Extreme-value statistics of intensities in a cw-pumped random fiber laser

We report on the extreme-value statistics of output intensities in a one-dimensional cw-pumped erbium-doped random fiber laser, with a strongly scattering disordered medium consisting of randomly spaced Bragg gratings. The experimental findings from the analysis of a large number of emission spectra are well described by the Gumbel distribution below and above the laser threshold, whereas the Fréchet distribution, typical of strongly fluctuating extreme events with heavy power-law probability tails, provides a nice support to the data near the threshold. We establish a close connection, relying on theoretical arguments, between the reported extreme-value statistics and the shifts in the statistics of intensity fluctuations, from the Gaussian to the Lévy distribution at the threshold and back to the Gaussian well above threshold.

Figure 1

Experimental setup of the Er-RFL system. (1) Fiber pigtailed semiconductor pump laser operating in the cw regime, (2) Er-RFL, (3) 1480 nm/1550 nm WDM, (4) power meter to measure the output power at 1480 nm, (5) RFL emission out to the spectrometer, (6) spectrometer, and (7) liquid-${\mathrm{N}}_2$ cooled InGaAs CCD camera.

Figure 2

Maximum intensity value $I_j$ (in arbitrary units) of the spectra $j=1,2,...,N$ ($=150000$) emitted by the Er-RFL system. Data are shown for four values of the excitation power $P$ (normalized by the threshold power $P_{\text{th}}=16.30$ mW): (a) $P/P_{\text{th}}=0.77$, (b) $P/P_{\text{th}}=0.88$, (c) $P/P_{\text{th}}=1.21$, and (d) $P/P_{\text{th}}=1.72$, corresponding, respectively, to the regimes well below, just below, just above, and well above the random laser threshold.

Figure 3

Maximum intensity value $I_{\text{max},n}$ (in arbitrary units) of the blocks $n=1,2,...,M$ ($=800$) in which the $N=150000$ spectra displayed in Fig. 2 were divided. Data are shown for (a) $P/P_{\text{th}}=0.77$, (b) $P/P_{\text{th}}=0.88$, (c) $P/P_{\text{th}}=1.21$, and (d) $P/P_{\text{th}}=1.72$.

Figure 4

PDF $P\left(I\right)$ of intensity values of the $N=150000$ spectra emitted by the Er-RFL system at (a) $P/P_{\text{th}}=0.77$, (b) $P/P_{\text{th}}=0.88$, (c) $P/P_{\text{th}}=1.21$, and (d) $P/P_{\text{th}}=1.72$. Experimental results from the analysis of the sequences $\left\{I_j\right\}$ shown in Fig. 2 are depicted in red circles. Dashed lines in (a)–(c) indicate best-fit curves to the Lévy $\alpha$-stable distribution, whereas in (d) an exponential best fit emerges. The Lévy statistical behavior with $\alpha =1.69$ is identified just above the threshold in (c). Gaussian regimes are present both well below the threshold, with the best-fit value $\alpha =1.98$, and well above, since the exponential PDF is governed by the $\alpha =2$ Gaussian statistics, according to the CLT. The onset of the transition from the Gaussian to Lévy behavior, characterized by the strong enhancement of intensity fluctuations, is observed as the threshold is approached from below [(b) $\alpha =1.88$]. The insets show details of the main plots.

Figure 5

CDF $F\left(I_{\text{max}}\right)$ of maximum intensity values of the $M=800$ blocks of spectra emitted by the Er-RFL system at (a) $P/P_{\text{th}}=0.77$, (b) $P/P_{\text{th}}=0.88$, (c) $P/P_{\text{th}}=1.21$, and (d) $P/P_{\text{th}}=1.72$. Experimental results from the analysis of the sequences $\left\{I_{\text{max},n}\right\}$ shown in Fig. 3 are depicted in green circles. Best-fit curves to the GEV and Gumbel distributions are indicated by solid red and dashed blue lines, respectively. A Fréchet CDF is identified just above the threshold [(c) $\xi =0.62>1/2$], consistently with the Lévy statistical behavior of intensity fluctuations. Gaussian regimes are found well below and well above the threshold, characterized by distributions with $\xi \approx 0$, either Fréchet [(a) $\xi =0.08$] or Gumbel [(a) and (d): $\xi \to 0$]. Just below the threshold [(b) $\xi =0.35$], the strong increase of the shape parameter toward the $\xi =1/2$ Fréchet limit signalizes the onset of the transition from the Gaussian to Lévy behavior. The insets show details of the main plots.