Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium

We present a general derivation of the frequency-domain volume integral equation (VIE) for the electric field inside a nonmagnetic scattering object from the differential Maxwell equations, transmission boundary conditions, radiation condition at infinity, and locally-finite-energy condition. The derivation applies to an arbitrary spatially finite group of particles made of isotropic materials and embedded in a passive host medium, including those with edges, corners, and intersecting internal interfaces. This is a substantially more general type of scatterer than in all previous derivations. We explicitly treat the strong singularity of the integral kernel, but keep the entire discussion accessible to the applied scattering community. We also consider the known results on the existence and uniqueness of VIE solution and conjecture a general sufficient condition for that. Finally, we discuss an alternative way of deriving the VIE for an arbitrary object by means of a continuous transformation of the everywhere smooth refractive-index function into a discontinuous one. Overall, the paper examines and pushes forward the state-of-the-art understanding of various analytical aspects of the VIE.

Figure 1

A simple single-body scatterer with a smooth boundary and without internal interfaces.

Figure 2

An example of a complex multibody multilayered scatterer with smooth interfaces. Each closed surface separates exactly two domains, one of which may be the external medium.

Figure 3

An exclusion domain for deriving the boundary condition for $\mathbf{E}\left(\mathbf{r}\right)$ consisting of two oblate rectangular prisms. The width perpendicular to the image (not shown) also equals $d$. Gaps between the prisms and the locally flat interface are shown solely for convenience. The dashed central line denotes the variation of $\mathbf{r}$ used to calculate the limit when approaching the interface.

Figure 4

Conjectured sufficient conditions for the existence and uniqueness of the solution of the scattering problem in a passive host medium, described as blank areas ($Z_2$ and $Z$) in the complex plane for (a) ${\varepsilon }_2$ and (b) $\varepsilon$, respectively. The shaded areas contain different kinds of resonances (see text). The dashed lines in (a) and (b) extend from the origin through the values of ${\varepsilon }_1$ and its complex conjugate, respectively.

Figure 5

An example of a multibody multilayered scatterer with piecewise-smooth boundaries and interfaces (having a finite number of edges and vertices). Each $S_i$ is a closed connected surface, but not necessarily a regular one; it separates at least two domains, one of which may be the external medium.

Figure 6

An example of the deformation of the integration surface to circumvent the scatterer interfaces (shown by dashed curves), from (a) to (b). The surface integral does not change for any integrand that is continuous across the interfaces, while the volume integral is the same for any integrable function.

Figure 7

An example of the exclusion of shape singularities from the interfaces and volume domains. See the main text for an explanation of the symbols; the dashed lines denote the parts of the original irregular surfaces $S_i$ (cf. Fig. 5) falling inside the neighborhood of singularities. Each $S_j^{\mathrm{r}}$ is a regular connected surface separating exactly two domains, one of which may be the external medium.