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Advances in Optics and Photonics

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  • Editor: Bahaa E. A. Saleh
  • Vol. 3, Iss. 1 — Mar. 31, 2011
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Modes of random lasers

J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste  »View Author Affiliations


Advances in Optics and Photonics, Vol. 3, Issue 1, pp. 88-127 (2011)
http://dx.doi.org/10.1364/AOP.3.000088


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Abstract

In conventional lasers, the optical cavity that confines the photons also determines essential characteristics of the lasing modes such as wavelength, emission pattern, directivity, and polarization. In random lasers, which do not have mirrors or a well-defined cavity, light is confined within the gain medium by means of multiple scattering. The sharp peaks in the emission spectra of semiconductor powders, first observed in 1999, has therefore lead to an intense debate about the nature of the lasing modes in these so-called lasers with resonant feedback. We review numerical and theoretical studies aimed at clarifying the nature of the lasing modes in disordered scattering systems with gain. The past decade has witnessed the emergence of the idea that even the low-Q resonances of such open systems could play a role similar to the cavity modes of a conventional laser and produce sharp lasing peaks. We focus here on the near-threshold single-mode lasing regime where nonlinear effects associated with gain saturation and mode competition can be neglected. We discuss in particular the link between random laser modes near threshold and the resonances or quasi-bound (QB) states of the passive system without gain. For random lasers in the localized (strong scattering) regime, QB states and threshold lasing modes were found to be nearly identical within the scattering medium. These studies were later extended to the case of more lossy systems such as random systems in the diffusive regime, where it was observed that increasing the openness of such systems eventually resulted in measurable and increasing differences between quasi-bound states and lasing modes. Very recently, a theory able to treat lasers with arbitrarily complex and open cavities such as random lasers established that the threshold lasing modes are in fact distinct from QB states of the passive system and are better described in terms of a new class of states, the so-called constant-flux states. The correspondence between QB states and lasing modes is found to improve in the strong scattering limit, confirming the validity of initial work in the strong scattering limit.

© 2010 Optical Society of America

1. Introduction

With the renewed experimental interest in random lasers came also a number of attempts to generalize laser theory to describe such a system. Early on a major distinction was made between conventional lasers, which operate on resonant feedback, and random lasers, which at least in some cases were supposed to operate only on nonresonant feedback [4

4. H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, and R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003). [CrossRef]

]. In the case of nonresonant feedback the light intensity in the laser was described by a diffusion equation with gain, but the phase of the light field and hence interference did not play a role. A key finding is that there is a threshold for amplification when the diffusion length for escape LDdL2l becomes longer than the gain length (here d=2,3 is the dimensionality). The spatial distribution of intensity above threshold would be given by the solution of a diffusion equation. In this approach there would be no frequency selectivity, and the amplified light would peak at the gain center. Clearly such a description would be inadequate to describe random lasers based on Anderson localized modes, as such modes are localized in space precisely owing to destructive interference of diffusing waves arising from multiple scattering.

In this paper, we present recent work, both numerical and analytical, which has shown that within semiclassical laser theory, in which the effects of quantum noise are neglected, definite answers to these questions can be given, without resorting to exotic scenarios. Sharp laser lines based on interference (coherent feedback) do exist, not only in strongly scattering random lasers where the localized regime is reached [25

25. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000). [CrossRef] [PubMed]

, 26

26. C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001). [CrossRef]

, 27

27. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002). [CrossRef]

], but also in diffusive random lasers [28

28. C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007). [CrossRef] [PubMed]

, 29

29. C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009). [CrossRef]

] and even for weak scattering [30

30. X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006). [CrossRef]

]. Numerical studies have shown that they are associated with threshold lasing modes (TLMs), which, inside the cavity, are similar to the resonances or quasi-bound (QB) states of the passive system (also called quasi-normal modes). The resemblance is excellent in the localized case [26

26. C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001). [CrossRef]

, 27

27. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002). [CrossRef]

] and deteriorates as scattering is reduced. A new theoretical approach based on a reformulation of the Maxwell–Bloch (MB) equations to access the steady-state properties of arbitrarily complex and open cavities allows one to calculate the lasing modes in diffusive and even in weakly scattering random lasers (lL) [31

31. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006). [CrossRef]

, 32

32. H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007). [CrossRef]

, 33

33. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643 (2008). [CrossRef] [PubMed]

, 34

34. L. Ge, R. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895 (2008). [CrossRef] [PubMed]

, 35

35. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]

]. A major outcome of this approach is the demonstration that although lasing modes and passive modes can be very alike in random systems with moderate openness, in agreement with the above numerical results, they feature fundamental differences. Their distinctness increases with the openness of the random system and becomes substantial for weakly scattering systems. Constant-flux (CF) states are introduced that better describe TLMs both inside and outside the scattering medium for any scattering strength. In addition this theoretical approach allows one to study the multimode regime in diffusive random lasers and get detailed information about the effects of mode competition through spatial hole burning, which appear to differ from those for conventional lasers.

In this past decade, different types of random lasers (semiconductor powders, pi-conjugated polymers, scattering suspension in dyes, random microcavities, dye-doped nematic liquid crystals, random fiber lasers…) have been considered in the literature. We will focus throughout this review mostly on 2D random lasers that consist of randomly distributed dielectric nanoparticles as scatterers. This choice makes possible the numerical and theoretical exploration of 2D finite-sized opened samples where transport can be made ballistic, diffusive (in contrast to 1D), or localized [36

36. P. Sebbah, D. Sornette, and C. Vanneste, “Anomalous diffusion in two-dimensional Anderson-localization dynamics,” Phys. Rev. B 48, 12506–12510 (1993). [CrossRef]

] by adjusting the index contrast between the scatterers and the background medium.

