# Levy Statistics and the Glassy Behavior of Light in Random Fiber Lasers
## 3 Results
### Theoretical Framework
Therefore, in this sense, a parameter \(q_{\rm max}\) can be defined to indicate the locus of the maximum of the distribution P(\(q\)), which is considered as the Parisi order parameter [84]. The value of \(q_{\rm max}\) thus signalizes a replica-symmetric paramagnetic or an RSB spin-glass phase, respectively, if the maximum of P(\(q\)) occurs exclusively at \(q_{\rm max}=0\) (no RSB) or also at values \(\,\,1\,q_{\rm max}\,|\
eq 0\) (RSB). In the photonic context, an analogue correlation function between modes can also be suitably defined [42; 79] (see details below). Its distribution of values determines, in a similar way, the presence or absence of the photonic RSB spin-glass phase. On the experimental side, the very first evidence of photonic RSB glassy behavior in an RL system arose in the 2D functionalized T\({}_{5}\)OC\({}_{\rm x}\) oligomer amorphous solid-state material [42]. Subsequent demonstrations appeared in 3D functionalized TiO\({}_{2}\) particle-based dye-colloidal [85] and neodymium-doped YBO\({}_{3}\) solid-state [82] RLs. Here, we highlight that the RSB spin-glass phase has also been characterized in the above-mentioned Er-RFL system [68; 69]. The emergence of such behavior in Er-RFL is actually justified since this system also presents the disorder and nonlinear ingredients necessary to induce the RSB glassy RL phase, as discussed above in the context of the effective photonic Hamiltonian (2). We now turn to the discussion on the statistical regimes of output intensities emitted by RL systems. We first review some analytical developments regarding the distribution of intensity values. Noteworthy, the set of Langevin equations, given by Equation (1), also provides the underlying theoretical basis for such analysis. Indeed, by writing \(l_{k}=c_{k}|a_{k}|^{2}\), a manipulation of Equation (1) yields [82]:  
\[\frac{1}{c_{k_{2}}}\frac{dl_{k_{2}}}{dt}=-2 Re\Bigg{\{}\sum_{\{k_{1}\}^{\prime} }g_{k_{1}k_{2}}^{(2)}a_{k_{1}}a_{k_{2}}^{*}+\frac{1}{2}\sum_{\{k_{1}k_{3}k_{4 }\}^{\prime}}[g_{k_{1}k_{2}k_{3}k_{4}}^{(4)}+g_{k_{1}k_{4}k_{3}k_{2}}^{(4)}]a_ {k_{1}}a_{k_{2}}^{*}a_{k_{3}}a_{k_{4}}^{*}+a_{k_{2}}^{*}F_{k_{2}}\Bigg{\}} \tag{3}\]  
The restricted sum in the quartic coupling generally involves three classes of mode: combinations [73; 86]: \(\omega_{k_{1}}=\omega_{k_{2}}\) and \(\omega_{k_{3}}=\omega_{k_{4}}\), \(\omega_{k_{1}}=\omega_{k_{4}}\) and \(\omega_{k_{2}}=\omega_{k_{3}}\), and the remaining possibilities satisfying the frequency-matching conditions, which have been usually disregarded [73; 86]. We consider the diagonal contribution in the quadratic coupling to dominate over the off-diagonal part. By expressing the optical white noise as the sum of additive and multiplicative statistically independent stochastic processes [87], so that \(F_{k}(t)=F_{k}^{(0)}(t)+a_{k}(t)F_{k}^{(1)}(t)\), and considering slow-amplitude modes \(a_{k}(t)\) (if compared to the rapidly evolving phase dynamics), we obtain the Fokker-Planck equation [81; 82; 87] for the probability density function (PDF) of the output intensity,  
\[\frac{\partial P}{\partial t}=-\frac{\partial}{\partial l_{k}}[(-d_{k}I_{k}-b_ {k}I_{k}^{2}+2 QI_{k})P]+2Q\frac{\partial^{2}}{\partial l_{k}^{2}}(I_{k}^{2}P) \tag{4}\]  
where the parameter \(Q\) controls the magnitude of the multiplicative fluctuations through: \([F_{k}^{(1)R}(t)F_{m}^{(1)R}(t^{\prime})]=2Q\delta_{k,m}\delta(t-t^{\prime})\), \(b_{k}=g_{kkkk}^{(4)R}/c_{k}\), and  
\[d_{k}=\sum_{n\
eq k}[g_{kkm}^{(4)R}+g_{kkmk}^{(4)R}+g_{nkkn}^{(4)R}+g_{nink}^ {(4)R}]I_{n}/c_{n}-2(\gamma_{k}-\alpha_{k}) \tag{5}\]  
The steady-state solution of Equation (4) is [81; 82; 87]:  
\[P(I_{k})=A_{k}I_{k}^{-\mu_{k}}\exp{(-b_{k}I_{k}/2Q)} \tag{6}\]  
with \(I_{k}>0\), \(A_{k}\) as the normalization constant, and \(\mu_{k}=1+d_{k}/2Q\). This PDF presents a power-law decay combined with exponential attenuation. Its second moment can be very large, though still finite, depending on the value of \(\mu_{k}\), mainly if \(b_{k}/2Q\ll 1\).