Self-starting nonlinear mode-locking in random lasers
Fabrizio Antenucci, Giovanni Lerario, Blanca Silva Fernandéz, Milena De Giorgi, Dario Ballarini, Daniele Sanvitto, Luca Leuzzi
SSelf-starting nonlinear mode-locking in random lasers
Fabrizio Antenucci , , Giovanni Lerario , Blanca Silva Fernand´ez ,Milena De Giorgi , Dario Ballarini , Daniele Sanvitto ∗ and Luca Leuzzi , ∗ Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France CNR-NANOTEC, Institute of Nanotechnology, Soft and Living Matter Lab., Piazzale Aldo Moro 5, I-00185, Rome, Italy CNR-NANOTEC, Institute of Nanotechnology, Via Monteroni, I-73100 Lecce, Italy. and Dipartimento di Fisica, Università Sapienza, Piazzale Aldo Moro 5, I-00185, Rome, Italy ∗ In ultra-fast multi-mode lasers, mode-locking is implemented by means of ad hoc devices, likesaturable absorbers or modulators, allowing for very short pulses. This comes about because ofnonlinear interactions induced among modes at different, well equispaced, frequencies. Theorypredicts that the same locking of modes would occur in random lasers but, in absence of anydevice, its detection is unfeasible so far. Because of the general interest in the phenomenologyand understanding of random lasers and, moreover, because it is a first example of self-startingmode-locking we devise and test a way to measure such peculiar non-linear coupling. Through adetailed analysis of multi-mode correlations we provide clear evidence for the occurrence of nonlinearmode-coupling in the cavity-less random laser made of a powder of GaAs crystals and its self-starting mode-locking nature. The behavior of multi-point correlations among intensity peaks istested against the nonlinear frequency matching condition equivalent to the one underlying phase-locking in ordered ultrafast lasers. Non-trivially large multi-point correlations are clearly observedfor spatially overlapping resonances and turn out to sensitively depend on the frequency matchingbeing satisfied, eventually demonstrating the occurrence of non-linear mode-locked mode-coupling.
When light propagates through a random medium, scattering reduces information about whatever lies across themedium, fog and clouds being everyday-life examples. The electromagnetic field is composed by many interferingwave modes, providing a complicated emission pattern as light undergoes multiple scattering. In the so-called randomlasers [1–12], this random scattering is used to reach the population inversion activating the lasing action. The randomlaser device is made of an optically active medium and randomly placed light scatterers. The medium provides thegain for population inversion under external pumping. The scatterers provide the high refraction index and thefeedback mechanism of multiple scattering, playing a role analogous to cavity mirrors in standard lasers, and leadingto amplification by stimulated emission. The same material can both sustain the gain and the scattering [1, 2, 5, 6],else two apart components with complementary functionality can be combined[7–13].Standard multi-mode laser theory has shown that that the dominant mode interaction above threshold is highlynon-linear[14], nonlinearity being represented by multi-mode couplings and characterized by mode-locking. In therandom laser case, couplings are predicted to be disordered, both from the point of view of the interaction networkand for what concerns the coupling values. Therefore, cross-mode interactions understanding is a very debated topicand fundamental questions still need to be answered: how strong are the mode couplings? What is their sign? Howmany modes are simultaneously involved in each interaction?Clearly, modes must spatially overlap to manifest mode locking [15–17]. This has been observed in experiments onspecifically designed random lasers, where pairwise (therefore, linear) interaction manifests as the consequence of atwo modes competition for sharing their mutual mode intensities within the same optical volume [18]. Spatial overlapis not, however, a sufficient condition for interaction, nor it provides any information about the coupling values. Atthe same time, the exact structure of the spatial distribution of the modes in commonly used random lasers is hardto be determined, which makes a quantitative analysis of the interacting parameters hard to be obtained. Therefore,we have developed a theoretical analysis - making use of statistical mechanics - of random systems of interacting lightmodes providing information about the mode-coupling constants.Thanks to this statistical physics analysis on the emission spectra of a GaAs powder-based random laser, we notonly experimentally demonstrate the non-linear coupling of spatially overlapping modes, but we also provide evidenceof its mode-locking nature. ∗ [email protected], [email protected] a r X i v : . [ phy s i c s . op ti c s ] M a r RESULTS
