Asymmetric lasing at spectral singularities

Scattering coefficients can diverge at spectral singularities. In such situation, the stationary solution becomes a laser solution with outgoing waves only. We explore a parity-time $\left(\mathcal{PT}\right)$-symmetric non-Hermitian two-arm Aharonov-Bohm interferometer consisting of three coupled resonators enclosing synthetic magnetic flux. The synthetic magnetic flux does not break the $\mathcal{PT}$ symmetry, which protects the symmetric transmission. The features and conditions of symmetric, asymmetric, and unidirectional lasing at spectral singularities are discussed. We elucidate that lasing affected by the interference is asymmetric; asymmetric lasing is induced by the interplay between the synthetic magnetic flux and the system's non-Hermiticity. The product of the left and right transmissions is equal to that of the reflections. Our findings reveal that the synthetic magnetic flux affects light propagation, and the results can be applied in the design of lasing devices.

Figure 1

Schematic of the coupled resonators AB interferometer. Uniformly coupled resonator array with the embedded three coupled resonator. The gain (pink) and loss (green) in the resonators are balanced. The passive resonators are in white. The primary resonators (ring shape) are coupled through the auxiliary resonators (stadium shape). The photons circling in the primary and auxiliary resonators are in opposite directions.

Figure 2

The coherent perfect absorbing and lasing. The circles indicate the non-Hermitian structures. The incoming (outgoing) waves are indicated by the blue (green) arrows. (a) Coherent perfect absorbing, (b) symmetric lasing, (c) asymmetric lasing, (d) unidirectional lasing.

Figure 3

Transmission and reflection amplitudes. (a)–(c) $\mathrm{\Phi }=\pi /2$ and (d)–(f) $\mathrm{\Phi }=0$. (a),(d) Symmetric transmissions $\sqrt{T}$; (b),(e) left reflection $\sqrt{R_{\mathrm{L}}}$; and (c),(f) right reflection $\sqrt{R_{\mathrm{R}}}$.

Figure 4

(a) Spectral singularities in the $\mathrm{\Phi }$ and $\gamma$ parameter space for $k=\pi /4$. The curves indicate the place where scattering coefficients diverge. Spectral singularities appear in the region $\sqrt{2-\sqrt{2}}\le |\gamma /\kappa |\le \sqrt{2+\sqrt{2}}$. (b) The lasing contrast ratio at spectral singularities.

Figure 5

Numerical simulations of lasing wave intensity for (a),(b) $\mathrm{\Phi }=0$, (c),(d) $\mathrm{\Phi }=\pi /2$, and (e),(f) $\mathrm{\Phi }=\pi$. (a),(c),(e) Density plots of the intensity $P(t,j)/h$, where $h$ is the left-traveling wave height as shown in (b), (d), and (f) on their left halves. (b),(d),(f) Configuration of the lasing wave intensity $P(t,j)$ at $t=500/\kappa$. The three coupled resonators embedded in the center of the resonator array are indicated by the red triangles. A Gaussian wave packet with $\alpha =0.02$ and a wave vector $k_{\mathrm{c}}=\pi /4$ centered at $N_{\mathrm{c}}=900$ (indicated by the red arrow) is initially traveling left. The trajectory of the wave-packet center is indicated by the dashed green line. An asymmetric wave is emitted after reaching the center at approximately $t=210/\kappa$.