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Scattering by a Sphere with a Dielectric Half-Space or on Another Sphere



Egon Marx


Particle contamination of dielectric or conducting surfaces can be detected by shining light on the surface and looking for abnormal scattering distributions.  This procedure can be simulated by computing the scattering distribution for a dielectric sphere on a dielectric half-space for an incident plane monochromatic wave.  Similarly, the effect of light scattering by particle doublets in smoke can be simulated by electromagnetic scattering by two tangent spheres.  We have used such computations to determine the corresponding extinction coefficients [1].  We also consider the problem of spheres above a dielectric half-space, partially buried in the interface, or inside the half-space.  The latter represents, for instance, scattering by pigments in a coating.  These are essentially three-dimensional scattering problems that can be solved by using integral equations derived from Maxwell's equations [2, 3], although the rotational symmetry can be used to simplify the problem.  Solutions of the exact Maxwell equations are needed when the dimensions of the scatterer are comparable to the wavelength of light.We minimize the number of integral equations by an appropriate choice of auxiliary fields and unknown tangential surface fields defined on the boundaries between the regions.  We have a choice between singular and hypersingular integral equations.  Although there is a region where one sphere is in contact with the plane interface or to the other sphere that produces a sharp edge or a cusp, we have determined by numerical experiments that the increase of the unknown fields near that edge is not troublesome, especially in the determination of the far fields.  Hypersingular integral equations are best used when fields near the sharp edge are of interest [4], but computation of these integrals is complicated and we here restrict ourselves to singular integral equations.  Even these equations become hypersingular if a surface divergence term is integrated by parts, a complication that we avoid by performing a numerical integration instead.We define the homogeneous fields that have to be subtracted to define scattered fields, auxiliary fields, and unknown tangential surface fields for the different configurations and present the resulting sets of integral equations for the surface fields.  These equations are reduced to linear algebraic equations by dividing the surfaces into patches, which may involve complications at the poles of the spheres.  On the other hand, closed integrals over the azimuthal angles can increase the accuracy of the results.  Once the boundary fields are computed, the far fields are determined by integration and they allow us to determine extinction coefficients.  We limit the region of interest on the plane interface to a circle, and we use spherical coordinates for the sphere and polar coordinates in the plane.  We also allow the surface to be deformed in the area near the sphere, as well as roughness on the surface.  In practice, these three-dimensional problems are severely limited by the computing resources, as the sizes of the arrays increase rapidly with the number of patches.We show numerical results of the far-field angular distribution for a number of cases of interest, as well as extinction coefficients for doublets.  We also show the behavior of the tangential surface fields near the interface.REFERENCES1. E. Marx, 1998 Digest of the IEEE Antennas and Propagation Society International Symposium, pp. 2198-2201, 1998.2. E. Marx, J. Math. Physics, vol. 23, pp. 1057-1065, 1982.3. E. Marx, IEEE Trans. Antennas Propagat., vol. 32, pp. 166-172, 1984.4. E. Marx, IEEE Trans. Antennas Propagat., vol. 41, pp. 1001-1008, 1993.
Proceedings Title
PIERS Meeting
Conference Dates
September 29, 2003
Conference Location
Pisa, IT
Conference Title
PIERS 2004: Progress in Electromagnetics Research Symposium


electromagnetic scattering, scattering by a doublet of spheres, scattering by a sphere and a half-space, singular integral equations


Marx, E. (2004), Scattering by a Sphere with a Dielectric Half-Space or on Another Sphere, PIERS Meeting, Pisa, IT (Accessed May 30, 2023)
Created January 1, 2004, Updated June 2, 2021