Mode-locking in ordered and random lasers
Despite the mode-locking phenomenon is known to be nonlinear and light modes are expected to be coupled, themechanism and nature of this nonlinear coupling in random lasers has never been experimentally tested. On the otherhand, the theory for stationary regimes in an active random medium under external pumping leads to a descriptionin terms of an effective stochastic non-linear potential dynamics for the mode slow amplitudes a ( t ) (more informationin Sec. A of Supplementary information) whose Hamiltonian reads H = − X k | FMC( k ) g (2) k k a k a ∗ k − X k | FMC( k ) g (4) k k k k a k a ∗ k a k a ∗ k + c. c. (1)where the acronym FMC on sums stays for the nonlinear Frequency Matching Condition | ω k − ω k + ω k − ω k | < γ ; γ ≡ X j =1 γ k j (2)where γ ’s are the linewidths of the resonances angular frequency domain. Besides this requirement, further complexityof the mode interaction is hidden inside the g coupling coefficients, g (4) k k k k ∝ Z V d r ˆ χ (3) ( r ; ω k , ω k , ω k , ω k ) · E k ( r ) E k ( r ) E k ( r ) E k ( r ) (3)where ˆ χ (3) is the nonlinear susceptibility tensor of the medium and E k ( r ) the slow amplitude mode of frequency ω k .In standard lasers, when the system is pumped above threshold, the FMC induces phase-locking [19, 20]. Thisis responsible for the onset of ultra-short pulses in standard multimode lasers [14, 21, 22], in which the resonatingcavities are designed in such a way that mode frequencies of the gain medium have a comb-like distribution [23–25].To reach mode-locking, nonlinear devices are employed, such as saturable absorbers, for passive mode-locking, ormodulators synchronized with the resonator round trip, for active mode-locking[14]. In random lasers no evidence hasbeen obtained so far about the occurrence of mode-locking in connection with mode-coupling. Indeed in the randomcase no ad hoc device is present in the resonator and even the definition of resonator is far from straightforward[26].Consequently, mode-locking would be , in case, a self-starting phenomenon due to the randomness of scatterers’ positionand the optical etherogeneity of the random medium.In principle, the direct way to identify a possible mode-locking in random lasers would be to look for a temporalpulse, composed by modes at different frequencies, with a non-trivially locked phase. However, such a putative pulsemight realistically be longer than typical pulses in standard mode-locking lasers and modes at different frequencieswould contribute differently and in an uncontrolled way [27]. Moreover, pulses do not form unless mode frequenciesare regularly separated and mode couplings g (4) take (mostly) positive values. Indeed, it can be theoretically provedthat even when FMC, cf. Eq. (2), is satisfied, continuously distributed frequencies [19] or a non-negligible fraction ofthe random mode interactions given in Eq. (3) [28], prevent the onset of laser pulses.Since the direct observation of mode-locking via optical pulses is neither practical nor conclusive, in this work wedemonstrate a different approach to detect mode-locking in random lasers, based on multi-point cross-correlationmeasurements.Using this method we prove that in a random laser modes at different wavelenghts interact nonlinearly. Moreover,their interaction is mode-locked, i. e., their frequencies satisfy the matching condition reported in Eq. (2). Data analysis
Our random laser is composed by a thin deposition of GaAs powder, cf. Methods. A Gaussian laser beam (780 nmexcitation wavelength) illuminates the sample propagating perpendicular to the deposition plane (x,y). The detectionline is along the z direction in transmission configuration (i.e., at the opposite side of the sample with respect to theexcitation line). The sample thickness is irregular in the z direction, though always thinner than 100 µ m. Negative optical response is supposed to occur in the so-called glassy random lasers [15, 17, 29, 30]
