THE UNIVERSITY OF MICHIGAN 7644-3-T Inverse Scattering Investigation Quarterly Report 1 April - 30 June 1966 by V. H. Weston and J. J. Bowman July 1966 Contract AF 19(628)-4884 Prepared for Electronic Systems Division, ESSXK AFSC, USAF Laurence G. Hanscom Field Bedford, Massachusetts 01731

THE UNIVERSITY OF MICHIGAN 7644-3-T ABSTRACT The approach to the inverse scattering problem based upon the representation of the scattered field in terms of plane waves is investigated. This technique is shown to have several advantages. If the scattered field is thought of as arising from a set of discrete sources, the field can be obtained everywhere outside and between each individual source, i. e. it is not restricted to the region outside the minimum convex shape enclosing the sources. This could have practical uses for investigating cavities or antennas mounted on the surface of the body. In addition, if the scattered field (phase and amplitude) is known only over some angular bistatic sector the near field (in the high frequency case) can be still obtained in certain regions. Thus, if it is assumed apriori that the body was a perfect conductor, then those portions of the scattering body giving rise to the observed portions of the scattered field can be found. For non-magnetic and non-perfectly conducting bodies, it is shown that the exact total field inside the body could be represented in terms of a plane wave expansion involving the far field quantities. This representation involves an appropriate split up of the far field data, and a fundamental problem still exists to uniquel determine the split up from the knowledge of the far field data alone. It is possible that additional information will be needed; perhaps knowledge of the complete scattering matrix for all frequencies. This is an important problem since its solution will yield both the shape and material of the body. I iii

I I I

THE UNIVERSITY OF MICHIGAN 7644-3-T TABLE OF CONTENTS ABSTRACT iii INTRODUCTION vi I GENERAL THEOREM 1 1.1 Determination of the Near Field from the Far Field 1 1.2 Additional Comments 4 1.3 Rotations of the Reference Axes 5 1.4 Relationship to Spherical Harmonics 8 1.5 Relationship to Aperture Problems 8 1.6 The Field Quantities 9 1. 7 Determination of Field in Free-Space Region 11 between Distinct Sources II ANALYTIC CONTINUATION 12 III HIGH-FREQUENCY SCATTERING 16 IV REPRESENTATION OF THE FIELD INSIDE THE SCATTERING BODY 21 REFERENCES 30 v

THE UNIVERSITY OF MICHIGAN 7644-3-T INTRODUCTION The approach to obtaining the near field from knowledge of the complete scattered field, based upon the plane wave representation is investigated for a fixed frequency. Such a technique has been used for particular direct scattering problems, and modified versions of it appropriate to the high frequency case, have been employed in geometric optics (Kline and Kay, 1965). vi

THE UNIVERSITY OF MICHIGAN 7644-3-T I GENERAL THEOREM 1.1 Determination of the Near Field from the Far Field. A problem of fundamental significance in inverse scattering theory is that of determining the electromagnetic field at all points in space from a knowledge of the field in the far zone. In this regard, an important representation of the electromagnetic field in free space may be obtained as a combination of infinite plane waves whose amplitude factors are given by the far-field and whose directions of propagation are, in general, complex. Although the representation discussed in this section is valid only at points outside the sources of the field, the extension to points within a source region is under investigation and will be discussed in Section 4. Consider the electromagnetic field produced by a given volume distribution of electric currents j varying harmonically with time (e ) and located in some finite volume V of free space. The field everywhere in space may be expressed in terms of the vector potential A given by, I ikR IA(_)=) = -L- R dx' (1.1) 4Jr R V where R=[x-x', and the far-field distribution has the form ikr A (r,,0) -A (0,0) (1.2) roor -o r -) co 1

