015224-4-I THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING Radiation Laboratory INTERIM SCIENTIFIC REPORT Grant No. 77-3188(D), 1 January - 31 December 1980 Thomas B.A. Senior Prepared for: AIR FORCE OFFICE OF SCIENTIFIC RESEARCH AFOSR/NM j> Building 410 Bolling Air Force Base Washington, D.C. 20332 15224-4-1 = RL-2278 December 1980 Ann Arbor, Michigan

015224-4-I ELECTROMAGNETIC SCATTERING This interim scientific report summarizes the work carried out under the Air Force Office of Scientific Research Grant No. 77-3188(C) during the year ended 31 December 1980. The focus of our efforts has continued to be the effect that the material properties of a body have on its scattering behavior. In many instances the effects can be simulated using an approximate boundary condition, e.g., the impedance (Leontovich) boundary condition for a solid body, and the resistive sheet conditions for a plate. Each of these can take many forms, and a survey of approximate boundary conditions in general has been prepared [l] as an invited 'mini-reviw' for the Transactions of the IEEE Antennas and Propagation Society. In some cases the change from an ideal, e.g., perfectly conducting, surface to a non-ideal one does not significantly affect the applicability of techniques available for the solution of the scattering problem, but in others the effect is quite profound. This is certainly true for a wedge subject to an impedance boundary condition or consisting of two resistive sheets in an otherwise homogeneous medium. The latter problem is of interest in the design of a leading edge treatment for a low cross section wing of an aircraft, and is also a feasible model of a homogeneous dielectric wedge for which there is no form of solution yet available. The resistive sheet wedge has been studied intensively this year. We had hoped that by using the Maliuzhinets and Kontorovich-Lebedev transform methods (which are related but superficially quite different) it would be possible to generate the exact solution. Unfortunately, we have not succeeded, although a number of interesting results have been obtained. In particular, it has been found possible to reduce the high order coupled difference equations that each method produces to a single, uncoupled second order equation. The analyticity properties of the solution may lend themselves to complex function-theoretic techniques which convert this difference equation into an integral equation of the singular type, similar to the conversion of differential equations into integral equations.

-2 - The theory of these singular equations is well-established, and has found application in several problems in potential and diffraction theory. These techniques are currently being explored. Finally, along with analytical attempts at determining the solution, the second order equation is amenable to a power series expansion in the resistivity R of the sheets which brings out the nature of the coupling between the interior and exterior region. Explicit results have been obtained for a rr/2 wedge. Another problem where the change in boundary condition profoundly affects the method of solution is the finite resistive plate of either uniform or non-uniform resistivity R. For a plate in the plane z = 0 a straightforward formulation leads to a pair of tightly coupled integral equations for the two components of the induced electric current, the kernels of which involve second derivatives of the free space Green's function. If R 1 0, the procedure which Rahmat-Samii and Mittra (1974) developed for a perfectly conducting plate for eliminating these derivatives is no longer applicable, and though there are several possible formulations of the resistive plate problem [2], none are attractive from the numerical point of view. Nevertheless, the problem is an important one. A resistive polygonal plate is a realistic simulation of a composite or fiberglass fin for a missile or aircraft, and a study of its scattering behavior would be a logical continuation of our previous (and current) work on scattering by resistive strips. We have commenced such a study, looking first at the numerical treatment of the singularities by considering the corresponding low frequency (or static) problem. Several formulations of the Rayleigh scattering problem for a perfectly conducting plate have been examined. Not all of these are effective from the numerical standpoint, but one has been shown capable of providing rather accurate values for the components of the dipole moments and, hence, the elements of the corresponding polarizability tensors. A computer program has been written and data,obtained for a variety of plate geometries. For the 'difficult' case of the normal component of the magnetic dipole moment, the accuracy appears better