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Peaks in the spectra are identified by performing multiple fitting with linear combinations of a variable numberof Gaussians (details are reported in Supplementary information). To avoid overfitting, the optimal set of curvesis chosen according to the Akaike Information Criterion [31]. An instance of the outcome of our fitting procedureis reported in Fig. 1, right panel, where we plot raw data compared to multi-Gaussian interpolating functions.Eventually, we build a complete list of all resonances for each spectrum produced in each one of the different dataacquisitions t = 1 , . . . , N spectra = 1000 in the series of measurements. Each intensity peak k of the spectrum t isdetermined by its frequency ω k , its linewidth γ k , its position x k and the FWHM ∆ x k in its position coordinate: I ( t ) k ≡ I ( t ) ( x k , ∆ x k ; ω k , γ k ). Strong correlation discrimination
Of each set of intensity peaks we compute the normalized fourth order cumulants of their intensities c ( ω j , ω k , ω l , ω m ),cf. Methods. Since there are many spurious effects that may contribute to a correlation among modes, in order toidentify anomalously large correlations we need a reference for the background correlation. One can observe thatthe largest correlation values (in the tails of the distributions) turn out to be attained always on those sets whoseresonances taken at the same time overlap in the x position (SOIR), with respect to NOIR and background correla-tions. Moreover, comparing the distributions in Fig. 2 as the number of acquired emissions increases, the dominionof possible values for the SOIR correlations extends its extremes in the low probability tails. This is not the case, onthe contrary, for NOIR and background c , insensitive to the change in acquisition time. We, thus, compute threekinds of multi-point correlations: ) c ) c x = 7, 143, 222, 313 - 1000msx = 7, 143, 222, 313 - 100msx = 7, 143, 222, 313 - 10ms bg (c ) c x=217 µ m - BG 100msx=217 µ m - BG 10ms FIG. 2. Normalized distribution of c for acquisition times 10, 100, 1000 ms. (a) c taken at the same shot, same x = 210 µ m.(b) c taken at the same shot, but different positions x . (c) c taken at the different shots, same x . ) σ bg c x = 217 µ m - diff shots (bg)x = 7, 143, 222, 313 µ mx = 217 µ m FIG. 3. Normalized distributions of c at acquisition time 100 ms for background correlations, non-overlapping resonancecorrelations and spatial overlapping resonance correlations. The vertical lines correspond to ± σ of the background correlationdistribution. • SOIR, the correlations of all quadruplets composed by possibly spatially self overlapping intensity resonances,i.e., occurring at the same planar position x ; • NOIR, the correlations quadruplets composed by non overlapping intensity resonances, i.e., occurring at welldistinct x positions in the same spectral data acquisition; • BG, the background correlations among resonances pertaining to different spectra, i.e., acquired after differentpump shots.In Fig. 2 (a) we display the probability distributions of the SOIR four-point correlation functions c for 100, 1000 and10000 shots. Sets of SOIR are candidate to be nonlinearly interacting. In the center panel of Fig. 2 we display theNOIR c . The latter set might still be composed by interacting modes in an extended mode scenario [32]. Finally, inthe right panel of Fig. 2 the BG c are plotted.With high confidence, we, finally, operatively identify nonlinearly interacting sets of modes as those whose multi-mode correlation is larger - in absolute value - than the 3 σ of the background correlation distribution. In Fig. 3 wesuperimpose instances of the normalized distributions for the background, the NOIR and the SOIR correlations foran acquisition time of 100 ms, clearly showing that the tails of the SOIR extend well beyond the 3 σ of the other two.The presence of these interacting sets of modes is our first result: spatially overlapping modes interact nonlinearlyin the random lasing regime. We recall that the spectra results from the integration over 10 µ m along the y direction and a deposition thickness lower than 100 µ mso that not all resonances at same x are guaranteed to be actually spatially overlapping, see Supplementary information. σ c ( ∆ ) ∆ SOIR x = 210NOIR x= 143, 222, 313, 379bg x = 210 σ c ( ∆ ) ∆ SOIR x = 217NOIR x: 143, 222,313, 379bg x = 217
FIG. 4. Mean square displacement of the four-point correlation c values vs. ∆ at 100 ms acquisition time for backgroundcorrelation (bg), non-overlapping resonances and space-overlapping resonances. Left: at pixel x = 210(7) µ m. Right: x =217(7) µ m. Frequency matching role in strongly correlated modes
