Lasing threshold of diffusive random lasers in three dimensions

F. A. Pinheiro1,* and L. C. Sampaio2

1Instituto de Física, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier 524, 20550-900 Rio de Janeiro, Brazil2Centro Brasileiro de Pesquisas Físicas—CBPF/MCT, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil

�Received 7 November 2005; published 31 January 2006�

We have numerically investigated the lasing threshold of three-dimensional diffusive random lasers com-posed of pointlike scatterers. The lasing threshold �T is determined by the mode with the smallest decay rate.The dependence of �T on the number of scatterers N �system size� is shown to follow the power law �T

�N−2/3. This result, which reflects the diffusion law for the lasing threshold, is shown to apply even when thesystem is very close to the Anderson localization transition. We suggest that this result corroborates thescenario in which extended modes are at the origin of random lasing in the diffusive regime, as revealed byrecent experiments.

DOI: 10.1103/PhysRevA.73.013826 PACS number�s�: 42.55.Zz, 42.25.Dd

The observation of laserlike emission in multiple lightscattering media with gain �1�, a phenomenon theoreticallypredicted in 1968 by Letokhov �2�, has boosted the researchon random lasers over the past decade �3�. In the so-calledrandom lasers with nonresonant feedback, one observes aremarkable narrowing of the luminescence spectrum to asingle peak above a well-defined threshold. The thresholdcorresponds to the situation where the gain due to opticalamplification compensates losses through the sample bound-aries. Random lasing with nonresonant feedback has beenobserved in several systems, such as solutions of micropar-ticles dispersed in laser dye �1�, ceramic �4� and polymer �5�systems, dye-doped cholesteric �6�, and nematic liquid crys-tals �7�. Light emitted from random lasers was demonstratedto be coherent as a true laser light �8�, a fact which agreeswith calculations that show that the emission spectrum abovethe threshold is Poissonian �9�.

Recent experiments have revealed that lasing modes inrandom media may exhibit extremely narrow linewidths�10,11�. These modes can be either localized �10� orextended �11�. Localized modes can provide the feedbackmechanism necessary for lasing by means of interferenceeffects generated by recurrent light scattering insidehigh quality “random cavities.” This scenario where laseremission is essentially due to Anderson localization wasobserved in disk-shaped zinc oxide powders �10� and is fre-quently referred to as random lasing with coherent feedback.Anderson localization is also the mechanism responsible forrecently observed low-threshold random lasing in one-dimensional �1D� amplifying layered media �12�. The inter-play between random lasing and Anderson localization hasbeen studied theoretically in 1D �13� and 2D �14�. For 3Dsystems, Anderson localization, and consequently randomlasing with coherent feedback, is much more difficultto achieve. The situation where extended modes are respon-sible for the observed ultranarrow peaks in the emissionspectra correspond to a completely different physicalscenario �11�. In this case, light transport is diffusive and

these extended lasing modes correspond to very long andrare light paths inside the medium. These paths probe alarger volume inside the sample and are thus subjected tohuge amplification �11�. It was also suggested that somerare localized modes �also called “prelocalized” modes� �15�,that are expected to exist even when light transport is diffu-sive, could be at the origin of lasing action in diffusivesamples �16�.

The lasing threshold of 2D random lasers was numericallyinvestigated by Patra �17�. This investigation is based on theanalysis of the decay rate statistics P��� of open disorderedslabs within the 2D Anderson model. An extensive numericalanalysis was used to derive P��� and then the dependence ofthe lasing threshold �T on the system size, both in diffusiveand in localized regimes �17�. Burin et al. have also numeri-cally investigated the influence of the system size on thelasing threshold of 2D media composed of poinlike scatterers�18�. They have shown that the dependence of the lasingthreshold on the system size follows a power law that reflectsthe increasing probability to form high quality random cavi-ties responsible for lasing as the system size increases.This result agrees with experiments on strongly scattering,close to the Anderson localization threshold, planar ZnOpowder random lasers with coherent feedback �10�. In 1D,the lasing threshold was shown to decrease exponentiallywith the system size, reflecting the fact that in such systemsall modes are exponentially localized �13�. For 3D systems,we are not aware of any theoretical or numerical study on thesystem size dependence of the lasing threshold in randommedia.

The aim of the present paper is thus to investigate theinfluence of the system size on the lasing threshold of 3Drandom lasers in the diffusive regime by means of anab initio numerical model. Our approach is based on thedipole model for wave propagation �19�. Within this ap-proach, based on the spectra of Green matrices for wavepropagation, it is assumed that the scatterers are dipolesmuch smaller than the incident wavelength. Using this ap-proximation and assuming a Breit-Wigner model for thescatterers, one can calculate the decay rates � of allmodes �20�. The lasing threshold is determined by the mode*Corresponding author. E-mail: faap@uerj.br

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with smallest �. We demonstrate that the dependence ofthe lasing threshold on the system size follows a powerlaw, a behavior that reflects a diffusive scenario for wavepropagation. We show that this behavior is valid even whenthe system is very close to the Anderson localization transi-tion. We discuss the origin of this finding as well as its con-nection with the mechanism that should govern random las-ing in the diffusive regime, as suggested by recentexperiments �11�.

