Scalable numerical approach for the steady-state ab initio laser theory

We present an efficient and flexible method for solving the non-linear lasing equations of the steady-state ab initio laser theory. Our strategy is to solve the underlying system of partial differential equations directly, without the need of setting up a parametrized basis of constant flux states. We validate this approach in one-dimensional as well as in cylindrical systems, and demonstrate its scalability to full-vector three-dimensional calculations in photonic-crystal slabs. Our method paves the way for efficient and accurate simulations of microlasers which were previously inaccessible.

Figure 1

(a) 1D slab cavity laser of length $L=100\mu \mathrm{m}$ with purely reflecting boundary on the left side and open boundary on the right side. The mode shown in red (gray) corresponds to the intensity profile of the first lasing mode at threshold. (b) SALT eigenvalues corresponding to the scattering matrix poles for a uniform and linearly increasing pump strength $D_0(\mathbf{x},d)=d$ applied inside the slab $[D_0(\mathbf{x},d)=0$ outside]. We use a refractive index $\sqrt{{\varepsilon }_c}=1.2$ in the slab $(\sqrt{{\varepsilon }_c}=1$ outside), a gain frequency $k_a=100{\mathrm{mm}}^{-1}$ and a polarization relaxation rate ${\gamma }_{\perp }=40{\mathrm{mm}}^{-1}$. The trajectories start at $d=0$ (circles) and move toward the real axis with different speed when increasing $d$. The first lasing mode (dash-dotted red line) activates at $d=0.267$ (triangles) with $k_1=115.3{\mathrm{mm}}^{-1}$. The trajectories end at $d=1$ (squares) where a second lasing mode (dashed green line) turns active and the two other nonlasing modes (blue dotted and yellow solid line) remain inactive. The values at $d=1$ coincide with the data in Fig. 2.

Figure 2

Single pump algorithm applied to the 1D edge emitting laser shown in Fig. 1. Here the lasing modes for the single pump strength $D_0=1$ are obtained within three iteration steps (see blue, red, and green colors, respectively). (a) In the first step, the eigenfrequencies of Eq. (10) are determined for $D_0=1$. Full and empty circles represent modes in the upper (nonphysical) and lower (nonlasing) part of the complex plane, respectively. The dashed ellipse indicates the boundary inside of which all eigenvalues are determined using the contour integral method. The dotted vertical line marks the most nonphysical mode which is used as a first guess of a lasing mode in the next step (b). This ansatz shifts not only the corresponding eigenvalue down to the real axis, but also the other eigenvalues are shifted downwards due to the resulting pump depletion (see modes indicated in red). (c) After including again the most nonphysical mode of the previous step as a guess for the second lasing mode (see red dot) and performing the corresponding iteration with Newton, we obtain the solution which coincides exactly with the data in Fig. 1 (see squares there), where two modes are active while all other modes are nonlasing.

Figure 3

Comparison between the laser output using SALT with exact outgoing boundary conditions and PML absorbing layers, on the one hand, and a full time integration of the MB equations using FDTD, on the other hand. We study the first and second TM lasing modes of a 1D slab cavity which is similar to the one above. The applied pump is uniform, $D_0(\mathbf{x},d)=d$, the cavity has a uniform dielectric $\sqrt{{\varepsilon }_c}=2$ a length $L=100\mu \mathrm{m}$, and gain parameters ${\gamma }_{\perp }=3{\mathrm{mm}}^{-1},k_a=300{\mathrm{mm}}^{-1}$. For the FDTD simulations additionally ${\gamma }_{\parallel }=0.001{\mathrm{mm}}^{-1}$ was used. The PML method is nearly as accurate as the outgoing boundary condition, but has the advantage of being easily generalizable to two- and three-dimensional calculations [20]. The times to reach $d=0.11$ are shown for the two methods (with identical spatial resolution). The FDTD computation was done on the Yale BulldogK cluster with E5410 Intel Xeon CPUs, while the SALT computations were done on a Macbook Air.

Figure 4

Output intensity vs pump strength in a 1D resonator with reflecting boundary on the left side and outgoing radiation on the right side; see Fig. 1. The cavity has length $100\mu \mathrm{m}$ with a refractive index $n=1.01$. The gain curve has its peak at $k_a=250{\mathrm{mm}}^{-1}$ and a width $2{\gamma }_{\perp }=15{\mathrm{mm}}^{-1}$. The output intensity is given by ${\left|\mathit{\Psi }\right|}^2$ evaluated at the right boundary $x=L$. The pump is constant in the entire cavity. Solid lines describe the results of our solution method. Comparing them to the solutions of the CF-state formalism with 30 (long dashed), 20 (dashed), and 15 (dash-dotted) CF-basis functions, one observes that the two approaches converge towards each other for a sufficiently high number of CF states being included.