THE UNIVERSITY OF MICHIGAN 7644-3-T with ___ -ik. x' A (0,0) = (x')e - e -dx' ~~~~~~~V ~(1.3) k = k (sin 0 cos0, sin 0 sin0, cos 0) The currents j may be thought of as equivalent sources for some scattered field or as real sources for some radiation field. ikR For points exterior to V the Green's function e /R can be expanded into plane waves. We shall employ the well-known integral representation due to Weyl (1919) (see also Stratton, 1941, p. 578) 2yr - -ico ikR r Y* 2 e ik ik.(x- x') eR _7 e (x - - 4sinacbd/ (z>z') (1.4) 0 0 where now k= k(sinacos/3, sina sin/3, cosca) is a function of the variables of inte1 gration a and 3 running from 0 to I2r-ico and 0 to 27r, respectively. It is seen ikR that in this expansion of the spherical wave e /R all possible plane-wave directions within the limits 0. <3 21r, 0 < a < (7/2) are included; values of a lying in (7r/2), a, r correspond to plane waves travelling in from infinity in the half-space z> z', and are, therefore, excluded. In addition, however, inhomogeneous plane waves with an exponentially decreasing amplitude in the z-direction (for z > z') are included in order to yield the necessary singularity at R-)O. These waves correspond to that part of the integration path running from a= (r/2) to a= (r/2)- i oo. An alternative representation valid in the half-space z. z' may be obtained by 2

THE UNIVERSITY OF MICHIGAN 7644-3-T selecting a different path of integration in the a-plane; thus, for example, we may write 0 2 +ico e - / / R 2r 0 ik (x-x')si e - - - sincrdcad (z.<z') (1.5) When (1. 4) is introduced into (1. 1) and the orders of integration interchanged, one obtains for the vector potential A the following result r r /'i0 ik 0 0 A(x_) = 2r 4 J (x /' e- (- - sinacrdjddx' V 0 0 7. 2 2 -ioo ik0 2 O0 O ik x o -ik x' e - (x)e -i d - sinadac, V J (1.6) and, upon recognizing in view of (1. 3) that the quantity contained in 4 ately above is merely A (a, 3), one finds -O immedi 7r ik A(x) = 27 0 0 ik. x e- - A (a,/3)sina da dc — o (1.7) provided x lies in the half-space formed by the portion of the z-axis above the source volume V, that is, z > z'. In this upper half-space, then, Eq. (1. 7) max 3

THE UNIVERSITY OF MICHIGAN 7644-3-T provides a representation of the near field in terms of the far-field data. For x lying in the lower half-space below the source region, z < z'., we have by virtue mm of (1. 5) 27r +ioo i 2ye A(x) -= / e'- A (a,/3)sina daed. (1.8) 0 Or The integrals (1. 7) and (1. 8) together give the field everywhere in space except in the region z',<(Zz' which sandwiches the sources. It is clear, however, that m max other paths of integration in the a-plane depending on the observation angle 0 may be selected to yield results even within z' i z z z', although the source region m in max must still be excluded. Choosing other paths of integration is tantamount to rotating the reference axes and will be discussed shortly. 1.2 Additional Comments. As we have seen, the integral representation of the near field in terms of the far field requires integration over a surface element dQ= sina da d! of the com plex unit sphere Q. It is interesting to note that integration over the real portion of the unit sphere yields a result which contains both incoming and outgoing waves. Thus, in view of the representation (Stratton, 1941, p. 410) 2r r sinkR k ik (x- x') R 4 e - -- sinadadl, (1.9) 0R -0! 4

THE UNIVERSITY OF MICHIGAN 7644-3 -T one finds 2k ikx sinR k ik -x Po sinkR 47 e - -A (ca,)sinadad, 3 = 7- I(x') R dx'. (1. 10) 47 - 4r R 00 V On the other hand, complex values of 3 as well as a may be included since, by a straightforward modification of the Weyl formula (1. 4), we have ikR e ik R 21r 7r 7 - 1ioo ioo — +ioo — +ioo 2 2 ik- (x-x'), e- -- sina dca dp, (1.11) and thus -— iao 2 A(x) = 3 2 eik X A (c, 3) sinc da d:. — +io 1. 3 Rotations of the Reference Axes. (1.12) The integral representations (1. 7) and (1. 8) taken together exclude certain portions of free space, namely the free-space points lying within the region z. z < z' sandwiching the source volume V. These points may be included mm m ax by means of a rotation of the reference axes. Consider the integral (1. 4) 7. 2 2 -- i co ikR. 2 e ik R 21r 0 0 ik (x-x') e - - - sina da d/3 (z > z') 5