Eventually, we are interested in testing whether the frequencies of interacting modes satisfy FMC. This is indeed,a signature for self-starting mode-locking in the GaAs random laser. To make FMC quantitative, cf. Eq. (2), weintroduce a “FMC parameter” against which we can straightforward test multi-mode correlations. In the case of the4-mode correlation, taking for illustrative purpose modes 1 , , ∆ ≡ | ω − ω + ω − ω | γ + γ + γ + γ (4)In Fig. 4 the mean square displacement σ c of the correlations among modes within a given ∆ interval are displayed.For correlations computed from a series of 1000 spectra, each one acquired in 100 ms, we plot σ c the distributions ofthe SOIR, of the NOIR and of the background correlations. It can be observed that no dependence on ∆ is shown forBG and NOIR correlations. On the contrary, the σ c ’s of SOIR correlation distributions depend on ∆ . In particular,non-trivially strong correlations only occur among the SOIR, and only for small ∆ , signaling that in those sets ofmodes - those most probably coupled - their interaction strength depends on how well FMC is satisfied.This behavior occurs at all acquisition times used in experiments, as it can be observed in Fig. 5. In particular, onecan observe that, decreasing the acquisition time, i.e., decreasing the number of recorded photon emissions after eachpumping laser shot, the threshold value of ∆ ML4 below which surely interacting modes can be neatly discriminatedfrom background correlation decreases.Ideally, to analyse mode-locking, for every single shot one would like to have the intensity of emission measured asa function of space ( x, y ) in the sample plane, to control the spatial overlap, and contemporarily resolved in angularfrequency ω , to check the FMC, Eq. (2).Although the integration over many shots (the emission intensity is too low to allow the resolution of the energy-space emission map of a single shot experiment), we can clearly observe the shrinking of ∆ ML4 with decreasing numberof shots, consistently with the upper theoretical limit of ∆ th4 = 1 for single shot experiments.The evidence that only SOIR show ∆-dependent correlations is a clear indication of the onset of nonlinear mode-locking in random lasers and our main result: sets of modes satisfying FMC are much strongly correlated than setsof modes not satisfying FMC. The former non-trivially large correlations is to be accounted for by the interaction ofthe modes. The latter cannot be distinguished from ∆-independent background correlations. There are actually three non-equivalent permutations of indices in Eq. (2). We always consider all three distinct permutations and takethe smallest ∆ . ∆ ML m s σ c ( ∆ ) ∆ x = 222 µ m, 10ms bg 10msNOIR 10 msaverage bg σ
0 1 2 3 4 5 6 7 ∆ ML m s ∆ x = 222 µ m, 100msbg 100msNOIR 100 msaverage bg σ
0 1 2 3 4 5 6 7 ∆ ML m s ∆ µ m, 1000msNOIR 1000 msaverage bg σ FIG. 5. Mean square displacements σ c of the distributions of four-resonance correlation values c at fixed ∆ intervals. ForSOIR quadruplets at x = 222 µ m we use large full points, for instances of background (non-iteracting) quadruplets we use smallempty points and for the instances of NOIR quadruplets small full points. The σ c ’s are plotted versus the FMC parameter∆ for acquisition times 10, 100 and 1000 ms, corresponding to a spectral integration over, respectively, 100 (red), 1000(blue) and 10000 (green) pumping shots. Dashed lines are parabolic interpolations of SOIR σ (∆) behaviors. Background andNOIR correlations do not show any dependence on FMC, whereas SOIR correlation distributions, with large tails at ∆ ’ ML4 at which SOIR σ ’s decreases to values of the order of background σ ’s depends on the acquisition time. In particular, ∆ ML4 decreases with thenumber of shots, towards the expected limit of ∆
ML4 (cid:46)
DISCUSSION
In the present work we considered multi-mode correlations among spatially overlapping intensity resonances (SOIR),spatially non-overlapping intensity resonances (NOIR) and background correlations among resonances in independentspectra (BG). Firstly, by compared analysis of sets of background correlations and correlations among NOIR wecannot appreciate any difference in the behavior of their distributions, as reported in Figs. 2 and 3. Since correlationsamong NOIR are not any larger than background correlations we cannot discriminate possible long-range non-linearmode-coupling with respect to noise.This is not the case for SOIR multi-point correlations. Calibrating P SOIR ( c ) by means of the background correlationdistributions, we identify interacting quadruplets as those composed by modes whose intensity fluctuations correlationlies in the tails of their distribution, actually extending beyond 3 σ BG of the Gaussian interpolation of the c BGdistribution, as shown in Fig. 3.The same