ESD-TR-68-215 INVERSE SCATTERING INVESTIGATION FINAL REPORT Vaughan H. Weston Wolfgang M. oerner April 1968 SPACE DEFENSE SPO (496L/474L/N) DEPUTY FOR SURVEILLANCE AND CONTROL SYSTEMS ELECTRONIC SYSTEMS DIVISION AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE L. G. Hanscom Field, Bedford, Massachusetts I This document has been approved for public release and sale; its distribution is unlimited. II. __. JL J (Prepared under Contract No. AF 19(628)-67-C-0190 by The University of Michigan, Department of Electrical Engineering, Radiation Laboratory, Ann Arbor. Michiaan)

FOREWORD This report (8579-1-F) was prepared by the Radiation Laboratory of the Department of Electrical Engineering of The University of Michigan. The work was performed under Contract F19628-67-C-0190, "Inverse Scattering Investigation," and covers the period 3 March 1967 to 3 March 1968. Dr. Vaughan H. Weston is the Principal Investigator and the contract is under the direction of Professor Ralph E. Hiatt, Head of the Radiation Laboratory. The contract is administered under the direction of the Electronic Systems Division, Air Force Systems Command, United States Air Force, Laurence G. Hanscom Field, Bedford, Massachusetts, 01730, by Lt. L. E. Nyman, ESSXS. This report was submitted by the authors on 30 April 1968. This technical report has been reviewed and is approved. Bernard J. Filliatreault Contracting Officer Space Defense Systems Program Office ii

ABSTRACT A brief review of the salient features of the theoretical investigation of the c.w. bistatic inverse scattering problem is presented. The effect of changing the origin of the coordinate system upon the convergent properties etc., of the spherical vector wave function representation of the near scattered field and the surface loci E x E= 0, is discussed. It is pointed out that a great deal of analysis remains to be done in this area. The determination of the surface of the scattering body from knowledge of the local total electric field is given. Emphasis is placed upon the generalization of the condition E x E* = 0 as applied to perfectly-conducting bodies, to scattering surfaces characterized by the impedance boundary condition. Properties of the Matrix inversion associated with the determination of the expansion coefficients from far field data are discussed. Some numerical results are presented, and restrictions upon the choice of aspect angles are deduced. iii

TABLE OF CONTENTS ABSTRACT iii I INTRODUCTION 1 n REVIEW OF INVERSE SCATTERING THEORY DEVELOPED FOR 2 C.W. BISTATIC SYSTEM III EFFECTS OF A DISPLACED ORIGIN ON CONVERGENCE PROPERTIES 16 IV ASPECTS ON CONTINUOUS WAVE INVERSE SCATTERING 20 BOUNDARY CONDITIONS FOR THE DETERMINATION OF THE SHAPE AND THE MATERIAL CONSTITUENTS OF A CONDUCTING BODY V MATRIX INVERSION FOR THE CASE OF END-ON INCIDENCE ONTO 40 A ROTATIONALLY SYMMETRIC SCATTERER REFERENCES 71 DISTRIBUTION LIST DD FORM 1473 iv

I INTRODUCTION A brief review of the salient features of the theoretical investigation of the c.w. bistatic inverse scattering problem is presented in section II. In section III the effect of changing the origin of the coordinate system upon the convergent properties etc., of the spherical vector wave function representation of the near scattered field and the surface loci E x E* = 0, is considered. In section IV, a discussion upon the determination of the surface of the scattering body from knowledge of the local total electric field is given. Emphasis is placed upon the generalization of the condition E x E* = 0 as applied to perfectly-conducting bodies, to scattering surfaces characterized by the impedance boundary condition. In section V, properties of the Matrix inversion associated with the determination of the expansion coefficients from far field data, are discussed. Some numerical results are presented, and restrictions upon the choice of aspect angles are deduced. 1

II REVIEW OF INVERSE SCATTERING THEORY DEVELOPED FOR C.W. BISTATIC SYSTEM 2.1 THE PLANE WAVE EXPANSION It is shown here that the total field produced by a plane wave incident upon a scattering body, can be expressed as the sum of two terms, the incident field and the Fourier transform of a quantity which is related to the scattering matrix. The resulting expression is valid for all points in space including the interior of the scattering body. To begin, the analysis will be restricted to non-magnetic bodies (although it could be easily generalized to include such cases) and the geometry of the scattering body will be limited by placing certain restrictive analytical properties on the surface S which encloses the volume V of the scattering body. The classes of surfaces chosen, will belong to class C defined by Barrar and Dolph as follows: 3 A surface S is said to belong to class C if there exists a finite number m of images, x = x (u, v), yV = y (uv), z = z (u, ), v = 1.2.3.. m of the 2 2 disk u + v < 1 that cover the surface S, and such that the third derivatives of x, y and z with respect to u and v exist and are continuous. Harmonic time dependence exp(-iwt) will be taken in which case Maxwell's equations become VAE = ik /c E H kl inside the body g~o YAH -i(E/k) - 1 L/E AH =^.l(E/- k2 outside the body 2 where k, the square of the propagation constant in the body, is given by 2 2 k1 L= Elo + iwt0o1. 2

The incident electric intensity will be expressed in the form E = (27)-3/2a eik - (2.1) where k is the direction of the incident wave, and a, the unit vector denoting polarization. The homogeneous body will be treated first. In this case, it follows from Barrar and Dolph, that the total electric intensity E induced by the plane wave Eq. (2. 1), incident upon the body, satisfies the following integral equation (which has a unique solution) _ _ -_ _ ( k2) fp f E(x) = E (x)+ 4 - PE dv- 2 V 0 0(E- n)dsl V S (2.2) where ikR = eR RR = (x - x') and E is the value of E obtained by approaching the surface from the interior of the body. If E is the value obtained by approaching the surface from the exterior of the body, it follows from the continuity relations that 2 - 2 + k2 E. n = k E n. (2.3) 1 The appropriate expression for the magnetic field is given by (k2 _ k2) f H(x)= H (x)+ ( 2 V A E 0dv (2.4)...._ 47irr W f V Before deriving a plane wave representation for E (x), the following vector will be introduced. 3

T (k'_ k) -ik'. x' ik' -ik'- x - e - - _ E(x', k) dv' - -= e - -' E n ds' {k2( (2.5) (2.5) From Eq. (2. 2), it is seen that when |xl - o, in the direction given by the vector k', then the total field becomes E(x), |Ixl-a>o E (x) + -- \(2 ik Ixl e x T (k', k) I _ (2.6) indicating that the vector T (k', k) is related to the scattering matrix. When Ik'l = I kl, T(k',k) is a measurable function being proportioned to the far scattered field in direction k', which is produced by a plane wave of frequency A ck, incident upon the body with direction of incidence given by k. The following theorem may now be proven. Theorem: If (k', k) =2 I e-ik'- x E If - (k',k) = ( 3/2 e -- - E(_,k)dx, (27)0 (2.7) then 4(k'.k) = 6(k-k')a T (k',k) lim 2 2 -- o k - k' +i (2.8) Proof: It follows from Eq. (2.2) that (k',k) = 6(k-k')a+ -- - _ _ _ _ 4 7r { 1 (2ir 3/2 E (x', k) V (-ik'. xd dv J _ -ik' *x 1 ek-'" 22- -— 2 k2 V 3(2)3'2 k (2r) 0 p (E- n) ds jS 4

On setting S (E-. n) ds one can show that -ik' x - V dx = lim R —4oo co S Oo -ik' x 0 ds+ ik' e - -' dx e rG/ dr~~~ when S is the surface of a sphere of radius R. Letting k have a small imaginary part, it is seen that the surface integral will vanish when R - co. co The resulting integrals can be reduced as follows 1 (27r)3/2 / -ik'- x 2 1 e - - 0dx = I- Jdp2dx y'7[ (27r )3 ip- (x-x') - ik'.-x e 2 k2 p - k - ie -i. x' e - 2 2 p - k - 1 6 (p- k') -ik'. x 2 e - e Combk2 k Combining the above expressions, one obtains the result (k'- k) = 6(k-k')a- lim E — o T (k',k) k2 _k2+i k -k' +!i As an immediate consequence of the above theorem, the total field everywhere in space including the interior of the body,can be expressed in terms of the incident field and the quantity T(k', k), as follows 5

E(x,k)= Ei(xk) + — 1 (27)3/2 e i. x p2 2 T (pk)dp. p-k (2.9) If the integration space p is expressed in spherical polar coordinates (p, p, 0p) where the range of the variables are - o <p < oo, 0 < < 7r/2, and < 0 < 27r, then in the above integral, the contour of the variable p bends above the pole at p = -k, and below the pole at p = k. It can be shown in a similar manner that the magnetic field can be expressed in a similar form _ ~i _~1 H (xk) = H (xk) + w Ao(27r32 0^Zr7 eiP x p2 2 A T(j,_k)dp p - k (2.10) The above results were derived for a homogeneous body. The results may be extended to include inhomogeneous non-conducting bodies, i.e. where 2 2 2 k = 0 E (a real function) and k varies continuously in the medium. In this case, the appropriate integral equation for the total electric field is (Barrar and Dolph) E = Ei+ 1 0(k k2) E dv - 0 - - 4w 7 - k - 4wrk2 I V S (k2- k (E * n) ds l V + - 47r V 2~ - 2dv kI Unfortunately from the standpoint of rigor, only the uniqueness of the solution of this integral equation has been proven. It's existance has not yet been shown. Additional relationships involving T (k', k) have been shown in 8579-4-Q (1968), among which is an integral equation involving T (k', k) 6

and the Fourier transform of the body. 2.2 ALTERNATIVE REPRESENTATION FOR POINTS EXTERIOR TO THE BODY For a fixed direction of incidence and frequency, the quantity T (k', k) is required for all values of k', in order to obtain expressions for the field everywhere in space. Unfortunately, for fixed k, T (k',k) is a measurable function (far scattered field), only for values of k' such that | k' = | k I. However it will be shown that restricting the requirements on the knowledge of T (k', k) to the values of k' which lie on the sphere k' \ = I k1, the total field can be obtained everywhere outside the minimum convex surface enclosing the body. To show this, let the plane z = z be the tangent plane of the body, such that the body lies in the half-space z < z. For points in the half-space -0 O z > z, the following representation of the near scattered field s I eP1 e - E (x) = 3/2 2 2 T (p,k) d (2.11) (27r) p -k can be reduced as follows: Set p = k sin a cos p = k sin a sin a p =kq when the domains of integration are 0 < 2< 27r, 0< a < i7r/2-ioo, and - oo < q <oo Expression (2.11) becomes 7r /2 - io O 2T P co Es = - -k-\ \ T IO( S0 _Jk) g(a,, q) cosasinodada d q, (27r)312 - O q- - cos - i g = exp ik sin a (x cos + y sin ) + iqzk. (2.12) 7

From expression (2.5), it can be seen that for z > z, the contour in the following integral ikzq e T (kdq 2 -o. q - cos2a- ie may be deformed, to yield the following 7r i eco T (k sin a cos 3, k sin a sin k cos a; k) Hence expression (2.12) may be placed in the form 7r/2- ioo r27r s ik ik'. x E = - e- - E (a,3) sin adad 3 (2.13) o0 where k' = k(sin acos, sin asin 3, cosa) and E is the amplitude and phase of the far scattered field in direction given -O by (0, 0), i. e. ikR E R E (0, ). (2.14) R -O Thus it can be seen that for c.w. bistatic scattering, where E (, 0) is a measurable function, the scattered field for all points in the domain z > z can be obtained from expression (2.13) provided that the analytic continuation of E (e, 0) can be computed for complex values of, from knowledge of real values of e. It has been shown (Weston, Bowman, Ar, 1966 ) that the analytic continuation to complex values of e is possible provided that E (e, 0) is -O 8

known over the complete domain 0 < 0 < r, 0 < < 2r, and a procedure for doing so is presented. By rotating the coordinate system, equivalent expressions to (2.13) may be derived which yield the scattered field everywhere outside a different half-space excluding the body. Since the union of all the half-spaces exterior to the body is bounded by the minimum convex surface enclosing the body, it can be deduced that the scattered field at any point outside this convex surface can be obtained from the knowledge of the far field bistatic data (measured over all angles). At this point, it should be mentioned that expression (2.13) is not the only expression that can be used to evaluate the near scattered field from far field data. It has been shown that expression (2.13) can be re-expressed in terms of an expansion of the spherical vector wave functions. For a fixed region of the coordinate system, the expansion will be convergent down to at least the minimum sphere enclosing the body. Alternatively it was shown (8579-2-Q) that expressions (2.13) could be placed in the form ikR E e -j- _ R E (O,0) R J -n n=0 where E are derived from the E by a recurrance relation (Wilcox expansion). -n -n- 1 2.3 EXTENSION OF THE DOMAIN OF CONVERGENCE AND THE CONCEPT OF EQUIVALENT SOURCES It was shown above that for a fixed direction of incidence (fixed transmitter position), one could derive the near field everywhere outside the minimum convex surface enclosing the body, from measurements of the phase, amplitude and polarization of the far scattered field made over all directions. This result has been extended. Considering representation (2.13) it was shown that it was convergent down to the tangent plane z = z, and represented the scattered field everywhere on the half-space z > z. However for perfectly-conducting 0 9

smooth convex bodies, it has been shown (Weston, Bowman,Ar, 1966) that this expression was convergent down to the plane z = z* where z*> z, where the value of z* depends upon the global geometry of the body. Since the true total field displays a discontinuity across the physical surface of the body, expression (2.13) combined with the incident field, cannot represent the true field in the intersection of the half-space z > z* and the volume of the body. Therefore in this region, expression (2.13) is said to represent the field arising from a set of equivalent sources. In particular it was shown that equivalent sources for a prolate spheroid lie on a line joining the focal point, and for a sphere, the point at the center. Thus for smooth convex bodies, the domains of convergence are not limited by the minimum convex surface enclosing the body, but by a much smaller volume, that is contained by the minimum convex surface enclosing the equivalent sources. As an example, for the prolate spheroid with its center lying at the origin of the coordinate system and oriented so that its major axis lies along the z-axis, representation (2.13) is convergent in the half space z > c, where c is the focal length, and both Wilcox's representation and the representation involving the vector spherical wave functions, will be convergent down to the sphere of radius c. The extension of the domain of convergence will hold for piecewise smooth conducting bodies. However the minimum convex surface enclosing the equivalent sources must enclose the curves or points of surface discontinuties such as edges. For cavity regions that penetrate the minimum convex surface that encloses the equivalent sources, alternative representation for the scattered or near field must be employed. Such representations will not be of the exterior type, the plane wave representation (2.13), the expansion in vector spherical wave functions, or Wilcox's expansion, which are based upon knowledge of the far scattered field as a starting point, but must be of interior type, i. e. based upon knowledge of the total field in a domain adjacent to the body. Such 10

representations are only convergent in a finite volume. One particular type of such a representation is given in 7466-1-F (Weston Bowman, Ar, 1966). 2.4 DETERMINATION OF THE SURFACE OF PERFECTLY-CONDUCTING BODIES. Given expressions for the total field (obtained from far field data) which can be computed in the vicinity of the surface of the body, the next step is to employ techniques which will locate the surface of the body. For perfectly conducting bodies, the tangential component of E (the total field) must vanish. This implies that at a point on the surface E = n t where n in the unit normal to the surface and & is a complex quantity. For a general point in space the total field can be decomposed into the real vectors I and R as follows I R E = +ig For a point on the surface IA n = 0 and R n = 0, which implies that I Al R == 0. Thus a necessary condition for a point to be on the surface of a perfectly conducting body is that, at that point, the total field must satisfy the condition E A E*= 0, when E* is the complex conjugate of E. The advantage of this condition is that it is a local condition, requiring only the calculation of the total field at the point in question. Since the above condition is not sufficient, additional requirements would be required, such that E x E* must vanish at adjacent points x and x +Ax, and that E (x). Ax lim 0 Ax-z.0 (Axl which implies that E is normal to surface. In addition, the resulting surface firmed by the set of points must be closed. However, even if a surface S o 11

