Inverse Design of a Layered Medium for Maximal Surface Localization

TL;DR

The paper presents an inverse-design framework for maximizing surface localization of a TM mode at the interface between a periodic layered half-space and a homogeneous half-space. It combines the transfer-matrix method with eigenvalue analysis to characterize localized modes and formulates a constrained optimization problem solved by Particle Swarm Optimization over the design variables $c_A$, $c_B$, and $\theta$. The authors demonstrate that carefully chosen layer speeds and thickness fractions yield exponentially decaying modes with small $|\lambda_1|$, exemplified by an optimal design achieving $|\lambda_1|=0.1742$. This approach provides a tractable route to engineer photonic interfaces with enhanced confinement and suggests extensions to more complex 2D structures and plasmonic systems. The work highlights a clear pathway from physical modeling to computational design for maximal surface localization in layered media.

Abstract

Electromagnetic wave manipulation plays a crucial role in advancing technology across various domains, including photonic device design. This study presents an inverse design approach for a periodic medium that optimizes electromagnetic wave localization at the interface between a layered half-space and a homogeneous half-space. The approach finds a maximally localized mode at a specified frequency and wave number. The mode propagates in the direction of the interface. The design parameters are the permittivity of the layered medium, their relative thicknesses, and the permittivity of the homogeneous half-space. We analyze the problem using the transfer matrix method and apply the particle swarm optimization to find a rapidly decaying mode that satisfies the design constraints. The design process is demonstrated in a numerical example, which serves to illustrate the efficacy of the proposed method.

Figures (6)

• Figure 1: For $x>0$, the medium is a periodically layered medium where each period consists of layers of materials A and B, the period is $p$. For $x<0$, the medium is homogeneous.
• Figure 2: This figure illustrates the combination of the first period $p = 1$ of the medium. The relative thickness of material A is $\theta$ and the relative thickness of material B is $1-\theta$.
• Figure 3: We fix the medium parameters $c_A = 2$, $c_B = 1$, and $\theta = 0.6$. (a) This subfigure illustrates the wave localization behavior across values of $(\eta, \omega)$. The orange and white regions indicate regions in which waves decay in $x>0$, with the white specifically denoting areas where the waves also decay in the negative direction, thereby localizing at $x=0$. A green circle highlights the specific parameters $\omega = 6.18$ and $\eta = 2$, under which the conditions for wave localization are met for the chosen medium combination. (b) This subfigure visualizes the wave amplitude $u(x)\sin \eta y$ corresponding to the parameters marked by the green circle in subfigure (a). In the right proportion of this subfigure where $x>0$, we use solid and dashed lines to exhibit the medium structure. Solid lines mark the boundaries of periodic cells, while dashed lines differentiate materials, indicating their respective volume fractions ($\theta = 0.6$ in this example). This subfigure effectively shows the wave localization at $x=0$.
• Figure 4: This figure illustrates the objective function values within a three-dimensional solution space, where the optimal solution is highlighted by a red star. The red star is located in the deepest color region, visually affirming its status as the most optimal point across the entire solution space. This 3D representation effectively underscores the thoroughness of the optimization analysis.
• Figure 5: (a) This subfigure displays the chosen wave parameters $\omega$ and $\eta$ for the optimized medium composition of $c_A = 2.15$, $c_B = 0.50$, and $\theta = 0.87$, denoted by a green circle in the white zone, indicating wave localization at $x=0$. (b) This figure visualizes wave propagation within this optimized medium configuration, facilitating an easy comparison with previous images and showing accelerated decay. The solid lines and dashed lines also exhibit the structure of the optimized medium, with dashed lines marking materials with optimized volume fraction $\theta = 0.87$.