behavior is found also if we change the pumping power, as far as the random laser is above threshold andclear resonances can be distinguished in the space-energy spectrum (Supplementary information).As observed in Section Results, cf. Figs. 4, 5, the frequency matching condition, Eq. (2), appears to play adeterminant role in the distribution of the c values of interacting set of modes. Moreover, we observe that theshorter the acquisition time, the smaller the range of values of ∆ at which large correlation occurs. According toEq. (2), interaction between modes would be allowed only to modes whose energies satisfy the FMC relationship. Interms of data reported in Figs. 5 this would imply that interacting mode sets should appear for ∆ (cid:46) O (1). Given thestatistical nature of (i) the modes identification and (ii) the interaction recognition from anomalously large multi-pointcorrelations, the outcome is strongly compatible with such a requirement.This observation is a strong evidence in favour of the occurrence of mode-locking in random lasers, that is, the samemechanisms behind the nonlinear mode coupling in standard, ordered, multimode lasers, though without any ad hoc device like a saturable absorber or a modulator. It is a self-starting mechanism induced by randomness.As a last remark we recall that in the ordered case mode-locking is responsible for ultra-fast pulses. On the contraryin random lasers, no train of pulses is present, because the distribution of frequencies is random, rather than comb-like[23], preventing the rise of a pulse even in presence of frequency matching and phase-locking. Indeed, the Fouriertransform in time does not produce a modulated signal with a short envelope [19]. Actually, also in presence of almostequispaced resonances it would be extremely difficult to identify a pulse shorter than the pumping laser pulse and, inthe subclass of optically random media displaying glassy random lasers, this might not be feasible at all [28]. To unveilthe self-starting mechanism beyond the just demonstrated locking of modes in random lasers mandatorily requires theidentification of mode phases. We believe that the presented results might be a significant step to stimulate and leadthe theoretical understanding and the experimental procedures necessary to provide a protocol to determine modephases in random lasers. MATERIALS AND METHODSSamples
The sample is made by grinding a piece of GaAs wafer - bathed in methanol - in a pestle and mortar, in order toobtain a paste with grain typically smaller than 10 µ m. The resulting paste is then highly diluted in methanol anddeposited by drop casting on a glass substrate. During the deposition, the glass substrate is placed on a heater to havea fast evaporation of the solvent. The packing density and sample thickness is increased by repeating 50 times thedrop casting process. Obviously, the sample is highly inhomogeneous. However, the thickness never exceeds 100 µ m.A second glass slide covers the sample, which is finally sealed with parafilm on the sides. Experimental setup and measurements
The laser source is a 30 fs pulsed laser at 780 nm with repetition rate 10 KHz. The excitation line is orthogonal tothe sample surface, while the detection line is along the opposite side of the sample (i. e., transmission configuration).Adjusting the spot size and excitation power, we can roughly control the number of active random lasers loops. TheGaussian spot size is tuned to about 300 µ m (FWHM); this spot size guarantees a total number of emitters thatare easily distinguishable when the real space emission map is projected on the CCD camera. The detection lineconsists of two plano convex lenses projecting the sample plane at the spectrometer entrance (so in focus on the CCDcamera). The spectrometer slits aperture allows the selection of a vertical slice of the emission space map and theenergy resolution of the emission spectrum. The experiments have been performed with a slit aperture correspondingto 10 µ m horizontal selection on the sample surface, in order to have the selection of a single emitter along thehorizontal axis. The overall numerical aperture of the detection line is 0 . Multi-point correlation definition
