Topological lasing in resonant photonic structures

We exploit topological edge states in resonant photonic crystals to attain strongly localized resonances and demonstrate lasing in these modes upon optical excitation. The use of virtually lossless topologically isolated edge states may lead to a class of thresholdless lasers operating without inversion. One needs, however, to understand whether topological states may be coupled to external radiation and act as active cavities. We study a two-level topological insulator and show that self-induced transparency pulses can directly excite edge states. We simulate laser emission by a suitably designed topological cavity and show that it can emit tunable radiation. For a configuration of sites following the off-diagonal Aubry-André-Harper model, we solve the Maxwell-Bloch equations in the time domain and provide a first-principles confirmation of topological lasers. Our results open the road to a class of light emitters with topological protection for applications ranging from low-cost energetically effective integrated laser sources, also including silicon photonics, to strong-coupling devices for studying ultrafast quantum processes with engineered vacuum.

Figure 1

Schematic view of the structured region: a 1D chain of resonant two-level layers (blue) in a homogeneous bulk (green) of frequency-independent dielectric function ${\varepsilon }_b$.

Figure 2

Real (green circles) and imaginary (blue squares) part of the left-edge-state frequency for $\beta =1/3, \eta =0.2/\pi , {\omega }_o=1.533$ eV, and ${\varepsilon }_b=12.25$. The dashed vertical lines show the $\xi$ values ($\xi /\pi =0,1/6,5/6,1$) where, for symmetry reasons, edge states do not exist. The continuous lines mark the boundaries of the gap.

Figure 3

Squared amplitude of the reflection coefficient $r_{\infty }(\varphi ,\omega )$ from the left side of the semi-infinite (a) US and (b) BS chain; reflection coefficient phase for the (c) US chain and (d) BS chain. Chern numbers $C$ and winding numbers $w$ are indicated.

Figure 4

(a) Real and (b) imaginary part of the left-edge-state frequency for $\beta =1/3, \eta =0.5, {\varepsilon }_a=1$, and ${\varepsilon }_b=12.25$. (c) Field intensity distribution inside the system for the configuration with $\varphi =0.7\pi$.

Figure 5

Field and population inversion ${\rho }_3(z,t)$ spatial profile for different observation times $t_i$ for the US structure with ${\xi }_{TC}=0.5\pi$. (a)–(c) Reference configuration with $\gamma =1\times {10}^{-29}$ Cm. (d)–(f) Topological configuration with $\gamma =1.4\times {10}^{-27}$ Cm.

Figure 6

(a) Field and (b) population inversion ${\rho }_3(x,t)$ spatial profile for different observation times $t_i$ for the BS structure with ${\varphi }_{TC}=0.7\pi$.

Figure 7

(a) Time-dependent output intensity in the left side of the BS chain; mode beating is observed in the time range $(1÷30$ ps). (b) Time evolution of its spectrum. (c) Snapshot of the electric field for $t=35$ ps.

Figure 8

(a) Time-dependent spectrum of the output signal in the left side of the US chain. Emitted spectrum for (b) ${\mathrm{\Gamma }}_o=5.89\times {10}^{-3}$ eV and (c) ${\mathrm{\Gamma }}_o=2.945\times {10}^{-2}$ eV.