# Levy Statistics and the Glassy Behavior of Light in Random Fiber Lasers
## 3 Results
### Theoretical Framework  
A great advance in the theoretical understanding of the combined effect of amplification, nonlinearity, and disorder in RL systems was put forward in a series of articles [73; 74; 75; 76; 77; 78; 79; 80; 81; 82] published within the last decade. In this subsection, we review the physical mechanisms and some analytical developments underlying the richness of photonic behaviors displayed in the phase diagram of RL systems, as a function of the input excitation power and disorder strength. In fact, in refs. [73; 74; 75; 76; 77; 78; 79; 80], a variety of interesting photonic phases emerge, which keep close analogies with some characteristic behaviors of magnetic systems, such as spin glass, paramagnetism, and ferromagnetism. Moreover, such richness is also present in the diversity of statistical regimes observed in the output intensity emitted by RL systems. Interestingly, as we shall see below, the same theoretical starting point that gives rise to the variety of photonic behaviors also explains the shifts in the statistical properties of the distribution of intensity values. In fact, as discussed in the following, the increasing of the excitation power promotes a sequence of changes in the distribution of intensity values, from a Gaussian regime below the RL threshold to a Levy-type behavior around the RL transition, and back to a second Gaussian regime well above the threshold. The theoretical basis [81; 82] for these statistical aspects of RL emission is also reviewed below. We start by reviewing the theoretical background [73; 74; 75; 76; 77; 78; 79; 80] underlying the diverse photonic behaviors displayed by RLs. In a photonic system with intrinsic disorder, arising either from the inherent optical random noise or from the presence of randomly-located light scatterers (e.g., micro or nanoparticles [12; 13; 14; 15; 16; 17], or dielectric nanocrystals doped with rare-earth ions [20; 21]), the amplitudes \(a_{k}\) of the electromagnetic modes generally present stochastic dynamics. The [PERSON] approach is thus a suitable theoretical framework to describe such dynamics through the following set of equations,  
\[\frac{da_{k}}{dt}=-\frac{\partial H}{\partial a_{k}^{*}}+F_{k}, \tag{1}\]  
where \(F_{k}\) represents a Gaussian (white) uncorrelated optical noise term, and in the slow-amplitude regime of the modes, the general complex-valued functional \(H\) is given by [73; 74; 75; 76; 77; 78; 79; 80] (closely following the notation of [78]):  
\[H=\sum_{\{k_{1}k_{2}\}^{{}^{\prime}}}g_{k_{1}k_{2}}^{(2)}a_{k_{1}}a_{k_{2}}^{* }+\frac{1}{2}\sum_{\{k_{1}k_{2}k_{3}k_{4}\}^{{}^{\prime}}}g_{k_{1}k_{2}k_{3}k_ {4}}^{(4)}a_{k_{1}}a_{k_{2}}^{*}a_{k_{3}}a_{k_{4}}^{*} \tag{2}\]  
The symbol \(\{\ldots\}^{\prime}\) implies the frequency-matching conditions \(|\omega_{k_{1}}-\omega_{k_{2}}|<\gamma\) and \(|\omega_{k_{1}}-\omega_{k_{2}}+\omega_{k_{3}}-\omega_{k_{4}}|<\gamma\) in the quadratic and quartic terms, respectively, with \(\gamma\) denoting the  
Figure 4: (**a**) Variance of the emitted intensity as a function of the normalized input power _P/P\({}_{th}\)_; (**b**) Standard deviation of the maximum intensity (normalized by its average value) of the CW semiconductor pump laser (as function of the output power \(P_{out}\); squares) and the Er-RFL system (as function of the normalized input power _P/P\({}_{th}\)_; diamonds). (Reproduced with permission from ref. [68]). finite linewidth of the modes. The disordered nature of the scattering medium directly affects both the quadratic and quartic interactions among the spatially-overlapping modes in Equation (2). Indeed, the physical origin of the quadratic coupling \(g_{k_{1}k_{2}}^{(2)}\) lies in the spatially inhomogeneous refractive index, as well as in the nonuniform distribution of the gain and effective damping contribution due to the cavity leakage.