A new single surface integral equation for light scattering by circular dielectric cylinders

A new single surface integral equation is derived for light scattering by circular dielectric cylinders. Without adopting the concept of equivalent electric or magnetic surface currents, our formulation is directly derived from coupled-surface integral equations by the property of commutative matrices of Green functions. Further development by such matrix equations leads to only one unknown function for circular dielectric-coated cylinders. In addition, numerical simulations show that even applied to elliptic scatters our equation still gives reasonably good approximate solutions in the sub-wavelength limit. 2007 Elsevier B.V. All rights reserved. For scattering problems in electrodynamics, integral approach, in contrast to differential methods, possesses the analytical characteristic of solution of Helmholtz equation with point source, namely, an integration of Green function. With the use of Green identity, the scattering field is obtained by the integral of the total field and its normal derivative on the enclosed surface of the object. In scatters of perfect conductor, the integrand reduces to only one variable, the normal derivative of field, since the field is zero on perfect conductor. So the integral becomes with one unknown of Neumann boundary condition. In dielectric homogeneous object, because none of the two variables vanishes; one single surface Green integral indeed cannot be solved with two boundary conditions. For finding these two unknown functions on the boundary, dual surface integral equations are indispensable when such integrals are numerically expressed in two sets of linear algebra equations [1,2]. For three-dimensional (3D) scattering of arbitrary-shaped body, the coupled vector integral equations require one to solve a set of unknown equivalent electric and magnetic surface currents [3–5]. The matching of 0030-4018/$ see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.030 * Corresponding author. E-mail address: [email protected] (C.-C. Tsai). boundary conditions between two media in scattering is accomplished by derivation of the fields from electric and magnetic potential via the corresponding equivalent currents. In two dimensional (2D) scattering, due to the advantage of decomposition of field into TE and TM components, the respective Helmholtz equation can be treated as scalar scattering problems and discussed separately; and the equivalent surface current is essentially equivalent to the normal derivative of field. The studies of 2D electromagnetic scattering by integral equation method are extensively found in literature [6–10]. Nevertheless, the double-loaded coupled integral equations seem unpleasant in attempting the solution by numerical scheme. The first effort to reduce these two integral equations into one in 2D problems was proposed by Maystre [9,10]. With the derivation from distribution theory, Maystre successfully expressed the boundary field and its normal derivative in terms of a single surface current function. Once the equivalent current is found, the fields on the boundary can be obtained through a conversion integral by substitution of the current function. Later, the same idea was applied and generalized to 3D scattering problems [11–13]. With all in common, the substitution of the real field with an equivalent surface current function is the core constituent in these formulations. 248 C.-C. Tsai, S.-T. Wu / Optics Communications 277 (2007) 247–250 In this communication, we derive a new single integral equation without the conception of equivalent principle for the cases of circular scattering in 2D scattering. That is, none of the surface current is necessary in our formulation. Our approach is based on the use of commutative matrices to simplify two coupled linear algebra equations into one. With the same technique applied to circular dielectric-coated cylinders, we successfully reduce a set of four integral equations into one single linear algebra equation. All these works with consistent numerical results are addressed as follows. Let us consider the scattering from a circular dielectric cylinder as shown in Fig. 1. In spite of the existence of the analytical solution [4], the coupled-surface integral equations for TE polarized wave are,