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Numerical analysis of resonances by a slab of subwavelength slits by Fourier-matching method. (English) Zbl 1496.78020

Summary: This paper proposes a simple and rigorous Fourier-matching method to study transverse-magnetic-polarized electro-magnetic resonances by a perfectly conducting slab with a finite number of subwavelength slits of width \(h\ll 1\). Since variable separation is applicable in the region outside the slits, by Fourier transforming its governing equation, a magnetic field can be represented in terms of its derivative on the aperture. Next, inside each slit where variable separation is still available, the field can be represented as a Fourier series in terms of a countable set of basis functions with unknown Fourier coefficients. Finally, by matching the two subdomain representations on the aperture, we establish a linear system of an infinite number of equations governing the countable Fourier coefficients; the unknowns are further rescaled to be in the standard \(\ell^2\) space. By the asymptotic expansion of each entry of the coefficient matrix, we rigorously show that its certain principal submatrix is invertible so that the infinite-dimensional linear system can be reduced to a finite-dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn leads to an asymptotic formula of accuracy \(\mathcal{O}(h^3\log h)\) for computing the resonance frequencies. We emphasize that the new formula is more accurate than all existing results and is the first formula for slits of number more than two to the best of our knowledge. Numerical experiments are carried out finally to validate the proposed formula and demonstrate its accuracy.

MSC:

78M99 Basic methods for problems in optics and electromagnetic theory
78A45 Diffraction, scattering
78A40 Waves and radiation in optics and electromagnetic theory
65T50 Numerical methods for discrete and fast Fourier transforms
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B34 Resonance in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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