was found such that it satisfied these conditions, it would not necessarily be the correct surface. Since for the enclosed volume formed by two closed surface, on which n x E vanishes, there is a discrete spectrum (in frequency), this implies that at a particular frequency, there may exist additional surfaces S for which n x E vanishes. However the geometry of the additional surfaces depend upon frequency, and these can be separated out from the proper surface by employing at least two different frequencies. For the illuminated region of the body, an auxiliary condition was developed which considerably helped discriminate the proper surface from the loci E x E* = 0. This condition given as I E - I Es = 0, yields an approximate surface, which approaches smooth convex portions of the correct surface in the limit of high frequency scattering. From the numerical standpoint where there are errors due to input information, etc., it was demonstrated both theoretically and numerically that the condition E x E* = 0, should be replaced by finding the minima of E xE*|. 2.5 FAR FIELD INFORMATION LIMITED TO A SOLID ANGLE. For most practical solutions, the far scattered field (phase, amplitude and polarization) is measurable only over a region of limited bistatic angles. In this case it is important to know what information can be determined about the body, where measurements of the scattered far field are confined to a limited domain of aspect angles. This in turn depends upon the accuracy to which the near scattered field can be computed using the limited far field data. A general discussion of this latter point is presented in 8579-1-Q. However, the employment of high frequency asymptotic results yields a far more illuminating picture (8579-2-Q). These results will be briefly presented here. When the scattered far field is measured over a limited aspect region (example: the cone 0 < 0 <0 ), the analytic continuation of quantity E (a(, j) given in expression (2.13), cannot be found for the complete complex a plane. In this case the 12

following approximation to expression (2.13) will be employed 27r r 8 ^s ik o iki. x E = 2- e- - E (a,3) sinadadfl. (2.15) 27r -o 0 0 For the case of the perfectly conducting sphere of radius a, illuminated by the plane wave i A -ikz E x e the geometric optics scattered far field has the form E (a) = 2 (CB) exp [-2ika cos (a/2)] where A 2 2 A e (,) = x co cos + cosa sin/3 cos - A - z sin a cos From relation (2.15) the near scattered field is given approximately by 27r e - ika ikf(aL3)A E(W) - ie f ( a, ) s sin adadj3 (2.16) w0 0h where f(a,3) = r [sin 0 sin a cos (- ) +cos 0 cos a - 2a cos 2 As k — co the dominant contribution to the integral arises from the vicinity of the stationary phase point (3=0, a = a0) where a satisfies the equation rsin (a - ) = asin(-i_) 0 13

provided that 0 < a < 0. By means of first order stationary phase evaluation - O0 we obtain immediately S DEo ) 1/2 A (a iks - ika cos ( ) E(x) = - D() e(a 0)e where D(s) = os %+; + +- cos ) 2 a a 2 The distance s given by the relation s = rcos(a -) -a cos ( ) o \2 / is the distance along the reflected ray from the point of reflection. The resulting A S expression for E (x) is the geometric optics near field expression. When the far field information is limited to the cone 0 < 0 < 0, the near field expression will be limited to a volume of space such that rsin(a - ) = asin(- ) 0< a <, (2.17) 0 - 2 - 0 - 0 where (r, 0, 0) are the coordinates of a point in this volume. In order to find the portion of the scattering surface which can be determined, set r = a, in which case it is seen that a 8 = ' O< < 0 2' - o This then implies that the near scattered field given by expression (2.15) can be found on the portion of the sphere given by the cone 0 < 0 < 0 /2. For points outside the volume of space given by Eq. (2.17), stationary phase techniques cannot be employed, in which case the asymptotic approximation to Eq. (2.16) is 14

obtained by alternative means. The above example illustrates the fact that in the case of high frequency, asymptotic evaluation of expression (2.15) is equivalent to tracing back the rays to the portion of the scattering surface from which they arose. The result is not confined to just perfectly smooth convex shapes but can be applied to piecewise smooth convex shapes, the far scattered field is decomposable into components of the form ik ~ (0,0) E (0,0) = ( A (e0,) e -n n and stationary phase evaluation of the expression (2.15) for each individual component, effectively traces backthe various rays to their originating portions of the scattering object. The result is applicable to flat portions, in which case the far field component approaches a delta function of the angular variables as k -* oo. The example of the flat plate given in 8579-2-Q illustrates this. 15

III EFFECTS OF A DISPLACED ORIGIN ON CONVERGENCE PROPERTIES In employing the Wilcox theorem or the vector spherical wave function expansion, it can be shown that the near field representation, derived from the far scattered radiation pattern, is uniformly convergent for all points outside the minimum sphere enclosing the equivalent sources of a smooth convex shaped body. From the knowledge of the near field everywhere outside this minimum sphere, it is sought to find the associated scattering surface by proper application of suitable boundary conditions. These Inverse Scattering Boundary conditions, in general, ought not involve any constituent parameters (surface normal n, surface locus S(x, y, z), relative impedance 7 ) of the unknown scatterer and must be given solely in terms of the known nearfield representations of the incident and the scattered field with respect to a fixed origin which lies within the scatterer. For a perfectly conducting surface it was found that the following two boundary conditions can be successfully applied: a) {I E'l- IESIJ = (3.1) where E denotes the incident field and Es the nearfield representation of the scattered field. This boundary condition represents the geometrical optics limiting approximation and in general, is applicable only within a narrow cone about the specular point (8579-3-Q, 8579-4-Q). b) T T191 T Ei s xE xE J=0, where E = E +E (3.2) This boundary condition was derived by V. H. Weston (Final Report 7644-1- F, Weston, Bowman, Ar). Although it is only a necessary but not sufficient condition, namely producing a family of concentric surface loci in addition to the proper one, its application together with |E il - IE |j = 0 proved to be 16

indispensable since that portion of the proper locus within the conical section exterior to the Wilcox minimum sphere can be determined with great accuracy for a minimum number of expansion terms, (8579-3-Q, 8579-4-Q). To be more precise, the combined boundary conditions will yield the proper surface locus only on those portions of the surface for which the chosen expansion for the scattered field is convergent, for a prolate spheroid, the boundary conditions applied to the complete vector spherical wave function expansion associated with the origin located at the center of the body, will yield portions of the surface outside the sphere r = c where c is the semi-focal length. In order to obtain the side portions one has to displace the origin in the plane passing through the center of the body and perpendicular to the axis of the body. Thus in using far field data one needs to know the domain of convergence of the expansion. Generally this can be only obtained if the complete expansion were known, which in practice is not achievable since the number of receiver locations will be a finite number. If the domain of convergence is not known, then the application of the boundary conditions to the expansion may yield some correct portions of the body, but what the correct portions are cannot be prescribed unless some additional condition is employed. However if it was known apriori that the origin of the coordinate system was located at the center of the body, and the portions of the body farthest removed from the origin where smooth and convex, then the application of the boundary condition will yield the correct surface locus (to within some numberical error) for these portions. Thus if the approximate location of the origin was not known, some additional criteria should be employed, such as the one described below. A possible additional condition would be to displace the origin of the coordinate system, and on employing the boundary condition to the scattered field expansion derived in the displaced coordinate system to obtain a new surface locus, one would then determine the shift in the surface locus due to the displacement. If the shift is negligible for portions of the surface, then those portions

may be the proper surface. For example, if the origin of the coordinate system located at the center of a prolate spheroid, was shifted slightly down the axis, the loci for the top and bottom of the prolate spheroid derived from the boundary conditions applied to the near field expansions, should be the same as before the shift. These remarks are strictly qualitative. A quantitative study should be made and is outlined below. Let x represent the points in a cartesian frame of reference the origin of which is located in the vicinity of the scattering body (its precise location was not established) Let a plane wave be incident upon the body producing a scattered field, the far field ikR s e E A i E (8, 0) R -o being measured at a finite number (N) locations (8, 0J. Let (x, N) be a finite expansion of the near field derived from the far field data (the expansion may in vector spherical harmonics, or a Wilcox type expansion) Suppose that the origin is displaced a distance d, such that if x' represents a point in the displaced coordinate system then x'= x + d In the displaced coordinate system, the far field directions {n, 0n remain the same, (i.e. for R' —oo, 0' = 0, 0' - 0), however the far field quantity E' (0', 0') is related to E (0, 0) by the relation -O -O E' (', ) = E (0, 0) exp(-ik d) where k is a vector in the radial direction. Let E' (x', N) be a finite expansion of the near field derived from far field data (N receivers) in the displaced coordinate system. 18

If x is a point in space where the complete (N = oo) expansions are convergent the difference E (x, oo) - E' (x + d, o) would be zero. However when a finite number of terms are used in the expansions (finite number of receivers) the difference will not be zero, except of course at the receiver positions in the far field. Thus it would be fruitful to determine quantitatively a number 6 (d) such that if lim 6 E d - — o where E is the incident field, the application of the boundary conditions on the total field at points x for which the above inequality holds will yield the correct surface to a specified degree of accuracy. Such a condition given above may have to be obtained through numerical examples. It will depend upon what expansion for the near field is chosen, and how the expansion is matched to far field data (i. e., by matrix inversion, least squares, or otherwise). To obtain the portions of the surface outside the domain of convergence (if the complete expansion were used or outside the domain given by the above inequality for a finite expansion, the origin of the coordinate system would have to be significantly changed. For this new location of the origin, the domain validity of a finite expansion would have to be chosen by displacing slightly the origin and employing the above inequality. Some initial investigations of the effects of displacing the origin for a finite expansion in vector spherical wave functions, upon the surfaces derived through the boundary condition E x E = 0, is discussed in internal memo (W. Boerner). However, this memo is just an initial attempt to understand the problem and will not be reported here. 19

IV ASPECTS ON CONTINUOUS WAVE INVERSE SCATTERING BOUNDARY CONDITIONS FOR THE DETERMINATION OF THE SHAPE AND THE MATERIAL CONSTITUENTS OF A CONDUCTING BODY 4.1 INTRODUCTION Although methods have been outlined on how to determine the material characteristics of a conducting body, employing the monostatic-bistatic theorem (7644-1-F, Chapter XI), no suitable method has been found so far which will simultaneously determine the shape and the associated material constituents of an unknown body, since the inverse scattering boundary condition E xE_ = 0, where ET =E +E, derived by Weston (7644-1-F, Chapter X) holds only for the case of a perfectly conducting body. This condition ET x ET = 0 j, though necessary but not sufficient, may however be considered as a first step of formulating a more general c.w. Inverse Scattering Boundary Condition. An attempt of generalizing this condition will be prescribed in this section. Before any C. W. I. S. boundary condition can be applied, a suitable near field representation of the scattered field from the measured far field radar data must be sought. This can be achieved by either employing a series expansion into proper vector wave functions and if so required their associated plane wave integral representation (7644-1-F, Chapter V; 8579-1-Q, 8579-2-Q) or an expansion method derived by Wilcox (1956) and MUller (1956) as discussed in 7466-1-F (Chapter IV). The associated expansion coefficients may be obtained from a matrix inversion technique (8579-2-Q and 8579-1-F). Assuming that such a sufficiently accurate nearfield representation of both the electric and the magnetic field vector is found, the question of how to derive suitable I. S. boundary conditions arises which as well can be applied to the determination of the characteristic parameters of conducting bodies and their shapes. In contrast to problems of direct scattering and diffraction for which the shape and the material constituents of the scatterer are assumed to be known a priori together with the prespecified incident field vector E as regards the 20

computational coordinate system and thus may be incorporated into the boundary conditions, in problems of inverse scattering such boundary conditions must be sought which in particular do not depend upon either the shape or the material properties of the scattering body and its enclosing surface, but allow to specify those characteristic parameters uniquely. And this solely from the near field representation of the electric and the magnetic field vectors, derived from the measured far field data as described above. Now the question arises as to how many and which characteristic parameters must be defined to uniquely determine the shape and the material constituents of the unknown scatterer. In the scalar case it is sufficient to employ the following parameters: i) the local surface normal n (R, 8, 0) of the proper surface, ii) a relative surface impedance r (R, 8, 0) which is a scalar quantity, or the interior propagation constant ki (R 0). int. Thus at least three independent characteristic equations, solely expressed in terms of the near field quantities E = E + E and H = H + H, must be found to determine the surface locus S (R,, 0) and the surface impedance l = rl exp i. Whereas in the vector case the relative surface impedance must be formulated in dyadic or tensor formulation, involving a further unknown parameter A A which may be described by a local polarization angle e (R, 0, 0; ki, ei) or A A a local depolarization angle 6 (R, 0,0; ki e ). These angles can be expressed A in terms of the surface normal n (R, 8, 0) as well as the unit direction vector * A A of the incident wave ki and its associated polarization vector e. Thus the impedance dyadic 1 will be a function of both the material properties of the scattering surface, the surface locus S (R, 0, 0) or its associated local normal A A A n (R., 0), and the prespecified properties ki, ei of the incident wave. This additional complication may ask for one or two more independent characteristic equations for the elimination of e or 6. 21

If such a set of independent scalar and vector equations exist which can be employed to uniquely determine the surface locus S (R, 0, 0) and the material characteristic, then one may argue that the I.S. boundary condition T T* E x E =0 may constitute the remaining part of such a set of independent equations for the degenerate case of r7 = 0. With this goal in mind more der T T* T i st tailed properties of Ex E = 0, E xE+ E will be discussed next, as well as properties of the Leontovich boundary condition and its complementary formulations. Then it will be shown that for the scalar case a set of independent scalar and vector equations can be derived, based upon the concept of describing the material surface properties by a Leontovich boundary condition, which will T T* in the limit as r1 = 0, degenerate into the condition E x E = 0 as derived by Weston in 7644-1-F (Chapter X). T T* 4.2 THE C.W.I.S. BOUNDARY CONDITION E x E = 0 FOR THE DEGENERATE CASE OF A PERFECTLY CONDUCTING SCATTERER, i.e. r = 0. In chapter X of 7644-1-F, the necessary but not sufficient condition T T* E x E =0 was derived intuitively based upon the physical properties that the T i s A T total field E = E + E must satisfy the boundary condition n x E = 0 on A the surface of a perfectly conducting body, and since the surface normal n is T T T* a real vector also E cannot be of complex direction, thus E x ET*= 0. The application of this boundary condition is verified for spherical and prolate spheroidal test cases in 8579-3-Q and 8579-4-Q, together with the condition {IEi J - I E_ = 0 as derived from the geometrical optics approximation. Both conditions yield the most accurate results if the incident polarization is parallel i s to the generators of the scattering body, since then E + E 0 identically. T T* Although E x E = 0 is only a necessary condition but not sufficient in that an infinite set of exterior concentrical (hyperbolic) surface loci, and for scatterers of larger electrical measure ka a limited set of interior pseudo loci(emanating from the associated interior caustic) in addition to the proper one are generated, this proper surface locus can be determined with least error. In fact, in 8579-3-Q 22

T T* and 8579-4-Q, it is verified that with E x E =0 the proper surface locus can be identified more accurately as with the condition )Ei - I EI = 0 for which only one locus is obtained, however with pronounced deviations from the T T* proper one (8579-4-Q). The family of resulting surface loci of E x E = 0 are equidistantly spaced, concentrical hyperboloids with AX = k AR = r /2 spacing along the axis of symmetry for rotationally symmetric bodies with end-on incidence. The resulting plots for one and the same scatterer versus the electric length are identical for different frequencies. If however, the resulting loci are plotted versus the geometrical length R = X/k, then it can be shown (8579-4-Q) that the proper surface locus is stationary whereas all the other interior as well as exterior additional loci will shift, since the proper surface locus is independent of the wavelength and all further loci correspond to a discrete set of eigenfrequencies (7644-1-F, chapter X). Thus the application T T* of the boundary condition E x E = 0 at only two different frequencies may be sufficient to uniquely determine the ahape of a smooth convex-shaped, perfectly conducting body in those regions which are exterior to the Wilcox minimum sphere. Those portions of the scattering surface within the Wilcox minimum T T* sphere can be determined with E x E = 0 as well, but not with the condition IE1/ - E Si= 0, if properties of displacing the computational origin from the geometrical center are employed at two different frequencies by matching the geometrical length of the displacement vector so that a stationary surface locus within the angular domain along the direction of the displacement vector is found, (8579-1-F ). T T* Thus the boundary condition E x E = 0 must be considered indispensable in computational problems of C. W. Inverse Scattering as applied to the identification of smooth, convex-shaped, perfectly conducting bodies. T T* Finally the question arises whether the boundary condition E x E O= may not bear more physical information and whether its generalizationto conducting 23

bodies may reveal a relationship with Maxwell's Stress-Energy Tensor at the bounding surface of a scatterer. It was therefore attemptedto employ the Leontovich boundary condition of the scalar case to study this question in more detail. 4.3 THE LEONTOVICH BOUNDARY CONDITION AND ITS VARIOUS COMPLEMENTARY FORMULATIONS. In problems of C. W. inverse scattering, the concept of an impedance boundary condition may be employed to its best, if such an impedance can be defined so that it describes the averaged local electromagnetic properties of the unknown scattering surface. The boundary condition (4.3. 1), known as Leontovich condition (Leontovich, 1948) suggests to offer the desired formulation for the scalar case: T _ TA T E- (n E )n = r Z nxH (4.3.1) where ET = E + Es the total electric field T i s HT = H + H the total magnetic field in the region surrounding the body, =o 1 Z = / = = 120 7r Q, the intrinsic impedance of o J ~ o o free space, n = n*, the unit outward normal as regards the surface which is a purely real vector., the relative impedance, a complex scalar quantity, designated as Leontovich impedance. The degenerate cases for which (4.3. 1) is not applicable are treated in detail in Senior (1959, 1962) and Weston (CAA-0020-10-TR) andwillbe excluded from further discussion. The direct application of (4.3. 1) would imply the a priori knowledge of n = n (R, 0, 0) and r7 = rl (R, 0, 0). Thus other formulations of 24

