Steady-state ab initio laser theory for fully or nearly degenerate cavity modes

We investigate the range of validity of the recently developed steady-state ab initio laser theory (SALT). While very efficient in describing various microlasers, SALT is conventionally believed not to be applicable to lasers featuring fully or nearly degenerate pairs of resonator modes above the lasing threshold. Here we demonstrate how SALT can indeed be extended to describe such cases as well, with the effect that we significantly broaden the theory's scope. In particular, we show how to use SALT in conjunction with a linear stability analysis to obtain stable single-mode lasing solutions that involve a degenerate mode pair. Our flexible and efficient approach is tested on one-dimensional ring lasers as well as on two-dimensional microdisk lasers with broken symmetry.

Figure 1

Results for a one-dimensional ring laser system with circumference $L=1, {\gamma }_{\perp }=1$, and ${\omega }_a=61$. The openness of the system is modeled by a lossy dielectric function $\sqrt{\epsilon \left(x\right)}=1+0.0002i$. (a) Sketch of the ring laser system, which can accommodate clockwise (CW) or counterclockwise (CCW) traveling modes. (b) Lasing mode intensity vs applied pump strength for the traveling-wave solution. Note that the gain parameters have been chosen such that only a single pair of degenerate modes reaches the lasing threshold. The traveling-wave solution in which the system lases has a frequency of $\omega \approx 62.65$ which stays constant while the pump strength is increased. (c) Stability analysis of the SALT results. Shown are the stability of this traveling-wave solution under variation of the pump strength $D_0$ and the relaxation rate of the inversion ${\gamma }_{\parallel }$. The results of the FDTD simulation are color coded with green (light gray) marking parameter combinations where the SALT solution is stable, whereas the red (dark gray) region represents unstable behavior and the solid black lines mark the border between those two regimes. Blue dashed lines show the independent results of the linear stability analysis, which provides an excellent estimate for the border between stable and unstable regions.

Figure 2

Stability eigenvalues ${\sigma }^j$ of the ring laser for $D_0=0.06$ and ${\gamma }_{\parallel }={10}^{-3}$ (blue circles), ${\gamma }_{\parallel }\approx 7\times {10}^{-3}$ (green crosses), and ${\gamma }_{\parallel }={10}^{-1}$ (red Ys). All other parameters are chosen as in Fig. 1. One can see that the single-mode lasing solution is only stable for the intermediary value of ${\gamma }_{\parallel }$ (green crosses), since for both other cases, a ${\sigma }^j$ with a real part larger than zero exists. The stability eigenvalues with $\mathrm{Im}\left({\sigma }^i\right)\approx {10}^{-2}$ are related to the competition between the nearly degenerate modes. The other visible clusters of stability eigenvalues at $\mathrm{Re}\left({\sigma }^i\right)\approx -{\gamma }_{\parallel }$ are related to damped relaxation oscillations observed when initially turning on the laser. The eigenvalues with $\mathrm{Im}\left({\sigma }^j\right)>10$ are related to other resonances of the system that could potentially also turn into active lasing modes.

Figure 3

Results for a ring laser system as in Fig. 1 with ${\gamma }_{\parallel }=0.01$, but with an additional scatterer added with $l=0.05$ and $\sqrt{\epsilon }=1.05+0.0002i$. (a) Sketch of the system. (b) Frequencies of single-mode SALT solutions. It can be seen that the two degenerate modes of the symmetric system from Fig. 1 have split into modes $A$ and $B$. Additionally, at $D_0\approx 0.003$, two predominantly traveling-wave solutions $C_1$ and $C_2$ branch off from solution $A$. These two solutions are mirror-symmetric images of each other. Dotted lines mark unstable, solid lines mark stable solutions. (c) Intensity distribution of modes $A$ and $C_1$ with the scatterer marked in gray. To better show the differences in the shape of the modes, both modes are normalized to a maximal intensity of 1. Mode $A$ is shown directly at threshold ($D_0\approx 0.0015$), mode $C_1$ at $D_0=0.1$. It can be seen that while $A$ is a standing wave with minimal intensity 0 at the nodes, the predominantly traveling-wave $C_1$ features a non-vanishing intensity everywhere in the cavity. The stability of mode $C_{1,2}$ is studied in Fig. 4.

Figure 4

Stability of the SALT solution for the broken-symmetry ring laser shown in Fig. 3. The analyzed solution is the one marked as branch $C_{1,2}$ in Fig. 3. While the shape of the stability region is very similar to the one observed in Fig. 1, the broken-symmetry system requires a minimal pump strength $D_0$ to compensate for the frequency splitting of the nearly degenerate modes before a stable predominantly traveling-wave solution can emerge.

Figure 5

Single-mode lasing states for a 2D cavity with a wedge. The disk has a radius of 1, a refractive index of $n=2+0.01i$, and the wedge on the right side of the disk has a depth of $0.05$ and a width of $0.05$. The gain parameters are ${\omega }_a=4.83$ and ${\gamma }_{\perp }=0.1$. (a) The cavity is outlined in white with arrows marking the position of the wedge. The spatial intensity pattern of two stable single-mode solutions are shown. Left: Standing-wave mode $B$ at pump strength $D_0=0.0773$ with regularly spaced nodes with intensity zero. Right: dominantly traveling-wave solution $C_1$ at pump strength $D_0=0.15$. (b) Laser frequencies of single-mode solutions to the SALT Eq. (4). Curves $A$ (orange) and $B$ (red) correspond to two standing-wave solutions, which have even and odd symmetry with respect to the $x$ axis. The two modes represented by curve $C_{1,2}$ (blue) feature a broken symmetry, but are mirror-symmetric to each other. Solid (dotted) lines denote a stable (unstable) solution for ${\gamma }_{\parallel }=0.01$. Note that the pump axis is separated into three regions of different linear scaling for the sake of clarity. (c) Stability diagram of the single-mode solutions shown in (b) with respect to the pump strength $D_0$ and the relaxation rate of the inversion, ${\gamma }_{\parallel }$. The color coding is as in Fig. 1 except that the stability is here solely determined by the linear stability analysis.