Quasi-bandgap behavior in non-Hermitian photonic crystals

Abstract

We investigate non-Hermitian photonic crystals in which the lossy and lossless constituents share the same real permittivity and differ only in their imaginary part. We characterize the complex band structure and reflection response of both one-dimensional (1D) and two-dimensional (2D) systems, and show that introducing even a small amount of material loss opens a quasi bandgap at the Brillouin-zone boundary. This quasi bandgap, absent in the lossless limit of the same structure, gives rise to sharp reflectivity peaks whose origin we explain through second-order perturbation theory. As an application of this behavior, we demonstrate a selective reflector combining a conventional photonic-crystal waveguide with a non-Hermitian photonic crystal, achieving wavelength-selective reflection with broadband absorption.

Figures (5)

• Figure 1: 1D non-Hermitian photonic crystal: (a) schematic diagram, (b) reflectivity for three values of $\mathrm{Im}[\varepsilon_2]$. A reflectivity peak of 0.57 is observed at a scaled frequency 0.35 $(2\pi c/a)$, (c) real part of the band structure computed assuming a real wavevector (colored lines) and a real frequency (thin black lines). (d) enlarged view of the first and second bands showing the quasi bandgap around frequency 0.35, (e) imaginary part of the band structure assuming real wavevector, (f) density of state / inverse group velocity $(d\omega/d \mathrm{Re}[k])^{-1}$ for the bands and frequency range examined.
• Figure 2: Perturbation theory prediction for the quasi bandgap size. (a) Imaginary bandgap size between the first and second band from $1^{st}$ order perturbation theory; (b) real bandgap size between the first and second band from $2^{nd}$ order perturbation theory; (c) imaginary bandgap size between the second and third band from $1^{st}$ order perturbation theory; (d) real bandgap size between the second and third band from $2^{nd}$ order perturbation theory.
• Figure 3: 2D non-Hermitian photonic crystal. (a) Schematic diagram of the 2D non-Hermitian photonic crystal with square lattice lossy rods ($r = 0.21a$, where $a$ is the lattice constant) with $\varepsilon_2 = 2 + 2i$ in a lossless background $\varepsilon_1 = 2$; (b) TM band structure from $\Gamma$-$X$-$M$-$\Gamma$ (dashed blue lines show the band structure for the corresponding Hermitian system: homogeneous permittivity of 2); (c) real part of the band structure at all combinations of $k_x$ and $k_y$; (d) imaginary part of the band structure at all combinations of $k_x$ and $k_y$.
• Figure 4: Reflectivity of the 2D non-Hermitian photonic crystal for TM mode at normal incidence. The reflectivity has a sharp peak at frequency 0.35 and a weaker peak at 0.71, analogous to the 1D case.
• Figure 5: A selectively reflective waveguide. (a) Schematic diagram of the reflector: the left part is a 2D triangular air hole photonic crystal with a linear defect; the right part is a non-Hermitian square lattice photonic crystal with a square array of lossy pillars ($\varepsilon = 13 +6i$) embedded in a lossless substrate with the same real part of permittivity ($\varepsilon = 13$). (b) Reflectivity and transmission of the combined selective reflector from $0.9\,\mu$m to $1.2\,\mu$m. (c) and (d) Out-of-plane electric field $E_z$ distribution at wavelengths $1.13\,\mu$m and $1.077\,\mu$m, respectively.