Energy budget and optical theorem for scattering of source-induced fields

We provide rigorous definitions of various components of the energy budget for scattering of source-induced electromagnetic fields by a finite nonmagnetic object. We use the classical volume-integral-equation (VIE) framework and define power rates in terms of integrals of the Poynting vector over various surfaces, enclosing some or all of the impressed sources, scatterer, and environment (such as a planar multilayered substrate). Thus, we generalize the conventional cross sections and obtain new interrelations analogous to the well-known optical theorem. We rigorously treat the strong singularity of the VIE kernel, but keep derivations accessible to a wide audience. The defined power rates are further related to the decay rate enhancement and apparent quantum yield of an arbitrary emitter, which are the core concepts in nanophotonics, surface-enhanced Raman scattering, and electron energy-loss spectroscopy. We also discuss the practical calculation of the power rates and decay rate enhancements in the framework of the discrete dipole approximation (DDA). In particular, we derive the volume-integral expression for the scattered power and use it to prove the automatic satisfaction of the optical theorem irrespective of the discretization level. Thus, the optical theorem cannot be used as an internal measure of the DDA accuracy.

Figure 1

Definitions of geometry and energy flows, when the sources are located near the scattering object. All symbols are defined in the text.

Figure 2

Illustration of the distant source ($r_{\mathrm{s}}\to \infty )$; $a$ is a characteristic scale (maximal dimension) of both the scattering object and the sources. $A_1-A_3$ are the same as in Fig. 1. $A_3^{\prime }$ is a part of $A_3$, which is a far-field limit of $A_1$. Blue arrows indicate that both incident and scattered waves are in the far zone near the object and the sources, respectively.

Figure 3

(a) Division of the integration sphere into the upper and lower halves allows one to avoid field discontinuities at the interface. (b) Extra definitions of integration surfaces for the partly immersed scatterer. (c) A multilayered substrate may contain a planar waveguide, leading to an additional channel of energy transfer. See the text for explanation.