Symmetry, stability, and computation of degenerate lasing modes

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.

Figure 1

Degenerate pairs of standing-wave modes in a laser. For this uniform dielectric disk (top), which has $C_{\infty \mathrm{v}}$ symmetry, the two eigenfunctions (of which only the real part is shown) are proportional to $cos\left(\ell \varphi \right)$ and $sin\left(\ell \varphi \right)$ (here, $\ell =9$). For a homogeneous dielectric square ($\varepsilon =5$ inside a square of length 1), the eigenfunctions are $\frac{\pi }{2}$ rotations of one another. Here, the order of the irreducible representation (irrep) is $\ell =1$, which is the only possibility for $C_{4\mathrm{v}}$. The $\frac{\pi }{2}$ rotation is an exact symmetry of the geometry, so there is an exact degeneracy even for the numerical grid. For $C_{6\mathrm{v}}$, the symmetry group of the regular hexagon, as in this example of TE modes (with the transverse magnetic field $H_z$ shown) in a 2D slab with air holes (described in further detail in Sec. 5b), the two eigenfunctions have no immediately obvious symmetry operation that transforms between them, but are in fact still degenerate (here $\ell =1$, and there also exists an $\ell =2$ irrep with its own degenerate pair). In all three cases (and in fact for all $C_{n\mathrm{v}}$), the standing-wave modes have mirror planes that are $\frac{\pi }{2}$ rotations from one another, and the two standing-wave modes have opposite parities across these mirror planes.

Figure 2

Lasing amplitudes (top) and frequency shifts (bottom) for 1D laser (periodic geometry with $0<x<1$) with uniform dielectric $\varepsilon \left(x\right)=1+0.3i$ and gain profile $D_0\left(x\right)=D_{\mathrm{t}}(1+d)[1+0.2cos\left(2\pi nx\right)]$. Here, we have chosen $n=5$, so the gain has $C_{5\mathrm{v}}$ symmetry, and the discretization had $N=150$ grid points. Data points were obtained by solving Eq. (12) (SALT) numerically using Newton's method [14], while theoretical lines were provided by Eqs. (18) and Eq. (19). For the standing mode, numerical results are independent of relative phase $\theta$, as predicted by Eq. (19). Agreement between numerics and theory is excellent for mode amplitudes [up to at least $d\approx 100$ (not shown in the figure), and possibly much higher] but only good at $d<0.005$ for frequency shifts. For the numerical data in the amplitude plot, only the magnitude of the ${\mathbf{E}}_{+}$ component (obtained by taking the normalized inner product of the lasing mode $\mathbf{E}$ with ${\mathbf{E}}_{+}$) of both the circulating and standing-wave modes are shown. For the ${\mathbf{E}}_{-}$ component, the circulating mode had magnitude zero and the standing mode had the same magnitude as the ${\mathbf{E}}_{+}$ component.

Figure 3

Stability eigenvalues for the circulating (top) and standing (bottom) lasing modes from Fig. 2. For each lasing mode, the four lowest eigenvalues obtained using the numerical procedure of Ref. [7] are matched against the four eigenvalues found in perturbation theory. In the top panel, a zero eigenvalue is clearly seen, coming from the global phase freedom. The real parts of the third and fourth values of Eq. (32) are equal, so two of the curves coincide. None of the other eigenvalues go above the real axis, indicating that the circulating mode is stable. For the bottom panel, there are two zero eigenvalues, in agreement with Eq. (33). One of the other two eigenvalues becomes positive, indicating that the standing lasing mode is not stable.

Figure 4

The incorrect (left) intensity pattern, correct (center) intensity pattern for the pair of low-$Q$ dielectric square modes, slightly above threshold (the pattern remains essentially the same even for much higher pump strengths), and degenerate passive mode of opposite chirality (right) of low-$Q$ dielectric square for lasing very high above threshold ($D_0=100D_{\mathrm{t}})$. The left intensity pattern was obtained by solving two-mode SALT without any interference effects, while the center pattern was obtained by constructing the stable linear combination predicted by symmetry and perturbation theory and then solving single-mode SALT. The correct pattern clearly has a chirality, which the incorrect pattern lacks. The profile on the right is not simply a mirror flip of the lasing mode, but is in fact degenerate with it, as explained in Appendix pp2.