The fourth order connected correlation function of intensity peaks I j ( ω j ) reads: C ( ω j , ω k , ω l , ω m ) = h I j I k I l I m i − h I j I k I l ih I m i − h I j I k I m ih I l i − h I j I m I l ih I k i − h I m I k I l ih I j i−h I j I k ih I l I m i − h I j I l ih I k I m i − h I j I m ih I k I l i + 2 h I j I k ih I l ih I m i + 2 h I j I l ih I k ih I m i +2 h I j I m ih I k ih I l i + 2 h I k I l ih I j ih I m i + 2 h I k I m ih I j ih I l i + 2 h I l I m ih I j ih I k i − h I j ih I k ih I l ih I m i where the average is taken over the statistical sample of all the combinations of the same set of modes displayedin experiments at fixed external condition and stable pumping. We, further, normalize C to the mean squaredisplacements of the intensities of the modes, i.e., c ( ω j , ω k , ω l , ω m ) ≡ C ( ω j , ω k , ω l , ω m ) σ j ( ω j ) σ k ( ω k ) σ l ( ω l ) σ m ( ω m ) σ j ( ω j ) = q h ( I j − h I j i ) i SUPPLEMENTARY MATERIALSA. Theoretical modeling
The dynamics of the electromagnetic field is suitably expressed in the slow mode decomposition E ( r , t ) = X k a k ( t ) e ıωt E k ( r ) + c.c. (5)where the modes E k ( r ) of angular frequency ω k are such that their complex amplitudes a ( t ) evolve slowly with respectto ω − and follow a nonlinear stochastic dynamics. In standard mode-locked lasers with passive mode-locking inducedby a saturable absorber, e. g., the phasors dynamics is given by the so-called Haus master equation [14], that, in theangular frequency domain [21], reads:˙ a k ( t ) = ( g k − ‘ k + ıD k ) a k ( t ) + ( γ + iδ ) FMC X k k k a k ( t ) a ∗ k ( t ) a k ( t ) + η k ( t ) (6)where g k , ‘ k and D k are, respectively, the frequency-dependent components of the gain, the loss and the dispersionvelocity. In the case of passive mode-locking, the real part of the non-linear coefficient γ represents the self-amplitudemodulation coefficient of the saturable absorber and the imaginary part δ represents the coefficient of the self-phase-modulation caused by the Kerr-effect. The acronym FMC on sums stays for Frequency Matching Condition, cf. Tab.I and Eq. (2) MT in the main text, arising in the approach leading to the master equation (6), after averaging out singlemode’s fast oscillations ∼ e ıω n t [14, 16, 20–22, 33, 34]. The white noise η n ( t ) is a stochastic variable representing thecontribution of spontaneous emission, linked to the thermal kinetic energy of the atoms through its covariance h η k ( t ) η n ( t ) i ∝ T δ kn δ ( t − t ) (7)where T is the temperature. For arbitrary spatial distribution of modes E ( r ) and heterogeneous susceptibility, i.e.,when the system is intrinsically random, starting from quantum dynamical Jaynes-Cumming equations, downgradingfrom quantum creation-annihilation operators to classical complex-valued amplitudes, taking into account spontaneousemission and gain saturation, a generalized phasor dynamic equation is recovered for random lasers [17]:˙ a k ( t ) = FMC X k g (2) k k a k ( t ) + FMC X k k k g (4) k k k k a k ( t ) a ∗ k ( t ) a k ( t ) + η k ( t ) (8)where we have considered only the first nonlinear term satisfying time reversal symmetry in the electromagneticphasor’s dynamics. Further terms would only perturbatively modify the leading behavior of the fourth order term.Odd terms like those related to the χ (2) optical susceptibility, occurring in non-centrometric potentials, can also beincluded theoretically but practically will play no role because of the usually limited wavelength dominion in theintensity spectra of the random lasers . The complexity of the mode interaction is hidden inside the g coefficientsin Eq. (3) MT of the main text. Finally, recognizing in Eq. (8) a potential Langevin equation, the effective phasorHamiltonian of Eq. (1) MT in the main text is derived. ω ’s indices FMC2 k ≡ k , k | ω k − ω k | < γ k + γ k k = k , k , k | ω k − ω k − ω k | < γ k + γ k + γ k k = k , k , k , k | ω k − ω k + ω k − ω k || ω k − ω k + ω k − ω k || ω k − ω k + ω k − ω k | < γ k + γ k + γ k + γ k TABLE I. Summary of all possible FMC’s for modes interacting via Eq. (1). Pairwise, three- and four-body terms are reported.
B. Observable definitions and data analysis
Identification of the intensity peaks of the modes by multi Gaussian interpolation
We herewith describe the fitting procedure composing the resonances identification step in Section Results in themain text. In a first series of measurements, because of a large refinement in energy with respect to the behavior ofemission spectra, we have first coarse-grained the position-energy grid binning four pixels in the λ direction. Thiscorresponds to a resolution of 0 .
15 nm in the wavelength (and 1 . µ m in position). We have 335 pixels in the wavelengthdirection, with the lowest extreme being λ = 843 . . λ , this means that λ [nm] = 52 . / ∗ (pixel −
1) + 843 . ∈ [1 : 335] (9)In the second series of experiments the wavelength pixels are only 672, with resolution already of 0 .
