A Fourier-matching Method for Analyzing Resonance Frequencies by a Sound-hard Slab with Arbitrarily Shaped Subwavelength Holes

Abstract

This paper presents a simple Fourier-matching method to rigorously study resonance frequencies of a sound-hard slab with a finite number of arbitrarily shaped cylindrical holes of diameter O(h) for h 1. Outside the holes, a sound field can be expressed in terms of its normal derivatives on the apertures of holes. Inside each hole, since the vertical variable can be separated, the field can be expressed in terms of a countable set of Fourier basis functions. Matching the field on each aperture yields a linear system of countable equations in terms of a countable set of unknown Fourier coefficients. The linear system can be reduced to a finite-dimensional linear system based on the invertibility of its principal submatrix, which is proved by the well-posedness of a closely related boundary value problem for each hole in the limiting case h → 0, so that only the leading Fourier coefficient of each hole is preserved in the finitedimensional system. The resonance frequencies are those making the resulting finite-dimensional linear system rank deficient. By regular asymptotic analysis for h 1, we get a systematic asymptotic formula for characterizing the resonance frequencies by the 3D subwavelength structure. The formula reveals an important fact that when all holes are of the same shape, the Q-factor for any resonance frequency asymptotically behaves as O(h −2 ) for h 1 with its prefactor independent of shapes of holes.

Mathematical theory for electromagnetic scattering resonances and field enhancement in a subwavelength annular gap

“…We first develop a vectorial analogy of the mode matching method originally proposed in [41,34] to reformulate the scattering problem (E) with trivial incident field. Before proceeding, we introduce the following bilinear form over H −1/2 (Div, A h ) × H −1/2 (Div, A h ) (see [25, P. 306]):…”

“…The matched asymptotic expansion techniques have also been applied to construct the solution of the slit scattering problem in [22,23,24]. The generalization of the above techniques to the studies of the acoustic wave resonances in three-dimensional subwavelength holes can be found in [16,28,34]. We would also like to refer readers to [1,2,3,4] and references therein for the mathematical studies of other type of subwavelength resonances, such as Helmholtz resonators and nanoparticles, etc.…”