Inverse scattering of periodic surfaces with a superlens

Abstract

We propose a scheme for imaging periodic surfaces using a superlens. By employing an inverse scattering model and the transformed field expansion method, we derive an approximate reconstruction formula for the surface profile, assuming small amplitude. This formula suggests that unlimited resolution can be achieved for the linearized inverse problem with perfectly matched parameters. Our method requires only a single incident wave at a fixed frequency and can be efficiently implemented using fast Fourier transform. Through numerical experiments, we demonstrate that our method achieves resolution significantly surpassing the resolution limit for both smooth and non-smooth surface profiles with either perfect or marginally imperfect parameters.

Figures (6)

• Figure 1: Geometry of the model problem. The slab is positioned with its lower boundary $\Gamma_h$ and upper boundary $\Gamma_b$ above the scattering surface $\Gamma_f$, and the measurements are taken on $\Gamma_b$.
• Figure 2: Sketch of the computational domain for the finite element method. A PML is placed above $\Gamma_b$ to absorb the scattered field, allowing $u = 0$ to be used as the boundary condition on the outer boundary of the PML. Periodic boundary conditions are enforced on the left and right boundaries, while the usual PEC boundary condition $u = 0$ is imposed on $\Gamma_f$.
• Figure 3: Numerical experiments for the smooth profile function depicted in Eq. \ref{['eq:smoothProfile']}. The red solid line represents the true profile, while the blue dashed line illustrates the reconstructed profile. The relative permittivity and permeability of the slab are specified as follows: Row 1: $\varepsilon = 1.0$ and $\mu = 1.0$; Row 2: $\varepsilon = 16.0$ and $\mu = 1.0$; Row 3: $\varepsilon = -1.0$ and $\mu = -1.0$; Row 4: $\varepsilon = -1 + 0.05i$ and $\mu = -0.97$; Row 5: $\varepsilon = -1 + 0.1i$ and $\mu = -1.06$. The cut-off frequency is specified as follows: Column 1: $N = 1$; Column 2: $N = 3$; Column 3: $N = 10$.
• Figure 4: The scaling factor $|\Upsilon_n|$ in Eq. \ref{['eq:scaling']} is plotted on a logarithmic scale for various values of $\varepsilon$ and $\mu$ in the numerical experiments presented in Fig. \ref{['fig:smooth']}.
• Figure 5: Numerical experiments for the nonsmooth profile function \ref{['eq:nonsmooth']}. The red solid line represents the true profile, while the blue dashed line corresponds to the reconstructed profile. Row 1: $\varepsilon = 1.0$, $\mu = 1.0$, $N = 1, 2, 4$; Row 2: $\varepsilon = 16.0$, $\mu = 1.0$, $N = 1, 4, 5$; Row 3: $\varepsilon = -1.0$, $\mu = -1.0$, $N = 1, 4, 8$.

Theorems & Definitions (1)

• Remark 1