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Bifurcation and Asymptotics of Cubically Nonlinear Transverse Magnetic Surface Plasmon Polaritons

Tomáš Dohnal, Runan He

Abstract

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the $H^1$-Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. Because of the TM-polarization the problem cannot be reduced to a scalar equation. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

Bifurcation and Asymptotics of Cubically Nonlinear Transverse Magnetic Surface Plasmon Polaritons

Abstract

Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the -Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. Because of the TM-polarization the problem cannot be reduced to a scalar equation. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of PT-symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.
Paper Structure (16 sections, 13 theorems, 200 equations, 6 figures)

This paper contains 16 sections, 13 theorems, 200 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Linear spectral problem
  4. Homogeneous layers
  5. Two homogenous layers
  6. Three homogeneous layers
  7. Bifurcation of nonlinear surface plasmons
  8. Proof of Theorem \ref{['mainthm_loc_bifurc']}
  9. Proof of Theorem \ref{['T:PT_sym']} (bifurcation under the $\mathcal{P}\mathcal{T}$-symmetry)
  10. Numerical Results
  11. Numerical Finite Difference Method
  12. Bifurcation Examples
  13. $\mathcal{P}\mathcal{T}$-symmetric example
  14. Non-$\mathcal{P}\mathcal{T}$-symmetric example
  15. Solvability of $L_k(\omega_0)u=r$ with $\omega_0\in \sigma_p(L_k)$ and $r\in Q_0 L^2(\mathbb R)$ for the case of two homogenous layers using the variation of parameters
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k \in\mathbb R$. Assume (A-E), (A-V), (A-T), and (A-Na). Let $\varphi_0 \in D(A)$ be an eigenfunction corresponding to $\omega_0$ normalized to $\|\varphi_0\|=1$ and $\varphi_0^*$ the eigenfunction of the adjoint $L_k^*$ with the eigenvalue $\overline{\omega_0}$ normalized to $\langle\varphi_0, where $\phi$ is a unique solution in $D(A)\cap {\mathcal{H}}^1\cap \langle\varphi_0^*\rangle^\perp

Figures (6)

  • Figure 1: Functions $d_j(\omega)$ with $j=-1,0,1$ from \ref{['cond_d_cD|M|cD']}.
  • Figure 2: Quantities $\alpha(\omega)$ and $|d_1(\omega)-\beta(\omega)|$ with $\alpha$ and $\beta$ from Propositions \ref{['3layers_T:pt-spec-1D']} and \ref{['3layers_T:disc_sp_1D']} respectively.
  • Figure 3: (a) The graph of the $\mathcal{P}\mathcal{T}$-symmetric $V(\cdot,\omega_0)$ given by \ref{['E:Vpm-ex']} with $\omega=\omega_0 \approx 1.7914$. (b) The corresponding bifurcation diagram for the bifurcation from the eigenvalue $\omega_0$. (c) Convergence of the approximation error $|\omega-\omega_0-\varepsilon \nu|$. For comparison a curve with a quadratic convergence is plotted.
  • Figure 4: The nonlinear solution $u$ at $\omega \approx 1.7167$. Recall that $E_1=u_{1}, E_2=u_{2}, H_3=u_{3}$. The linear eigenfunction $\varphi_0$ (normalized to have a similar amplitude to that of $u$) is plotted for comparison.
  • Figure 5: (a) The graph of the $V(\cdot,\omega_0)$ given by \ref{['E:Vpm-ex-2']} with $\omega=\omega_0 \approx 0.4679-{\rm i}~0.061$. (b) The corresponding bifurcation diagram for the bifurcation from the eigenvalue $\omega_0$. (c) Convergence of the approximation error $|\omega-\omega_0-\varepsilon \nu|$. For comparison a curve with a quadratic convergence is plotted.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Remark 1
  • Remark 2
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 25 more