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

TL;DR

The paper develops a dynamical formulation of stationary scattering (DFSS) in two and three dimensions by expressing transfer matrices as time-ordered evolutions of effective non-Hermitian systems. A fundamental, operator-valued transfer matrix $\widehat{\mathbf{M}}$ in 2D (and its 3D analogue) admits a Dyson-series expansion, yielding explicit leading and next-to-leading terms in a low-frequency expansion of the scattering amplitude $\mathfrak{f}$ in powers of $k\ell$ (and $kL$ in 3D). The authors derive closed-form expressions for the expansion coefficients in terms of Fourier components of the permittivity perturbation, validate the approach against exactly solvable problems (and exact Born limits), and demonstrate a practical cloaking scheme based on thin bilayer coatings. Extending the method to 3D preserves the structure of the 2D results and offers analytic tools for designing and analyzing low-frequency scattering in realistic slab configurations, with potential applications in wave control and cloaking in optics and acoustics.

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, $e^{-iωt}ψ(\mathbf{r})$, with $ψ(\mathbf{r})$ satisfying the Helmholtz equation, $[\nabla^2+k^2\hat\varepsilon(\mathbf{r};k)]ψ(\mathbf{r})=0$, $ω$ and $k$ being respectively the angular frequency and wavenumber of the incident wave, and $\hat\varepsilon(\mathbf{r};k)$ 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.

Figures (8)

• Figure 1: Schematic views of an infinite planar slab containing an isotropic inhomogeneous material with possible regions of gain and loss.
• Figure 2: Schematic views of the scattering setup for the scattering of left- and right-incident waves (respectively on the left and right) by an infinite planar slab containing an isotropic inhomogeneous material with possible regions of gain and loss. The green dashed lines correspond to $x=\pm\infty$ where the detectors are located.
• Figure 3: Plots of the real and imaginary parts of the scattering amplitude $\mathfrak{f}(\pi/3)$ as a function of $k\ell$ for $\hat{\varepsilon}$ given by (\ref{['epsilon=']}) and (\ref{['ex1']}) with $\theta_0=4\pi/3$, $\mathfrak{z}=0.1$, $\alpha=500/{\rm mm}$, $\ell=1~\mu{\rm m}$, and $L=10~\mu{\rm m}$. The dotted and dashed curves respectively correspond to the first and second-order low-frequency approximations. The solid curve represents the exact expression which is valid for $k\ell\leq\alpha\ell=0.5$, i.e., in the region colored in yellow.
• Figure 4: Schematic view of a slab with a Gaussian permittivity modulation depicted in shades of blue and the corresponding low-frequency bilayer invisibility cloak. The pink and green regions represent the first and second layers with permittivities $1-\mathfrak{z}_0(k)$ and $1+0.4\mathfrak{z}_0(k)$, i.e., $\mathfrak{z}_1(k)=-\mathfrak{z}_0(k)$ and $\mathfrak{z}_2(k)=0.4\mathfrak{z}_0(k)$, respectively. The relative permittivity of the slab is given by \ref{['finite-slab']} and (\ref{['g=eq1']}) with $L=2\ell$. The distances are measured in units of $\ell$.
• Figure 5: Schematic view of the scattering setup with the source of the incidet wave located at $z=-\infty$. The blue planes represent the distant planes where the detectors are located. $\mathbf{r}$ marks the position of a detector screen that is placed at $z=+\infty$ and depicted as an orange ellipse.