Dynamical formulation of low-frequency scattering in two and three dimensions

Abstract

The transfer matrix of scattering theory in one dimension can be expressed in terms of the time-evolution operator for an effective non-unitary quantum system. In particular, it admits a Dyson series expansion which turns out to facilitate the construction of the low-frequency series expansion of the scattering data. In two and three dimensions, there is a similar formulation of stationary scattering where the scattering properties of the scatterer are extracted from the evolution operator for a corresponding effective quantum system. We explore the utility of this approach to scattering theory in the study of the scattering of low-frequency time-harmonic scalar waves, $e^{-i\omega t}\psi (\mathbf{r})$, with $\psi (\mathbf{r})$ satisfying the Helmholtz equation, $[{\nabla }^2+k^2\hat{\epsilon }(\mathbf{r};k)]\psi (\mathbf{r})=0$, $\omega$ and $k$ being respectively the angular frequency and wavenumber of the incident wave, and $\hat{\epsilon }(\mathbf{r};k)$ denoting the relative permittivity of the carrier medium which in general takes complex values. We obtain explicit formulas for low-frequency scattering amplitude, examine their effectiveness in the study of a class of exactly solvable scattering problems, and outline their application in devising a low-frequency cloaking scheme.

28 pages, 8 figures, slightly expanded version, accepted for publication in Phys. Rev. A

References in corpus (20)

• PT-symmetric laser-absorber
• Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at real Energies
• The transfer matrix: a geometrical perspective
• Optical Spectral Singularities as Threshold Resonances
• Transfer Matrices as Non-Unitary S-Matrices, Multimode Unidirectional Invisibility, and Perturbative Inverse Scattering
• A Dynamical Formulation of One-Dimensional Scattering Theory and Its Applications in Optics
• Optical Spectral Singularities and Coherent Perfect Absorption in a Two-Layer Spherical Medium
• Spectral Singularities and Whispering Gallery Modes of a Cylindrical Gain Medium
• Spectral Singularities in the Surface Modes of a Spherical Gain Medium
• Low-energy potential scattering in two and three dimensions
• Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
• Universality of low-energy scattering in (2+1) dimensions
• Perfect Broadband Invisibility in Isotropic Media with Gain and Loss
• Approximate T-matrix and optical properties of spheroidal particles to third order in size parameter
• Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti-PT-symmetry
• Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions
• Low-frequency scattering defined by the Helmholtz equation in one dimension
• Dynamical formulation of low-energy scattering in one dimension
• Exactness of the first Born approximation in electromagnetic scattering
• Can N-th Order Born Approximation Be Exact?

Cited by in corpus (2)

• Low-Frequency Scattering of TE and TM Waves by an Inhomogeneous Medium with Planar Symmetry
• Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility