Condensation in Disordered Lasers: Theory, $3\mathrm{D}+1$ Simulations, and Experiments

The complex processes underlying the generation of a coherent emission from the multiple scattering of photons and wave localization in the presence of structural disorder are still mostly unexplored. Here we show that a single nonlinear Schrödinger equation, playing the role of the Schwalow-Townes law for standard lasers, quantitatively reproduces experimental results and three-dimensional time-domain parallel simulations of a colloidal laser system.

Figure 1

Electromagnetic spectrum as obtained after a wideband excitation; the left inset shows the energy density in the middle section ($\lambda =586.5\mathrm{nm}$, asterisk); a sketch of the system is also shown on the right.

Figure 2

$3\mathrm{D}+1$ Maxwell-Bloch simulation of RL: (a) energy distribution in the sample middle section; (b) spectrum for $N_a=2\times {10}^{24}{\mathrm{m}}^{-3}$; (c) as in (b) for $N_a=5\times {10}^{24}{\mathrm{m}}^{-3}$.

Figure 3

Theory: Peak spectrum (a) and spectral waist (b) for various $\kappa$ versus the pumping rate in dimensionless units. Inset: Spectral profiles for $E=1.1$, 10, and 20.

Figure 4

Experimental results: (a) spectra at energies $20\mu \mathrm{J}$ (thin line) and $1000\mu \mathrm{J}$ (shaded area is a Gaussian fit), inset: corresponding enlarged central spectral region; (b) unitary-area averaged spectra (100 shots) vs energy.

Figure 5

(a) Left scale, experimentally retrieved laser peak spectrum versus pump energy, where the line is a best fit from theory; right scale, nonlinear eigenvalue (dashed lines correspond to $E=1$). (b) Linewidth vs energy; the continuous line is the best fit from theory; the FDTD Maxwell-Bloch simulations (diamonds) are also shown.