Mode Locking of the Hermite-Gaussian Modes of a Nanolaser

Mode locking is predicted in a nanolaser cavity forming an effective photonic harmonic potential. The cavity is substantially more compact than a Fabry-Perot resonator with a comparable pulsing period, which is here controlled by the potential. In the limit of instantaneous gain and absorption saturation, mode locking corresponds to a stable dissipative soliton, which is very well approximated by the coherent state of a quantum mechanical harmonic oscillator. This property is robust against noninstantaneous material response and nonzero phase-intensity coupling.

Figure 1

Mechanical analogy of (a) Fabry-Perot resonator and (b) harmonic oscillator implemented in a chirped DBR; (c) Parabolic dispersion in a DBR; (d) Hermite-Gaussian modes in a DBR with parabolic potential. Green area: gain and absorber region.

Figure 2

Laser behavior for instantaneous gain and absorber saturation with common widths equal to $5x_{\mathrm{\Omega }}$. (a),(b) Transient evolutions of (a) normalized intensities and (b) relative phases between the modes after the simulation is started from random mode amplitudes. (c) Evolution of the intracavity intensity in steady-state regime. (d) Positions of the coherent state (black dashed line) and the soliton (solid red line) and soliton width (solid cyan line) normalized to $x_{\mathrm{\Omega }}$. (e) Amplitude (left axis, solid red line) and phase (right axis, solid green line) of the soliton at a fixed time and corresponding coherent state (dashed line) with amplitude $2.2I_g$.

Figure 3

(a) Phase diagram: different steady-state regimes vs unsaturated gain and absorption normalized to ${\gamma }_0$. White hatched region: different unlocked mutimode regimes. (b) Corresponding false color plots of the laser intensity spatial distribution vs time, for different regimes: (0) below threshold; (1) solitonlike pulse; (2) mode $n=1$ alone; (3) mode $n=2$ alone; (4) simultaneous oscillation of modes $n=0$ and $n=2$.

Figure 4

(a) Percentage of occurrences of the different regimes, when the simulation is run 40 times with random initial conditions for each value of the gain and saturable absorption window width $w$. Other parameters are $R_I=5$, $g_0=5.5{\gamma }_0$, $a_0=9{\gamma }_0$. (b) False color plot of laser intensity vs $x$ and $t$ in two examples of regimes labeled (1) and (2) in (a), obtained for $w=6x_{\mathrm{\Omega }}$. They, respectively, correspond to oscillation of two or three pulses inside the cavity.

Figure 5

Laser behavior for slow gain and absorber saturations. (a) Phase diagram: different steady-state regimes vs unsaturated gain and absorption normalized to ${\gamma }_0$: (0) below threshold; (1) single-mode $Q$-switched operation; (2) $Q$-switched ML operation; (3) continuous-wave mode locking. (b),(c),(d) Stable ML behavior. Same as Figs. 2 corresponding to $r_g=70$ and $r_a=10$ in area number (3).