In order to investigate the dependence of the lasingthreshold on the system size, we employ the scalar pointdipole model, introduced by Rusek and Orlowski �19,21�.This approach is based on the analysis of the spectrumof the Green matrix, that describes light scattering from ran-domly distributed pointlike dipoles �i.e., particles muchsmaller than the wavelength of light�. For an incident planewave �0�r� in a system composed of N identical dipoles withscattering matrix t, the field acting in the dipole at ri is givenby �19,21�

��ri� = �0�ri� + t�j�i

N

G�rij���r j� . �1�

The complex-valued N�N matrix G�rij� describes lightpropagation of the wave scattered by the dipole at ri to thedipole at r j. Since the eigenvalues �M of M�I− tG and �Gof G are related by �M =1− t�G, and t depends on frequency� via the scattering phase shift ���� �19�,

t��� = a�exp�2i����� − 1� �2�

�with a a real-valued constant�, an eigenvalue �G withRe �G=−1 facilitates an appropriate choice for ���� suchthat �M =0. This would correspond to a genuinely localizedstate somewhere inside the random medium �19�. Since thesystem is open, there are no bound states but resonances orquasimodes with a finite frequency width �or decay rate� � ata position �. We assume that the scatterers have an internalstructure with one sharp Breit-Wigner resonance of width �0at the position �0. In this case the phase shift ���� has theform

cot � = −� − �0

�0. �3�

The total scattering cross section k2�=4 sin2 � can then bewritten as

k2� =4�0

2

�� − �0�2 + �02 . �4�

Using this model of Breit-Wigner pointlike scatterers it ispossible to obtain, in a good approximation, the resonancewidths � via �G �21�

�

�0 1 + Re �G. �5�

In order to determine the lasing threshold �T, we assumethat it is given by the mode with the lowest decay rate.This assumption is intuitive since in this mode the conditionfor lasing, namely, when the gain due amplification becomes

larger then the decay caused by leakage through the systemboundaries, is achieved faster than in other modes. Wealso assume that the distribution of gain inside the sample isuniform so that �T is given by the lowest decay rate modeof the passive medium. We consider that this simplification,already used in Ref. �22� to determine the threshold,should not qualitatively affect our results. It is also importantto emphasize that we do not make any a priori assumptionabout the nature of the modes responsible for the lasingthreshold, i.e., they can be either localized or extended.However, in both cases these modes correspond to thelowest decay rate modes: whereas in the former this isdue to recurrent scattering inside high quality randomcavities, in the later this occurs because light experienceslong residence times associated with long paths inside thesample.

Whether extended or localized, the modes that determinethe threshold of random lasers have different decay rates fordifferent statistically identical systems of finite size. As aresult, an investigation of the influence of the system size onthe threshold of random lasers requires a statistical treatment.Thus we determine the lasing threshold �T as the average of

the smallest decay rates �̃i over M realizations of thedisorder

�T = �i=1

M�̃i

M. �6�

In our simulations we have randomly distributed Npointlike scatterers inside a cube of size L with uniformphysical density of particles per wavelength cubed.This specific geometry was chosen motivated by the cubicsamples utilized in recent experiments on 3D diffusiverandom lasers �11�. The incident pumping radiationwavelength �0 was chosen to be in resonance with the inter-nal degree of freedom of the scatterers so that the totalscattering cross section is given by �=�0

2 /. Scatteringis stronger at resonance which facilitates the reduction ofthe threshold. As we are interested in the size dependenceof the lasing threshold �T, we have increased the numberof particles N while keeping constant the physical density .This is equivalent to increase the size of the system.

Figure 1 exhibits the dependence of the lasing threshold�T on the number N of resonant scatterers for two differentvalues of the �uniform� physical density, =1 and =10 scat-terers per wavelength cubed. The values =1 and =10correspond to localization parameters kl*=22 /20 andkl*=22 /2, respectively. Here k is the wave number andl* is the transport mean free path. In order to estimate thelocalization parameter we have used the fact that for point-like scatterers l* is, in a good approximation, equal to themean free path l=1/n�, with n the density of particles. Thisvalues for indicate that light transport in our simulations isglobally diffusive since in 3D disordered systems the onsetof Anderson localization is expected to occur for kl*�1�Ioffe-Regel criterium� �23�. The decay rates were computedusing Eq. �5� and �T was determined as in Eq. �6�, where wehave considered up to 1000 realizations of the disorder.For each realization, this involves the diagonalization of the

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N�N Green matrix G. Due to computational limitations wewere not able to treat systems with a number of particlesgreater than N=5000. From the analysis of Fig. 1, we noticethat the lasing thresholds corresponding to the higher density=10 are approximately one order of magnitude smaller thanfor =1, as expected since the degree of disorder is higher.In addition, we observe that the lasing threshold �T decays asa power law �T�N−2/3 as the number of particles �systemsize� increases for both values of the density considered,=1 and =10.