Figure 5

Splitting in degeneracy due to discretization error for even-$\ell$ modes of dielectric cylinder versus the resolution $1/h$ of the discretization, where $h$ is the distance between adjacent grid points. The oscillations, which are due to the discontinuous interfaces between dielectric and air that “jump” when the resolution is changed, could in principle be smoothed by using subpixel averaging techniques for the discretization [37].

Figure 6

$\ell =8$ threshold modes [$\mathrm{Re}\left({\mathbf{E}}_{\mathrm{e},\mathrm{o}}\right)$] for a cylinder with uniform dielectric $\varepsilon =5$ and radius $r=1$. Unlike in the odd-$\ell$ case (Fig. 1), however, the discretized modes are not $\frac{\pi }{2}$ rotations from each other. Consequently, there is an unphysical splitting, due to discretization, of $0.11\%$ in $\mathrm{Re}({\omega }_1-{\omega }_2)$ (for a resolution of 14 pixels per wavelength) and $11.5\%$ in $\mathrm{Im}({\omega }_1-{\omega }_2)$ at zero pump strength [the latter being larger only because these are high-$Q$ modes and $\mathrm{Im}\left({\omega }_{\mu }\right)$ is already very small at zero pump strength]. A difference in imaginary parts also means a splitting in the threshold pump strength $D_{\mathrm{t}}$.

Figure 7

Dielectric perturbation $\delta \varepsilon$ obtained by solving QP for threshold modes with $\ell =8$. The real part (left) has a dependence $cos\left(2\ell \varphi \right)$, while the imaginary part (right) is a more complicated function.

Figure 8

Relative splitting in threshold pump strength $D_t$ and frequency ${\omega }_t$ for even-$\ell$ cylinder modes after QP iterations. The relative splitting in frequency is defined in the usual way as $2\left|\frac{{\omega }_1-{\omega }_2}{{\omega }_1+{\omega }_2}\right|$, and similarly for the pump strength. Only two iterations of QP were required reduce the splitting in both the threshold frequency ${\omega }_t$ and the threshold pump strength $D_{\mathrm{t}}$ to ${10}^{-10}$ or smaller.

Figure 9

$L_2$ norm of resulting $\delta \varepsilon \left(\mathbf{x}\right)$ function obtained from QP procedure versus discretization resolution $1/h$ for nearly degenerate even-$\ell$ modes of the cylinder, where $h$ is the spacing between adjacent grid points. The same resolutions as in Fig. 5 were used, and the oscillations resemble the curve for splitting very closely. This is because the larger the splitting ${\omega }_2-{\omega }_1$, the larger the $\delta \varepsilon \left(\mathbf{x}\right)$ function needed to enforce the degeneracy. The fact that ${‖\delta \varepsilon ‖}_2^2$ appears to be going to zero as the resolution increases indicates that our QP procedure is convergent.

Figure 10

Above-threshold splitting in real and imaginary parts of $\delta {\omega }^{\prime }$ after performing QP procedure for even-$\ell$ modes. The magnitude is very small because the intensity profile is very close to rotationally symmetric.

Figure 11

Dielectric perturbation obtained from QP procedure for hexagonal cavity. Since the mode is TE ($\mathbf{E}=E_{\mathrm{even}}\widehat{\mathbf{x}}+E_{\mathrm{odd}}\widehat{\mathbf{y}}$), we have allowed the perturbation to be a diagonally anisotropic tensor, as in Eq. (D4). Shown here are the real (left) and imaginary (right) parts of $\delta {\varepsilon }_{xx}$. The $\delta {\varepsilon }_{yy}$ looks similar except rotated by ${60}^{\circ }$.

Figure 12

Intensity pattern for stable lasing mode for hexagonal cavity. The pattern appears to be sixfold symmetric, which is expected. Unlike in the right panel of Fig. 4, however, the chirality is not significant enough to be visible since the hexagonal cavity is not as lossy as the square cavity in Fig. 4. In the ideal system, the second pole $\delta {\omega }^{\prime }$ stays degenerate with the lasing eigenvalue $\delta \omega$, and this linear combination stays stable for all pump strengths above threshold. In the discretized system, there is not a true $C_{6\mathrm{v}}$ symmetry, so there is a small splitting similar to that of the even-$\ell$ cylinder modes. Again, this splitting is too small to affect physically meaningful results of the simulation, but can be removed using the QP procedure if desired.