(4.3. 1) are required which, however, must contain identical information such that the surface locus S (R, e, 0) and r (R,,0) can be found solely from T T E and H. The first additional equation is found by applying a vector product operation of n onto (4.3. 1), yielding E xn = r Z [ -(n. H)n] (4.3.2) which merely represents an alternative statement of (4.3. 1) and may be rewritten with e = 1/r into TA T A A T H H n = - Y (nx E) (4.3.2a) O T T and thus corresponds to (4.3.1) under the transformation E -T H T T Z H - - Y E ) and ri -e A, where e denotes the relative admittance. Senior (1962a) has shown the affinities of this transformation with Babinet's principle and has proven the invariance of this transformation, attributing rto the material constituents of the body. In addition to (4.3.1) and (4.3.2) its conjugated formulations will be introduced as: T* A T* A A T* E -(n* E )n = ri Z nxH (4.3.3) and E xn = r Z HT* (n T* Exn H (n - H)j (4.3.4) or [H*-(n- HT)n = -L*Y (nxE). (4.3.4a) The validity of statements (4.3.3) and (4.3.4) must strictly assume that all implied fieldT and HTas well as the relative impedance that all implied field quantities E and H, as well as the relative impedance 25

= rl (R, 0, 0) are analytical functions, and n = n (R, 0, 0) is piecewise continuous, satisfying the set of linear equations (4.3. 1) and (4.3.2) which in turn satisfy Maxwell's equations. The particular constraints on rl and n for which this is not the case are discussed by Weston (CAA-0020-10-TR), and scatterers of these particular features will be excluded. The conjugation of (4.3. 1) and (4.3.2) solely implies the reversal of the reactive character of all implied electromagnetic quantities, thus the pair (4.3.3) - (4.3.4a) bears similar affinities as does the pair (4.3. 1) - (4.3. 2a), now however, for a surface of reversed reactive character. Assuming that the electromagnetic behavior in the vicinity of a scatterer satisfies the Leontovich boundary condition (4.3. 1), the question must be answered whether a set of scalar and vector equations can be derived from the complementary set (4.3. 1) to (4.3.4) of the Leontovich boundary condition which may be employed to determine the surface locus S (R, 0, 0), the surface normal A (R, 0, 0)3 the relative surface impedance rl (R, e, 0) solely from the T T knowledge of the near field representations of E and H. This may be achieved by proper scalar and vector product operations of (4.3. 1) to (4.3.4) onto one another which must result in the following set of independent equations: i) A vector or scalar equation which ought to be independent of rl and may thus determine the surface locus S (R, e, 0) of the unknown target. This equation may possess a discrete set of additional solutions, resulting in a family of concentric surface loci. ii) A set of independent scalar equations necessary to define both the amplitude I T7 and the phase rt//r7/= exp i b of the relative surface impedance. These equations should result in a pair of quadratic equations such that a fourfold complementary solution is obtained, corresponding to the complementary character of equations (4.3. 1) to (4.3.4). iii) A closed form expression for the surface normal n, which must yield a purely real vector quantity and ought not depend upon r, otherwise the correct solution of rT, satisfying (4.3.1) cannot be found. 26

To this end the scalar formulation of the Leontovich boundary condition was considered only which in general cannot be employed for problems of inverse scattering as applied to the determination of the shape and material constituents of closed convex-shaped bodies. The vector wave nature of the inverse scattering problem requires the definition of a dyadic formulation of the relative impedance, where the following equations must be satisfied on the surface of a non-perfectly conducting body ET (n ET )n Z (nxH ) (4.3.5a) 0 T TA TA E xn = Z r (HT (n ) ). (4.3.5b) O The impedance dyadic is symmetric and will contain all terms except for the case of plane wave incidence onto a planar surface with invariance along one axis normal to the plane of incidence for which the diagonal terms are involved only. In all other cases a local angle of polarization as well as a local angle of depolarization together with the radii of curvature will be involved. These properties however may not allow the formulation of the required independent equations since the impedance dyadic will become a function of the radiant dependence along the outward normal (Morse and Feshbach, 1953). Before any further assumptions can be made, the scalar case will be investigated in detail. 4.4 FORMULATION OF A SET OF INVERSE SCATTERING BOUNDARY CONDITIONS FOR A SCATTERING SURFACE, SATISFYING THE SCALAR LEONTOVICH BOUNDARY CONDITION (ET - (fn ET) n) = rl Zn x HT. Applying scalar and vector product operations of equations (4.3. 1) to (4.3.4) onto one another a complex set of interdependent scalar and vector equations results which are given in the appendix. Inspecting these equations it was found practical to employ the following three pairs of vector expressions as fundamental vectors for further analysis (where E = E T and H = H ) 27

A_ = A+A = -, II -1 -2 -I -I E x E)_ + rl n* (H x IP' (4.4.1) (4.4. la) (4.4. lb) A * -A - _I Thus A and A are purely imaginary vector quantities -I -II B, = 1 B2 = [(E*x H)+ *1(E x H*1 (4.4.2) -I = -B -I: (4.4.2a) B* = B -II -II I, II C1 + C2 E x H) + r Hl(E'x H*) -, -01 - (4.4.2b) (4.4.3) (4.4.3a) C* -I -I C* = C -II -II (4.4.3b) In addition to these three pairs of fundamental vectors, it was found useful to introduce the following notations: DI Dn EI EI -I = A x B D* -I -I -I (4.4.4) = A xB = - D - - - II (4.4.5) = A xAI = E* = 2A xA = 2 r*(E x E*) x (H x H) -II -I -1 -2 (4.4.6) B xB - E = 2 B xB =2 *(E*xH*) x(E xH*) (4.4.7) -I -II -II -1 -2 - (4.4.) 28

E- = C xC = -E = 2C xC - 2rr*(ExH)x(E*xH*) -III C I - E 2 C xC 2 r (E xH ) x(x H) (4.4 8) F = -F A xB (4.4.9) F II= -F - II=B xA (4.4.10) - -I-I -I -U*( Employing properties of the derived set of scalar and vector equations as given in the appendix, three orthogonal vectors out of the thirteen vectors defined must be found to which all the remaining vectors can be simply associated. Inspecting equations(4. 6. 6a, 9b, 13) of the appendix, two of the required vectors may be found in A and B I since n. A =0 (4.4.11a) -I n B = 0 (4.4. 1 lb) -I A B nxA (4.4.11c) -I -I A A -nxBI (44. 11d) -I Thus the purely imaginary (4.4. la, 4.4.2a) vector quantities A I and BI are perpendicular to each other (4.4. 1 lc, 4.4. lid) and in fact tangent to the local scattering surface (4.4. 1 la, 4.4. 1 lb). Furthermore, it can be shown that AI. A =(BIx) x A,-A (Bx) ~ A (BixA) = -I -I i i - I i I = B* (n I = B B B (4.4.12a) and A BI 0 (4.4.12b) Since A and B are tangent to the local scattering surface and perpendicular to one another, its cross product DI (4.4.4), apurely real vector quantity, must -I 29

be directed along the local outward normal n of the scattering surface, where DI = A x BI = -(AI A)n = (B B n = [n (A xB (4.4.13) and with equation (. ), where it was shown that: [ ~(A xB )] = - 2 C1 C2 -2 (A A2) + B ) (4.4.14a) Di = -2 C1 * 2 C = -2 (A A2+B B 2 (4.4.14b) Since no further triplet of orthogonal vectors was found and the remaining vector quantities as defined in (4.4. 1) to (4.4. 10) can be uniquely decomposed along the directions of A I B I and D I the following theorem can be formulated: Theorem: If the electromagnetic behavior in the vicinity of a scatterer satisfies the Leontovich Boundary condition T A T A t T ET n EnE n = r7Z nxH then the following two purely imaginary vectors T T*, T T* A = (E x E )-*r (H xH ) and T* T T T' I = r (E xH ) - (E xH are orthogonal vector quantities which lie in the local tangent plane of the scatterer and its cross product, a purely real vector quantity, D = A xB = n I [ (A x B, DI = AI -I n -I -I is directed along the outward local normal of the scatterer. The absolute values of these three vector quantities are identical, where A- AI = B- B = ( ) -I -I -I -I 30

and AI, B I DI constitute a right-handed orthogonal vector triplet. These peculiar properties result in the following characteristic equations from which a set of inverse scattering boundary conditions may possibly derived. Since with (4.4.13) AI AI = BI Bi = (n * D) = +2C1 C2 =2(A *IA_+B *B) 1 -2 -1 -2 two characteristic equations for the determination of the amplitude and the phase of the relative impedance rl are found: (Ex E*) (E x E*) + (r7 r)2 (H x H*). (H x H*) - 2 Y r1*(E*x H) ~ (E x H*) (44. 15a) and r (E x H) (E*x H) + rr (E x H*) (E x H) = 2?r rl*(E x E*). (H x WI). (4.4.15b) For the derivation of (4.4. 15a) and (4.4. 15b), equation (4.4. 12b) was not used, thus the orthogonality condition of A and B may be employed as the characteristic equation for the determination of the surface locus, i.e. A B =0 or (4.4. 16a) [E x E*) - r *(H x H*)]. [r (E*x H) - r*(E x H*)] = 0 which can be rewritten into [(E - E*) - r1*(H e H*)J [r * (E* H) + n* (E - I)] = = r (E. H) [(E* * E*) - *2 (HI H*)] + (E* - H*) _E * E) - rl (H H)] (4.4.16b) Although (4.4. 16) has not been employed in the derivation of (4. 4. 15a, b), these three equations may be linearly dependent which still must be investigated. The fourth characteristic equation for the determination of the surface normal, then is given by 31

t D A x BI A IxB 4 -I -I (4 -7b ~ " -[AxB = 2 C1 C2 (4.4.17b) n - DI fi x1 -2 - { (Ex E*) - n r*(H x [ (E*x H) - *(E x H*)]} 2 r r'(E x H) * (E*xH*) which is dependent upon r) and so far could not be formulated solely in terms of the field quantities. Inspecting these characteristic equations it is found that a fourfold solution for the relative impedance rl results as was predicted, furthermore these ri represent a set of four complementary solutions. For the degenerate case r} - 0, equation (4.4.15a) degenerates into the inverse scattering boundary T T* condition E x E =0 which was derived by Weston in 7644-1-F (chapter X). To determine the proper amplitude and phase of r, the formulation (4.3. 1) of the Leontovich equation must be satisfied which would require a characteristic equation for the surface normal n being independent from r, otherwise a unique solution may not be found from the orthogonality condition (4.4.16a). Namely were rl a function of frequency then also n may become frequency dependent. However it may be possible to express DI in terms of cross procuts of A 2' B 2 or C 2 such that the normal becomes i, 2' 1, 2 i, 2 a closed form expression in terms of the field quantities only. Although the characteristic equations (4.4.15, 16 and 17) cannot be applied in the general problem of inverse scattering for the determination of the constituent parameters S (R, 0, n (R, e,0) and r (R, 0, 0) due to the inherent vector nature, the presented derivation suggests that such a formulation of local inverse scattering boundary conditions may exist. A more sophisticated analysis, employing the vector form of the Leontovich boundary condition, may yield the desired characteristic equations. 32

4.5 SUMMARY It was shown that it is possible to derive a set of scalar and vector equations from the complementary set of the four formulations of the Leontovich boundary condition which may be employed as local boundary conditions to the problem of continuous wave inverse scattering. Although the derivations are based upon the formulation of a scalar Leontovich impedance, this first approach reveals a rather important result. Namely inspecting the characteristic equation (4.4.15a) it can be shown that the associated scattering surface locus can be found if solely the absolute value of the impedance is known. Thus if the material character of the surface is known, the following local inverse scattering boundary conditions may be applied to determine the surface locus S (R,, 0), i.e. the shape of the scatterer: a) ri = 0 (perfectly conducting scatterer): T T* T T* (E xE ) (Ex ) = 0 (4.5.1) A) r7 = oa (perfect magnetic conductor or scatterer with plasma coating): T T* T T* (H xH )* (H xH ) = 0 (4.5.2) y) 0 < r7 < oo, rl} I 1 (conducting scatterer for which only the absolute value,of the relative Leontovich impedance has to be known): T T* T T* T T T T* (E x E )E (TEx EH) + (rY) (HT x H ) -2 T (ET T T * = - 2 nrtn* ()E x H ). (4.5.3) These boundary conditions are necessary but not sufficient,however they are local conditions which permit a point by point determination of the surface locus and its application of two or more different operational frequencies may result in the unique specification of the proper locus out of the infinite set of the obtained loci as was discussed in detail in section (4.2). 33

Furthermore the properties of the formulated theorem of section (4.4) suggest the formulation of a more general set of local inverse scattering boundary conditions for which the a priori knowledge of the relative Leontovich impedance is not required. Such a formulation requires first of all the proof that the characteristic equations (4.4. 15a), (4.4. 15b) and (4.4. 16a) are linearly independent. Then a formulation of the surface normal must be sought which is independent of rj so that the impedance corresponding to the proper plane can be selected. So far no explicite physical interpretation on the nature of these boundary conditions was presented, preliminary investigations however suggest that a relationship between the electromagnetic stress tensor (J.A. Stratton, 1941) and the properties of the above state theorem exists. It is therefore proposed to investigate such existing relationships in all detail. 4.6 APPENDIX T For convenience, the total magnetic field vector H = H is normalized T with respect to the total electric field vector E = E by the intrinsic impedance of free space Z = 120ir 2, where Eqs. (4.3.1) to(4.3.4) will assume the following form {[E- n E) n] -(nxH)} 0 I (Ex )-r CH- (n H) n]=0 II [E- (n * E') n] -, (nx H)} =0 III ( (E*x n) - r [H- (n * H,) ] = 0 IV The scalar and vector product operations of these equations onto another result into the following relationships: 4.6.1 SCALAR PRODUCTS I- I or II- II 2 (H H) E2 2 ( H)2 (4.6. (E ~E) - rl (H H) = ( E) 2 2 (4.6.la) 34

77 (n x H) = (E * E)E) = r [H (E x ) = 2 2 2 r7 (H H) - r (n. H) (4.6.lb) or 2yj n. ~(H Ej = [(E * E) + r (H * H)] [( E)2 + 2 ( H)2] (4.6. lc) I*'. P or IP*. II (E* E*) -r ( ( n - H *)2 = ( E r)2 n r* [n (H*xE*)] = r N [E* (x H*)] = n* [HI* (E*x n) = (E* E*) ( *) 2 ( H*- H*) '- 2 (n H*)2 or (4.6.2a) (4.6.2b) 2rl" n * (H'x E'] = [(E* - E') + 2' (H'* A le ct1 E,)2 + r2 (n (4.6.2c) I II and PI* IIP r (E H) = rl(n E) (n H) r7*(E* H*) = r( * E*) (n * H*) I * 1* ( x E) ( x E*) - ( x H) * (n x H) = 0 or (E * E') - r7*(LH H*) = (n E) (n * E') - r77'(n H) (n H*) II IP* (4.6.3a) (4.6.3b) (4.6.4a) (4.6.4b) n 7(E* x H) - r*(Ex H)} = (4.6.5) 35