149 nm (and0 . µ m in position) and there was no need for binning.We, then performed multi-Gaussian interpolations of spectra I ( x ; λ ) at fixed position x = 1 , . . . , N x . We denotethe number of Gaussians employed by N G and we estimate two parameters for each Gaussian n = 1 , . . . , N G , mean No chance of second harmonic generation, for instance. ¯ λ n and variance σ n , as well as a relative weight w n , for a total number of parameters K = 3 N G . At each position x ,and for each N G , we compute the log-likelihoodln L (cid:0) λ |{ w n } , { ¯ λ n } , { σ n } (cid:1) = ln N G X n =1 w n exp " − (cid:0) λ − ¯ λ n (cid:1) σ n (10)The best fit at each x is the one whose parameter estimators for { w n , ¯ λ n , σ n } maximize the log-likelihood ln L or,equivalently, minimize the so-called the Akaike parameter A = 2 K − L built with the least number of Gaussians. i.e., the least K . This is the Akaike Information Criterion to avoid overfittingand underfitting.To each experiment t of the series, a grid of mode intensities is associated: I ( t ) ( x, ∆ x ; λ ; ∆ λ ), where ∆ x and ∆ λ arethe Full Width Half Maximum of the interpolated distributions around, respectively, x and λ (∆ λ n = 2 √ σ n ).We consider the same single mode n as present in the spectra of two different experiments t A and t B of the series if x { t A } n = x { t B } n | λ { t A } n − λ { t B } n | < δλ λ An − ∆ λ Bn ∆ λ An + ∆ λ Bn < ¯ δλ where the absolute and relative uncertainties δλ and ¯ δλ are chosen depending on the resolution required. ∆ x is notconsidered in the analysis but it is of the order of 6 pixels. I. e., 7 µ m in the first series of experiments and 4 µ m inthe second one.In both series of experiments we used δλ = 1 . δλ = 0 .
1. For spectra of lower resolution and less intensepeaks we also considered a rougher coarse graining, where two parameters in the mode identification are δλ = 4 . δλ = 0 .
3. We resorted to this less precise approximation exclusively when the modes’ statistics is too low toprovide clear behaviors of the distributions of c values. This is the case, e.g., in some of the data with acquisitiontime of 10 ms and low pumping energy in the second series of experiments (see Sec. D). Localization of resonances in the monochromator vertical direction y Having nearby x coordinates for the modes is a necessary condition to have spatial overlap of the intensity peaks.Indeed, mode’s extensions should also overlap in the y direction, the horizontal direction of the slit, and in the z direction, the thickness of the sample. Knowing the total range in y and z , we can estimate the probability that iffour modes have the same x they will be effectively mutually overlapping also in y and z . Let us call L y the total y -range, equal to the slit opening ( L y ’ µ m) and ∆ ≡ ∆ y ’ ∆ x = 7 µ m the typical extension of the most peakedresonances that we analyzed. Let us, then, take as indicative average depth of the sample L z ’ µ m, the total z -range. The probability that four modes of roughly the same extension ∆ at the same x share a non-zero spatialintersection along the y (or z ) coordinate, in a total range L y (or L z ) (enough larger than ∆), can be estimated as p ( y,z )4 (∆ w , L w ) = 8 ∆ w L w ; w = y, z (11)This is independent from the coordinates of the modes. For the setup and the sample we used the probability ofhaving a four mode overlap at given x position is, thus, p ( x )4 = p ( y )4 (7 , × p (7 , ’ .