THE UNIVERSITY OF MICHIGAN 7644-3-T where k==k(sina cost, sina sinB, cosa). This integral is invariant to a rotation of the reference axes, thus change to new variables a', 1' defined by direction cosine relations sina'cos'= sina cosO cos (3 - ) - cosasin0, O O O sina' sinj'= sine sin (3- 0), cosa' = sina sin0 cos (3-0 ) +cosacos 0 oo o (1.13) where 0, 0 are arbitrary (real) angles. The inverse transformation, defining the old variables (a, 3) in terms of the new variables (a,',,'), is sine cos3= cos c' sin e cos 0 + sina' [cos 0 cosl3' cos 00- sin3' sin ], sina sin = cosa' sin0sin + sin 0 + sina' [cosos0 s sin0 +sin' cos0 cos a = cos a' cos 0 -sin a'sine cos3'. (1.14) o o Further dQ = sina da d3 = sina' da' do'= d2' (1.15) and the limits of integration may remain the same. The integral representation is essentially unchanged in form: 7f 2, 2-100 ikR ik ae R 21ik J i 0 0 ik- (x-x') sn d e- -- sina' da' (8' (1.16) where the directional cosines of k are to be obtained from the inverse transformation relations given in (1. 14). The integral, however, now converges at a' =(Tr/2)-ico 6

THE UNIVERSITY OF MICHIGAN 7644-3-T provided i (x-x')sin 0 cos0 + (y-y')sin sin0 + (z-z) cos o > 0, (1. 17) O0 0 O that is, provided A A X X > X' X - 0 - 0 (1.18) where x is the unit vector o x = (sin0 cos0, sin0 sin0, cos ) O0 0 0 0 0o (1. 19) We have, therfore, 27 r ik A(x) 2 0 0 0 ik' x ei- -A (, 1) sina'dc' dP' -o (1.20) provided A x x > max (xt- ). - 0 0 (1.21) This representation is thus valid for all x lying in the half-space formed by the porA tion of the x axis not containing the sources. Since x is an arbitrarily directed O o vector, it is clear that portions of free space within z' z z', which were mm max previously excluded, may now be included. In particular, by rotating the x vector 0 we may generate the field everywhere in the space outside some minimum convex shape surrounding the sources. 7

THE UNIVERSITY OF MICHIGAN 7644-3-T 1.4 Relationship to Spherical Harmonics. Assume the far field is known as an expansion in spherical harmonics A (0,0) = a pm (cos O)eim -o i nm n n m (1.22) Then, because of the integral representation derived by Erdelyi (1937): 2r -iaoc 2 inh(l)(kr)Pm(cos 0)e~i 2 - n n 2 0 0 ik-x m +i. e P Xm(cos a) e sin dad3p, n (1.23) we have immediately from (1. 7) A(x) = ik7 an ilh() (kr) Pm(cos 0)eim L-..j L.* i nm n n n m thereby giving the field as an expansion in spherical wave functions. 1. 5 Relationship to Aperture Problems. (1.24) Assume the source currents j(x') are confined to the plane z'=0 and denote the directional cosines of k by u, v, w where u = sin a cos 3, v = sin a sin 3, w = Cosa. (1.25) The far-field amplitude (1. 3) may be written in the form * 8

THE UNIVERSITY OF MICHIGAN - 7644-3-T /u f -ik (ux' + vy') A (u, v)= 4r (x', y t) e dx'ddx' dy' (1.26) and the spherical wave (1. 4) may be expressed as 0 e ikR ik ik [u(x-x')+v(y-y')+w(z-z')] dudv (127) R 27r J e w -00 since sinadad/3 = (1/w)dudv. Thus, remembering z'=O, we have /, ooikR _(X ^0o (.,Y') eR dy ik eik(ux+vy+wz), ( dudv A(x) - j(x y e - dx'dy) — e A (u,v). -co (1.28) When z=0 this leads to the well-known result that polar diagrams and aperture distributions are related by two-dimensional Fourier transformations (see e.g. Bouwkamp, 1954). 1. 6 The Field Quantities. The electromagnetic field is derived from the vector potential by means of the relations 1 H = curl A, O i (1.29) E = curl curl A. 0 e 0j _ 00o 9

