SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION The mode-loking transition of random lasers Maro Leonetti, Claudio Conti, 2 and Cefe Lopez Instituto de Cienia de Materiales de Madrid (CSIC) and Unidad Asoiada CSIC-UVigo, Cantoblano 2849 Madrid España. 2 Dep. Moleular Mediine and CNR-ISC Dep. Physis, University Sapienza, P.le Aldo Moro 5, I-85, Roma Italy (Dated: July 25, 2) EXPERIMENTAL SUPPLEMENTARY INFORMATION Sample preparation Our newly oneived, disordered laser devie onsists in a self assembled luster of titania partiles embedded in dye doped visous liquid. The solution is prepared by mixing equal parts of ethanol and ethylene glyol to whih we added.5% volume of rhodamine B and.5% volume of sattering powder (rutile, average size < µm refrative index n=2.7). A small drop (2µl) of the solution is aged between 2 mirosopy square overslips separated by 5 µm thin plasti spaers and sealed by yanoarylate glue, obtaining a entimeter sized squeezed drop, in whih many sparse lusters aggregate. The single luster (average thikness µm) lyes on the bottom overslip due to high speifi weight. SLM BE Spetrograph Sample Mirror DT P2 F BD Experimental setup The shape of the pump pulse (generated from a frequeny doubled Nd:YAG pulsed laser, frequeny 532 nm, maximum pulse energy 25 mj, pulse duration 9 ns), is ontrolled by a spatial light modulator (Haloeye, LC-R 8, 92x2 pixell) in amplitude modulation onfiguration [] (see setup sheme Figure ). Light is olleted by a 5 µm fiber plaed in the image plane of an enlarging (x) telesope in transmission geometry, thus olleting light from 5 µm area of at the enter of the sample. A olored filter eliminates residual pump light. Spetra are obtained by a 33mm foal length spetrograph (Andor, Shamrok 33) onneted to a low noise Charge Coupled Devie array (Andor, idus Spetrosopy CCD). Definition of the degree of spikiness S and orrelation C We haraterized the spikiness of a random laser spetrum by its Fourier transform normalized to a unitary area (FTS). After defining a utoff K=.2 nm in the horizontal sale of the FTS, we alulate S as the area of the FTS lying in the high period part from K (i.e., orresponding to periods greater than K). S returns a value lose to one for very spiky spetra, while a value P 532 nm FIG. : Experimental Setup. The vertially polarized laser beam is first refleted by an high power dihroi polarizing mirror (P) to lean residual unpolarized light and then passes in a 5x beam expander (BE) before impinging on the spatial light modulator (SLM). A demagnifying telesope (DT) images the spatial light modulator on the sample after a polarizing dihroi mirror (P2) selet horizontal polarization, while residual vertially polarized light is absorbed by a beam dump (BD). Emitted light is olleted by a mirosopy objetive (numerial aperture, working distane 6 mm), to be imaged by an enlarging telesope (x) on a 5 µm ore fiber, after being filtered by an edge filter(f) utting residual green laser light. Spetrum is then obtained by a CCD onneted to the spetrograph. lose to is retrieved for smooth spetra. Inter-mode interation is haraterized by the orrelation in intensity flutuations. First we measure the ensembles relative to modes at λ and λ 2 of intensity values obtained from N= shots: [I(λ )] and [I(λ 2 )]. The pair nature photonis

2 orrelation C p is defined as : N i= C p = (I(λ ) i I(λ ))(I(λ 2 ) i I(λ 2 )) N i= (I(λ N ) i I(λ )) 2 i= (I(λ 2) i I(λ 2 )) 2 () where I represent average over the ensemble. If two modes show a C p lose to their flutuations are unorrelated. Modes that are spatially overlapped may share or ompete for energy resulting in positive or negative values of C p. We alulate the average orrelation C (figure 2 main text) as the average of C p over 5 wavelength pairs Stati Disorder One the lusters are assembled we observed by optial imaging that their position remains fixed for days. The RL spetra is ritially dependent on luster stability that we further investigated by monitoring flutuations in the spekle pattern obtained by shining a single luster with a ontinuous Helium-Neon laser before and after 5 shots from the pump laser, in a similar way to referene [2]. The two spekle pattern in figure 2 show a orrelation of about.85 orresponding to a stati disorder. a) b) FIG. 3: C (irles) and S (squares) as a funtion of Θ in the fixed volume onfiguration. Pumping is. nj/µm 2 a larger irumferene, but in this ase for every value of Θ we also adjusted the value of the radius R of the external irumferene in order to onserve the area of the illumination surfaes. In pratie an inrement of Θ orresponds in a lowering of the value of R to preserve a fixed total laser spot of.6 mm 2. Figure 3 shows the relative measurement of C and S as a funtion of Θ for this fixed volume onfiguration. This measurement allows to rule out the effets of volume or energy dependent terms, like amplified spontaneous emission. It is also worth mentioning the fat that S and C do not depend on how many wedges are used to ompose the pumping area. Although we show results for only two wedges any number an be used to obtain similar results. FIG. 2: Passive spekle pattern from a single luster obtained before (panel a) and after (panel b) 5 pump shots. The orrelation between the two images is.85. Intensity flutuation of the random lasing spetra Measurement at fixed volume The parameter that ontrols the transition between unorrelated and mode loked random lasing is the angle Θ that defines the possible input diretions on the luster. To inrease Θ we add to the inverted area angular setions of a disk. Although this hoie learly defines the input diretions, a variation of Θ does not onserve volume of the illuminated part of the sample and total energy impinging on the sample. To onfirm that the hange in regime is due the quality of the inter-mode interation, we performed a measurement at fixed volume and energy. The measurement is very similar to the previous one with a entral disk plus the angular setors of FIG. 4: C (irles) and Σ (squares) as a funtion of Θ for the luster C. To ensure that the behavior of C at large Θ is due to interation and not to a lowering of the overall flutuations, we monitored the single frequeny flutuations as a funtion of Θ. Figure 4 show C together with relative flutuation Σ defined as σ(λ)/i(λ) (where σ(λ) is the standard deviation of the intensity for the frequeny λ 2 nature photonis

3 over shots) averaged for the 5 more intense modes of luster C. The fat that Σ is nearly onstant for all the values of Θ investigated onfirms the mode loking piture. a regime the eletromagneti field an be expressed as a superposition of modes and their amplitudes obey the oupled mode theory equations (see, for example, [5] and referenes therein) Time resolved measurements da n dt =+iδ n a n α n a n + m n κ n,ma m + g(δ n ) a n +γ n a n, 2 (2) The behavior of C and S reported in the main artile may be obtained also with a pioseond pump pulse (we used a 3 ps pulse frequeny doubled ND:YAG laser with 6 mj/pulse maximum energy) that also allows to resolve temporal behavior of emitted light [3]. Light is olleted from a single luster with a photodiode ( nanoseond risetime and ns falltime) and a 3MHz osillosope. Full width at half maximum of the peak in intensity is obtained by averaging 8 single shots time domain measurements. Measurements for luster C are reported in figure 5. Lasing in the orrelated regime (Θ=4 ) shows shorter pulses with respet to the unorrelated (Θ=8 ) random laser for all the 5 lusters. The measured average shortening is about 3% for a fluene of. nj/µm 2. The same figure 5 also reports temporal shape of fluoresene obtained pumping the sample only with the entral disk (Θ= ). FIG. 5: Intensity as a funtion of time for the luster C pumped with wedges of different aperture. In the ordered ase pulse duration sales as /N [4], onversely in the disordered ase is not at all expeted that the pulse duration dereases substantially with the number of modes; on the other hand, it is reasonable that some shortening happens beause the modes at synhronously, e.g., they tend to turn at the same time, beause as one of them gets energy, it also feeds the others (see figure 9). COUPLED MODE THEORY (CMT) EQUATIONS AND THEIR NUMERICAL SOLUTION Introdution A luster of dieletri partiles sustains several eletromagneti resonanes, whih are haraterized by a distribution of finite life-times and speifi frequenies. In suh with δ n = ω n ω, with ω n the angular frequeny of the resonane, or mode, with index n, and ω the transition frequeny of the resonant system. In previous work[6, 7] the ontinuous limit of suh equations has been shown to desribe the emission linewidth from random lasers, when it appears as a single smooth peak. Here we show how eqs.(2) an desribe the transition for regime in whih the spetrum is omposed by various peaks to a regime in whih there is a single olletive line. This transition is aompanied by a proess of mode-loking between the modes, i.e., the fat that the phases of the resonanes get loked in their evolution. In the following we detail the parameters in (2): Resonant frequenies and their losses are given respetively by the parameters δ n and the inverse deay times α n. Various previous investigations have shown that in general frequenies tend to be statistially equispaed, i.e., the mode spaing has a peaked distribution due to level repulsion. In addition the deay time distribution α n is known to have a wide bell shaped distribution, enompassing strongly loalized modes (smaller α n ) and deloalized ones (larger α n, see e.g. [8] and referenes therein). In our numerial approah we onsidered various sets of parameters, those we disuss in this manusript are a representative ensemble. The parameters in (2) an be saled by an arbitrary fator time-sale t, by letting t t/t ; all the frequenies and loss parameters are given in units of /t. In our ode we generate N = 5 modes suh that the frequeny spaing is Gaussian with mean value ω/n, with ω =. and relative standard deviation.; similar results are obtained by varying the number of modes and the standard deviation of the frequenies. The deay times are taken Gaussianly distributed with mean value α =. and standard deviation.; we also onsidered the ase of uniformly distributed deay times with very similar results. Gain The linear gain linedwidth an be approximated by a paraboli profile: [ ( ) ] 2 δ g(δ) =g (t), (3) where g is the gain at the referene frequeny ω, taken at the peak of the fluoresene signal, and orresponding to δ =. δ g measures the gain bandwidth. The gain saturation oeffiients is γ n. As the amplitudes of the modes δ g nature photonis 3