060 (12)This means that once we know that four resonances at different wavelength of the acquired spectra are at the samepoint x , they will be actually overlapping in 3D space only a fraction p ( x )4 of the times. In the results shown in thispaper, we always have at least ∼
100 quadruplets of resonant modes at given x coordinate, so that we can expectwith high probability that in all cases there are at least few spatially overlapping sets for all the cases shown. Atthe same time, the fact that the probability of overlapping is relatively low ( (cid:46) C. Multi-set mode correlation vs. statistics
We further considered how finite sample size may affect the results shown in paper. In particular, note that typicallythe number of possible quadruplets decreases rapidly as a function of ∆ . We therefore performed a further test toverify whether a high (low) correlation c at a given ∆ value might be related to the presence of a large (small)number N q of sets of modes for the same frequency combination ∆ . In Fig. 6 the rescaled histograms of N q asfunction of ∆ are plotted together with the average value of the modulus of the correlation | c | at the same ∆ : nospecial correlation can be observed. ∆ x=210 µ m -- 10 ms -- P( ∆ )x=210 µ m -- 10 ms -- |-c— |-( ∆ ) 0 0.5 1 1.5 2 2.5 3 3.5 ∆ x=210 µ m -- 100 ms -- P( ∆ )x=210 µ m -- 100 ms -- |-c— |-( ∆ ) 0 0.5 1 1.5 2 2.5 3 3.5 ∆ x=210 µ m -- 1000 ms -- P( ∆ )x=210 µ m -- 1000 ms -- |-c— |-( ∆ ) FIG. 6. Normalized histogram P q (∆ ) of the number of SOIR quadruplets (at position x = 210 µ m) whose frequencies yielda given ∆ compared to a average value of the modulus of the correlation c . To be quantitative, we, further, computed the Pearson correlation coefficient r between | c (∆ ) | and N q (∆ ) forboth the SOIR and NOIR cases, as reported in Fig. 7. We stress that the NOIR correlations are typically evaluated ona much larger number of quadruplets than the SOIR ones, as in the former case one considers all possible quadrupletsamong modes in four different positions while in the latter only among modes at one fixed position. The BGcorrelations are instead always evaluated with the same statistics as the SOIR ones at the same position, for an easiercomparison. We observe that the value of r is (i) always relatively small (cid:46) .
3, (ii) without a definite value overdifferent experiments and (iii) of the same order of magnitude in the two cases SOIR and NOIR. This observationreinforces the conclusions of the previous sections, showing that our result of a dependency of c from ∆ cannot beexplained as a small sample size effect. -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 NOIR SOIR
10 100 1000 ms 10 100 1000 msfull symbol ∆λ = 4.5 nm empty symbol ∆λ = 1.5 nm Correlation of P q ( ∆ ) and |-c— |-( ∆ ) FIG. 7. Pearson correlation between the average | c (∆ ) | of the four-point correlation values and the number of quadruplets P q (∆ ) over which the correlations are computed. Values obtained using data from NOIR and SOIR quadruplets are compared.Outcome from experimental series at well different acquisition times (10 , , λ = 4 . λ = 1 . D. Mode-locking dependence on pumping power
Eventually we have carried out our analysis on the same sample for different powers of the pump laser and exploredif and how non-linear mode-coupling depends on pumping. In Fig. 8 we compare the probability distributions ofthe background 4-peak correlations and the SOIR correlations at given resonance locations at two different pumpingpowers, 25 and 60 mW. The same effect of long tails and non Gaussianity is observed at all powers for which clearintensity resonances can be resolved in space and wavelength and, once the mode resonances are there, no furtherdependence on the external power is observed. ) σ bg pixel x = 440(6) c BG 25 mWBG 60 mWSOIR 25 mWSOIR 60 mW
FIG. 8. Comparison between P ( c ) of SOIR and BG correlations at different pumping power P = 25 mW and 60 mW atthe same resonance at the same point x = 400(5) µ m for acquisition time 100 ms (1000 shots). At both powers the SOIRdistributions have wider tails than the background ones. Both distributions do not appear to change with the pumping power. References [1] Cao, H. et al.
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The authors thank D. Ancora, G. Gradenigo, M. Leonetti, A. Marruzzo for useful discussions. The research leadingto these results has received funding from the Italian Ministry of Education, University and Research under thePRIN2015 program, grant code 2015K7KK8L-005 and the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation program, project ElecOpteR Grant Agreement No. 780757 and projectLoTGlasSy, Grant Agreement No. 694925.
Author contributions
G.L. and B.S.F. performed the measurements on the random lasers; G.L. prepared thesamples; G.L. and D.S. designed the experimental setup; D.S. coordinated the experimental work; F.A. and L.L.proposed the theoretical framework and performed the data analysis; L.L.. prepared the manuscript with input fromF.A., D.S. and G.L. All authors contributed to the discussion of the data and to the final draft of the manuscript.