THE UNIVERSITY ( 7644-3-T OF MICHIGAN In the far zone these equations give ikr H (r, 0, 0) e H (0, 0) ikr E(r, 0, 0) - E (0, 0) r -0 ikr =e -)kAA (0, ), r A Q -o ikr e -i =e (-i )kAkAA (0,0). r A Wo - - o o o (1.30) When these relations are applied to the integral representation (1. 7) of the vector potential, one finds 2r ik H(x) = 20 0 7r. 7. r2 -ioo 2 ik.x ( i e - (-) kAA (aC, 3) sina dac d3 /A - - O i 2r ik 2wr (1.31) ik. x,-i- H (r, [3)sin a dad3, -o and similarly for the electric field -ik -- 2ir 7r 2 — ioo 2 2r-1 O O *00 ik. x e - -E (a, ) sin a d 3. -o (1.32) The near fields may thus be represented directly in terms of the electromagnetic fields in the far zone. 10

THE UNIVERSITY OF MICHIGAN 7644-3-T 1. 7 Determination of Field in Free-Space Region between Distinct Sources. If the original source region V can be separated into a number of disjointed volumes V. in free space, then it may be possible to determine the field in the space between these distinct source regions. For example, let the currents j be located in two finite volumes V1 and V2, and further assume that V lies within 1 2' 1 the range z < z < z while V2 lies within the range z < z < z with z < z ~1 2~ 2~ 3~ o4 2 3 Also, let HW(0, 0) and H[2O, 0) denote the far-field amplitudes due to the sources 0 in V1 and V2, respectively. Then the field in the free-space region z2 < z < z between the two volumes V1, V can be represented in the form 1' 2 2 — ioo 2r -+ ioA -(-) 2IX4 eikH~ (CUx3)sinadkd- P X eA ik, X 2](3)sinadaci3 H(x)= - - H (, sina c - e -- O O 0 0r (1.33) The first integral above converges for z > z2 while the second converges for z<z3; hence, this representation is valid in the desired region z2< z< z. This has an immediate application in providing a means of separating out distinct sources of the scattered field that may occur, such as an antenna or other protuberance mounted on a smooth surface. I 11

THE UNIVERSITY OF MICHIGAN 7644-3-T II ANALYTIC CONTINUATION The field H (0, 0) in the far zone is measurable only for real values of -o 0, 0 in the ranges 0 < 0 < ir, 0. 0 < 2r. However, in order to obtain the near field by means of the integral representation discussed in the previous section, it is necessary to know H (a0, 0) where a = 0+ii. Therefore, we need an extension into the complex a-plane based upon the measured quantity H (0, 0). Now H (0, 0) is immediately known for the range - -r <0 <r. This follows 0O from the definition i -ik. x' Ho(e,) k A j(x) e — dx (2.1) 0 4ir A V and the fact that k as a function of 0 and 0 satisfies k(-, 0) = k(,0~7r); (2.2) hence H (-0,0) = H (0,0~ir). (2.3) In addition, H is periodic with period 2r in both 0 and 0. — O To obtain an expression for H (a, 0) in the complex a-plane, we observe -O from (2. 1) that H (a, 0) is a harmonic function in the variables 0 and ~; that is (a2 2 2 +.) H (0+i,0) =0. (2.4) ae a I 12

-- THE UNIVERSITY OF MICHIGAN 7644-3-T As such, H (a, 0) may be expressed in the following form -o 00 Ho(a 0) = a nein(O+ii) n= - n n= -Goo (2.5) where the coefficients a are derived by means of the relation -n a - -= e i H (0', 0)d0e' -n 27r -o — i'. (2. 6) I This provides an extension into the complex a-plane. The series (2. 5) may be partially summed and put into closed form as follows: 7r i( +i H) o, ________________ 1 in(O+i~) -in0' H (ca, 0)- =..- dO' +- e H (O,)e dO' -O0 2' - i(O+ i) iO' 2r -o e - e ~ n=l (27) -2r -itn (2.7) for g < 0, and I H (a, 0) ='10 2-x. e 0 9 0 ~. - H (e' d) e' H1( o, 0' d&+ E -in(O+ig) H (,,)e do' r i(t +i) 2ir n+1 (2 -ol e -e n= ^ nlr — - (2.: 8) for > 0. To investigate the convergence of the series (2. 5) we examine the behavior of a as n-4oo. Now -n 13

