Tunable two-dimensional laser arrays with zero-phase locking

Two-dimensional (2D) laser arrays are shown to be achievable at a large scale by exploiting the interplay of higher-order topological insulator (HOTI) physics and the so-called non-Hermitian skin effect (NHSE). The higher-order topology allows for the amplification and hence lasing of a single mode protected by a band gap, whereas the NHSE is introduced to compete with the topological localization of corner modes. By tuning the system parameters appropriately and pumping at only one site, a single topologically protected lasing mode delocalized across two dimensions emerges, with its power widely tunable by adjusting the pump strength. Computational studies clearly indicate that the lasing mode thus engineered is stable, and the phase difference between all lasing sites is locked at zero, even after disorder is accounted for. The total power of the designed laser array is proportional to the area of the 2D lattice accommodating a HOTI phase. Based on existing experiments, we further propose to use coupled optical ring resonators as a promising platform to realize large-scale 2D laser arrays.

Figure 1

Schematics of a 2D lattice of coupled ring resonators accommodating a TQI phase, made possible by engineered couplings. The unit cell is boxed by the dashed line, which consists of four sites (AA, BA, AB, BB). Along each direction, there are $(2N-1)$ lattice sites. The intercellular coupling $t_2^{\pm }=t_2{e}^{\pm g}$ is nonreciprocal, with the couplings $t_2{e}^{+g}$ pointing towards the left and the bottom. Note also that along the ordinate, the signs of the intracellular coupling and intercellular coupling alternate.

Figure 2

(a) Spectrum of the Hamiltonian in Eq. (1) vs $\xi$ in the linear limit $\eta \to \infty$. A TCM always exists within the band gap. When $\xi$ is not sufficiently large, the TCM energy has a negative imaginary part and hence cannot be significantly stimulated. (b) The density profile of the TCM for $\xi =2$ with significant state densities extended over all the BB sites. (c) Total saturated intensity $I_{\mathrm{tot}}$, which is calculated at the steady state with $\eta =1$, vs $\xi$ for $g=1$ and $g=0$. $\xi \approx 1.3$ is seen to be the threshold value for the TCM to be stimulated. Beyond this threshold, the total intensity with NHSE ($g=1$) increases much faster than the case without NHSE ($g=0$). (d) Density profile of the steady state after a long-time evolution with $\xi =2$ and $\eta =1$. The other system parameters here are set at $N=3, t_1=1, t_2=e, g=1$, and $\gamma =0.5$.

Figure 3

(a) Time evolution of $|{\psi }_n^{BB}|$, the magnitude of the state amplitude over all the BB sites, which reach almost the same amplitude in the end. (b) Time evolution of the phase difference between two neighboring BB sites $|\mathrm{\Delta }{\varphi }_n^{BB}|$, always stabilized to zero, with a random state for the initial state. (c) and (d) The same as (a) and (b), but with weak disorder of the strength $(W=0.3)$. The eventual phase difference is still locked at zero.

Figure 4

Total saturated intensity $I_{\mathrm{tot}}$ vs $N$. Dots show the data obtained from numerical simulations, and lines represent a fitting function, with the orange line showing $I_{\mathrm{tot}}=1.90N^2$, the pink line showing $I_{\mathrm{tot}}=1.09N^2$, the light blue line showing $I_{\mathrm{tot}}=0.276N^2$, and the grey line showing $I_{\mathrm{tot}}=0.693$. The output power scales with $N^2$ with NHSE, also enhanced linearly by $\xi$.