Nontrivial point-gap topology and non-Hermitian skin effect in photonic crystals

We show that two-dimensional non-Hermitian photonic crystals made of lossy material can exhibit nontrivial point-gap topology in terms of topological winding in its complex frequency band structure. Such crystals can be made of either lossy isotropic media which is reciprocal or lossy magneto-optical media which is nonreciprocal. We discuss the effects of reciprocity on the properties of the point-gap topology. We also show that such a nontrivial point-gap topology leads to the non-Hermitian skin effect when the photonic crystal is truncated. In contrast to most previous studies on point-gap topology which used tight-binding models, our work indicates that such non-Hermitian topology can studied in photonic crystals, which represent a more realistic system that has technological significance.

Figure 1

(a) Reciprocal complex band structures in 1D lead to trivial windings while (b) nonreciprocal complex bands can give rise to nontrivial windings.

Figure 2

(a) Unit cell of a photonic crystal where the lattice periodicity is $a$. Here, the background is air and the triangle has refractive index $n_{\text{tri}}=2-1i$. (b) The lowest three bands of the real part of the eigenfrequences in the first Brillouin zone for the photonic crystal geometry in (a). The dashed black lines at fixed $k_y=\frac{-2\pi }{3a},0$ and $\frac{2\pi }{3a}$ indicate the path traced for the reduced 1D $k_x$-space band diagrams plotted in Fig. 3. (c) Finite stripe geometry that is periodic along $y$ and has finite $N=20$ unit cells in the $x$ direction. The ends in the $x$ direction have PML that are matched to air.

Figure 3

The lower part of each panel shows the band structure of the infinite 2D photonic crystal, plotted on the complex frequency plane, as well as the spectra of the finite stripe geometry. The band structure is plotted in black and the arrows give the direction of winding as $k_x$ varies parametrically from $k_x=-\pi /a$ to $\pi /a$. Clockwise winding corresponds to $W=-1$ and the enclosed area is shaded red, whereas anticlockwise winding corresponds to $W=1$ and the enclosed area is shaded blue. The eigenfrequencies of the finite stripe geometry is colored by mean $\left|E\right(x,y\left)\right|$ -field position, where red signifies an electric field distribution localized at the right edge and blue signifies an electric field localization localized at the left edge. The upper panels represent the electric field distribution of the eigenmodes of the stripe geometry at various complex frequencies indicated in the bottom panel. Panels (a)–(c) correspond to $k_y=\frac{-2\pi }{3a},0$ and $k_y=\frac{2\pi }{3a}$, respectively, and (A–I) are the normalized electric field eigenstates.

Figure 4

Same as Fig. 3, except with $k_y=-0.36\pi /a$, and at a range of a real part of frequencies that are higher as compared with Fig. 3. Point B is the Bloch point.

Figure 5

The real part of the eigenfrequencies as a function of $k_y$ for the finite stripe geometry shown in Fig. 2, where the triangle consists of a reciprocal material. The color indicates the mean position of the electric field for each eigenmodes.

Figure 6

(a) Same as Fig. 3, except with a nonreciprocal photonic crystal. (b) The real part of the eigenfrequencies as a function of $k_y$ for the finite stripe geometry shown in Fig. 2, where the triangle consists of a nonreciprocal material. The color indicates the mean position of the electric field for each eigenmodes.