I THE UNIVERSITY OF MICHIGAN 7644-3 -T where k=k (sin0 cos 0, sin 0 sin 0, cos 0) (2. 10) k- i -iO * Write k in the form k= -(e t+e t ) (2.11) 2 with t = (-icos0, -isinl, 1), (2.12) then -ik -* (x>\ -e i(n-1)0-ik.x' d +-i(n+l) 0-ik*x' d a -- (xA t e -- d+t e -- dO dx' n 162 ].... V -r -7r (2.13) But k- x' may be written in the form k x' = kr' cos cos0' + sin sin0' cos(0- 0') (2. 14) = kr'p cos(0-~), where 2 2 2 2 p = cos O'+sin O' cos (0-0'), (2. 15) tan - = tan 0 cos(0-0'), and, in view of the integral representation 14

THE UNIVERSITY OF MICHIGAN 7644-3-T I.n 1 -imo/ 1 e J (kr'p)=- e m 2a -t -imO-ikr'p cos (0-I) dO' ~~~dO6, (2.16) we have the closed form expression te - J+l(krp)'p-t e-inJ dx te n - I - n+ (2 (2.17) V As n tends to infinity the dominant contribution is due to the first term within the braces in the integrand ONa -r (-i) -n n —>oo 8o/f Nk -i 11n-1 J(n).i(x')A t 2( e V Constant (kR\ /' (n) 2 2 (2.18) (2.19) hence a n — ) oo where R represents the maximum value of r'. A similar result holds for a |. The convergence of the series (2. 5) is therefore secured for all a= 0+ii because of the gamma function in the denominator of (2.19). 15

THE UNIVERSITY OF MICHIGAN 7644-3-T II HIGH-FREQUENCY SCATTERING It will often happen at high frequencies that the far scattered field from a body can be characterized either by a rapidly varying phase function (e. g. specular scattering) or by a rapidly varying amplitude function (e. g. scattering by a flat plate). Under these conditions the contour integral representation of the near field in terms of the far field is particularly convenient because the powerful methods of asymptotic analysis are then at one's disposal. As an application of the integral representation at high frequencies we shall here consider the geometrical optics field produced by a plane wave incident on a perfectly conducting sphere. The incident field is taken in the form i A -ikz E =xe (3.1) i CIo A -ikz H=- ye and the scattered field is given by 2r Es( ik H0 20 r HS(x)= 2Fr. 2 0 2 i 0 eik- Es (ea, 3) sina da d3 -o (3.2) eik- H (a,3) sina da d, -o 16

THE UNIVERSITY OF MICHIGAN 7644-3-T The geometrical optics field in the far zone may be written as Es (a a a ^ (3) e-2ika cos (/2) E (ca 13)= —,e 0 2 E <C P - (3.3) s o a A -2ika cos(a/2) H (,3)=- -h(o, 13)e -0 2 A where e(ca,f) and h(a,0) are the unit vectors e(a, 3)= x(cos acos 3+sin 13)-y(1- cos a) sin:3 cos3 -zsin cos ^A.2 2 A (3.4) h(a, x)= -x(1- coso ) sinr3 cosa3+ ^ cosa )-sin+ cos - z sin sin The exponential behavior of the integrands in (3.2) is thus governed by the factor ikf(a,3) e (3.5) where f(a, /)= k * x -2 a cos = r sinOsina cos(0-3)+cos cosa -2acos2 L ~i 2 (3.6) Upon examining the convergence of the integral as a —(Tr/2)-ioo, one finds that the integrand decays exponentially so long as r cos 0> 0, that is, z > O. When z= 0, however, the integrand grows exponentially as a-)(r/2)-ioo and therefore diverges. Hence the representation (3.2) with (3. 3) is valid for all z> 0, or what is the same, e<0 < (r/2). 17