I' nIT" n. (ExE*) - 1ri(HxH*) =0 (4.6.6a) (nxE) * ( x H) + r'(nx ) * (nxH*) = 0 (4.6.6b) or - n n - E) n - H,-',c) (4.6. 6c) r (E' * H)+ n ' (E * H = n (n * E') + i'(n * E) (n * H*) (4.6. 6c) 4.6.2 VECTOR PRODUCTS I x I and II x II 2 (E x E) r7 (H x H) 0 (4.6. 7a) ), x r'' and IIP x II*.i.2 (E''*xE) = r'2 (HI x H'") = 0 (4.6.7b) Ix II and I' x II n (E x H) r= [n (H x E)] - (E E) + rn (H H n+ [(n E) E + rl (n H) H =[E{' E (H - H) E- (n -)n E) E + 2 (n H) H] [(E E) - 2 (n H)2]n+ [(n E) E + r2 (n H) H1 (4.6.8a) (E"'x H') = [71 (H''x EH)] - [(E' E)+ r,2' (H' * H-* n + + (n * E4) E+ 2 (n. H ]*) H] = [nrl(H* H*) - (n * E) n E + [(n. E') E''+ r(n Hi) H*] = [(E: - (n * H)2]+ [(n E) E'+ r2 n H') H*] (4.6.8b) 36

I x P' and II x I: n [(Ex E*) + rl7'(H x H*) = [r"*(n x E) (n) n E) ( x H) (4.6.9a) since (n x E) (n xH) = n. (E x E:) = - rn. (H x H') (4.6.9b) or r*(n xE) '(n xH) = n. (E x E*) = r7rn. (H x H*) (4.6.9c) thus n ExE(E E) -rr,(H x H) = 0 (4.6.9d) and n (n x E*) (n xH) + n (nxE x E) H =0 (4.6.9e) I xI n E) (n E) *(n H) ~ (n x H*)] = 0 (4.6. 10a) n (n x E) (n x E*)+,':(n x H) (n x H:)} = r* n x [(E x H*) n] (n H) X (E H) (4.6. 10b) II x P n (n x E) (n x E*) + n(n x H) (n x H*) = = rl(n x H:) x(n x E)- r (E*x H) + n x [(E*x H)x n ] (4.6. 10c) (I x *') - (IIx Ir) Ln (E* x H) - l' (E x H*) = n x H*) x(n x E) - (n x H) x(n x E) + 7n x [(E_'x H) x n r''n x (Ex Ix) X n = =n x (Ex Ex ) - nr '(H_ x H)] (4.6.11) 37

Equation (4.6. 11) constitutes an important result namely that a set of orthogonal vectors is found which are tangent to the local scattering surface since by (4.6.5) and (4.6.6a) n * r (E 'x H) - r'(E x H)] = 0 and n [(E x E ') - rn (H x H*)] =0 Verification of (4.6. 11) [n(E''x H) - r(E x H') = E'x (E x H) - Ex (E'xn)] + nn (n * H) [(n x H) x n - (n H') (n x H) x n] = = [(E,. n)E - n (E. E) - (E n) E'+ n (E. E*')] + r [(H n) H - (n. H) (n. H) - (.n * n n. n H) n] [(Ex E) x n] + rr (H x H) x n ] =nx [(ExE )-x.1 (HxH ) ] - =n x [(E x E*) - nl (H x H*)] There exists another formulation of (4.6.11) given by [E x E,') - rr','(Hx x H -n x (E:'x H) - nl*(E x H,')] (4.6.12) which may be verified as follows: [(E -x E*]) - nn*(H x H'') = [E x (n xx E iH)] - n [ x(E" x n)] A A A +{(n ) E x n] -nr (n * H') H x nA] = = n (E H) + r (E* H) - ( E)(n H') + n ( E) n H) = 0, according to (4.6.6c) +{[1 (n (* E"') H - r7 (n * H) E'j + [r(n * H) E - (E * n) H1 = = n[nx(HxE'j] + nx(E x HI) = - nx r (E'x H) - (E x H' )] q.e.d. 38

This concludes the derivation of scalar and vector product operations which resulted into the following identities: [(E x E*) - rr* (H x )] + n x [ (E* x H) - 7* (E x H0)] } O (4.6. 13a) or f[(E x E'*) - r7r*(H x H)] x n + [r (E*'x H) - rl*(E x IH)] =0 4.6.3 DERIVATION OF ADDITIONAL VECTOR IDENTITIES It can be shown that D n (D * ) can be reformulated as: n.D = n (A x B) = -2 *(n. E) (n E*) (H. H*) + r''(n * H) (n H'*) (E * E_*)] - [,*(n. H) (n * E*) (E * H*) + rr*(n * H') (E * n) (E* * H)]j= (4.6.13b) = - 2 nn* [(E x H) * (E* x H*) ] = - 2 nr r[(E x E*_) * (H x H*) + (E*x H) (E x H')] = - 2 r7*c [(E * E*) (H * H') - nrn(E* * H) (E 4 * H.) (4.6.14) 39

V MATRIX INVERSION FOR THE CASE OF END-ON INCIDENCE ONTO A ROTATIONALLY SYMMETRIC SCATTERER 5.1 FORMULATION OF THE MATRIX In problems of C.W. inverse scattering it is assumed that for a given incident field the measured far scattered field can be obtained for a sufficiently large number of bistatic angles. If the direction of the incident wave is chosen along and in direction of the negative z - axis of a spherical coordinate system A and the polarization vector et of the transmitted wave is parallel and in direction of the positive x - axis, then the scattered field may be represented by a series expansion into vector spherical wave functions: s+ r 1 E (Rc, 0c ) = ei)ni+ aemn Mmn +(i) b nNemn (5.1.1) n=l m=0 where the vector spherical wave functions are defined by: Memn(R, 0,) = + hf(1)(kR) Sn(e) sin (m ) - o n n cos n sin - h( )(knR) Rm) cs (5.1.2) Nemn (R,.), (n) = h(kR) Pm(cos e) '~O(ms ) R + kR dR R h (kR) Rn (cos 0) sin (m0) d (1) m sin A kR dR R h (kR) S (cos ) (m0) 0 (5.1.3) with S (e) i pm(cos 0) = cos 0 [(n-m+1) (n+m) Pm- 1(cos 0) + n sin c0 n 2 n + P+1m (cos 0) + m sin 0 P ( cos 0) (5.1. 4) n n' 40

a pm(cos 0) Rm(O) = nmO) = 2 (n- m) P os )m+l (Cos) m- ) (5.1.5) n a 2 n nLn (5.1.5) For a perfectly conducting sphe re the expansion coefficients a emn,; b emn bear the following relationship with the expansion coefficients given by Stratton (1941), where m = 1 for nose-on incidence onto a rotationally symmetric body: n+l (2n+l) a on = i(-_+ (2n+, a (5.1.6a) oln (an+ l)n n b = i(l)n (2n+1) b. (5.1.6b) eln (n+1) n n Employing the asymptotic approximation of the Hankel functions f (1) \ O n+1 exp ix (5..7a) lim th (x) = (-i) -s — (5.1.7a) n x x - 00 O x x x -~ 00 drd h ] (_i)n expix (5.1.7b) and extracting the factor (expix ) from the measured field components, the far scattered field is given by N E (0cos S + b R ( cos cos )+b (5.1.8a) 0 c c c n= c oin. n c eln n c c n=l N E (0 p ) = - sinc0 [a R (cos 0 )os (5.1.8b) 0 c c n c oln n c eln n c c n=l where N denotes the total number of receiver aspect angles, and for the c receiver the field components are given by 41

ES (0 0 ) ES (, ). c ~c ' c c c c For the formulation of a suitable near field representation, the unknown expansion coefficients aoln, bln must be computed, To do so, the field se 1 1 components E, E, and the inherent terms S (e), R (e), cos 0 must c c be arranged in a matrix formulation so that the most suitable matrix representation results which satisfies stability criterions, encountered in the inversion of such a matrix. TnaTetbig s Eqa,& (, L Sah whbida re'gresent tb habasits at frther analysis, these equations may be rewritten into the formulation E = AX (5.1.9) where the transpose E of E is given by: N* * * EN N); E1 (, E01 (e 2) 0 2) * * E 0 N O N' P(5.1.10a) ~.. EN(O N)] (5.1.I T which consists of 2N complex elements, and so does the transpose X of X which represents the unknown coefficients: T X =(a be a1 b. a b ) (5.. 10b) oll' ell' o12' e12' oin' eln With this arrangement the matrix elements of A are defined as well, where the rth row for 1 < r < N is given by S (cos ) cos, (cos 0 ) cos S (os o ) cos 0 R (cos 0 ) cos 0 *r r r r 2 os os r a S (cos e ) cos0 R (cos 0 ) cos 0 (5.1. la) N r r' N r r 42

and the (N+ r)th row for 1 < r < N is given by 11 -R(cos 0 ) si 0 S (cos r0) sin 0 - R (cos r) sin 0 S (cos ) sin.,- R(cos 0 ) sin r - SNcos 0 ) sin 2 r orN r r SN r rj The resulting matrix A can however be decomposed into the product of two square matrices: A (ec, 0,c n) = B(c) C(e, n). (5.1.12) The properties of this matrix must be investigated in detail to gain more information about stability criterions. 5.2 PROPERTIES OF MATRIX A (c' 0c' n) = B(0) C(0, n) Matrix A is a square matrix with 2N x 2N purely real elements for real angles 0, and 0. Matrix B(0 ) is diagonal, solely incorporating the 0 - dependence and its elements are given by: br = cos0 for 1< r< N (5.2.la) rr c - - btrN+r =-sin0 forN<N+r< 2N. (5.2.lb) bN+-r, N+r c The determinant of B is always smaller than unity, where. 1N N Det {B} ( - sin 20. (5.2.2) 2 c = 1 This matrix will become singular for 0 = O+m 7/2, m= 0,1,2.. (5.2.3a) c and is maximum for 0 = /4+ m 7/2 (5.2.3b) c 43

and it can be seen that otherwise no restrictions are imposed, e.g. the choice of 0 = 01 = 02 =.. = N is permissible. However, it may be advisable to extract the 0 - dependence entirely from the field components before inverting the matrix, since the 0 - dependence solely reduces the magnitude of determinant Det A Matrix C (0), solely incorporating the 0 - dependence in terms of c Legendre functions, needs further thorough investigation. Inspecting Eqs. (5.1. 8a, b) it is recognized that always two of the 2N x 2N elements are identical, and for even N matrix C can be decomposed into the sum of two matrices each containing non-identical coefficients, where for N = 2 we find: |11 12 13 14 11 ~ 13 O c21 c22 c23 c24 0 c22 0 c24 C C + C2 + = c31 c32 c33 c34 1 2 c31 0 ~33 0 c41 42 43 0 42 43 44 c44 0~ 12 0 c14 ~21 0 c23 0 0+ Oc Oc ~ c.(5.2.4) 0 C32 0 ~34 41 O c43 0 where 1 1 c = c32 = S( ) = 1 c3 =34 = S (0) = 3 cos 01 r O 1c c 3 1 21 = 42=S ) 1 c23 =c44 =S2(0) 3cos 2 2 = C31 = R1(e) cos C14 =33 = R2(0) = 3 cos 2 0 (52.5) c22 =c41 = R(2 cos 2 C24 =43 R2(2) = 3 cos 2 02 Furthermore, it can be shown that 44

since Det C1 = Det Cj 2 = TC1R (5.2.6a) (5.2.6b) where 0 1 0 0 T = 1 0 0 0 0 0 0 1 0 0 1 0 Loo1o0 0 R 1 LO 0 1 0 0 0 1 1 0 0 1 0 0 (5.2.6c) This property may be employed to derive the determinant of C more easily, where where -1 -1 C C +C C (I+C C ) = (C C2 +I) C eC = Det Det + C1 2 2 Det{C = Det{CjDetLI+C1 c2. (5.2.7a) (5.2.7b) For the chosen case N = 2, the determinants result into: Det{C = -(c12 c13 - c11 14) (c22 23- c21 C24) = - (3 sin 0 ) (3 sin20 ) 1 2 (5.2.8a) Det I+C1 C2} [2(cos 1 - cos 0 2) (5.2. 8b) thus Det {c3 N=2 = - 2 23 sin] 2 (cos0-cos0 - sin2O1 i[3 sin 212 (Cos 0 -Cos 02].(5.2.8c) 45

Employing this scheme, the determinants for the cases N = 4 and N = 6 were evaluated. There it was found convenient to formulate the determinants into a continued fraction expansion for increasing N, where DetI N DetfC}N N N 2 = C sin e, 1 1 11 C2 sin 02cos 2 - cos 01] -1 ---- C3 sin2 0[os 0 - cos 0]cos 0 - cos os0 21 C4 si 0 cos os 0 2cos 0 - cos 022 s c os co 01]2 1 C5sin 0 5[cos 05-cos 0 [os 5 -cos 36[cos0 -cos e cos0 -cos 0 5 I.1 1....1 C sin 20cos06-cos0 2...... os 0 -cos 0]2 646 etc. CN sin0N[cos0Ncos0Nll 2........... Cos0N0 Cos 8,N (5. 2. 9a) 46

The constant multipliers were found to be: C1 = 1 = (-1)1 + 1 2 1 2 ~1 C 2 2 (-1)2+ C 32 4 2 3.5 3+ 1 C = =(-1) 2 2 2 1 1.3 2 2 22 1 *2 2 2 2 4~1 C =-2 2 5. 7 =(-1) 1 4 34 54.72 5+1 5 6 2 6 2 2 C6= -3 7 11 = (_1)6 C6 4= 2 C =(-1) c+l [2c-1)'] c [(c-i 2 ~~~~(c _ ~ ~1)! 2 2 22. 2 12 32. 52. 72.92. 2 1.32 52 72.92 2 2 2 7 2 12. 22. 2. 2. 52 22 2 2 - C 62 (5.2.9b) (2c-1)" = 1- 3- 5- 7- 9 -- (2c-1) where computational results are given in Table V-l. Inspecting (5. 2.9) it can be seen that Det{C (0, 2N x 2N) can be decomposed into the product of two N x N determinants: C(e, 2Nx 2N)= D(sin 0, NxN)- E (cosO, NxN) c c c (5.2.10a) D is a diagonal matrix with diagonal element of the form I 2 d =(l)c+l 1 (2c- 1).] 2 2 2 dcc - c sin2 E(c- 1),}]2 (5.2.10b) 47

and E is of the Vandermonde type, where Vandermonde's determinant is here given by: V=E= lcos cos 1 '2 lcos0 2cos20 2 2 2 lcos0 cos 03 icos0 N. N N-l N - 1 cos 1 N- cos 0 N-c cos03 N-i cos0N (Cos -Cos 0 )(cos 0 -cos 0 ). * (cos 0-cos 0) 2 1 3 1 N 1 (cos 0 -cos 0 )(cos0 -cos ).-(cos -cos e2 cosN cos N 2 (cosN - cos 0-) I r N TT N>r>s 1 N (cose -cos ) = TT r s N N>r>s> 1 0 +0 0 -0 2sin( s r)sin( s r).(5.2.10c) 2 2 Thus instead of the continuous fraction expansion a closed-form solution for Det{C(0, N)j can be given, where C Det C(9c, N) = TT (-1 -(i )L c sinr}2 NT c=l N> r> sl 1 (cos0 -cos0 )2 = r s N,o N 0/ \ +0 0 -0 -2 = nTT (-1pcsi.^ s). cl [ N> sin 2 (2s+) sin( )sin( s r (5. 2. 11) N Inspecting (5. 2. 11) it is found that for a set of N aspect angles R = C c roots 2 c=i will be encountered, where N of these roots are the sin 0 terms and the of the Legendre functions, for all mutations of the N polar angles 0. Thus the c following restrictions must be imposed onto the 0 -dependence: C 48

i) 0: 0 or wr c ii) 0c 0C1 c =2,3,.... N iii) The distribution of the polar angles 0 must be chosen so that the C determinant neither becomes too much nor too large but of the order of unity where for each different number N an individual optimization procedure must be employed. 5.3 SUMMARY The obtained results may be summarized in the following theorem: Theorem: V-1 If an expansion of the scattered field into spherical vector wave functions is employed for the first dipole case (m = 1), i.e., end-on incidence onto a rotationally symmetric and perfectly conducting scatterer, then the determinant Det A( c 0; N)} Det B(0c)3 Det (0 N) = Nc c (2c- )'' (-1)N T 1 )c+I c c (2 (cos e -cos )2 N2i(c I1) sin2 c=l N>r>s>l of the associated scattering matrix A(0c, 0; N) will become singular for i) 0 = p7r0 +, 2 3 ii) 0 =qr q =0, 1 c iii) =0 c =1, 2, (N- 1) c c+l 49

