Non-Hermitian skin effect in a two-dimensional Lieb photonic crystal

Abstract

In this contribution paper, we construct a two-dimensional non-Hermitian (NH) photonic crystal (PhC) to prototype its NH skin effect for experimental proposal. Based on the tight-binding model for Lieb lattice with NH coupling, a nontrivial spectral winding number is pinpointed for certain eigenstates, which translates to geometry-dependent skin modes in a PhC slab with tilt boundaries. For ease of implementation, complex refractive indices are employed for the Lieb unit cell of PhC to emulate the NH coupling. Validated by full wave simulation, our work under scores the boundary dependence of the geometric skin effect, and provides a concrete prototype design of NH skin effect easily implemented in classical wave systems by state-of-the-art of topological metamaterial platforms.

Figures (10)

• Figure 1: (a) Schematic for the Lieb lattice, where A, B and C represent three sublattices. The unit cell is marked by an orange dashed line, and the inset represents the first BZ of such a unit cell. (b) Band structure of the Hermitian system with $\gamma=0$, the energy is purely real. (c)-(d) Real and imaginary band structure of the NH system with $\gamma=0.8$. Parameters: $t_1=t_2=1$, $t_y=1.2$.
• Figure 2: (a) Different routes in the first BZ, the magenta points represent the EPs, and the red point is the reference point we chose with $k_r(0.35\pi/a, 0.8\pi/a)$. (b)-(f) Complex energy spectra of route 1 to route 4. All the parameters are same as those in Fig. \ref{['fig:fig1']}.
• Figure 3: (a) and (d) Spectral area under square and parallelogram shaped structures under OBC. The red points represent the spectral area under PBC. (b) and (e) The real spectra of two structures, the orange line represents the ratio of the probability density of the wave function at the boundary to that over the entire structures. The blue dashed line represents the degree of localization $\eta_b/\eta_t=80\%$ of the probability density at the boundary for each eigenstate. (c) and (f) Spatial distribution of all eigenstates $W(x)$ distributions of two shaped structures, both consist of $20\times20$ unit cells. The boundary is defined as having a thickness of two unit cells.
• Figure 4: (a) A model for verifying the impact of refractive index on the energy propagation of electromagnetic waves. (b) The distributions of the reflection (R), transmission (T), and absorption (A) rates of the wave field with respect to variations in the refractive index $\rm{Re}(n)$ and (c) $\rm{Im}(n)$ of the rectangular dielectric media. (d) A PhC model based on the Lieb lattice, which is analogous to the tight-binding model. Parameters: $n_0=3.28$, $n_1=1.87$, $n_y=2$, $n=1.87+0.1i$ and $d=0.05a$.
• Figure 5: (a)-(b) The real and imaginary parts of the band structures for the Lieb PhC shown in Fig. \ref{['fig:fig4']}(d). (c) The first BZ of the Lieb PhC, and we select three closed straight line routes, corresponding to the complex spectra of the three bands are shown in (d)-(f).