I THE UNIVERSITY OF MICHIGAN 7644-3-T As k-moo the dominant contribution to the integral arises from the vicinity of the stationary phase point 0(=0, a=a ) where a satisfies the equation r sin(a -0)= a sin(a /2). o o (3.7) I The physical interpretation of this equation is shown in Fig. (3-1). The quantity p = a sin(a /2) may be interpreted as the impact parameter associated with an incident ray, and this is precisely the incident ray that reaches the observation point P after being reflected at the surface according to the laws of geometrical optics. The angle a is twice the angle of incidence. By means of a first order stationary phase evaluation we obtain immediately sin a ES(x)v a o [ ( )a s ]rsinO cos (a - 0)- cos- r sin 0 0 2 ] ikf(a,0) e (a,) )e (3.8) and similarly for the magnetic field. If we let s denote the distance along the reflected ray from the point of reflection s= rcos (- 0)- a cos(a0/2), (3.9) rsin0=ssina + a sin(a /2) (39) 0 0 then the result (3.8) may be written in the form 1/2 E -(x-)" - _s) f s,, r,o ^^-bdtJ~S iks- ika cos (a /2) e(ao, 0) e (3. 10) 18

THE a o incor z - FIG. 3-1: ] UNIVERSITY OF MICHIGAN 7644-3-T C E PHYSICAL INTERPRETATION OF STATIONARY PHASE POINT ro FOR HIGH-FREQUENCY (GEOMETRICAL OPTICS) SCATTERING BY CONDUCTING SPHERE. 19

THE UNIVERSITY OF MICHIGAN 7644-3-T where D(s)= ( os + ) (+ cos. (3.11) The magnetic field takes the analogous form H x)u - r /\1 ^ iks -ikacos(a /2) H (x) ^j - J- [k s (a, 0) e (3. 12) The amplitude factor [D(0)/D(s 1/2 in the equations above accounts for the divergence of the rays after reflection at the surface. This factor has been derived on the basis of geometrical optics for reflection from an arbitrary convex body by Fock (1948), and it is easy to verify that the quantity (3. 11) agrees with the expression given by Fock in the case of the sphere. The phase of the field is also in agreement with standard geometrical optics considerations. The ordinary stationary phase evaluation fails in the vicinity of the caustic given by the equation a a o r cos(a -0) - cos =0 (3. 13) o 2 2 however, it must be emphasized that the behavior of the field near the caustic may still be examined by applying a modified asymptotic analysis to the integral representation in (3. 2). The elegance and simplicity of this representation for application to high-frequency scattering is evident.! 20

THE UNIVERSITY OF MICHIGAN 7644-3-T IV REPRESENTATION OF THE FIELD INSIDE THE SCATTERING BODY In the previous sections the scattered field was represented in terms of a vector potential involving currents that were physical or otherwise; i. e. the fact that the scattered field arose from induced sources was not prescribed, only that it arose from some current distribution. Outside the source region the scattered field was then expressed in terms of an integral operator acting on the far-zone scattered field components. In this section the possibility of obtaining an expression for the total field inside the scattering body, in terms of the far-zone scattered field is examined. As a preliminary, the derivation of the total field in terms of a vector potential relating to the actual induced currents (conduction and polarization) is reviewed. It will be assumed that the scattering body is contained in a finite volume V. The material of the body will be taken to be non-magnetic (i. e. P=, ), and S O characterized by the relative permittivity E' which may be complex allowing for conductivity. For present purposes the conductivity will be taken to be finite (but can be extremely large) thus ruling out the mathematical concept of a perfect conductor. Let the incident field be generated by a current source J outside the body. The source will first be taken a finite distance from the body, then later allowed to go to infinity, to account for plane wave incidence. Maxwell's equations become V H= 0 (4.1) we V -'E= iV. J (4.2) o- - - -o VAE= iup H (4.3) VAH = ice'E + J (4.4) -- o0 - o 21

