Symmetry, stability, and computation of degenerate lasing modes

Abstract

We present a general method to obtain the stable lasing solutions for the steady-state ab initio lasing theory (SALT) for the case of a degenerate symmetric laser in two dimensions (2D). We find that under most regimes (with one pathological exception), the stable solutions are clockwise and counterclockwise circulating modes, generalizing previously known results of ring lasers to all 2D rotational symmetry groups. Our method uses a combination of semianalytical solutions close to lasing threshold and numerical solvers to track the lasing modes far above threshold. Near threshold, we find closed-form expressions for both circulating modes and other types of lasing solutions as well as for their linearized Maxwell-Bloch eigenvalues, providing a simple way to determine their stability without having to do a full nonlinear numerical calculation. Above threshold, we show that a key feature of the circulating mode is its "chiral" intensity pattern, which arises from spontaneous symmetry breaking of mirror symmetry, and whose symmetry group requires that the degeneracy persists even when nonlinear effects become important. Finally, we introduce a numerical technique to solve the degenerate SALT equations far above threshold even when spatial discretization artificially breaks the degeneracy.

Topological insulator laser: Theory

“…In passive settings (no gain or loss), for reasonable experimental parameters, one can consider the CW and CCW as decoupled (8,12). For a topological insulator laser based on this system, one must take into account that the CW and CCW modes inevitably interact with one another through the nonlinear effect of gain saturation and backscattering, both of which naturally occur in active media (43).…”

Is single-mode lasing possible in an infinite periodic system?

“…26,27 In fact, however, we show that singlemode lasing is possible even in infinite periodic structures for range of powers above threshold, by applying a Bloch adaptation of linear-stability analysis to the full Maxwell-Bloch equations. 19,20 We consider examples for both 1d DFB-like lasers and 2d BiC-based lasing, 10,11,28 and validate our result against brute-force time-domain simulations. 29,30 Using perturbation theory (in the supplementary material), we also obtain a simple condition for stability near threshold of low-loss resonances and confirm it numerically: the sign of the laser detuning from the gain frequency should match the sign of the a) Electronic mail: medbenz@mit.edu Stable ⇔ 𝜎 * (𝛿𝑘) < 0 ∀𝛿𝑘 .…”

“…Analytical details are shown in the supplementary material, using methods similar to those developed in Ref. 20. In the case of small loss, we obtain a simple approximate condition for stability near threshold: the band curvature Re d 2 ω dk 2 and the laser detuning (ω t − ω a ) should have the same sign at threshold.…”

“…Given this steady-state solution, one can then apply linear-stability analysis to the full Maxwell-Bloch equations, linearizing arbitrary aperiodic perturbations X = X k + δX, for X ∈ {E, P, D}, to determine whether perturbations δX exponentially grow (unstable) or shrink (stable). [18][19][20] Here, our key point is that, because the linearized equations for the perturbations δX are periodic (for a Bloch-mode steady state), we can apply Bloch's theorem 32 to decompose the perturbations themselves into Bloch-wave modes δE δk , solving a separate linearstability eigenproblem for each wavevector δk.…”

“…1(right)] must decay rather than grow. [18][19][20] At first glance, such stability may seem unlikely: any resonance in a periodic system is part of a continuum of resonances at different Bloch wavevectors with arbitrarily close lasing thresholds, and this seems to violate typical assumptions for stable lasing. [21][22][23] A finite-size structure discretizes the resonance spectrum and hence may suppress this problem, but instabilities have been observed in large enough finite periodic lasers where the resonances become very closely spaced.…”