In addition, pseudo-singular behavior may be encountered even though (i), (ii) and (iii) are not satisfied if the computational aspect angles exclusively lie within narrow cones about the z-axis of the computational coordinate system, thus reducing the value of the determinant to almost zero. These properties are illustrated in Figs. 5-1 and 5-2 for transmitterreceiver configurations which may occur in practice most frequently. For both cases the receiver aspect angles are assumed to be well distributed within a narrow cone whose invariant axis a is oriented perpendicular to back scatter direction. For simplicity it is assumed that 0 = 7r/4 c or 0c = 7r/4 + r, and main attention will be attributed to the dependence of the polar angle ce c The determinant associated with the configuration of Fig. 5-1 will become pseudo-singular, whereas the second configuration constitutes the optimum choice as regards the orientation of the conical section relative to the computational z-axis. This properly is demonstrated by the added sin 0 and 2 cos 0 - curves in which the respective Q2 = sin 0 and A = (cos 0 -cos ) c c pr p r are plotted. In general, the optimum distribution of the aspect angles (0c' 0 ) depends upon the given number N of receiver locations, and must be determined for each individual N separately. Yet from the general structure of determinant C(0c, N) it may be concluded that such an optimum distribution will be obtained if the aspect angles are placed in the now identical maxima of the Legendre functions P (cos0) for respective N. N 5.4 COMPUTATIONAL RESULTS With the evaluation of the first four determinants as described in section 3. 2, the properties of matrix C (0, N) were established, where Det C(O N) - Q P () (5.4.1a) c' N N c N QN = T1 CC (5.4. b) QN = 1 50

A A z = a Unit Sphere -= 225~ c -- 0 =45 A = (cosO -cos ) Q = sin 0 p p FIG.5-1:"PSEUDO-SINGULAR" DISTRIBUTION OF ASPECT ANGLES (Invariant axis a of conical section (0 ) identical with the computational z - axis) 51

A z 416 Unit Sphere a '4'l 2 C4 03 4O a+~~ -~~A = 2(cos - cosO ) II I ~,e cos 0 Scatterer = (cos -cos ) Q = sin20 k. A FIG.5-2:STABLE DISTRIBUTION OF ASPECT ANGLES. (Invariant axis a of conical section (0 ) oriented perpendicularlly to the backscatter direction). 52

N N 2 P( )= TT sin O 1 (cos -cose ) (5.4.1c) c= 1 N>r>s>l r > th where C denotes the c constant multiplier of Eq. (5.2.9a). The derivation c of a general algorithm for the constant multipliers C required additional c computation of the next higher order determinants Det C (0, N) together with the corresponding PN ( ). In table V-1 numerical results are presented, where Det C(O, N) was computed directly from the matrix formulation as derived c from (5. 1. 8a) and (5. 1. 8b). Apart from negligible round-off errors the computational results verify the assumed expression of C to its best, where c _[(2_-),,]2 2 C =c (5.4.2a) N N 2 t- ]2 N Tc = 1 c c cQN C 1 [(c 1)] 2 (5.4.2b) C=l c =l [(C-l) ]I For the spherical test case of electric measure ka = 2 (8579-3-Q) the matrix inversion was tested for a varying number of N of aspect angles and for different choices of the sets of aspect angles (0, c, C = 1, 2,.., N). In the following tables V-2 and V-3, test results for N = 4 and N = 6 are presented to verify properties of stability as they can be predicted from the determinant given by (5. 2. 9). To do so the far field components for the chosen sets of aspect angles are computed from known expansion coefficients as given by Table III- of 8579-3-Q with (5.1.6), where the coefficient ( kR ) is extracted. kR) Employing the matrix inversion technique, as described above, the expansion coefficients are computed. For a computational check the far-field components were recomputed with these coefficients. It was observed that the minimum deviation occurs for those chosen sets of aspect angles (0c I = 45, C = 1, 2,...,N) for which the magnitude of the determinant is of the order of unity. for which the magnitude of the determinant is of the order of unity. 53

TABLE V-1 COMPUTATIONAL DERIVATION OF THE CONSTANT MULTIPLIERS C c C/N 1 2 3 4 5 6 7 8 9 0 c sin 0 c cos 0 c log{Det[C(0, N)]} log{PN ( c} log{QN}Computed logCC Computed 900 750 105~ 60~ 120~ 45~ 1350 300 150~ 1.000000.000000.000000.000000.965926.258819.352170.79586-2.866025 -.258819 1.28048.01976-3 4.26072 2.70441.866025.50000 2.76864.81779-6 7.95090 3.69018.866025 -.500000 5.15498.61572-8 12.53926 4.58836.707107.707107 8.05340. 08254-10 17.97086 5.43160.707107 -. 707107 12.05830.86038-13 24.20789 6.23704.5000000.866025 16.20901.98609-16 31.22292 7.01503.500000 -.866025 21.59388. 59889-18 38.99499 7.77207.000000 1.55631.000000 1.55631 {C c Computed 1 22 32 2.3 2- 3 5 22 522 3454-72 26 36 72 2 4 2 24 2 2 42 2 2 82 2 2 2 32 74 112 13 3.52 112 132 38 52 112 132 172 28 22 2-4 ~~~~~~~~~~~~~2 - 4 2c - )' ]2 c 2) - 1C [(c-1)1]'2 ) [(c c-1) ]2 2 2 2 4 2 1 2 1.3 2 2 2 1-3-5 2 3.5 1-3 5-. 2 — 2' (li2325 =2 2 2 =.3 1-2.4) 1 1 2 -- (1-325-7-9 52 (13-25-7-911 6)2 (1 3579-1113 2 911 133158)2 (911 13 15 17 9)2 1- 2- 3-4 1 2-3-45 1-2.3-4-5-56 2-4- 6 2~4'6-8 '.000000 1.55630 2.70439 3. 69020 4.58839 5.43159 6.23707 7.015037 7.772061

TABLE V-2-la: COMPUTATIONAL RESULTS FOR N = 4. I - K. 1 It- A 1.. 1.78',4 1 - O.B04 199 1? 1 ~ 0471 99 PH1 I \J. t 'r f 4iUUU 0.7854000 NR = Using parametric number of alsn~ft ancrli 1 3-.30899 9 0.7854B0003 1 4..570800 0.7854000 OF = 0 NR = 4 NT = 1 Computation of Matrix Elements T = 1 R = 1 THETA(t,R) = 0.7854000 PHI(T,R) = 0.7854000 1- -..707 — TO- r05l.. 70 1 0 8 I OT - 2 0.24Q9967 1.49999c9 1.'500005 3 -0. J7, of T O 1 *50 9 RI i -8 3 '85. 0 4 -0,406251.2 0.6249819 9,374q77 T= -- - - --- 2 ETA(T -,'RI- -.-047199 PHi R --- P t -— T7 5-78 0o1 0.4 99985 0.8660263 0.0' 2- 072 -O. 0129903 —.257T-04 3 -0.437.C003. 0.3247495 5.6 7499 4 -O.?9q059 — 6 -1.3'3173 14.+1 8595 T = 1 R = 3 THETA(TR) = 1.308999 PHI(TR) = 0.7854000 e-3-s U RRTTO ----. B9i? 6 00 --- — 0 D ---6? -0.3995207 0.7499944 2.799041 3 -0,34-4 882 -.-T61 I- 95 4 0.1434330 -1.581923 -3.716404 T = 1 H( 4 I1Hb-Al, = I-.b(U U - H 1 U ~R) i 0U.854000 1 -0.3500772E-05 1.000000 0.0 '2 -T- C5 rOU-oO. -0.1050_22 9E-04 - 3.OX00 u 3. 52511 40E-05 -1.500000 -0.5251140F-04 + 0..750000 - 0.625570E-04 -7.500000. A MATKRIX (A is defined by equation (5.1. 12)) T.70 Y57 3' 0.4'99 9T 1-.-4-9' 2 -0 —. — 8765?8D-05 1.5 90974 -2.625011 0.6249781 -6.187173 2 7nO.71T7T05-.. 73535517 - l.. 70n60 —4 — O006 ---4 0.2653563 -3.R44R84 -1.104856 -3.535487 30. 071055 O.1R30100 0.5490327 -1.83711 7 -0.70541 0M -2.743846 -1.158044 2. 328167 4 C^,7T -O5 ~ — 0-;.-2- 475-400-05 0.74 2622 5-0-05 -.- 21 3 16 --.1.060658 0.4084420D-04 0.19565540-04 5.303291 { l-C 9 I9- I - - 0.7071 8078 0.-a —O 6 5 6 O-O 5 -—. 949 9 R —F2.62? 5021 -l. 90980 6.187195 -0.6249803 6 -0.35-o355 5 ---- 0. 0'' O'?10-0 I-.06066 8....-1.060658-" 3. 844897 -0.2651573 3.53550T 1.104860 7 O. 1830T1 ]15 0.7071 O8l UJ —.3T -74 -O-.5-5490 -3a4T7 _.743855.0.7054130 -2.32R176 1.1 58048 'B 0.747541 701-05 -0 U-70T I-O2 i. TT27 4 0J.-74762 570-.7-05 -0.40844340-04 1.060661 -5.303310 -0.18565610-04 Al MATRIX = B( )'C (0 ) as defined by (5. 2.1) and (5. 2.4) 1 0.7071053 0.o 1.49999? 0.0 Y7 -5O 9T?0;7-,-6 O. 590974 2 0.0 0.3535517 0o0 -1.060664 0.n.. -3.R4484 o. _ _ -3.535497 I 3. 7071 055 4 0.0. 0.0 -0.247 4080-05 0.5490327 0.0 04 0.0 0.0 -2.1 o1-3 -2.121316 - n --- n r^-.-'- -~,. n __ - F' - -ro —w e - n- - - IJ.r U-. n —.3K 72 Y' I OU UV t UCcUl J V uV V.u ) 3V U 3.' 1 55

T4BL. V-.2- lb_ COMPUTATIONAL RESULTS FOR N=4. 5 -0.4999998 ----- ^. W.\ 0.0 —.. a T — ' 0. 7665600-05 61 8H7195 C -c 6 0..0 -0.7071,080 0.0.....UT 5 7 3 f'0-.-2b"575 73 7 -0.1830115 0.,0 1.837124 C 2. 743R '5.5 -_ 73281 76 -( ). 0 ). 0 '1.060658 ' 1.104860 ). 0 ).0 - --- ------ ). 74262 52D- 05 8 0.0 -0.70'108.1 0.0 C U 0 U I I I 0 060661 OM -0 o I 656561 1-04 -~~ = B().C 2(p) (see 5.2.6) -.i. —0.., —',-:- -'99797-0.0-0 0.0,',. —.625011 0.0 — Z O. 707i.053 -.. ' T....- -1.0,0654....... 0.2651563 0.0 -1. 10456 0 -3 T0-J F -. I —;-83o1'09 a- - 0.0 '-?.743846 0.0 — 4 C T07T055 —. —.0 ~ — -O. T476?725D-05- C I r a r & cic. n I. 8T6 652 8 0 —'05 -6.187173.0 )o 0 2.328167 )'.'0~ ~ —' 'I rn -. UO l 0 J~eU,..1 'J-J U) J.1 tl +y/' U e U 5 0r"O - 0.7071078 -n0 r-1.49999' 0.0 -1.590980 0.0 -0.6249803 — 6r^O. T^^^ --- -— i 1,060668.0 -3.844897' 0.0 3.535501 0.0 - '7- 0 --- -T.- -0,.700-81 0 -0,sz0-)51-^ 3-4r-T Ol.,.. 0.7054130 0.0 1.158048 8 'O. 47541 ft-05 0.0 ' 2.121324 0.- 0 ___ -0.4084434n-04 0.0 -5.303310 0.0 DETA = 8.199224 Det B Detf} (defined by 5.1.12) ___ A. INVEPSE 1 -24.1.8691 43.94923 -32.83723 10.47621 T7 ---- 58,9R1 Fl - 4- 4.* 71,:2 95 -7'-37-5 9357 9;78'. 2 4-2 5 2 24.95R91 -44.14315?2. 59373 -9.782481 — * — 1. 6 81'. -- -94-9-04 32. 372 -70'7 — 47616 - 3 P. 626573" -15.34219 1.: 5?0?7 -3.555206 "" —7.91 H. — -13.74Rl'6.....9.759778....-2.51'7213 4 -7.916098 13.74R22 -9.759924 2.517228 - 8-.76-Z265-4 — _.7'4 1 2-. - I T — 520-1521 3-.-551 85 -....5 -2',468427 ' 4.499455 -3.373?34 1.006895 --- 2 *- -.-6-747s 4-.T3717 5- 7 —.-2~5576 T3 3 0.-5l51-4-3i6?.46.34R5 -4.137193. 2.557626 -0.5151468.2.4'68416. -.4.499433 3.137_1" -1.UU0068B9 7 0.6154678.-1.078557 0.7737976 -0.2'207796 0 —. 4351 979 -0.53-2716 0.20T-0-2O68 0. 1675255D-05 8 -0.4351990 0.5392729 -0.2002686' -0.16008140-05 -0.6154646 1.-078551 -0.7737928 0. 2207780 OETAl = -8.5577144 = Det' BB } Det-tCl -A TNV ERPSEI — 8-.147164- 0.0 -10.547817 0.0,. '-4.11410R 0.0 -3.499777 0.0 2 0.0 0.7542 295 0.D - 1.131351-. 0.0 ' -,3.01.6924 0.0 1.97.9821.3-7. 7372744 ~ 0.0- ' 11.07329 0. ---- - 4.042033 0.00 3.254776 0.0 -4.-.0-....-1.077481 070-.. -1.279510 56

TABLE V-2-lc: COMPUTATIONAL RESULTS FOR N = 4. 0.a0? h93675 0.0 -3.232408 4;44003 o ----4 Un ------- -6. 5U47642- 00 -?. '2_983'7 0.0 -1.682486 0.0.60 0 — * 502821 0. ----T 0 -T 75423- 0 0.0 -.011284 0.0 2.262690 7 — T.2725 -1 6 TI- -0"7 9i..T5 9 --- — 0 0 ---O. 047444 0.0 0.4309940 0.q -mO;t- 0 -.n —. -..-...(.9.58 aTOO -u'.s 323 24 71 -—. 0 0 1.077484 0.0 -1.292979 DETA2 = 8.557728 DetB Det C A2 INVEPSE_ I 0.0 3.016933 0.0 -1.979826 -— tO --- 0 — Or 7544263 — I-.-1.31347 7 2 4.114134 3 0.0 a n 4 -4.04?058' 7.3T273WB 5 0.0 n- r n —f 6 *.?29321 -4+.44U031 7 0.0 R -0. 042488 8 -0.8042489 0.0 '0; O n.n - n-?.693683 -.017 (46 7 0.0 —.-n —?.011290 -' o.502z1i 7 0.0 -1.077487 *0.043040 0.0 3.4'99o00 - — 10.-1547 86 0.0.(. U0 -3.254797 0. 0.-O 1. 682497 b6. 5UU66 0.0 -0. 43099'69 0. 0 3.232417 1. Z r505 0.0 u.U -7.262696 -.752i349Z 0.0 0.0 U 1.292983 0T.3 -245 0.0 - 1.-7-'5 35 9 q. 0.0 -— 1.905594 0 - 0 DEI = 0(.95ff1058 -= Det(I+ C11C} IrN VE -----. —?4. l rR-O- 4-;.-94923 -3?.7 37/23- 1-0.-47621 — -24.95881 44..14295 -3?.59357 9.782425 Z.2. 4.'55q —R-9 '-44.1 4315 323.5,73 - r - -9. 7824R1 24.186 81 -43.94904 3 2.83707 -10.47616 3 - 7-73- - -3- T. — — 34 21 9 1 520 7 -3. 555206 -7.916068 -13.74816 9.759778 -2.517213 — 4T- -. 60W98 -- — 3-.74 827 ---. 7 59 84 --- 7.51 72?2 — -8.626534 15.34212 -11. 52021 3.555185 5 — 2.468?1 7.., 4499455. - 3 '. 3: 737 1. c 00685 -2.463475 4.1'37175 -2.55761.3 0.5151431 ' -6.-4-6 85..7-.-'-7IJ Z ---- -.55626 2 -0 — -0- 5515 T —468 -2.468416 -4.499433 3.373217 -1.006889 7 0-6146T-54T-67 -T1. 07855 7 0-.737T976 -- " -.-2?-07796 ---- 0.4351979 -0 5392716 0.2002681 0.16752550-05 — 8 -0.4351990 T0.5392729 — -.'O686 U-. 16 14D-05 -0.6154646 1.078551 -0.7737928 0.2207780 R T ER F — _R ----— Rea! Part —; Imaginary Pat) --- —------- -- 1 _1 0.478 2311 -0.360n460 1 2 I 0.787246 - I - 3 1 1.075998 -0.1207969 ( 4 1 — 1.72995 --- —.T53R 5 1 -0.2931354 0.5000026) _ 6 1 -0.4874914 0 -327 --- —

