 2/3 2 a' 0 and.0 ~4.71) i2 r 1l/3 ( l KW)^ e 3 [2(pl)J ^ 1 a1 1 13 /3 K+ 12(p+1)]- 3aJ T+ (2 a a 2(p+l) 1 + 2 -1 i "12 +2 3:9 ^ 621l 2(p1) - 3o 8 [1 2(ptl)] j +0Lr]j) (4.72) and and

91 KW(_ ) 1 + P 2 1+ ( r o ---- --- 1 -'P+12( 1 r +.+ 2 ) - -1 1 - p. [ 2(pt -... (+ Q - 2 [2p)]] + ( [ 12(417)3 (o LJ o 2aO (.2(pFl)P - 22 8 [,2(P21)]2 ] -2(P1) [3+ 2(P+t1[ +p 2(P+1)]> (4273) 12(p+) 4[lp2(p+ 1)] 2 all valid for a 0. By substituting (4.69) through (4.73) into (1. 58) and (4. 58), M -2 (ka) and M V/(ka) are obtained asymptotically in r. Then with T = t/m2 and keeping terms to O [n ]: M(t) -2p + C(t) (4.74) 10ka and Mt)~ 3-2p + C(t) (4.75) 10 ka result, where C(t) v e P ) t + o [ w2 [w(4)] -w, [o(3) w [ (l)1 - W1wand]2 ( and

92 2 7 Ct0 el 3 1 w )1/3 (1 2 t [m ] W( - 4e (1) + [m4]) -1i - W(1) W(2) mr- [ (2 ] )W m [(t [W -.4r -'W(1)[()] W(2 )])-1 + 3 [l+0(m )](JQ [w(li)]w)[w()] - 2m a - wl, [) ] w(, )[ )])- 1 ) (4.77) 1+ p2(p l) - where a3 = 2a L2P+ 2(p+) 0 (4.78) P + 12p With the expressions given by (4.76) and(4.77) the first few zeros t, and t can be found approximately by solving the following equations numerically (1) (t_) 2p - 3 = - mC(tm ) (4.79) (1)(tj 20m 2 and w(1) T 2p -3 m c(t) (4. 80) i)(l), 20m2 Then the approximate contribution in the backscattering direction due to the first few creeping waves is given by (1.56), where the explicit derivatives ac(t) ar(t) a(t) and at( can be determined from (4.76) and (4.77). It must finally be noted that in (4.76) and (4.77) the asymptotic expressions for w(1) and w(,3) are given by

93. T _ 3 t2 41 (1) 3 e - t- 2 + 0 (m 4) (4.81) (p+ 1 /3 60m2 and 3(p_1) r [ 2(p+l) a2/3 +m2 t + (P( L' H J 3a0 ('2a24 2 } + (2 1 ) t +o( -4)} (4.82) + 3 -9 (4.82) \ o a m For large b, it is to be expected that the creeping wave contribution is small compared to the reflected field contribution. Also it must be mentioned that by observing the coefficients al, a2 and a3, those values of f3 must be excluded for which the denominator of these coefficients becomes zero, i.e. 1 4 1 and p - -1.

94 CHAPTER V CONCLUSIONS In brief, it has been shown that the monostatic cross-section of perfectly conducting spheres is enhanced or reduced, when coated with radially inhomogeneous dielectrics, depending on whether the radially inhomogeneous dielectric is of the converging or diverging kind. It has been verified in the case of the Nomura and Takaku radial inhomogeneity that the greater the gradient of divergence of the coating, the greater the reduction of the radar cross-section of the perfectly conducting sphere. Furthermore, the new class of radially inhomogeneous dielectrics has been determined to be important in analytical studies for radar crosssections, because it can present converging or diverging properties depending on the choice of the parameter y, and because it reduces the two differential equations (1.16) and (1.17) essentially to one. When this new class of radially inhomogeneous media is considered as the coating of a perfectly conducting sphere, it has been found that when 0 < 7 < 1 it enhances the cross-section, whereas when 7> 1 it reduces it. However, it must be mentioned that the computations for o when 7 is very close to 3, based on geometrical optics, N1 are not very reliable since the condition 0 < Imk < < 1 is not taken into consideration. This is verified, if it is recalled that when 7 = 1 the rigorous asymptotic theory to O[(ka)2] predicts a very large reduction of the crosssection, whereas the geometrical optics based computations for aN predict enhancement of the cross-section for 7 = 0. 99I. The introduced error in computing the radar cross-section by using the geometrical optics solution for the reflected electric field instead of the solution obtained by rigorous asymptotic theory to 0 [(ka) 2, has been found to be insignificantly small, except for the case where 7= 0. 99 3 and / near unity. In this latter case, the error is as large as 75 0/o due to the fact that the asymptotic solutions for the radial eigenfunctions are no longer valid since the condition |2ka(l- y)|~>> 1 is violated.

