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Generating customized field concentration via virtual surface transmission resonance

Yueguang Hu, Hongyu Liu, Xianchao Wang, Deyue Zhang

Abstract

In this paper, we develop a mathematical framework for generating strong customized field concentration locally around the inhomogeneous medium inclusion via surface transmission resonance. The purpose of this paper is twofold. Firstly, we show that for a given inclusion embedded in an otherwise uniformly homogeneous background space, we can design an incident field to generate strong localized field concentration at any specified places around the inclusion. The aforementioned customized field concentration is crucially reliant on the peculiar spectral and geometric patterns of certain transmission eigenfunctions. Secondly, we prove the existence of a sequence of transmission eigenfunctions for a specific wavenumber and they exhibit distinct surface resonant behaviors, accompanying strong surface-localization and surface-oscillation properties. These eigenfunctions as the surface transmission resonant modes fulfill the requirement for generating the field concentration.

Generating customized field concentration via virtual surface transmission resonance

Abstract

In this paper, we develop a mathematical framework for generating strong customized field concentration locally around the inhomogeneous medium inclusion via surface transmission resonance. The purpose of this paper is twofold. Firstly, we show that for a given inclusion embedded in an otherwise uniformly homogeneous background space, we can design an incident field to generate strong localized field concentration at any specified places around the inclusion. The aforementioned customized field concentration is crucially reliant on the peculiar spectral and geometric patterns of certain transmission eigenfunctions. Secondly, we prove the existence of a sequence of transmission eigenfunctions for a specific wavenumber and they exhibit distinct surface resonant behaviors, accompanying strong surface-localization and surface-oscillation properties. These eigenfunctions as the surface transmission resonant modes fulfill the requirement for generating the field concentration.
Paper Structure (14 sections, 9 theorems, 140 equations, 4 figures)

This paper contains 14 sections, 9 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Background discussion and scientific novelties
  4. Surface transmission resonance
  5. Transmission eigenvalue problem and its spectral patterns
  6. The geometric patterns of transmission eigenfunctions
  7. Artificial generation of customized field concentrations
  8. Herglotz approximation
  9. The nearly vanishing property of scattered field
  10. Proof of Theorem \ref{['11']}
  11. Numerical experiments
  12. surface-oscillation of transmission eigenfunctions
  13. Auxiliary results
  14. proof of Theorem \ref{['oscillating']}

Key Result

Theorem 1.1

Consider the scattering problem sys1 with a $C^{1,1}$ inclusion $\Omega$ in ${\mathbb R}^N,N=2,3$. We denote by $\Gamma \subset \partial \Omega$ any subset of the boundary $\partial \Omega$. Then the gradient of the total field $\nabla u$ can blow up in the following sense: for any given large numbe where $M$ is only dependent of the incident field $u^i$ and $\Gamma_e(\Omega,\epsilon)= \{x |\; \ma

Figures (4)

  • Figure 1: Schematic illustration of field concentration generator $B$ near $\Omega$.
  • Figure 2: The total field $u$ and the magnitude of gradient $\nabla u$ of an ellipse domain.
  • Figure 3: The total field $u$ and the magnitude of gradient $\nabla u$ of a square domain.
  • Figure 4: The total field $u$ and the magnitude of gradient $\nabla u$ of a kite domain.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 9 more