THE UNIVERSITY OF MICHIGAN 7644-3-T The field quantities H and E will be represented in terms of a vector potential A and a scalar potential 0 as follows MO H =VAA (4.5) E = iwA+ V (4.6) Equations (4. 1) and (4. 3) are automatically satisfied by the potential representation. Equations (4.2) and (4. 5) become icu icV- e' A+V-' V =V2 V J (4.7) 2 2 k V2A - V(V A) + k2 e'A- i V(W e 0) = J (4.8) 0 - 00 00 Since in place of A one could have used A+Vq where q is arbitrary, still automatically satisfying Eqs. (4. 1) and (4. 3), one can impose an additional condition on the potentials in terms of a gauge transformation. The particular choice will be taken as follows wo e o = i V. A (4.9) Equation (3. 8) reduces to 2 2 V2A-(1-')V(V A) + k EA= A J (4.10).... O 00o Taking the divergence of this equation, one obta!' s E- (4. 7) automatically. Thus, it is seen that with condition (4.9),.o- vector potential A must satisfy Eq. (4. 10). Outside the body'= 1, and this reduces to the free space Helmholtz j 22

THE UNIVERSITY OF MICHIGAN 7644-3-T equation operating on the components of A. The above equation can be placed in a different form useful for deriving an integral expression for A. Eliminating the term (1-c')V(V- A) from Eq. (4.10), with the help of relations (4. 6) and (4. 9), one obtains 2 2 V A+k A = U J -iPu E (1-e') E (4. 11) - 0 O 0-0 0 0 It follows that A can be expressed in the form PU ~ ikR -A(x) j j+J e dx' (4.12) V where R= x-x' I and J = iE (1-') E. (4. 13) This can be represented in the form /4 ikR A () = A (x) +- J(xI') dx' (4. 14) 47r R V s S where A (x) is the vector potential of the incident field. The magnetic field is thus given by 1 eikR H(x)= Hi() + 1 J(x')A -V' d. (4.15) V S I 23

THE UNIVERSITY OF MICHIGAN 7644-3-T The source J giving rise to the incident field H can now be taken to in-o finity, in which case Hi will represent an incident plane wave. The current J(x)= iWE (1-') E (x) is the current induced in the scattering body, being composed 0 of conduction and polarization currents. Both the vector H and A will be continuous everywhere, due to the assumption that p =p everywhere, and that E' is finite. In the limiting case when the body is a perfect conductor, H is then discontinuous. It can be shown that in the limiting case when Im E-loo, (i. e. a perfect conductor), the volume integral in (3. 15) reduces to a surface integral, and expression (3.15) reduces to ~/ ~ ikR H(x)- H () + 47- (rn*H)AV' - ds (4.16) S where S is the surface of the conductor and n is the unit outward normal. Having considered the above preliminary work, we are now in a position to discuss the possible representation of the field inside the body in terms of the far scattered field. The notation A will be used to represent that part of the vector potential which results from the induced currents, i.e.: u ikR AS(x) - J() e- dx. (4.15) 47.. R V s The scattering body designated by the volume V will be split up into the following s parts V (r), V () and V (c), where V (r) is the intersection of V and the half+ 6 + s space z< <, V () is the intersection of V and the half-space z {, and V ({) s 6 the intersection of V and the slab 6 -6z< z ( + 6. The decomposition is displayed s in Fig. (4-1). 24

THE UNIVERSITY OF MICHIGAN 7644-3-T 1 z V_ (z+6).-yy ^ —^ 7 = )O _ -.,,,,,., Z Z'r'r Z Z Z Z J& - Z, - S -r I a 9 Z = -- X% N A X - -1 -lk x x x x % x % Is -6?~- 6 V6 V+(z-6) -l x FIG. 4-1: DECOMPOSITION OF THE SCATTERING BODY Associated with the above, the following vector potentials will be considered, A+s ( 4op A (x)= - -+ - 4,. ikR J(x') e dx' -- R - (4. 16) V+(z-6) A (x) =4 -- - 47, ikR J(x') e dx' R (4. 17) v (z-6) A A (x) -6-4ir ikR J(') R dx' -- (4. 18) v6(z) 6 25

