Frequency-dependent capacitance matrix formulation for Fabry-Pérot resonances. Part I: One-dimensional finite systems

Abstract

We study scattering resonances of finite one-dimensional systems of high-contrast resonators beyond the subwavelength regime. Introducing a novel tridiagonal frequency-dependent capacitance matrix, we derive quantitative asymptotic expansions of the hybridized Fabry-Pérot resonant frequencies in terms of the material contrast parameter. The leading-order shifts are governed by the eigenvalues of this matrix, while the corresponding eigenmodes are approximated, to leading order, by trigonometric functions on selected spacings between resonators. Our results extend the use of discrete approximations as a powerful tool for characterizing the resonant properties of a system of high-contrast resonators at arbitrarily high frequencies.

Figures (12)

• Figure 1: A chain of $N$ resonators, with lengths $(\ell_i)_{1\leq i\leq N}$ and spacings $(s_{i})_{1\leq i\leq N-1}$.
• Figure 2: Resonant intervals (red) and non‑resonant intervals (blue) for the example.
• Figure 3: Illustration of the correspondence between the integer intervals $\mathcal{I}_j = \llbracket a_j, b_j \rrbracket$ and the principal submatrices $C_j$ of $\mathcal{C}(k_0)$ defined in Section \ref{['sec: generalized capcitance \nmatrix']}. The large box represents $\mathcal{C}(k_0)$. Colored blocks indicate the positions of the submatrices $C_j$; adjacent gray squares ($1 \times 1$) denote possibly nonzero entries outside the blocks, while all remaining entries are zero. In the chain diagram above, blue segments are non-resonant and red segments are resonant.
• Figure 4: Comparison of theoretical asymptotic expansions with numerical results for the resonant frequencies. The parameters are $N=6$, $\bm{t} = (1, 2, 1, 0.75, 1.25, 1, 2, 2, 1, 1.5, 0.5)^{\top}$, $k_0 = \pi$, $r = 1$, and $v = 1$. The nonzero eigenvalues of $\mathcal{C}(k_0)$ are $\lambda = 1$ (with multiplicity two) and $\lambda = 0.25$ (simple). For branches (a)--(f), the refined expansion \ref{['equ: omega^i,pm \nexpand']} gives $\omega(\delta) = \pi + c_1 \delta^{1/2} + c_2 \delta + \mathcal{O}(\delta^{3/2})$, with coefficients $(c_1, c_2)$ indicated in each panel. Branch (g) corresponds to a resonant frequency satisfying $\omega(\delta) = \pi + \mathcal{O}(\delta)$ as in \ref{['eq:eigenfrequencydelta1']}.
• Figure 5: Log-log plot of the error between numerical resonant frequencies and their asymptotic expansions. The theoretical slopes of $\mathcal{O}(\delta^{3/2})$ (for branches (a)--(f) in Figure \ref{['fig:all']}) and $\mathcal{O}(\delta)$ (for branch (g) in Figure \ref{['fig:all']}) are confirmed by the linear fits, matching the predicted asymptotic orders.

Theorems & Definitions (29)

• Remark 2.1
• Example
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4: (Eigenmodes for a simple resonance)
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof