Lévy exponents as universal identifiers of threshold and criticality in random lasers

Critical excitation in random lasers under picosecond and nanosecond pumping was experimentally studied. The resulting emission intensity statistics were analyzed using fits to $\alpha$-stable distributions. We find that the transition value of $\alpha$, the tail exponent of the stable distribution, is a clear indicator of the threshold of random lasing. We discuss this exponent as an identifier of the threshold. This definition is compared with the conventional definitions for the threshold, namely, the probability of random lasing in the case of coherent random lasers and the intensity enhancement and bandwidth collapse for diffusive random lasing emission. We find a universal applicability of the $\alpha$ exponent as an identifier of the threshold, and hence the criticality, in random lasers.

Figure 1

(a) Typical spectra from samples at pump energies below (lowermost blue spectra) and above (upper red and black spectra) lasing threshold. The transport mean free path ${\ell }^{*}$ was 190, 350, and 1500 $\mu \mathrm{m}$, as shown in the three panels. (b) Symbols depict the distribution of emission intensity ($I_{\lambda }$ at $\lambda =557$ nm) at $E_p$ where $\alpha$ was minimum. The solid lines are the fit to the $\alpha$-stable law.

Figure 2

(a) Statistical behavior of the random laser with ${\ell }^{*}=$ 1500 $\mu \mathrm{m}$. Solid blue circles: $P_{RL}$, probability of random lasing. Solid black squares: tail exponent $\alpha$ of intensity distribution at $\lambda$ = 557 nm. Solid red triangles: fluctuation coefficient $\Phi \left(I_{\lambda }\right)$. A coincidence of $E_p$ (0.6 $\mu \mathrm{J}$) at which $P_{RL}$ becomes nonzero and $\alpha$ falls below 2 can be observed. (b) Threshold pump energy $E_{th}$ estimate using $P_{RL}$ (blue circles) and the $\alpha$ exponent (black squares).

Figure 3

(a) Typical spectra from systems with ${\ell }^{*}=$ 330 and 1500 $\mu \mathrm{m}$ under nanosecond pumping. Lowermost blue curves: subthreshold ($E_p\sim$ 15 $\mu \mathrm{J}$) spectra. Upper red and black curves: spectra at $E_p\sim$ 70 $\mu \mathrm{J}$, above the random lasing threshold. (b) Parameters for a representative system with ${\ell }^{*}=$ 800 $\mu \mathrm{m}$ as a function of $E_p$. Blue circles: bandwidth collapse. Black squares: tail exponent $\alpha$. Red triangles: $\Phi \left(I_{\lambda }\right)$. Orange stars: emission intensity $I_{\mathrm{out}}$. (c) Comparison of threshold energies $E_{th}$ estimated using the input-output curve and $\alpha$ statistics.

Figure 4

Threshold of random lasing as estimated using the $\alpha$ statistics across the spectrum from systems with ${\ell }^{*}=350$ $\mu \mathrm{m}$ (black circles) and ${\ell }^{*}=1500$ $\mu \mathrm{m}$ (red squares). Open symbols: picosecond pumping. Solid symbols: nanosecond pumping.