TABLE V-2-ld: COMPUTATIONAL RESULTS FOR N = 4. 7 1 -0.6937167 0.10'A299 E. ~-.....) -O.R67,94.....O.1699qY7J --- -.. -p --- _ - - _ ___ (Real Part) (Imaginary Part) -I ----l-7 -0 7.76 q q — 0?9 09160"4-l h6 2 1 o.708078 0. 49 1''? 1 7 a 7 ] ~ 0,A 46 07 a =7oln ~ —.4..]," OW96/-70 0n 7 1 677. 7t-, l.....4 1 -0.37?677h 5 -0. 23039? -5. 7 ---- -2 382 5-7-F''-; ---O' 4.9t'67E-03 6 1 0.41743 1 2 E-0 1 0.300n?' 857-02 7 - 4- 1706R 6 E — 0 8 -.2 -OZ 0.0 4 9? 3 7 C0 F-'5 b' e'nr1 8 1 -0 1 96 97 76 F-02 -0. 61762 4E-05 R T FRC FIC -----— (Real Part) ---- maginary Part --- 1 1 0.4782311. -0n.3604 60 — 2 -T —. —7 -4-6*t -0 —2 0 4O-50 I E5 3 1 l.07.r997 -0.] 207969q 0 -4 TI... 1 9?qr1 ' ' -0.1571ba -58- / — - 5 1 -0;. 293'13 53 0. 4000024 --- -- -.874'TT 0 --- — 0-.3 - Z'2 (Z E 7 1 -0.6937160 0.103298 0 'I'.. —1 —.79339 - 0.16O-' 7 I? 58

TABLE V-2-2: COMPUTATIONAL RESULTS FOR N = 4. T R THFTA P HI 1 1 1. I0471 OS 1?. 094394' ---- 1.3089'9 1 4 1. 5707Q'+ 0.78 5 979 0.78'3979 0.7R''8979 0.78t5 79 _......'. *.7.7 nFTA - 410.7537 = Det B Det C (see 5.1.12) -.. DETAl 1.075'1 4 = Det BDet C (see 5:2.1 and 5.2.4) nETA2 = l.0175?* _ _ = Det B.Det C (see 5.2.1 and 5.2.6) n!.,'.0071557 _6. -2 ~,,,i.~...................... DET =.. 3R 1. 9136 = Det I + C, * C, (see 5. 2. 7b) R T 1 3 4 5 1. 1 1 1.. FR (Real-Part -- 0,7R.?4..' '1.0711,61 1. 1075q97 I.?4 77 95 -n. 4R74 7.4 F I nimaginary art) -0.20 40 975 ) _0.59-04Sa ( E -0o.1707Q776: 0 -0o.153'40 ) - o.3275740 o 6 1 -0. 9387797 -0.796017 E 7 |1 -0O. Q 27113 O^ -0. I 363 3 6 e81. -O.-8679?(7? -0.16 o5909 J R T (Real Part) (Imaginary Part) 1 1. 737?74'CR9 -0.500)141 9 2 1. 0.7087083 0C 0.4 146110 63 1 -0. 20 9Oq 9 0. 56 761 2F -01 oln 4 1 -n.37268'35 00230333 3__ 5 1 0..?372309F-091 -.97664F-8 '3 6 '. 41 75 04'E- -01 0,300?90P2F-O?. 7 I -0,14?0165E-02 0.4491291E-05 en 8 I -0^ 1.9700? 5 F-02 -O* 863()775r-05....5 R.T EQC -IC (Real Part) (Imaginary Part) I?,_. -0,?. 047' 975! 2 l 1.e071. 1t6O -0'.'545808 94 E ~? 1 1.07 9 06 -0.1207976 > 4 1' 1..72994 -0. 153 140 J M 5 1 -04).487a873 0,3,785?39 \ 6 1 -0.93 8795 -0.795 6010 E6 7 _12 -0.693711.8 0.1035333 B 1 -0n8679290. -0*1695908. 59

TABLE V-2-3: COMPUTATIONAL RESULTS FOR N = 4. T P. THFTA PHI I 1.1'78097 0.78R5379 '' ' 1? 1. 089g5, _ 078.5 79 1 3 1.43'A496 0.7853979 1_ 4 1 570794 0_..78_'979 __7 DFET A 0. 117??65F-01 = Det B Det C (see 5.1.12) =T+ Pn * 'R + (c / 29 1 tnd 5.2.4) nETAlt 0.1375974.- 'l-,.., D %,1 1 \% -< v. L. 1 CL OETA? = 0. 1 3t 5988 = Det B Det C2 (see 5.'2.1 and 5.2.6) nFT = 0.8519518F-"01 = Det[ + Cj. C2} (see 5.2. 7b) R T.F rT (ReaTPiart) {(Imaginary part) 1.....1 0.~Th_3.5? -0. 150f9-c ) 2 1 1. 075997 -0.?207977 E 3 1 l1.)9241 ' -0.121 0306 E 4 1 l.?72995 -0 153 1l40 ) 4 5 1 -0.59156h0 0.?225537 \ 6 1 -0.69;71 lq 0.1O' i 3 3 E 7!_ -! '0.7879701, -0.? 756677-01 8 1 -0.86792? -0. 1.6909 R r XQ Rl (Real Part) (Imaginary Part) I 1 0.7'?7408 -O. qC'C416 2 ____ 1 O. I') ~7~3fe7^q 90 __. s (, a 4 1 -0.37?68F5 -0.?? t3 1 39l 4 1-.5 1 0.2382? 34F-0 -0.974 54671r —03 A I 0.4 74596E-01 O 3002?943c- 02 b beln 7 1 -0. 41 9 9O 0-02 0.4560921F-05 P. '1 -0. 1701RF-02 -0.85905r)5E-05 ~~...... R T FPRC.-(Real Part) 1' Ot, 0.9354351 - 2, 1.n759.q, - 3 1. 1. 19.. 92410 4 1 1.27299'+ 5 1 -0. 591.5629 6 1 -0.6937119 7 1 -0.7879701 - 8 1 -0.8679291 - r — I ('. (Imaginary Part) 0.1;nq95q i -0.120 7976 -0.121 0306 -0. 1 1 40 0.2225536 0.1035333 0. 2765677F-0' -0 1695908 60

TABLE V-2-4: COMPUTATIONAL RESULTS FOR N = 4. T THF T AFTA Pil I 81 "' 1 0.5235986.077853 79 1 _2 0.7RS3979 0.7853979 1 3 1.047195' 0.7853979.1.4 1.3(0995.0.7853979 nETA = 0. 3n53473 OETAl = 0.6143691. nETA2 0.614-35434 -DET = 0.4970094 P T FR FTI 1 1. 0.22732?4 -.,531 -0 70 n 2 1 0.4 78F2 7 -0. 3603,48.3. 0.7822435 -0.?o0475 4 1 1.075907 -0.1 207976 5 1. -0.130o1. PI 0.61P5058 6 1 -0.2931336 0.5000023 7 1 -0.4874874' 0.3285240 8 1 -0*6937119 0.1035333 R T XP. Xl i 1 0.7327479 -0.9099605 2 1 0.6708288R 0.4lQO3t 3 1. -0.2.09950! 0.567 189R-01O 4..1 -0.3t?6825 -0O.2?303446 5 1 0. 23822 9 RF-01. -0. 9756h 9E-03 6 1 0.417445RE-01 0. 30041 37E-0? 7 1 -0.14?0120F-02 0.46P99246F-05. ~ R 1 -0.1969R86E-02 -0.87R3497FE-05 R T. FRC, FIC 1* --- 1 1..?27?245 -0.5,16,700 2 1 0.47R229R -0.:60A4,9? - 3.1 0.78?2435. -0.?040975 "4 1 1,0759Q. -0. 1207977 5 1t -0.13991 R 0.6185057 6. 1~ -0. 2931336 0.5000023 7 1 -0.4874873. 0.3285?39 8. 1 -0.6937118 0.1035333 61

TABLE V-3-la: COMPUTATIONAL RESULTS FOR N = 6..... —. = -'THFI' A' ~^1 -Ti --- 1 -V1I-.-7T8540(-0 0.7 8 0 00 - 1 2 1.0471.o 0.7854000 NR - 3 --- —-.. —...P.3- 7 999 -r -70-O.-I — 78- -00 -- 1' 4 1.570800 0.7854000 " I...0.13O0 ' 9-00 0' ', --- 1 ' 6 0.5236000. 7854000 DF = 0 NP = ) NT = 1 Computation of Matric Elements T = 1 THFTA(TR) =.0.7854000 PHI (T,R = 0.7854000 7'1 -0. 707105. ----7-800 2 0.2499967 1.499999 1.500005 -— 3 0. 177801T - 1.590981 -1.-5.O330 - ' 4 -0.4062?52 0.6249819 9.q374077 5 —0.3'7564R 1 7.00-43'8RR- ---.2867- -—. 6 -0. 1,84324 -2.296881 1.640434 T 5 ' R — 7 THEI(AT,R )' =- '1 471 0t99 PHII T 'R). H854000' - 1 0,4999985 0.8660263 0.0 Z —'0.1250023 --- --- -.2 99035 ---- -?5On 3 -0.4375003 0n.3247495 5.6?4091 4- — 0.2890596 - 1.3531 73- -- 21.86q9 --- - --- - 5 0.8984733F-01 -1.928251 -4.921061.... 0.327 342 - -.49 -1T47261 8 -14. 503i9 T = I R = 3 THFTA(T,R) = 1.308999 rHI (T,R) = 0.7854000 --— 0,. 70.-.- ' -0.965 265. 6. -.... —. — O. 0 2 -0.3995207 0.7499944?.799041 3 --- 0. 344 P R83 - -- 0.9636111 --.22-6 2?1 9 5 4 0.14'4310 -1.581923 -3.716404 "5 D 0'.",a2776C-) - 0-.2.833007 -- 102 990 6 0.4309556F-01 2.059617 -0.706279R T T =- -- - 4.... T TFfET'A'TTT R)..= 1..570'800. PHI (T-RT — O - 78bDOOO — 1 -0.3500762F-05 1.000000 0.0 — 0.' 50000000... -105022 9E -04-. 3.000000 -- - 3 0.5251 ]40E-05 -1.500000 -0.525 1.40F-04 '4 '0.3750000 '-0.767557Q0E-04 -7.500.000 ' s -0.6563923?-05 1.875000 0. 1837899E-03 6- 0-. 31l75000( -.-n.'4594748E-0 9 — 13.1 5040".... *T-= I R = 5 THETA(T,R) = 0.3927000 PHI(TR) = 0.7854000 1 —. 9- 2 3 87Q9 -- - 3 8 6 8 4 7 0 ---0 -._ — 2.0.7803288' 1.060661 0.4393414 53 ' 0.562 8.1. 875781. Z. 2491. 4 0.3615928 2.629439 5.464164. 5 o.31-37l ff6 - — 3I21 167T- 1n —.n8 56Q — 6 -0.7636195F-01 3.188541 18.60'277 T -- -T-= - F THET R)- = -0 523000 --- PH r T-R1 — 0 7854000- - 1 0.8660249 0.5000010 0.0 7Z a,62499.1 1 9. 9',Fq1qO9 U."f/UU0U2') ---3 0.3247566 2.0624.98 3.247604 4 —G.3- 431iE- 7. 4I 6 E — 7' 2. -435 - -- - -T — - 5 -0.2232749 2.167948 14.20821 — T -- 7402 4. I4 -2074 -, -894 q7- A MATRIXequation (see equation 5. 1.2) 1 -1 -0790q3-7 -1. 590974 ~~ -O.< 3r~ U. 49999 9 -,.625011, —r.a65w58 1.4999y 0.6249781 — 2.Z96-8-66 -0. 877665280-05 --6. ]37173 -—.7840'84. —.-. _632_

TABLE V-3-lb: COMPUTATIONAL RESULTS FOR N = 6. 2 0.7071055 0.35355517 1.060654 -1.060664 0. 7651563. -3.944884. —. 04-3.53 5487 --.1. 574405,2,693146 -0.40601?0 9.802808 3 —0-707105- 0- 1 30.1 09 — 0. 5490327. ---1.37117- -0.7'054!,05 ___ -?.74}846 -1.158044 2.328167 X- 7-07? 89-. --- '7216644 ---., 507739 - 0. 8R9427. 4 0.7071055 -0.74754080-05 -0.74262250-05 -2.121316. ---'1,'.060.:+ O "70 W44 —" 70r-04,-0."1 o..W-J n —4.... 303 70 ^ 1 -1. 3?5823.-0.1346002D-03 -0.32489710-04 -9.290760 -- -0.7071 054' — 0.6 537800 " 1.9'59938 — 1.49 993 - 3.465977 1.767078 4.858549 0.6249722 5.7671 27 -— '.'510626.. 5.. R91 6, 0 -—. — 7710974 - 6 0.707105.2 0.61 3707.1;-3711 1.00653." l.t9U.... 00. 7 296?3717 3. 444sr67 - -Z.6316T7 3.06)59?2 -7.!91532 1.7078R5 -12.58701 -17.;o0.4 99 q --— 0. 7071 0718 — 0. 87 665600'5 — -1.499 98 2.6250 21 -. 90980 6.187T195 -0.6249803 -7. 2 65574 — 0. 099 43876-?.784094 ' -2.2 6875 - 8 -0.3535529 -0.7071 080 1.060668 -1.060658 3'.T4'.. -0. 26515 3 3.5 55 01 — 9 - 1.1048160 -7. 60o 155 1.574410 -9.802843 0.4060135 Y - 0.18 3011 1 — -—. 7071081 --- 1.83l 7124 "-0.5 49034 7 2.7438'5 0.7054130 -2.328176 1.158048 -7.21 6670 -0'. 2073 90 1 - 0. 9 9 64 54. ---- 1". 50 745 10 0.24754170-05 -0.7071081 2.121324 0.74262520-05 -'0.4'H44340-04 IT.06066'1 -- -5.30331 0 "0-. -0156561iD-04 0. 1 3460060-03 -1.325828 9.280793 0.3248983D-04 — IT — 0.653?824 -0.7071080 -'1.49990 -- 1 -.959846 - -1.767085 - -.465989 -0.6249 45 -4.858566 -7. 51 0634 — --.76-T48 -- -- 7.71 1 001-. -5-.891642 --- 12 -0.6123729 - 0.7071078 -1.060657 -1.837117 ' -0.2296245 -2.916812. 2.6"7 1681 -3.444579 7.391559 -3.065937 12.58705 -1.707891 Al MATRIX B (0) 1 0.7071053 0.0 1.499992 0. 0 1.590974 0.0. 00.62749781 0.0 -0.9943840. 0.0 -2.296866 0.0 T.- 0. —3 53 5 517-T -- 0..0 - -1.06066 4 --- 0.0 -3.844884 0.0 -3.535487.0, O —_ — -7.-'693146 -.O 9. 80280 RO 3 0.-7071055. 0.0 0. 5490327 0.0 -0. 70541 05. 0.0.. - -1.158044. O 0 0. 073893 O. 0 1.507739 0.0 40'.0,. --,-?-7- 8- 475 40 0 0 0')- -2. 1-21 316 0.0. 0.4084420D-04 0.0 5.303291 0.0 - 0.1 346'00 ----- - - -9. 280760 5,0.7071054. 0.0 1.959838 0.0 3.465977 40.0 —0,' 4.858549 0.0 5.767127 0.0 5.891620 0.0 6. 0..0 O -. T71 6-27 7.0 6 5..0.0 0. 296237 0.0 -2.651671 o-. 0 - _7. 391 gl -5 -~0.0.7~50~7 -52 T01-2Y-"7 -0.4999998 0.0 0.87665600-05 0.0 2.625021 l0.0- ~ 6.187195 0.0 7. 265574-. 0.0 2.784094 0. 0 0- _0.7 071 08 0.0 -106065 8 -0.0 -. -0. 2651 573 0.0 1.104860 0*0 1 *57441 0 0,0 -. -. -0*4060135 63

