An Open Waveguide with a Thin High Contrast Core Layer: Asymptotic Analysis and Inverse Detection Problem

TL;DR

This work analyzes the Helmholtz equation in a 2D open waveguide containing a thin, high-contrast core layer and derives an explicit, thin-core asymptotic expansion of the Green function $G$, decomposing it into guided and continuous parts and further into symmetric and antisymmetric components. The analysis reveals resonant frequencies at which the core layer dramatically affects wave propagation, enabling an inverse problem: recovering the core's location, thickness, and index from multifrequency measurements on a bounded screen. The authors propose a three-step reconstruction algorithm that exploits resonance behavior and the asymptotic structure of $G$, and validate it numerically under noisy data, showing robust recovery of the first resonance frequency, core position, and mean index, with thickness estimable from resonance peak widths. The results provide a rigorous framework for detecting and characterising thin laminated structures in applications such as seismology and photonics, where high-contrast, ultra-thin layers play a critical role in waveguiding and scattering.

Abstract

We investigate the Helmholtz equation in a two dimensional open waveguide with a thin and high contrast core layer. We develop an asymptotic analysis of the Green function of the problem, and through it we identify and characterize the appearance of resonant frequencies. For waves originating outside of the core, the waveguide response at these resonant frequencies is vastly different than the response at non-resonant frequencies. Using this phenomenon and multifrequency measurements containing the first resonance, we propose, theoretically analyze, and numerically validate a reconstruction algorithm to identify the location, thickness and index of refraction of the core layer.

Figures (11)

• Figure 1: Non-resonant (first row) and resonant cases (second and third rows) for an open waveguide with a thin high contrast core layer. The waveguide parameters are $h=0.005$, $n_{cl}=1$, $n_h=\frac{\pi}{2h}$, $k_1^*=1$. We compare the modulus of the Green functions \ref{['eq:def_G']} (first column), their zero order approximations in $h$, described in Theorem \ref{['thm:total_G']} and written in terms of the Hankel function (second column), and the corresponding error (third column).
• Figure 1: Position of the point source, core layer and screen. The angle of rotation $\alpha$, the distance $x_0$ from the point source to the core (dashed line), the thickness $2h$ and the index of refraction $n_h$ of the core layer are unknowns. Measurements of the Green function are available on the screen $S$ for a range of frequencies $k$ (see Figure \ref{['fig:plotG']}).
• Figure 1: Successive estimations $\widehat{k}^{*(1)}_1$ (left) and $\widehat{k}^*_1$ (right) of the first radiating resonance $k^*_1$ from measurements with $5\%$ noise. It also shows the estimated thickness $\delta^*_1$ of the first peak, which is used in Step 3 strategy (ii) of the algorithm.
• Figure 1: Graphs of $y^2$ (blue line) and $y^2\sec^2(y)$ (green line). Given a height $L^2$ (red line) the red X's mark the roots of $y^2\sec^2(y)=L^2$ at which $\tan(y)> 0$, the blue * marks the root of $y^2=L^2$ and the black X marks $y=(2p-1)\pi/2$ for $p$ equal to the number of red roots. The last red root is in the interval $((2p-3)\pi/2,(2p-1)\pi/2)$ if and only if $(p-1)\pi< L \leq p\pi$, i.e. $p-1< L/\pi\leq p$ and $p=\lceil L/\pi\rceil$.
• Figure 2: Measurements on a screen $S$ for the geometry setting of Figure \ref{['fig:geometry-main']} for an open waveguide with parameters $h=0.005$, $n_{cl}=1$, $n_h=\frac{\pi}{2h}$, $k_1^*=1$. Comparison of $||G||_{L^2(S)}$ (red line), $||H||_{L^2(S)}$ (blue dotted line), and $||2H_a||_{L^2(S)}$ (blue line) in terms of frequency. Notice the change of behavior (narrow peaks) at the resonant frequencies $k_p^*$, $p=1,2,3,4,\ldots$.

Theorems & Definitions (23)

• Lemma 2.1
• Proof 1
• Remark 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Remark 3.5
• Lemma 4.1
• Proof 2