In order to understand the origin of the power law behav-ior �T�N−2/3 in 3D diffusive systems, it is useful to comparethis result with the prediction for the size dependence of thelasing threshold in a random medium in the absence ofAnderson localization �2�

�T �D

L2 . �7�

In Eq. �7� D=vEl* /3 is the diffusion constant in 3D, with vEthe energy transport velocity of light inside the medium. In arandom medium, vE was shown to be affected by the timedelay �d associated with microscopic resonances that occurin each scattering event �24�

vE c0

1 + �d/�MF, �8�

with c0 the velocity of light in the vacuum and �MF the meanfree time �MF= l /c0, the average time between two scatteringevents. Since �d=1/�0, with �0 the decay rate of a singleresonant scatterer, vE in Eq. �8� is approximately given byvE l�0. Using this estimative for vE, Eq. �7� assumes theform

�T

�0

1

3

2

N2/34/3 , �9�

where we have used the relation l=LN−1/3−2/3 and thatl* l for pointlike scatterers. The result �9� explains our nu-merical findings for the dependence of the lasing thresholdon the number of particles N �system size�, as exhibited inFig. 1, for both values of the density considered, =1 and=10.

It is not surprising that we have found exactly the predic-tion �7� for size dependence of the lasing threshold for thelowest density consider �=1�. Indeed, in this case, whichcorresponds to kl*�20, the system is very far from theAnderson localization transition in 3D, expected to occur forkl*=1 �23�. This same diffusive behavior �7� has been re-cently reported in numerical simulations within the 2DAnderson model far from the localization threshold �17�.On the other hand, one could expect deviations fromthe diffusive behavior �7� of the lasing threshold for =10�which corresponds to kl*�2� since in this case the system isvery close to the Anderson transition. In spite of the factthat for kl*�2 wave transport in the random medium isdiffusive on average, the probability of finding a �rare� “pre-localized” mode for such degree of disorder is expected toincrease �16�. As a result, the presence of these modesshould, in principle, affect the size dependence of the lasingthreshold. The fact that we have not observed any deviationfrom the diffusive behavior �7�, even when the system isvery close to the Anderson transition, suggests that othermechanisms than localization should govern the lasing. Inthis sense, we conclude that a scenario where extendedmodes are responsible for lasing seems to be more appropri-ate to describe the behavior of the random lasing thresholdin the diffusive regime. These modes correspond to verylong light paths inside the system, as recently observedexperimentally �11�. In this case, the power law decrease ofthe lasing threshold as the system size increases, as exhibitedin Fig. 1, should reflect the increasing probability that lightexplores a bigger volume inside the system as its size in-creases, enhancing amplification and thus reducing thethreshold.

Although we cannot completely exclude the existence of“prelocalized” modes from the present study, one of the pos-sible reasons why we do not observe their influence on thesize dependence of lasing threshold is the absence of corre-lations in disorder, as discussed in Ref. �16�. Since we didnot introduce correlations in the disorder, the typical timerequired for an excitation to leave the system should domi-nate over the lifetime of high-quality modes correspondingto “prelocalized” states. This leads to the diffusive law de-pendence �7� for the lasing threshold. The effect of disordercorrelations in “prelocalized” modes and their impact on thelasing threshold should be tested in further numerical simu-lations within our model. This model seems to be adequatefor this purpose since it does not exhibit lattice-specific “pre-localized” states as the Anderson model �25�, within whichthe quest for “prelocalized” states has been performed so far�26,27�.

In summary, we have numerically investigated the

FIG. 1. Dependence of the lasing threshold �T on the number ofscatterers N �size of the system� for two values of the uniformdensity: =1 �filled triangles� and =10 �empty circles� scatterersper wavelength cubed. The value of �T was normalized by thedecay rate of a single scatterer �0. The dashed lines correspond tothe diffusion law for the lasing threshold, Eq. �9�.

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dependence of the lasing threshold of 3D diffusive randomlasers on the system size. By means of an ab initiomodel based on pointlike scatterers, we have calculated thelasing threshold using the resonance widths distribution.The lasing threshold �T was shown to follow a powerlaw dependence �T�N−2/3 on the number of particles N�system size�. This behavior, which was shown to bevalid even when the system is very close to the Anderson

localization transition, reflects the diffusive law for the lasingthreshold. We conclude that this result qualitatively agreeswith the scenario where extended modes, and not localizedones �“prelocalized” modes�, are responsible for lasingaction in diffusive random media, as recent experimentsindicate �11�.

We thank P. C. de Oliveira for discussions. This work wassupported by CNPq/Brazil.

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