TABLE V-3-lc: COMPUTATIONAL RESULTS FOR N = 6. 9 -0.1830115 0.0 1.837124 0.0 2 -.71438R 's —.0-0 - - 2.32 8176 0.0 - -7.7216670 0 n -0. 8996454 0.0 -0-)- 0. 0.......- 0. 70171 081 — 0.00. 7 4 262 5 2)-05 0.0 1I.060661 0.0 _ -0.1 8R5561r-04 -'0.0........3.! 32582'8 3 3 --- —--. H4f89Rln-04 11 -0.6532R24 0.0 -1.499998 0.0 "' I.7670P. O-.-C — - 0.' 974 5 -....?.510674 0.0. 7.711001.0 T —2- 0.0 -0. 7071078 ' --— O.0 ---- - - - -1.837117 0.0 -7 916812 0.0 '-3.444579.. ---. 00 -—. -- - 6.. -3.06 5937' -.0 - ' - 1 1.707891 A27 MATRIX B(c) C2(0 ).0. 4999979 2. c --— 1....0.o....O. 4q9997q O. 0 0.0 -2.625011 0.0. —..... 0 ' —.-.-7. 765F54~.- -..0.0 - 0.87665280-05 -6. l 7173. - -- 7.7340 84 2 C.7071. 055 0.0 1.060654 0. 0 n.765 5. --- —----— o -1'r. 1 04O5 - -1.5744Cs 0.0 -0.4060120 0.0 -— 3. 0.0" n Y.O 18R'010 9 -.-0.0....R 37117 o0.. -'.743846 0.0 2. 281 67.- 0.0 -- 7.216644. — 0.n 0.0- -0.8896422 4 0.7071055 0.0 -0. 7426?25r-05 0.() -1i.0606'.,.165 b540-04 r -- 1.325823 0.0 -0.32489710-04 0.0 5 O. —n 5 —0. ~ 0.65322800 0. 1. 4 9993 0 1!.767078 0.0 0.6?49722 --- 0.0.. -?. 7510626 '..0 -. -7.710974 —.. 6 0.7071052 0.n 1.837111 0.0 - 2.916801.' O.IO 3.444,567 O UOO 3.065926 0.0 1.707885 0.0 70. ---7 - -0 —. -0.7071078 - 0.0 - -1.4998". 0. 0 -i.. 90980 0.0 -0.6 2/9 03 o-. - - - 0. C943876 -. ---O0. —. 2 296 P75 - 8 -0.3539529. 0. 1.060668 0.0 3. B449, R 0. 03. 35 51 7 O. 0 -?.693155 0.0 -9. 802843 0.0 9 ---.0..-. —o.. 70 -01.0 — 0- -.5490347 - - C.O 0.7054130 0.0 1.158048 o —0.0n-..- --— O. -0.2073901 -- 0.0 - -1.1507745 --- 10 0.24754170-05 - 0.0 2.171374 0.0 -0.4I4084434D —04 0.0 -5.30310 a 0.0 0.1346006D-03 0.0. q80793 0.0 I 1 - ',...t.-. 7N- 0 ' ---— 1. 598-46 — 0.0 -3.465989 0.0 -4.858566. —.0 0.0 -—. — - 76-71-48 - 0 -.0- 5. 891647 -'12 -0.6123729 0.00 -1.060657 0.0 -0. Z96?45 0.0 2.'L516 O0l6 0 7.391559 0.0 12.58705 0.0 DETA = -7.86464776 Det LB} Det{C c__ A INVERSF I -076.057 8 24.8329 -235.4175 36.47884 - 4829-4 -99 —1- 54.67 ---- 2-0 1. I97 7. — - — 8 '-3. 2 379 -- -22.4+470 34.48322 -4822.696' 5227.653 --- 2?- 06-.- 9 — -13238 72. 9 -407- -334..48125 --- 64

-~__..._ TABLE V-3-ld: COMPUTATIONAL RESULTS FOR N = 6. 482?.681 -57?7.'630 2076. O-, -824.P3R8 5235. 42:3...-3."48045 -- 4R9.960..-542-~ --- 3.773. 0667 -106.8711 87.52779 -13.40418 -- T79 8.77 --- -2. 2 4 1 -756'.12 3 6 -2 93. 461 9 80.44323_' - 1. 23036 178Q.705 -19.4.655 4-775'6.1 076:' " 03.4531.80. 4395 --- 1.22946 - -1789.769 1o?4.6?4 -773,0571 306.8698 Pi 7. '5 7 3 6,... 9........ 2 - 7.7 9 8. 7 3 ---r~,-: 5 -287.7437_ '1 114.1203 -32.43706 4.892121 - -6'69.8868 - 727.0248 -— 74.'096' ---- 103.4?2 87 --- -26.7q751 3. 7'37342 -662. 7319 712 7.876 -5- '74.'085' - "10,'3.4'42 2 - 26-. 79549 -- -3. 33691 -- 662.7093. l'712.7620 287.7342 -114.1178: 4,36 tB- -'.9 20 —w. ----o'6 669~ 616- - - 726~9986.... 7 89, 14685 -35.27639 9.953992 -1.476195 -207; 711. l -275.4 1 82 --- 81 1 86103.-? 9.D 89 81I 6.845014 -0.7152721 203.4543 -216.9684 -81'. 17q51 — q79'.-08693 -- -6.84404 ". 71 5086? - -20 3. 440r; 21. 9531 -89.14116 35.27445 —. 99.9^ T -.4.6 1 5-75 -rl.r. T5425 2 5 40(.7 -9 -19.R8575 7.822536 -2.183500 0.3186573 - 4646.4694- " SO.39694 -9-16.69391-.5.44681 - -1.09!01. 0.81014900-01 -44.75387 47.01113 -TO-6. 691 38 -.-445745 1.-. 90693- -0. 8095460D-01 -44.74829 -47.00520 19.88293 -7.821692 Z. 183Z95 -, -1 86,61+46. 41 I — -I0.39080.... 11 2.400715 -0.9341004 0.2574637 -0.3707457D-01 5. 675387 — 6-.1.36266.- 1D.695998 -- D 4660073 -- 0. 662072R8-01 0.83647740-04 5.242401 -5.312328 ---.6,95'39 - O.4657 577 - 0-.6613'2 7-0; 271- -0. 9 67)9 58 D-<04 --5.241025 5.310824 -2.400137 0.9338818 --- -.2ZSf40 65.-0.3 6BZI)-U-O1 — 5,673994t 6 1. 34767 ETAT' =I - A 3.l. —3"97594 Det _B D_-et.{C} -AI-NVFRSE-.. ----.. — -. —. ---1 ' 0-53.6156.0 -- - 107.8239 0.0 262.1271. 0.0 -204.3067 0.0 -- 1-7 —31-0. - 272 '389 0 -- 0.0 — 2 0.0 1. 4. 992 75 9.0 3-. 550754 a-O.D " ----- O~ 9 —.1- --— Y 1 4'.0071' - o'_0.0_n _ 16.22107 0.0 -1 4.23709 3 604.6294 0.0 115,.9178 0.0 -284.6938 0.0 _ 221.2133 0. O l-R *6274090 -2.- 29T~-51 --- — 4 0.0 -17.46460 0.0 -4.013707 0.-0 - i- 3.143-344339 -- O. --- r1-7.4B2 -- 0.0 -1 9.40593 0.0 16.49383 5 -435.3287 0, 0. -80.60011 0.0 203.6143 Q.O _ -157.3388 0.0 - T1-. 51176 - 214.11 89 o0. —0 - 6 0.0 13.98429 0.0 3.021337 - O —O'. 0.96?94 0 -0 I-4372-5 - 0.0 15.64476 0.0 -13.53111 7 215.4509 0.0 ' 37.30367 0.0 -99.46158. 0 76. 8499 0. 0 5.571632' 0. 0. 105.9758 0,8 0,0 -8.123169 0.0 -1.569300 01 —0 -6.514678 0.0 - a.802'4-6w 65

TABLE V-3-le: COMPUTATIONAL RESULTS FOR N = 6, 0.0 -8.723768 0.0 7,.82831 ~ -noo. rmin y U. ' - 1 )J.1 fi D ) 0 3a. 95625.00 _ -23. 36369, 0.0 1'. 5241 T --- -.0 3 33. R64 5.' 0 0. 10 0.0 3.191295 0.0 0.5233298 -o —'-. - 7.66795q9- 070- 4.,027597' 0.0 3.110316 0.0 -— 3.231770 r ---1, 0.'6.......0... i-.-1 ' —. ----0 -4. 652137 0.0 3.456405 0,0... O 0.1 967.558 W0-~ -.0 - --- 5'.36BI9Z62 -- 0.0 '12 0.0 -0.6498930 0.0 -0. 8706933n-l0 -- — o. n- 0.6 98 902 3 - -.n -1. 038577 0.0 -0. 5493498 0.0 O.7001620 DFOT^2 -3.9751 76 DetB Det JB ____c A2 INV.FRSE 1 0.0 -1 4.'00701 0.0 -16.22097 — '........14.-37 00O — '..... -14.99255 — 0.0 -3.550709 0.0 -11.44183 -2 '204.3484 -- -0.0 - - 17T7966V' -- 0.0- 272.4440 0.0 553,7254 0.0 - 107.B45t-. 0. - -2-7-1788 - -0. 0. 3 0.n 17.46228 0.0 19.40581 --- - I '1 46436 0.0 4.013655 0.0 13.44321 4 ---221.-2587 — ~ 0 0. --- -.-1. 63176 --- 0.0 '. -297.2960 _0.0 -604.7488 0.0 -, 115'.9408 ---.0 -9 414.7w -— o.0o -5. - 'b.nb -h__ ~+ ___.972'' n.T) -15.64467 —. - 0, ------ ' ''1 3. 5 3 1'0 3 0. 0 - 1 3. 9 8 4 1 1 0.0. -3.021296 0.0 -10.96280 6 — 15T73714 0 0.0.- 12. —545 —4 0.0. --- 214.1626 0.0 ' 435.41.45 0.0 80.6166 3...O. --- - 23-6547r 0 0.6 5 4 0. 7 0.0n.802417 0.0 8.723711 0.0 1.569277 0.9 6.514594 ---— 76.-30 0 -00 0.0 -5.572988 0.0 -- -105. 9974 0.0 -215.4931 0.0 -3-7311 80 0.0 -1.o -9 --- 99 — 8 14 ' - 0. 9 0.0. -4.027582 0.0 -3.110293 ---- ^.75 --- —----— *A. 231 753-4 --- (T."0 --- —-3.191253 0.0 -0."5233213 0.O -2.667926 T0 23 — 73W74 0. 0 1.5- 455 -4 0. 0 - 33.69325 0.0 68. 39530 O. 0 0743 -003 — - 3 0 0-0 30.-962-53- - 0.0-. 11' 0.0. 1.038575 0.0 0.5493452 -- 0.0 ' -0". 700i590 0.0 06498849 — 0.0 0.87067720-01 0.0 0.6498855 -12 -3. 457 7 o Y- - 0.196-8276 0.- D - -5.363034 0.0 -10.68115 0.0 -. 42..0-9 4 3. - 4(653328.0.DET 1.97ROR7 = DetlTC1 G.2- W-AI NVERS. 1 - 2076 -7 57 3-6-.-47 8 84 - Z4+. 5Z9 -35.41 75 66

TABLE V-3-lf: COMPUTATIONAL RESULTS FOR N = 6. -4?q, 994 59242.467 -206.q7 813.2371' -- 4 _ q.g-j -— ' — Z? --- 7.*7- -2-, q 6 --- —- '27-7- -63.-2 '. 44 74 ', 22 2.061.93R. -813.2238 279.4407 -34.48125 ---- -4'87.; - 6-1 5-'27. 630 2076.-0 55'- -824.838 P8 2?5. 4?23 -36.48045 48V9. 160 -5242.41H -3 773. 0667 -306.871 R — 87.57?79 -1 3.40418 -1798.773 -1952.241 756.1236 -?293.461 9.80.4R,+437. — 3 ' 177.79.9. 5- 5 4 -756.1076 ~?9.4531 -R80.t4^5 1'1.22946 - -1789.76n-. -'1 934. 624 --- — 773. 0571 - 306. R698 --R7.?367 13.40464 -17488.739 7 1952.208 5-7- 87.74 32 -" 1 14.1 203 -— 3 2.4 70 --- 4. 2 1 1 -669. R86R 777.0248 -274.0968 103.4287:-Z, ---_-2. ~3,b7T2 --- —62...... —6'T.. 73iT 9 7 T — 7.7 6 274.0893 -103.4?3? 26.70549 -3.336910 6 ---. 7703' - -712.7620.-. 287.7342- - 1 14. 1 178 -32.43678 -4.89?206 669.8616 -726.9986 7 --— 89.146:P:-...-..*7..276.39 - — '9. 9' 2 C19- 1.47615, — 207.7711 -2?5.4182 81.18603 -29. O0891 6 - 46.8ni4 "- 0.71527772.. 203.. 454 -216.79647 8 -R1. 17951 9.086 93 -6.844090 0.71 59063 -203. 4405 -- 21. 531 -8. 14116- 35.;7445 - -9.95P588 _ 1.476175 — 207.7565 225.4027 q — -19. 885'25 --- 7.822536 -..2. 1835(0 "' 0. 3186573 -46.466 4 0.39694 -16.693A91 5.44681 -T, 0 31- "13. 8TT17 01490D-01 -44,753817 o47 —IrT33.10 16.,9138. -5.445745 1.090693 -0.80954600-01 —. 44.74829 -- -47. 00520.19.88293-t -7.821692 2. 1 8375 -0.3186' 61 46.46171 -50.39080 -1-.- 2' 40071 5 - 0.-9341 004 ' 0.5 574637 -0.37074571-01 -5.675387 -6.1 36266 1. 695998 -0.4660073 0,66207?o0-01i 0.,83647640-04 5.242401 -b.312328 12 -1.695390 0.4657577 -0.66132270-01 -0.960S5800-04 --- 57;102-5 - 5T31-O 824. - 2.-0- 1 T -37 338 81 -0.2574065 0. 37067280n-01 -5.673994 6.134767 --, ~- ------ - '- ~............ -...... R T FR FI 1 1 n.4'777118 -0. 60 4 61 _ _ -2 - -7 —1 0.7 8? 3? 6 3 -—.?04 049:3 ]. 076478 -0.1 20'7?68 / E 47 —4-. -7 — 1. 273 0 79' --- 0 -.1 57 1. 7. -5 " 1. 0. 1342068 -0.6048777 _ L^ --- —1 O. 08 -, --- —7 1 -0.2977000( 0O.000026 - E -8 —:- 1-~-;~ 751-67 - —.- i- -'>2 -- ".- -.8 7-1. -0.9417 32 2 o93,.9 1 -0.6941013: 0.1035..9 0n 1 ----— 0-D.-R8n3 7 9 - 0.1-0. 5977 t1 1 -0 ).84?2738RE-0l, 0.6589078 712 1 -0,139749T 0.6 185065 " —R '-.. -XR -- --- I 1 T 0T.73?-4 65-O. 91 OCT) 4 57 2 1 0.6-706334 0.41 7143 ).3.1 -0.71.0 - ' - 0'.5680h-01? -' -" 4 1 -0.37261 1 -0.2301776 -5 'I~ - 0. 3850o65V —o0 -0. 9-&E 2 --- - 6. 1 0.4171939-01 0.3016903E-02/. b— 7 1 -I -0.T-42-84 E-02 O8 7837E-05 - el 67

TABLE V-3-lg: COMPUTATIONAL RESULTS FOR N = 6. 8 1 -O. 162751 E-02 -0.12877?FRE-004 I —i- O0.53RMMIl')E-OU -O.ilU+l4o)F'-o5 10 1 O0. 3R6307E-04 0.)979858F-06 - T= —1.Tf i '0. iI-g 0 -E5 2 -1- 4? E- 06 12 1 -0.141 965E-'05 -0.10461 OF.-06 R T FR C FIC 1 1 0.4'777117 -0.3h601460 72-' " -1-.7B '.7 8 ---—,.'0.-9?.4'-oq.-..4.- — 4 3 1 1.076477 -0.1 207967 _:4 - i'-.....l'.'2 73 2 7R: 7F; 0; 5 S1..... -4 —1-Tl.27?07R — 0O.1538157-7 / 5 1 0.1342067 -0. 6049777 -.I.i-.?770l Rh6....0.53166M0.... 7 1 -0.?q29699 0.5000025 -— I-R.4 Bn'-,48 - 7 5I'873 7R'52.73 9 1 -0.6q41013 0,1905299 E '10 TT. B ---'~37' T -.;T'8 -977 f --- 11 1 -O.oP427376F-01 0.6589078 lz~4S 1 1U0 bIt 68

