Random lasing from Anderson attractors

We introduce and study a two-dimensional dissipative nonlinear Anderson pumping model that is characterized by localized or delocalized eigenmodes in a linear regime and in addition includes nonlinearity, dissipation, and pumping. We find that above a certain pumping threshold, the model has narrow spectral lasing lines generated by isolated clusters of Anderson attractors. With the increase of the pumping, the lasing spectrum is broadened even if narrow lasing peaks are still well present in the localized phase of linear modes. In the metallic phase, the presence of narrow spectral peaks is significantly suppressed. We argue that the model captures the main features observed for random lasers.

Figure 1

Dependence of the space-averaged steady-state field power $P=\langle |A_{x,y}{|}^2\rangle$ on the active media parameter $\alpha /\eta$. Here, the parameters are $W=8, \beta =\sigma =1$. For $\eta =0.1$, the steady state is obtained at $t_e={10}^4$ with an averaging over time interval $\mathrm{\Delta }t={10}^3$. The lattice size is $128\times 128$. The values of the linear damping are $\eta =0.2$ (blue points), $\eta =0.1$ (red points), and $\eta =0.05$ (black points).

Figure 2

Anderson attractor in the DINAP model for a 2D lattice with a linear size $N=128$, a disorder strength $W=8, \beta =\sigma =1$, and $\eta =0.1$. The different panels show the lasing power distribution $w(x,y)=|A_{x,y}|$ for the pumping strength $\alpha =0.09$ (a), (d), $\alpha =0.11$ (b), and $\alpha =0.13$ (c). The lasing power distributions are shown after an evolution time $t_e={10}^4$ (a), (b), (c) and $t_e={10}^5$ (d). Panel (d) shows the attractor stability for long evolution times. The color bars give the values of $w(x,y)$. The $w(x,y)$ distributions are averaged over a time interval $\delta t={10}^3$.

Figure 3

Spectrum $Z\left(\omega \right)$ of random lasing in the DINAP model for a 2D lattice with a linear size $N=128$, a disorder strength $W=8, \beta =\sigma =1$, and $\eta =0.1$ (same parameters as in Fig. 2). The pumping strength is $\alpha =0.09$ (a), (d), 0.31 (b), (e), and 0.91 (c), (f). Panels (a), (b), and (c) show the lasing power distributions $w(x,y)$, and panels (d), (e), and (f) show the spectrum $Z\left(\omega \right)$ of the random lasing. The integral of the spectral lasing power is $Z_{\text{tot}}\simeq 8.7\times {10}^{-8}$ (a), (d), $6.0\times {10}^{-5}$ (b), (e), and $4.5\times {10}^{-4}$ (c), (f). Initial conditions and random realizations are the same as in Figs. 2 and 2.

Figure 4

The lasing clusters present in the red, green, and blue squares in Fig. 3 are shown in panels (a), (b), and (c), respectively. The frequencies activated by these clusters are shown by the red, green, and blue curves in panels (d), (e), and (f), respectively. The spectrum $Z\left(\omega \right)$ of the whole attractor [Fig. 3] is drawn in the background of panels (d), (e), and (f). Here, the random lasing spectrum $Z\left(\omega \right)$ is normalized by the integral $Z_{\text{tot}}\simeq 8.7\times {10}^{-8}$.

Figure 5

Same as in Fig. 3 but for a disorder strength $W=3$. Here, $Z_{\text{tot}}\simeq 2.41\times {10}^{-5}$ for $\alpha =0.09$ (a),(d), $1.9\times {10}^{-4}$ for $\alpha =0.31$ (b),(e), and $7.5\times {10}^{-4}$ for $\alpha =0.91$ (c),(f).