Indexed singular value bounds on scattering operators: How many channels can a photonic device support?

Abstract

Spectral properties of scattering operators, and their dependence on geometry, are of crucial importance to photonic design, enabling low-rank approximations and improved understanding of achievable power and information transfer. Here, we develop a method to bound indexed singular values (channel amplitudes) of the Green operator, $W$-operator, and proposed $P$-operator, for arbitrarily structured linear media. The approach yields computable upper bounds on the $n^{th}$ singular value, for any given $n$, that capture the complexity of multi-channel tradeoffs and competing scattering effects. As illustrations of the framework, channel bounds are provided for multi-wavelength three-dimensional ``mediating'' volumes (up to $64\,\lambda^3$, mimicking communication waveguide-like and metasurface-like configurations), power transfer between $9\,\lambda^3$ source and receiver volumes, and applied to elucidate the performance of a planewave angle discrimination problem (bounding the smallest singular value, or condition number, of a fixed input space). In addition to these exemplary uses, the approach is directly applicable to bounds on information theoretic objectives such as Shannon capacity and Fisher information, as well as computational guarantees, such as error limits for reduced-order models.