TABLE V-3-2: COMPUTATIONAL RESULTS FOR N = 6. T *P THFTA PHI 1 1 1'.5707 4". 1 2 1*^41 9R9 n. 7R853,979 0.753g979 t1 1 ns..309 ".,.... 0.78,A979 1 4 1. 1.1 7P09f )0.7P53979 1 5.04715 ',0.7S,979 1 6 0O'. 162976.0.7853979 DFTA = -(.2R 404F-0-? = Det B et C Det B Det C nfTAl = -0.5461.611F-0 DetB DetC1. Det B Det C OETA2 = -'0.?46.816F-0? = Det B et C 2 OFT g 0-45433>55'_ 6= Det I+ C C2} P T -FR Fc (Real Part) (Imaginary Part) 1. 1 1. 7'30)79 -0. 1 rni40 \ 2 1 1.1s2777 -0.1210t3n( 3.. 1.076477 -0.1?07Q75 4 1 0. 9.; 80 -0.1 509", 5 1.078R2327 -0,.?C04975.. 6 1 ".,A?A66?4R -0.27A5667 7 1 -OR6R80?RO -0.151 6 9591 8 -]. 7R? -0,71R2-050 -'. O.1 9 1 -0. f: 940970. 10 533 3 _ 10 1. -0.50!8370 ' 0.222537 I 11 _, ' -0,n 7517; ~ n.32n~, 41 12' 1. -0.3862768.0.4210207 R T XQ XI (Real Part) " (Imaginary Part) 1! c. 7327083 -0. 91 r)00752? 1. 0,.670P6PO.0.41,93\53 1 -0, 2 o)~ '+?' 4, 0, (.7 9-l =-0 I a 1 l-0.3?_0_~q aol n 4 1 -0.37369?^ -0,2 30)3743 -oln 5..L- l 3.- 3 1 6 71 — -n.1 -o.. 098F4743f-0~ 6 1. 0,41 747 PTr F-01 0.30 1 -97 9- -r)02 7 1 -0.1417519E-02 0.7195473E-05 8 t1 -0..!696q6F-0? -0.14??.3 71 —04 - 0.I O,.5133791F-04 -0.579q?,7F-06 b 10 1 0. 65t 0074E-04 0. 150R981 8.-05 eln 1 1 1 -'. I 1 17? nOE-05 -0. 5R81608.-07 1 2 1 -0. 1?21 33E-05 '-0,372 1 1 85F-06 _ R T FRC FIC (Real Part ). (Imaginary Part) 1 ~ 1..730 7q -O.1 5l r 1, C 7 1 1.19277h -0.121 )30A ___ _ 1 1. 076476 -0'. 1 207975 a, 4 1 O. 95R800r -0(.15 ()0 59 oln 5_ O. 0.78232?'7 -0O.?0'^9775 6 1 0,67I66;4R -0,. 76S667 7 i -0A.R680279. -0,(169C910 8 1 078H -0.788?84 -027656767-01 b 9 1 -0., 6940969 O* 1035333 eln!0 1 '-0,591 8363 0,2?255'7 69

TABLE V-3-3: COMPUTATIONAL RESULTS FOR N = 6. T R THFTA PHI 1 1. n,1?089s (). 785S7 1 2,. 61 t7931. 0.787rc179 1 '3 0. 39f69P9 0.78R59.79 1 4 O.r2359n6 0.78r51979 5 O.6544987 0.7853979 _1 _6.0o78R;3970.l 0.78.3979 rFTA =-0, 10 6,76 5 F- 6 DETIl -=-0. 2?4.567F-1 '5 DFTA2 = -0.6205844F-1 __ fET = 0.7784.1 E-01 R T. FR Fl 1 t. \.? '6479q'F-()1 -0. '- 7 904? 1. n. 666733,;F- -0. 61 6 7 3 I 0. 1 34 70? 6 -0. ~604P,7a 4 1 n.?7-701 1 -0.531 /r70 5 1 (. 34?'27 Q5 -0.447n709 6 1 0.477710nr -0.360, 83 7 1 - 20.0046GE-01. 0.70' 730_ 8.1 -,L.441 071 I.-01 0.6 7, ( 7 9 I - 0. O8427?98E-0I1 O. 65 9Cr O 10 1 - 30.139748' 0.61H~0.8 11 1 -0,.20c272?3q. 0.56t6968 1? 1 -0. 92698'9. '.5 00')0?3 P T Xq XI '1.1 ().7536437 -0.7004206 2... I 0. 6 0. F 8?'. 2.0 07h6Q3 3 1 -O.2177f88 -0.083 3 13r-01 4 _1 '-0.3684531 -0.16302770 _ 5.', 0.26601 POF-0]1 0. P1 9,7E-01 6. 1 0.41 0556, E- -0,1 8 0 ". 36V -01 7 - 1 -0.217998?F-0? -0.51,? 8 (0 —0 R8 1 -0.?060676F-02 0.51 S4, 1 F-n? n 1. 198Q400 F-03 0f810780PR=-03 10 1 0.12039?5FE-03 -0.847.415F-03 t1 1 -0.148852?oF —04 -0.650022?9F-04?12 1 -0.9q4c403E-05 0.68151 R7F-04 _ T FR'C F I ___1 1 0)?48802?65E-01. -0.698?1f3? 2.1 o. 65 2 5 rE-01 -0.661 473 ___' 1 0. 133161 78 -0.6_3_? 4 1..66036. -0.5307584 5 1.0.3474911 ' -0.446f,1 1 6 1. 4.775 945 -0. 357487 7.' 1 -0.191.19n7F-01 0.704o143 8 1..-).43 08 81 F-01 0,6P77364 -0.8338946E-01 0.65PqOF8 1 0 1 -0.13R8899 0*6181347 70

TABLE V-3-4: COMPUTATIONAL RESULTS'FOR N= 6. _ ~ ~ ~ ~ ~ ~ ~ r a..... _ it.. ~u 7 p THETA PH I =1 1' 1.570704/ 0. 7RP F 979?2 1\. 3)8 9 I O 78,7R3 979 I_ 1 __ '4 1. 04'71r n 0.785F 979 =1 --- _ O_.7f.5,q79 I 6 O. 785? 9 7. O0.7P51979 DE.TA = -1 7668R9q. nrTA1 -. -.647Q8RFr-o0 DETA2 = -O.164P34F-'O OFT = 0.10721t3Fr 10 R T FR rt 1 1 1. 27307) -n(. R140 7 1. 1. ()764.77 -0. l?0797r 3 1 1. 2 4O (. PlOQ1 4 1 *7**?3,1? ( -0.?04 975 5. 1. n7Oq P -0. ) 9tR 95 t. -() 1.90790 0. 103 t33 9 1 -,. 96055 5 -<O. 40770909.10 -0. 4P'751?.7 0.323 '41 i 1 1 -0,.93 5 23?. -0. 796 o 37 12 1 -,I9'9. 9 r9 0. 7 500r0 3 Q T XR Xi 2 __ l0.670R37. 0, 6 7.41 7 097 07 -0.1419.418F-?? 0.41014V?P-05 13 0.. 055 )7F- 4., 511 ^ 7-) 41. 1 -0. 2076?7.E-05..666 10 06-08 R T XR Xl I 1 * I 1.?7107R -0.153^140 6 * 1.4.076476 -0.107976 1. 1 0.7893??p -0.?040o76 7 1-.. 1 4 1,07087 -. 5-09. 5p6sF 6 1 0.4,777105 -0.3603481 78 1 - 6P. P10 797'C -O0.160Q or-q 8 1-0. 6940970. 0.10,5333 9 1 -0. 9605554 -0.45 77-:0908 10 1. t o0.418756 12 1- 0.328524s- 0 7 1!I -0. 11.O t 2(7 -02 7 '0 0.4.,.5I 4t O, — O 5 12 1'-.'1'7't 7:.E-05. R n10276TR'-07.q 10 t. ~? O.Z.5 5n61F." 7 1.1.0764 776 -. 1 707906 12 ', ' 1; 2 8 AX 3 -. ' 1 6 3 8 1-2 t F.C793? -. FOIC0 76 e; 1 1 '._? 0 (70n -7....0. 19^53a140?_ 1.'I..06 771' -0. 1. 2074 9 7 t, -r' l,.q2 797 -.0 1400s7s6. 1.-0,69'0970 0.107533, 9, 1.0' 960555,. -0 ~4770908 To 1 -n',.48751'21 0.3285240 71

TABLE V-3-5: COMPUTATIONAL RESULTS FOR N = 6. T P TH-FT A 1 1.1 7R907 1? 1.. 047105 1 3 1,ql1n?97'1 1 4 0.' 7953 97 I " 0.6 5449? I r% Ir ft C f, n ' 0. 78TR970 0. 7Q r 9 79 0.)78 3q979 0 * 7 8 3q.t 7 q I h ' ~.? ' ots _ '., U u.' 0*7, f '- - nFT4 -0 -n103'R30F-0 O OFTA = -0.94754t37F-06 OFTA? = -o.P46935Sr-06 _ OFT = 0.1087894 _ P T FR F P 1. ] 'Q)qCR3801 O -0.1 5o 0Qq) 2 1. 0.1. 78:37 - -0.?O4975 3 0. O.6?6670zS -0.7765667 4 n. 4 7771N ~ -(1 n n 4A 5 1 0. 3/ 73 05 -0.447 70g9 S6 '1 O.2270 I1PI -.0.531 6702 7 1 - ).ql 370__.?? t37__ 8 1 -0. 4875 1 2 0.32?P q41 q9.1 - F.38 6 7 6 8 0. 4 2 1. L 07 10 1 -0.?2o698? 0.50n )0?3 11. -0 720972 38 Q 56.6S 8oAR 1? 1 -O. 13974982 0.61 85 o R T XD XI I 1 0.7 I580O -0.91 2691 2 O (). r'7241 ' O.41 70nr80 3 1 -o.2nql977 0.57'-a462 -91 4 1 -0.27i157 -0.?311934 5 1 0.? 3 703 79F-01 -0. 2 1 9 5V -7 16 _1 0n.4l 8 6 8F-01 0.-. 30Q79() 7 -7 1 -. 3921 34F-02 0O.637l36F-04 8 __ 1 -0.201 4?6F-0? -0.q 5630R8F-04 9 1. 0.47834P8E-04 -0.)2f17P-05 10 1 0.741 171F-04 0.171q6R5F-04 1 ] 1 -0. 10 0560F-05 0.610 9376E-06 12 1 -0.75?34C1F-05 -0. 188R,11ir -05 R T FPRC F IC 1__ _ 2 1 0.93P80%F __-0. 1501'q7 3 1. 66678"^3 -0.7?O0567 3 13.. 6f.62'56 -0.?7&6 -67.F 4 6 7 _ 9_ 10' 1 1 1.1 1 0./ 777114 0.0347404 0.2 7010n -0.5918363 -0.4751 16l -0. 3P63769 -0. 2?96981 -0. r'60 4 q -0.44, 77 7 -0.51. 6709 0. 22?530 0.3? 8 5? 0.4210?00 0.500001 8 72

REFERENCES Barrar, R. B. and C. L. Dolph (1954) "On a Three Dimensional Problem of Electromagnetic Theory," Journ. Rat.. Mech. and Anal., Vol. 3, No. 6, 725-743. Leontovich, M.A. (1948) Investigations of Propagation of Radio Waves, Part II, Moscow. Morse, P.M. and H. Feshbach (1953) Methods of Theoretical Physics, Vol. II, McGraw-Hill Book Co., New York. Muller, C. (1956) "Electromagnetic Radiation Pattern and Sources," IRE-TR-AP-4, No. 3, 224-232. Senior, T. B. A. (1961) "Impedance Boundary Conditions for Imperfectly Conducting Surfaces, " Appl. Sci. Res., Section B, Vol. 8, 418-436. Senior, T.B.A. (1962) "A Note on Impedance Boundary Conditions, " Am. Journ. of Physics, Vol 40, 663-665. Stratton, J.A. (1941) Electromagnetic Theory, McGraw-Hill Co., New York. Weston, V. H. (1963) "Theory of Absorbers in Scattering Theory," IEEE-TR-AP, Vol. 11, No. 5, 578-584. Weston, V. H., J.J. Bowman and Ergun Ar (1966) "Inverse Scattering Investigation - Final Report, 1 Oct. 1965 - 30 Sept. 1966, " The University of Michigan Radiation Laboratory Report No. 7644-1-F, AF 19(628)-4884. Weston, V. H. (1967) "Inverse Scattering Investigation - Quarterly Report No. 1, April 1966, " The University of Michigan Radiation Laboratory Report No. 8579-1-Q, ESD-TR-67-517, Vol. I. Weston, V. H. (1967) "Inverse Scattering Investigation - Quarterly Report No. 2, July 1967, " The University of Michigan Radiation Laboratory Report No. 8579-2-Q, ESD-TR-67-517, Vol. II. Weston, V. H., W. M. Boerner and C. L. Dolph (1967) "Inverse Scattering Investigation - Quarterly Report No. 3, Nov. 1967," The University of Michigan Radiation Laboratory Report No. 8579-3-Q, ESD-TR-67-517, Vol. III. Weston, V. H., W.M. Boerner and D.R. Hodgins (1968) "Inverse Scattering Investigation - Quarterly Report No. 4, Feb. 1968, " The University of Michigan Radiation Laboratory Report No. 8579-4-Q, ESD-TR-67-517, Vol. IV. Weston, V.H. "Theory of Absorbers in Scattering',' Report CAA-0020-10-TR, Conductron Corporation, Ann Arbor, Michigan. Wilcox, C.H. (1956) "An Expansion Theorem for Electromagnetic Fields," Comm. Pure Appl. Math., 9, 115-134. 73

DISTRIBUTION Electronic Systems Division Attn: ESSXS L.G. Hanscom Field Bedford, Mass. 01730 27 copies Electronic Systems Division ESTI L.G. Hanscom Field Bedford, Mass. 01730 23 copies

Unclassified I Security Classification DOCUMENT CONTROL DATA - R & D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 N/A 3. REPORT TITLE INVERSE SCATTERING INVESTIGATION FINAL REPORT 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Final Report (3 March 1967 to 3 March 1968) 5. AUTHOR(S (First name, middle initial, last name) Vaughan H. Weston Wolfgang M. Boerner 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS April 1968 73 14 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) F19628-67-C-0190 ESD-TR-68-2I5 b. PROJECT NO. c. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. 8579-1-F 10. DISTRIBUTION STATEMENT This document has been approved for public release and sale; its distribution is unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Deputy for Surveillance and Control Systems, Electronic Systems Division, AFSC, USAF, L G Hanscom Field, Bedford, Mass. 01730 13. ABSTRACT A brief review of the salient features of the theoretical investigation of the c. w. bistatic inverse scattering problem is presented. The effect of changing the origin of the coordinate system upon the convergent properties etc., of the spherical vector wave function representation of the near scattered field and the surface loci Ex E =0, is discussed. It is pointed out that a great deal of analysis remains to be done in this area. The determination of the surface of the scattering body from knowledge of the local total electric field is given. Emphasis is placed upon the generalization of the condition ExE =O as applied to perfectly-conducting bodies, to scattering surfaces characterized by the impedance boundary condition. Properties of the matrix inversion associated with the determination of the expansion coefficients from far field data are discussed. Some numerical results are presented, and restrictions upon the choice of aspect angles are deduced. DD ~~~~~~~~~~~~~~~~~~FORM 1473 Unclassified~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I DD 0NOV 651473 Unc lasssifiied Security Classification

Unclassified Security Classification 14. LINK A LINK B LINK C KEY WORDS R|. _. ROLE WT ROLE | WT ROLE |WT Inverse Scattering Electromagnetic Theory 1 I - I - I I - m - Unclassified Security Classification

UNIVERSITY OF MICHIGAN Ii3 9015 037 40l I 3 9015 03527 4870