Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility

Abstract

The propagation of the transverse electric (TE) and transverse magnetic (TM) waves in an effectively two-dimensional (2D) isotropic medium is described by Bergmann's equation of acoustics. We develop a dynamical formulation of the stationary scattering of these waves and explore its application in the study of the low-frequency behavior of the scattering data. Specifically, we introduce a suitable notion of fundamental transfer matrix for TE and TM waves in 2D. This is an integral operator $\widehat{\mathbf{M}}$ that carries the information about the scattering properties of the medium and admits a Dyson series expansion involving a non-Hermitian Hamiltonian operator. For situations where the inhomogeneities of the medium are confined to a layer of thickness $\ell$, we use the Dyson series for $\widehat{\mathbf{M}}$ to construct the series expansion of the scattering amplitude in powers of $k\ell$, where $k$ is the incident wavenumber. We derive analytic expressions for the leading- and next-to-leading-order terms of this series, verify the effectiveness of their application to a class of exactly solvable models, and use them to study low-frequency invisibility. In particular, we develop a low-frequency cloaking scheme which is applicable for both TE and TM waves. Our results have immediate applications in the study of low-frequency scattering of acoustic waves in a 2D fluid as these waves are also described by Bergmann's equation.

Figures (7)

• Figure 1: Schematic views of the scattering setup for the scattering of TE and TM waves by the inhomogeneities of a generic effectively two-dimensional isotropic medium that are confined to the region given by $0\leq x\leq\ell$. The coloring of this region in different shades of brown in a yellow background represents the inhomogeneities of the medium. The left and right panels correspond to the incident waves with incidence angle $\theta_0$ satisfying $\cos\theta_0>0$ and $\cos\theta_0<0$, respectively. Dashed green lines represent the planes given by $x=\pm\infty$ where the detectors are located.
• Figure 2: Schematic views of the scattering of a TM wave with wavenumber $k\in(\kappa_0,\mathfrak{K}]$ by a grating given by \ref{['ws-exact']}. A density plot of the modulus of the relative permittivity of the grating, i.e., $|\hat{\varepsilon}(x,y)|$, is used to color the region representing the grating. The arrows mark the incident and scattered wave vectors. $\theta_{0\pm}$ and $\theta_{1\pm}$ are the angles the scattered wave vectors make with the positive $x$ axis.
• Figure 3: Graph of $\theta_{1\pm}$ as a function of $k/\mathfrak{K}$ for $\mathfrak{z}_1$ and $\mathfrak{z}_2$ given by \ref{['specs']}. The two curves meet at $k/\mathfrak{K}=\kappa_0/\mathfrak{K}=0.510$.
• Figure 4: Graphs of the real and imaginary parts of $\tau_{0-}$ as a function of $k\ell$ for $\mathfrak{z}_1$ and $\mathfrak{z}_2$ given by \ref{['specs']}, $\ell=100~{\rm nm}$, and $\mathfrak{K}=\pi~\mu{\rm m}^{-1}$. The dashed and dotted curves respectively correspond to the outcome of the first- and second-order low-frequency approximations, while the solid curve corresponds to the exact calculation of $\tau_{0-}$.
• Figure 5: Graphs of $|\tau_{1\pm}|^2\times 10^4$ as functions of $k\ell$ for $\mathfrak{z}_1$ and $\mathfrak{z}_2$ given by \ref{['specs']}, $\ell=100~{\rm nm}$, and $\mathfrak{K}=\pi~\mu{\rm m}^{-1}$. The yellow region corresponds to $\kappa_0<k\leq\mathfrak{K}$. The dashed and dotted curves respectively correspond to the outcome of the first- and second-order low-frequency approximations, while the solid curve corresponds to the exact calculation of $\tau_{0-}$.