Numerical Analysis of Resonances by a Slab of Subwavelength Slits by Fourier-Matching Method

Zhou, Jiaxin; Lu, Wangtao

doi:10.1137/21m1397532

Cited by 4 publications

(11 citation statements)

References 29 publications

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“…To prove I −A h has a bounded inverse for h 1, we no longer justify the diagonal dominance of the infinitedimensional matrix {δ m m − c m m } ∞ m ,m=0 as was done in [33], since now c m m is challenging to accurately approximate, although we conjecture that this property remains true 1 . To resolve this issue, we convert the study of the invertibility of I − A h to the study of the well-posedness of a closely related boundary-value problem in a semi-infinite cylinder Ω − = G 1 × R − , as was done in [5].…”

Section: Invertibility Of I − a Hmentioning

confidence: 96%

“…for some sufficiently small constant 0 > 0 and some sufficiently large constant K > 0. Similar as in [33], we consider even modes and odd modes symmetric about x 3 = −l/2.…”

Section: Single Cylindrical Holementioning

confidence: 99%

“…and {φ mn (x; h) = φ m (x 1 ; h)φ n (x 2 ; h)} ∞ m,n=0 form a complete and orthonormal basis in L 2 (Γ h ). In this case, following [33], we in fact can get by the method of Fourier transform the following identity,…”

Section: Two Examples and Quality Factormentioning

confidence: 99%

“…In this section, we study resonance frequencies when the slab contains N cylindrical holes {V i,h } N i=1 . As in [33], we begin with two holes to clarify the main idea.…”

Section: Multiple Cylindrical Holesmentioning

confidence: 99%

“…Nevertheless, much fewer theories have been developed so far to rigorously study resonances of 3D structures [18]. In a recent work [33], the authors developed a Fourier-matching method to study resonances by a slab of finite number of 2D slits. Unlike existing methods, the Fourier-matching method does not use the complicated Green function in a slit so that the overall theory becomes more straightforward.…”

Section: Introductionmentioning

confidence: 99%

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

Lu,

Wang,

Zhou

2021

Preprint

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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.

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Section: Invertibility Of I − a Hmentioning

confidence: 96%

“…for some sufficiently small constant 0 > 0 and some sufficiently large constant K > 0. Similar as in [33], we consider even modes and odd modes symmetric about x 3 = −l/2.…”

Section: Single Cylindrical Holementioning

confidence: 99%

Section: Two Examples and Quality Factormentioning

confidence: 99%

“…In this section, we study resonance frequencies when the slab contains N cylindrical holes {V i,h } N i=1 . As in [33], we begin with two holes to clarify the main idea.…”

Section: Multiple Cylindrical Holesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Lu,

Wang,

Zhou

2021

Preprint

Self Cite

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Mathematical Theory for Electromagnetic Scattering Resonances and Field Enhancement in a Subwavelength Annular Gap

Lin

Zhang

2023

Multiscale Model. Simul.

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A Fourier Matching Method for Analyzing Resonances in a Sound-Hard Slab with Subwavelength Holes

Lu,

null,

Zhou

2024

CSIAM-AM

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This paper presents a Fourier matching method to rigorously study resonances in a sound-hard slab with a finite number of narrow cylindrical holes. The cross sections of the holes, of diameters O(h) for h ≪ 1, can be arbitrarily shaped. Outside the slab, a sound field can be represented in terms of its normal derivatives on the apertures of the holes. Inside each hole, the field can be represented in terms of a countable Fourier basis due to the zero Neumann boundary condition on the side surface. The countably infinite Fourier coefficients for all the holes constitute the unknowns. Matching the two field representatives leads to a countable-dimensional linear system governing the unknowns. Due to the invertibility of a principal submatrix of the infinite-dimensional coefficient matrix, we reduce the linear system to a finitedimensional one. Resonances are those when the finite-dimensional linear system becomes singular. We derive asymptotic formulae for the resonances in the subwavelength structure for h ≪ 1. They reveal that a sound field with its real frequency close to a resonance frequency can be enhanced by a magnitude O(h −2 ). Numerical experiments are carried out to validate the proposed resonance formulae.

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Numerical Analysis of Resonances by a Slab of Subwavelength Slits by Fourier-Matching Method

Cited by 4 publications

References 29 publications

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

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

Mathematical Theory for Electromagnetic Scattering Resonances and Field Enhancement in a Subwavelength Annular Gap

A Fourier Matching Method for Analyzing Resonances in a Sound-Hard Slab with Subwavelength Holes

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