-3 - than for most of the results present in the literature. We have also examined the analogous formulation of the resistive plate problem, and have begun looking into the problem of a dielectric platelet whose thickness t is small compared with its lateral dimensions. Such platelets are of interest in atmospheric scattering and cannot be treated as resistive plates alone. Dielectric platelets are one aspect of our renewed investigations of low frequency (Rayleigh) scattering by dielectric particles. An application of this theory is to the design of obscurants, where the objective is to maximize the absorption of the incident radiation. Apart from the obvious dependence on the material properties of the particle, the absorption also depends on the particle shape, and for a particle of given volume and dielectric constant, it is of interest to consider how the shape should be chosen. This is discussed in [3], where it is shown that for an arbitrarily oriented spheroidal particle, a thin oblate spheroid or disk is the optimum shape. Some considerations relating to aligned particles are given in [4]. An important application of our work on low frequency scattering by dielectric particles is in propagation through the atmosphere, i.e., radiation transfer, where the field traverses a distribution of particles. For relatively sparse distributions of aligned particles, the attenuation per unit length is simply Na where N is the number of particles per unit volume and a is the absorption cross section of a single particle. The latter can be found from a knowledge of the (electric) polarizability tensor. For denser distributions it is customary to employ one of the many radiative transfer programs where the attenuation is arrived at as the end product of long and involved computations, but it seems possible that, to some degree, adequate results could also be obtained by treating the particle distribution as an artificial dielectric. For a sparse distribution, the equivalent permittivity of the dielectric is e' = e(I + N),

-4 - where c is the permittivity of the surrounding medium, I is the identity tensor, and P is the polarizability tensor of a single particle. As the density is increased, it becomes necessary to take into account the modification to the field to which each particle is exposed due to the presence of other particles. This can be done by means of the Clausius-Mossotti formula or, for ellipsoidal particles, using the alternative and self-consistent formulation proposed by Cohen et al (1973). The result is an improved effective medium from whose permittivity c' the attenuation can be found [5]. Although the theory is still, in essence, a single scattering one, it appears that the data obtained are accurate even for quite dense particle distributions. In the numerical solution of a scattering problem an alternative to the moment method which is applied to an integral equation is the finite element method applied to the differential equation. In principle, this could provide an effective means for solving the plate problem, and to examine this possibility, we have used the finite element method in the two-dimensional problem of a resistive strip illuminated by an E-polarized plane wave incident in a plane perpendicular to the edges. A key objective is to compare the efficiency with that of the moment method for strips of increasing electrical size. From the results so far obtained it appears that the moment method is superior, and though there are still some steps that can be taken to improve our finite element program, it does not seem that it will offer a viable alternative to the moment technique for this type of scattering problem.

-5 - Publications resulting from this year's Grant [1] T.B.A. Senior, "Approximate boundary conditions", submitted to IEEE Trans. Antennas Propagat. [2] T.B.A. Senior, "Scattering by a finite resistive plate", talk presented at the North American Radio Science Meeting, Quebec, Canada, 2-6 June 1980. [3] T.B.A. Senior, "Effect of particle shape on low frequency absorption," Appl. Opt. 19 (2483-2485) 1980. [4] T.B.A. Senior and H. Weil, "Particle shapes for maximizing low frequency absorption", Proceedings of the Conference on Obscuration and Aerosol Research, Aberdeen, MD, 21-25 July 1980. [5] T.B.A. Senior, "Absorption by a distribution of dielectric particles," talk presented at the National Radio Science Meeting, Boulder, CO, 12-15 January 1981. References Cohen, R. W., G. D. Cody, M. D. Coutts and B. Abeles (1973), "Optical properties of granular silver and gold fibers", Phys. Rev. 8 (8), 3689-3701. Rahmat-Samii, Y. and R. Mittra (1976), "Integral equation solution and RCS computation of a thin rectangular plate," IEEE Trans. Antennas Propagat. 22 (4), 608-610. Senior, T.B.A. (1976), "Low frequency scattering by a dielectric body," Radio Sci. 11 (5), 477-482.