4 (a) n = Normalized Wavelength.5 Spetrum (arb.units) n = Normalized Time FIG. 8: Phase of 5 modes versus normalized time (the pumping duration is in the region [., ]) for three different number of ouplings. n = n =3 n =6 n =2 n =4 n =6 n =8 Normalized Wavelength FIG. 7: Average spetra over shots for various number of ouplings n. an be saled by a normalizing fator, γn an be taken as dimension-less number and we assume γn =. Similar results are obtained by taking γn as a random number Gaussianly distributed with unitary mean value. In our dimensionless units we take δg =.3 (orresponding to a fluoresene linewidth of the order of the onsidered ω as in the experiments). The pulsed pump is represented by a Gaussian pulse 2 g (t) = g exp{ [(t tp )/tg ] }; results presented here 4 are for g = and tg = and tp = 2 4 (units of t ). Coupling oeffiients The oupling oeffiients will be taken all equal and an order of magnitude smaller to the mean inverse life-time. Suh parameters are very diffiult to be estimated, but they are expeted to be smaller than the mean inverse life-time, suh that the oupling matrix of the linearized equations (2) are positive definite. In the following we vary the number of oupled modes and let the oupling oeffiient onstant (i.e., κn,m an be either or κ, κ =.α ). We have made some numerial simulations by taking eah κn,m as Gaussianly distributed, but we did not find a substantial hange in the results disussed below. Numerial solution We onsider a set of N modes starting from Gaussianly distributed amplitudes with mean modulus., muh lower that the value reahed at osillation. We onsider the ase in whih eah mode is oupled with a number 4 8 FIG. 6: Spetrum obtained after averaging numerial solutions of the CMT equations (2): (a) no ouplings n = ; (b) as in (a) for n = n = n =6 (b) Phases (deg).8 Energies (arb. units) Spetrum (arb.units) Normalized Time FIG. 9: Energies an 2 of 5 modes versus normalized time (the pumping duration is in the region [., ] in the adopted units of 5 t ) for three different number of ouplings. 2 n of other randomly-hosen modes. In figure 6, we show two emission spetra obtained by the numerial solution of (2), and averaged over shots (for eah shot the deay time and the initial onditions are randomly extrated). We start omparing the ase of no-oupling (n = ) with the ase n =, in figure 6. In the ase of nooupling (figure 6a) the spetrum exhibits many peaks orresponding to the ωn, while for strong oupling (n = ) one obtains a smooth gaussian spetrum [6, 7] as in figure 6b. In figure 7 we show the average spetra obtained when inreasing the ouplings. As the number of ouplings inreases the spetrum tends to the a smooth Gaussian shape. This kind of transition orresponds to the onset of a synhronous regime between the modes, whih beomes more and more pronouned as the number of ouplings grows. The transition from a spiky to ontinuous spetrum is hene due to the phase-loking between modes. This is shown in the following figure 8 where we superimpose the time-dynamis of the phases. The modes tend to synhronize when the number of ouplings inreases (see also figure 4 of the main text). nature photonis

5 In figure 9 we show the energies a n 2 for different values of ouplings: the turning on of the modes beomes synhronized at large n while the onset of emission happens at different times in the ase n = ; for n > as the modes that first turn on furnish energy to the oupled modes, the onsets result synhronized. S Number of ouplings n In the figure, we also show the S parameter defined as in the main text, whih reprodues that experimentally measured. S orresponds to the area of the high-frequeny region of the Fourier transform (FT) of the spetrum. As the spetrum beomes smooth, S dereases. FIG. : Degree of peakness alulated after the Fourier transform of the spetrum, averaged over shots. Finally we show in figure the alulated C parameter showing that it reprodues the trend experimentally obtained and desribed in the main text..8 C Number of Couplings FIG. : Correlation between the spetra at two different wavelengths alulated in shots. [] N. Savage, Nat Photon 3, 7 (29). [2] S. Mujumdar, V. Türk, R. Torre, and D. S. Wiersma, Phys. Rev. A 76, 3387 (27). [3] M. Siddique, R. R. Alfano, G. A. Berger, M. Kempe, and A. Z. Genak, Opt. Lett. 2, 45 (996). [4] A. Yariv, Quantum Eletronis (Saunders College, San Diego, 99). [5] L. Angelani, C. Conti, G. Ruoo, and F. Zamponi, Phys. Rev. B 74, 427 (26). [6] C. Conti, M. Leonetti, A. Fratalohi, L. Angelani, and G. Ruoo, Phys. Rev. Lett., 439 (28). [7] M. Leonetti and C. Conti, J. Opt. So. Am. B 27, 446 (2). [8] S. Gentilini, A. Fratalohi, L. Angelani, G. Ruoo, and C. Conti, Opt. Lett. 34, 3 (29). nature photonis 5