Radiation scattered by two touching spheres

During the manufacture of powder metal the size distribution of the metal spheres can be determined to some extent by the distribution of light scattered by the spheres while they are streaming by a laser beam. Micrographs show the presence of chains of spheres and of spheres with a number of smaller spheres attached to them. To model the scattered light we assume that the spheres are sparse enough to scatter light independently, an assumption that is obviously invalid for agglomerates of spheres. It is thus of interest to compare the pattern of the radiation scattered by such agglomerates with the pattern produced by isolated spheres. We use an integral equation to compute the fields scattered by two perfectly conducting touching spheres when a plane monochromatic wave is incident from an arbitrary direction. The preliminary results indicate that a small sphere attached to a large sphere changes little the RCS and that there is a significant change for two equal spheres.<<ETX>>


INTRODUCTION
During the manufacture of powder metal the size distribution of the metal spheres can be determined to some extent by the distribution of light scattered by the spheres while they are streaming by a laser beam. Micrographs show the presence of chains of spheres and of spheres with a number of smaller spheres attached to them. To model the scattered light we assume that the spheres are sparse enough to scatter light independently, an assumption that is obviously invalid for agglomerates of spheres. It is thus of interest to compare the pattern of the radiation scattered by such agglomerates with the pattern produced by isolated spheres.
We use an integral equation to compute the fields scattered by two perfectly conducting touching spheres when a plane monochromatic wave is incident from an arbitrary direction. We have validated the program by symmetry considerations, for instance, for top incidence, and by its reduction to the single sphere, whose results can be compared directly to those obtained from the Mie formulas. Integral equations that solve the Maxwell equations exactly are especially useful in the resonance region, where the size of the spheres is comparable to the wavelength. The method is not practical for spheres much larger than the wavelength because memory limitations do not allow for a sufficient number of patches per wavelength. The use of approximate methods carries the risk of ignoring special features of the touching spheres, such as the region near the point of contact which resembles a sharp wedge.
We choose the z-axis through the z A centers of the spheres with the origin at the point of contact and the x-axis so that the propagation vector is in the xz-plane, as shown in Fig. 1. We then consider TM and TE polarizations, with Gin and zin along the y-axis, respectively. At each surface point we define a triad of orthonorm+ vectors formed by the uqit nozmal n and the tangent vectors q and t. The spheres have radii al and az. The incident wave is defined by the wave vector gi and its time dependence is exp(-iwt).
The magnitude of ci is k, which in terms of the wavelength X or the circular frequency w is k = 2n/X = o/c, Fig. 1. Geometry of the where c is the speed of light. scatterer.

INTEGRAL EQUATIONS
The surface current that is established on the surface of a perfect conductor satisfies the integral equation [l] where 3 = ; -z', R = 1x1, and F(R) is the derivative of the outgoing free-space Green function G(R) divided by R/2, that is, (2) The dash through the integral sign in (1) means that the self-patch contribution has been taken out. This is an integral equation of the second kind, which is generally better behaved than the integral equation of the first kind that can be derived from th+e continuity equation for the electric field. The surface current J, is tangential to she suTface and is best expressed in terms of the tangent vectors 'p and t by setting thereby reducing the+ storage requirements and building in the tangential nature of J,. The integral equation (1) takes the form
The integral equation ( 4 ) is reduced to a system of linear algebraic equations by the point-matching method. Once the components of the current density are obtaiped &y solving these equations, the far fields in the direction k = r are computed by integration from -* A where k = kr. The corresponding intensity is proportional to the scattering or radar cross section (RCS).

THE AVERAGING PROCEDURE
The fields are actually measured by using concentric diodes centered on the forward direction of the incident beam. We also assume that the agglomerated particles present themselves in all possible direction to the beam, whence we have to average over the possible directions of incidence and sum over the surface of the circular diodes. In the program, the discretized matrix 4 is computed and factorized only once, since it does not depend on the direction of incidence. When the radiation is coming in along the z-axis, the values of the outgoing polar angles are determined by the geometry of the detector and the values of the azimuthal angle are distributpd uniformly in the range of 2r.
Otherwise, the components of k are determined by rotating the vector by an angle B i about the y-axis.

PRELIMINARY RESULTS
We have used the method described above to compute the angular distribution of the far-field intensity for a few examples. We consider a plane wave of laser light of wavelength X = 0.6328 pm. In Fig. 2 we show a polar diagram of the angular distribution of scattered light or RCS of a single sphere, 1 pm in radius. In Figs. 3 and 4 we show the RCS of two touching 1-pm spheres for incidence along the z -and x-axis, respectively. The scales are not the same and the maximum intensities, I, are indicated in the figures. In Fig. 5 we show similar results for a 0.2-pm sphere attached to a 1pm sphere for incidence along the z-axis. The surface current density near the point where the spheres touch tends to be large, but increasing the number of strips on the sphere has little effect on the RCS. Here we divide the polar caps into spherical triangles, and the accuracy of the integration over 'p is helped by the fact that the integrand is periodic in (o [2].

CONCLUDING REMARKS
These preliminary results indicate that a small sphere attached to a large sphere changes little the RCS and that there is a significant change for two equal spheres. More precise conclusions will be drawn from computations of the light measured by the concentric diodes. Since at the frequencies of visible light metals are not perfect conduccors, we also need to address the problem of calculating the radiacion scattered by dielectric spheres with complex dielectric constants. Storage requirements can be minimized by using the method of the single integral equation