THE UNIVERSITY OF 7644-3-T MICHIGAN Using the relations ikR e ik R 2ir 0 0-io ik (x-x') d - - - sina d dp3 Z> Z' ik 2w +ico / ik-(x-x') n d l e - - - sinadcr d 3 z< z' 0 x A (x) and A (x) become 1 A (x) 2z ik 2r 0 x Po -ik * x' - J(x') e - - dx' sina dad/ 41 — - V (z-6) (4.19) A (_ - ik A (x) = -2 2v 2 w 0 +i 7 0oo ik.x e — x'o - ik- x' J(x') e - -dx' 4w - sinrad d/3 (4.20) v (z+6) provided that J(x') is absolutely integrable, allowing the order of integration to be interchanged. If J is bounded, it follows that each component of A (x) is bounded ~~~- -^~6 - 6 |A5 ( )| 4 4^ M a + a2 + 2d 0 0-6 26

THE UNIVERSITY OF MICHIGAN 7644-3-T where p =(x-x') + (y-y'), and a is the maximum value of p such that the cylinder p = a encloses V. It is easily seen then, that when 6 —0 6 A (x) — 0. (4.21) 6 - The above condition that J be bounded may be weakened, by allowing certain types of integrable singularities. However, these cases will not be considered at the present time. Letting 6-)0, the vector potential A can be expressed as follows A=Ai+ A (x) +A (x) (4.22) 2~ -ioo where 2 A (x)= -r e -k x A (a, 3)sinadadf (4.23) -+ 2z -o 0 0 and 2 2yr -+ioo ik ik-x - AS (x)= _ e - -A (a, 3)sincadadc (4.24) 0 X with the vector k(a,3)=k(sinra cos3, sinasin3, cosa). The quantities A (c,(3) and A (ca, ) are the far field components + Po -ik. x' A (Co,) — J(x')e dx' (4.25) -o 4r V(z-0) I 27

THE UNIVERSITY OF 7644-3-T MICHIGAN /I A (a, /)= -o 4w -ik-x' J(x) e - -' dx v (z+0) (4.26) arising from an appropriate decomposition of the quantity A (a, ) ='-O 4 J( X -ikx' J(x') e -- V S dx' (4.27) defined previously, i. e. A (c, /3) A+ (a,3) + A (a,3). (4.28) -0 -o 0 It can be shown that the same results hold for the magnetic field, in which I case i s s H-=H +H +H (4.29) where 2w -k H (x) =2. / -+ - 2v, 1 0 o I. 2 r ik x + e- - kAA (ca,3) sinadad/3 - _O 0 (4.30) x — +k H (x.) = 2, o 2r 0 -+ io 2 ik e ik - kAA (ca,3) sinadoad. (4.31) 7r 28

THE UNIVERSITY OF MICHIGAN 7644-3-T From the above it is seen that it is possible to obtain the magnetic field inside the body (composed of non-magnetic material with finite conductivity), from a knowledge of the far field data. This follows from the results in Section 2, which indicate how k(a, 3)A A (cq, 3) may be determined for complex values of a where a= +it, from the knowledge of the far field quantities A ) and A(0, measured int, the raknowlge o f the far field qutities e, ) and A(,de te) measured in the range 0.<, 0, 0.. 22r. However, if the body is inside the slab z2 < z < zl, the appropriate split up of k A A must be sought. The key problem remains of determining a method of uniquely performing this decomposition from knowledge of the far field data alone. Additional knowledge will most likely be required, such as, knowledge of the scattered field for all frequencies, or all angles of incidence. 29

THE UNIVERSITY OF MICHIGAN 7644-3-T REFERENCES Bouwkamp, C.J. (1954), "Diffraction Theory," in Rep. Progr. Phys., 17, 35-100. Erd6lyi, A. (1937), "Zur Theorie der Kugelwellen," Physica, 4, 107-120. Fock, V.A. (1948), "Fresnel reflection laws and diffraction laws, " Uspekhi Fiz. Nauk, 36 308-319. Kline, M. and I. W. Kay (1965), Electromagnetic Theory and Geometrical Optics, (Interscience Publishers, New York). Stratton, J.A. (1941), Electromagnetic Theory, (McGraw-Hill Co., Inc., New York). Weyl, H. (1919), "Ausbreitung elektromagnetischer Wellen'ber einem ebenen Leiter," Ann. Physik Ser. 4, 60, 481-500. 30

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