Figure 5

Output intensity vs pump strength in a laser system of two 1D cavities, each of length $100\mu \mathrm{m}$ and an air gap of size $10\mu \mathrm{m}$; see inset. The refractive index of the cavity material is $n=3+0.13i$ and the gain curve is centered at $k_a=94.6{\mathrm{mm}}^{-1}$ with a width of $2{\gamma }_{\perp }=4{\mathrm{mm}}^{-1}$. For the pump parameter in the interval $0<d<1$ the pump is linearly increased in the left resonator from zero to $D_{\max }=1.2$ (the intensity pattern of the mode lasing at $d=1$ is shown in the inset). For $1<d<2$ the pump in the left resonator is kept at the value of $d=D_{\max }$ and the pump in the right resonator is increased from zero to the same value as on the left. The output intensity here is given by the sum of ${\left|\mathit{\Psi }\right|}^2$ evaluated on both open ends. As a result of this pump trajectory, a nonmonotonous evolution of the total emitted laser light intensity is observed with a complete laser turnoff at around $d\approx 1.6$.

Figure 6

Movement of the complex SALT eigenvalues (resonance poles) along the pump trajectory realized in Fig. 5. Solid lines represent the eigenvalues computed with our solution method and dashed lines represent the solutions in the absence of the nonlinear spatial hole burning. Colors (red/blue) are chosen in correspondence with Fig. 5. Our results show that the laser shutdown can be associated with an avoided level crossing of the SALT eigenvalues in the complex plane. Details are shown in the top panel (a), where upward triangles mark the eigenvalues where the first mode starts lasing a second time in the course of the pump trajectory. Downward triangles mark the eigenvalues where the second mode starts lasing for the first time. In the main panel (b) circles label the eigenvalues at the starting point of the pump trajectory $\left(d=0\right)$ and squares label the positions of the eigenvalues at the first threshold.

Figure 7

Validation of the 2D Newton solver based on FDFD against the CF-state approach (using 20 CF basis states) in a circular cavity with radius $R=100\mu \mathrm{m}$ and dielectric index $n=\sqrt{{\varepsilon }_c}=2+0.01i$. TM-polarized modes are considered and the following gain parameters are used: ${\gamma }_{\perp }=10{\mathrm{mm}}^{-1}$, $k_a=48.3{\mathrm{mm}}^{-1}$. Increasing the strength of the uniform pump $D_0(\mathbf{x},d)=d$, we encounter strong nonlinear modal competition between the first two lasing modes with the result that for sufficiently large pump strength the second lasing mode is found to suppress the first one (see top panel). The internal intensity is defined as the integral over the cavity ${\int \left|\mathit{\Psi }\left(\mathbf{x}\right)\right|}^{2}d\mathbf{x}$. The real part of the lasing mode profile $\mathit{\Psi }\left(\mathbf{x}\right)$ at the first threshold is shown for both the exact Bessel solution $\left(\mathit{\Psi }\sim e^{-im\theta }\right)$ and for the finite difference solution (see bottom panel, where blue/white/red color corresponds to negative/zero/positive values). As the pump strength is increased, this profile does not change appreciably apart from its overall amplitude.

Figure 8

3D calculation of a lasing mode created by a “defect” in a photonic-crystal slab [12]: a period-$a$ hexagonal lattice of air holes (with $a=1\mathrm{mm}$ and radius $0.3\mathrm{mm})$ in a dielectric medium with index $n=\sqrt{{\varepsilon }_c}=3.4$ with a cavity formed by seven holes of radius $0.2\mathrm{mm}$ in which a doubly degenerate mode is confined by a photonic band gap (one of these degenerate modes is selected by symmetry; see text). The gain has ${\gamma }_{\perp }=2.0{\mathrm{mm}}^{-1}$, $k_a=1.5{\mathrm{mm}}^{-1}$, and nonuniform pump $D_0(\mathbf{x},d)=f\left(\mathbf{x}\right)d$, where the pump profile $f\left(\mathbf{x}\right)=1$ in the hexagonal region of height $2\mathrm{mm}$ in the $y$ direction, and $f\left(\mathbf{x}\right)=0$ outside that region and in all air holes. The slab has a finite thickness $0.5\mathrm{mm}$ with air above and below into which the mode can radiate (terminated by PML absorbers). The inset shows magnetic field $H_z$ $\left(\sim {\partial }_x{E}_y-{\partial }_y{E}_x\right)$ of the TE-like mode at the $z=0$ plane.