Various single source integral equations (SSIE) of electromagnetics (EM) have been introduced in the past for solution of the scattering problems on homogeneous and piece-wise homogeneous penetrable scatterers. Unlike traditional formulations of the surface integral equations (IE) for dielectric scatterers such as Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) and Muller IE, which feature two unknown surface vector functions, the SSIE formulations have only one such function. The Method of Moment (MoM) discretization of the SSIE, hence, leads to the matrix equations of half the size of the traditional formulations. This benefit of SSIEs comes at the cost of them being significantly more complicated and featuring many additional matrix-matrix products when MoMmatrix is solved directly and matrix-vector products when it is solved iteratively.
Previously known SSIEs have been derived from the traditional surface IEs by expressing electric current density in terms of the magnetic one or vice versa. In this work we discuss a new SSIE which is obtained from the volume IE (VIE) formulated for the homogeneous dielectric scatterer. The field inside the scatterer is represented as a superposition of the waves emanating from its surface and weighed with the unknown surface current density tangential to the surface. This unknown surface current density does not have an apparent relation to the electric and magnetic surface current densities and only has the meaning of the weighting function. As each of the waves emanating from the conductor surface satisfies the wave equation, so does the field represented as their superposition. Substitution of such field representation followed by enforcement of the VIE at the scatter surface and for the tangential component of the total electric field only reduces VIE to a SSIE with respect to the unknown surface current density. The new SSIE features the product of two integral operators one of which translates the surface current density to the volume current density and the other translating the volume current density to the tangental component of the scattered electric field on the surface. The new SSIE has been termed the Surface-Volume-Surface IE (SVS-IE) due to the nature of field translations in the formulation. The MoM solution of the proposed new SSIE has been previously demonstrated for the solution of scattering problems in two dimensions. Both the scalar case of TM waves scattering (A. Menshov, et.al., IEEE T-MTT, no. 1, vol. 61, pp. 341-350, 2013) and vector case of TE waves scattering were previously reported. In this work we discuss generalization of the proposed equation to three dimensions (3D). Two alternative MoM formulations for the discretization of the SVS-IE in 3D are discussed.