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Spectral topology and edge modes for one-dimensional non-Hermitian photonic crystals

Junshan Lin, Hai Zhang

Abstract

This work investigates edge modes in non-Hermitian photonic crystals with broken spectral reciprocity. In such systems, the spectra of the underlying operators generally form closed loops over the complex plane with nontrivial spectral topology, which gives rise to the so-called skin effect characterized by edge modes localized at interfaces. For discrete lattice models, the skin effect can be understood through the spectral theory of Toeplitz matrices. However, this mathematical framework no longer applies to continuous wave models, where finite-dimensional approximations break down. In this work, we employ a transfer matrix approach to describe wave propagation in one-dimensional periodic media and introduce a new spectral topological invariant based on the eigenvalues of the transfer matrix. The new topological invariant is equivalent to the winding number of the non-Hermitian spectrum and it enables the characterization of edge modes in one-dimensional non-Hermitian photonic crystals. The mathematical theory provides the theoretical foundation for the skin effect in continuous wave models.

Spectral topology and edge modes for one-dimensional non-Hermitian photonic crystals

Abstract

This work investigates edge modes in non-Hermitian photonic crystals with broken spectral reciprocity. In such systems, the spectra of the underlying operators generally form closed loops over the complex plane with nontrivial spectral topology, which gives rise to the so-called skin effect characterized by edge modes localized at interfaces. For discrete lattice models, the skin effect can be understood through the spectral theory of Toeplitz matrices. However, this mathematical framework no longer applies to continuous wave models, where finite-dimensional approximations break down. In this work, we employ a transfer matrix approach to describe wave propagation in one-dimensional periodic media and introduce a new spectral topological invariant based on the eigenvalues of the transfer matrix. The new topological invariant is equivalent to the winding number of the non-Hermitian spectrum and it enables the characterization of edge modes in one-dimensional non-Hermitian photonic crystals. The mathematical theory provides the theoretical foundation for the skin effect in continuous wave models.
Paper Structure (18 sections, 12 theorems, 81 equations, 5 figures)

This paper contains 18 sections, 12 theorems, 81 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Mathematical model for the one-dimensional photonic crystals
  4. Organization of the paper
  5. Band structure of photonic crystals
  6. General band theory and transfer matrix
  7. Band structure of the Hermitian photonic crystals
  8. Floquet-Bloch theory
  9. Hermitian photonic crystals with spectral reciprocity
  10. Hermitian photonic crystals breaking the spectral reciprocity
  11. Band structure of non-Hermitian photonic crystals
  12. Spectral topology and edge modes in non-Hermitian photonic crystals
  13. Topological index for the spectral gap
  14. Homotopy invariance of $\hbox{Ind}(D)$ for the principal gapped component
  15. Jump of $\hbox{Ind}(D)$ across the dispersion curve
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\mathcal{T}(\omega; z)$ be the transfer matrix associated with the ODE system eq:ODE_model2 and $\mathcal{T}(\omega; 0) = \mathcal{I}$. Then $\hbox{det}(T(\omega; z))=1$ for any $z\in\mathbb R$.

Figures (5)

  • Figure 1: Band structures of the three-layer periodic media which attain spectral asymmetry. The permittivity and permeability are given in the form of \ref{['eq:para_three_layers']}, with $\delta=6$, $\varphi_1=0$, $\varphi_2=0.8$, $\alpha=\beta=0.5$. Left: Hermitian photonic crystal with $\varepsilon_0=13$, $\tilde{\varepsilon}_0=1$; Right: Non-Hermitian photonic crystal with $\varepsilon_0=13+5{\rm{i}}$, $\tilde{\varepsilon}_0=1$.
  • Figure 2: Point gap (left) and line gap (right) for the spectrum of non-Hermitian operators.
  • Figure 3: Spectrum of non-Hermitian operators over the complex plane: a trivial gap when $\omega_n(k)=\omega_n(-k)$ (left) and a non-trivial gap when the spectral symmetry is broken (right).
  • Figure 4: A schematic plot of dispersion curves $\gamma_1$, $\gamma_2$, $\gamma_3$, $\cdots$.
  • Figure 5: The domains enclosed by $\gamma_n$ are not Jordan domains.

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 3
  • Definition 4
  • Lemma 5
  • Definition 6
  • Lemma 8
  • Theorem 9
  • Corollary 10
  • ...and 7 more