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On the lack of external response of a nonlinear medium in the second-harmonic generation process

Fioralba Cakoni, Narek Hovsepyan, Matti Lassas, Michael S. Vogelius

TL;DR

The work analyzes second-harmonic generation in a bounded nonlinear medium by formulating a coupled semilinear elliptic system for the $\omega$ and $2\omega$ fields and examining the possibility of nonlinear non-scattering. It introduces generalized transmission eigenvalues and studies their existence/nonexistence using Lyapunov–Schmidt projection, Fredholm properties, and radial/1-D reductions, under real-valued, frequency-independent coefficients and small $|\omega|$. The authors establish a nonexistence regime for small frequencies, derive discrete eigenvalue structures in degenerate cases, and prove existence results for forced problems via a two-step approach that first solves a linearized interior problem and then applies a Banach fixed-point argument, with careful handling of $\omega$-dependent invertibility. Together, these results provide a rigorous mathematical framework for nonlinear SHG scattering and the associated transmission-like eigenvalues, clarifying when nonlinear interactions can be invisible to external observers and outlining robust strategies to address forced or perturbed scenarios.

Abstract

This paper concerns the scattering problem for a nonlinear medium of compact support, $D$, with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency $ω$, generates additional waves at frequency $2ω$. The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequency $ω$ and $2ω$. We investigate whether there are situations in which the generated $2ω$ wave is localized inside $D$, that is, the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in $D$ with non-standard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.

On the lack of external response of a nonlinear medium in the second-harmonic generation process

TL;DR

The work analyzes second-harmonic generation in a bounded nonlinear medium by formulating a coupled semilinear elliptic system for the and fields and examining the possibility of nonlinear non-scattering. It introduces generalized transmission eigenvalues and studies their existence/nonexistence using Lyapunov–Schmidt projection, Fredholm properties, and radial/1-D reductions, under real-valued, frequency-independent coefficients and small . The authors establish a nonexistence regime for small frequencies, derive discrete eigenvalue structures in degenerate cases, and prove existence results for forced problems via a two-step approach that first solves a linearized interior problem and then applies a Banach fixed-point argument, with careful handling of -dependent invertibility. Together, these results provide a rigorous mathematical framework for nonlinear SHG scattering and the associated transmission-like eigenvalues, clarifying when nonlinear interactions can be invisible to external observers and outlining robust strategies to address forced or perturbed scenarios.

Abstract

This paper concerns the scattering problem for a nonlinear medium of compact support, , with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency , generates additional waves at frequency . The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequency and . We investigate whether there are situations in which the generated wave is localized inside , that is, the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in with non-standard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.
Paper Structure (20 sections, 1 theorem, 240 equations, 3 figures)

This paper contains 20 sections, 1 theorem, 240 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formulation of the Problem and our Main Results
  3. About generalized transmission eigenvalues in the 1-D case
  4. About generalized transmission eigenvalues in the 3-D radially symmetric case
  5. Solvability result in the presence of a force term
  6. Generalized transmission eigenvalues in the degenerate case $\chi_1=0$
  7. Proof of Lemma \ref{['LEM real case']}
  8. The Fredholm property of linearization
  9. Proof of Theorem \ref{['THM 1']}
  10. Lyapunov's projection equation
  11. The main idea of the proof and some auxiliary lemmas
  12. Concluding the proof of Theorem \ref{['THM 1']}
  13. Existence in the presence of a force term
  14. The main idea and the outline of the proof
  15. The equation $F_\omega(p) = f$ and invertibility of $F'_{\omega, p}$
  16. ...and 5 more sections

Key Result

Corollary 2.6

Assume M and chi2 and q assumption hold. For any $r>0$ there exists $\delta = \delta(r, Q, M,\chi_2)$, such that if $0<|\omega|<\delta$ and $0 < \|\bm{u}\|_X < r$ then

Figures (3)

  • Figure 1: The scattering of a monochromatic time-harmonic incident wave by a second-harmonic generation nonlinear optical medium of bounded support $D$.
  • Figure 2: The plot of $\omega^{10} \zeta(\omega)$ for $q=\chi_2 =1$ and $f=x$.
  • Figure 3:

Theorems & Definitions (25)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.5
  • Corollary 2.6
  • Conjecture 2.7
  • Remark 2.11
  • Remark 3.1
  • ...and 15 more