Topological phase transition induced by gain and loss in a photonic Chern insulator

We present a design of a non-Hermitian photonic Chern insulator whose band topology can be tuned via gain and loss. It is found that, by continuously varying the gain and loss parameter, a topological phase transition can take place in the photonic crystal, accompanied by the closing and reopening of a Dirac cone. We demonstrate this phenomenon through numerical simulations of the bulk and the edge dispersions, and steady field distributions excited by a point source. Our results show the possibility of non-Hermitian control over band topology in a Chern insulator setting and may find applications in active topological photonic devices.

Figure 1

Design of the 2D non-Hermitian photonic crystal with on-site gain and loss. The red and blue rods are made of gyromagnetic materials with relative permittivity ${\epsilon }_{+}=15+i\gamma$ and ${\epsilon }_{-}=15-i\gamma$, respectively. Here, $\gamma$ is a positive real number that controls the gain and loss in each rod. The dashed rectangle denotes the unit cell, whereas ${\mathit{a}}_{\mathit{1}}=[a,0]$ and ${\mathit{a}}_{\mathit{2}}=[0,a\sqrt{3}]$ are the lattice vectors. The remaining parameters are the lattice constant $a=17.5$ mm and the diameters of the rods $d_1=0.462a$ and $d_2=0.338a$.

Figure 2

Plots of bulk band diagrams against the non-Hermitian parameter $\gamma$. The first row [panels (a) to (d)] shows the bulk band structures of different $\gamma$ at $k_y=0$ with band gaps shaded in yellow. From (a) to (d), the sizes of the band gap are 0.1 GHz, 0.06 GHz, 0 GHz, and 0.1 GHz. The second row [panels (e) to (h)] shows the corresponding bulk band structures near the Dirac cone, where the band gap closes at $\gamma =3.05$. The eigenfrequencies $f$ are in units of GHz and the color scale in each panel indicates the imaginary part of the eigenfrequencies.

Figure 3

Plots of edge dispersions against the non-Hermitian parameter $\gamma$. Each panel is obtained by computing the eigenfrequencies of a supercell with periodic boundary condition along $x$ direction and 20 unit cells along $y$ direction. The top and bottom edges are covered by perfect electric conductors. The eigenfrequencies $f$ are in units of GHz and the color scale in each panel indicates the imaginary part of the eigenfrequencies.

Figure 4

Simulated out-of-plane electric field ($|E_z|$) distributions in finite photonic lattices with different values of $\gamma$. Panels (a) to (d) represent the field distributions with $\gamma =0$, 2.2, 3.05, and 3.9, respectively. The cyan star in each panel denotes the point source operating at 4.68 GHz. The right boundary is chosen as a perfect electric conductor, while the three remaining boundaries are set as scattering boundaries. Chiral edge modes are observed at $\gamma =0$ and 2.2 (topological phase), whereas electric field decays rapidly away from the source when $\gamma =3.05$ (transition point) and 3.9 (trivial phase).