95 Since this research has been confined to considering bodies whose radius is much larger than the wavelength of the incident electromagnetic field, particular emphasis has been placed upon the study of the reflected portion of the field. The creeping wave contribution is much smaller than the reflected field, since these waves radiate as they travel around the scatterer and since in actuality the dielectric coating presents some losses. Other possible contributions to the backscattering direction, such as lateral or evanescent waves are not taken into account. These contributions are waves with algebraic or exponential decay, respectively, and they are expected to be much smaller than the reflected field. Such kind of contribution is given in terms of branch-cuts of S(j)/ () and T(j1 () in the complex vplane; for example, it is seen from equation (2.14) that two branch points occur at v = + ka(1-T). Finally, it must be mentioned that from the practical point of view the research in this dissertation has possible applications to the study of the monostatic cross-section of space vehicles during their re-entry flight in the atmosphere. In particular, the black-out phenomenon may possibly be explained by the formation of a plasma coating around the body, whose index of refraction behaves as a radially inhomogeneous dielectric of the diverging type.

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA R & D (Security classification of title, boldy of abstract i lad indexi.ng nntottion nmut be entered when the overall report Is classifled) 1. ORIGINATING ACTIVITY (Corporate author) 20. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108__ 3. REPORT TITLE ELECTROMAGNETIC SCATTERING FROM CERTAIN RADIALLY INHOMOGENEOUS DIELECTRICS 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Scientific Interim 5. AUTHOR(S) (First name, middle initial, last name) Nicolaos Georgiou Alexopoulos 6. REPORT DATE 7a1. TOTAL NO. OF PAGES 7b. NO. OF REFS November 1968 98 37 8a. CONTRACT OR GRANT NO. 9o. ORIGINATOR'S REPORT NUMBER(S) F19628-68-C-0071 1363-3-T Project, Task, Work Unit Nos. 5635-02-01 Scientific Report No. 3; ^DoD Element 61102F.h!)h. OTHER REPORT NO(S) (Any other numbers that may be assigned DoD Element 6102F Ilis report) Dod Subelement 681305 AFCRL- 6810. DISTRIBUTION STATEMENT Nr. 1 Distribution of this document is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general public. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Submitted in partial fulfillment for Doctorate Air Force Cambridge Research in Electrical Engineering, The University of Laboratories (CRD) L. G. Hanscom Field Michigan, Ann Arbor, Michigan, 48108 Beo Massacses 01730 13. ABSTRACT In this report, the phenomenon of electromagnetic wave propagation through, and scattering from, radially inhomogeneous dielectrics is studied for very high frequencies. The dielectrics are considered lossless, radially inhomogeneous in the spherical coordinates system, and of the converging or of the diverging type. The lens problem is studied by the geometrical optics technique and the radar cross-section of perfectly conducting spheres coated with radially inhomogeneous dielectrics is investigated. By assuming a plane wave as the incident electromagnetic field, the contribution in the backscattering direction due to the reflected field and the creeping waves is determined by applying asymptotic theory. This necessitates the use of the WKB and/or Langer's method for thesolution of the pertinent differential equations, depending on whether there exist transition points in the range for which the solutions are required. Also, the integrals of Scott(1949) are needed in order to determine the reflected portion of the field. Such a study is interesting not only from the theoretical but also from the practical point of view, in that it lends itself useful to the understanding of radio wave propagation in radially inhomogeneous dielectrics and of the effect of coating perfectly conducting spheres with radially inhomogeneous media. It also has applications to problems of wave propagation in the ionosphere and around the earth. DD DFORM73 147 —- XUU i~~N~~O ~~~Vo UNCLASSIFIED S't'u irilt C"liCssi..it'.tiO-,,

TTNCT ASSTnWT1EnTh Security Classification 14. YLINK A LINK 0 LINK C KEY WORDS RO L E WT ROLE WT R OLE WT High Frequency Electromagnetic Scattering Asymptotic Theory Geometric Optics Scattering Cross-Section I UNCLASSIFIED StrI1I1Itv ('l.a s iti; ii7

II -I IIli Ciii* ^ Esryo 3 9015 02493 7669 THE UNIVERSITY OF MICHIGAN DATE DUE 4ri t o A