The outline of this review is as follows: in Section 2 we review early numerical explorations of localized and diffusive random lasing demonstrating the existence of TLMs in all regimes. In Section 3 we present recent numerical work based on a time-independent model, which indicates the difference between passive cavity resonances and TLMs, discussing only single-mode random lasing. The following section will explain why, in principle, QB states cannot describe TLMs. Section 5 will introduce the concept of CF states and describe the self-consistent time-independent approach to describe random lasing modes at threshold as well as in the multimode regime.

2. Early Numerical Explorations: Time-Dependent Model

2.1. Localized Case

In this subsection, we will consider case (2). Localized modes in a disordered scattering system are quite like the modes of standard optical cavities, such as the Fabry–Perot [38

38. The idea of considering localized modes for random lasing can already be found in P. Sebbah, D. Sornette, and C. Vanneste, “Wave automaton for wave propagation in the time domain,” Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, 1994), p. 68.

]. Hence, one can expect that in the presence of gain the lasing modes in this regime of strong disorder will be close to the localized QB states of the passive system without gain, in the same way as the lasing modes of a conventional cavity are built with the QB states of the passive cavity. To verify that this is really the case one must have access to the individual modes of both the passive system and the active system. Experimentally, such a demonstration has not been achieved yet, essentially because the regime of Anderson localization is difficult to reach and to observe in optics. Besides, until recently there was no fully developed theory describing random lasing modes and their relationship with the eigenstates of the passive system. The easiest way to check this conjecture has been to resort to numerical simulations.

Historically, most of the early numerical studies of random lasers were based on the diffusion equation (see references in [4

4. H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, and R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003). [CrossRef]

]). However, it is not possible to take into account under the diffusion approximation the interference phenomena that are at the heart of Anderson localization. This is why Jiang and Soukoulis [25

25. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000). [CrossRef] [PubMed]

] proposed to solve the time-dependent Maxwell equations coupled with the population equations of a four-level system [40

40. A. E. Siegman, Lasers (University Science Books, 1986).

]. The populations Ni, i=1 to i=4 satisfy the following equations:
dN1dt=N2τ21WpN1,
(1)
dN2dt=N3τ32N2τ21(Eωa)dPdt,
(2)
dN3dt=N4τ43N3τ32+(Eωa)dPdt,
(3)
dN4dt=N4τ43+WpN1,
(4)
where Wp is the rate of an external mechanism that pumps electrons from the fundamental level (1) to the upper level (4). The electrons in level 4 relax quickly with time constant τ43 to level 3. The laser transition occurs from level 3 to level 2 at frequency ωa. Hence, electrons in level 3 can jump to level 2 either spontaneously with time constant τ32 or through stimulated emission with the rate (Eωa)dPdt. E and P are the electric field and the polarization density, respectively. Eventually, electrons in level 2 relax quickly with time constant τ21 from level 2 to level 1. In these equations, the populations Ni, the electric field E, and the polarization density P are functions of the position r and the time t.

The polarization obeys the equation
d2Pdt2+ΔωadPdt+ωa2P=κΔNE,
(5)
where ΔN=N2N3 is the population density difference. Amplification takes place when the rate Wp of the external pumping mechanism produces inverted population difference ΔN<0. The linewidth of the atomic transition is Δωa=1τ32+2T2, where the collision time T2 is usually much smaller than the lifetime τ32. The constant κ is given by κ=3c32ωa2τ32 [40

40. A. E. Siegman, Lasers (University Science Books, 1986).

].

Finally, the polarization is a source term in the Maxwell equations,
Ht=c×E,
(6)
ϵ(r)E/t=c×H4πPt.
(7)
The randomness of the system arises from the dielectric constant ϵ(r), which depends on the position r. This time-dependent model has been used in random 1D systems consisting of a random stack of dielectric layers separated by gain media [25

25. X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000). [CrossRef] [PubMed]

] and in random 2D systems consisting of a random collection of circular particles embedded in a gain medium (Fig. 1) [26

26. C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001). [CrossRef]

]. In both cases, a large optical index contrast has been assigned between the scatterers and the background medium to make sure that the regime of Anderson localization was reached. The Maxwell equations are solved by using the finite-difference time-domain method (FDTD) [41

41. A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 1995).

]. To simulate an open system, perfectly matched layers are introduced at the boundaries of the system [42

42. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1995). [CrossRef]

]. The pumping rate Wp is adjusted just above lasing threshold in order to remain in the single-mode regime.

In one dimension, the QB states of the passive system were obtained independently using a time-independent transfer matrix method [43

43. X. Jiang and C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E 65, 025601(R) (2002). [CrossRef]

]. In two dimensions, the Maxwell equations were solved without the polarization term in Eq. (7), again using the FDTD method. First, the spectrum of eigenfrequencies was obtained by Fourier transform of the impulse response of the system. Next, QB states were excited individually by a monochromatic source at each of the eigenfrequencies.

Finally, in 1D systems [43

43. X. Jiang and C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E 65, 025601(R) (2002). [CrossRef]

] as well as in 2D systems [27

27. P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002). [CrossRef]

], lasing modes obtained by the full time-dependent model with gain and localized QB states of the corresponding passive system without gain were compared and found to be identical with a good precision. This was verified for all modes obtained by changing the disorder configuration. An example of a 2D lasing mode and the corresponding QB state of the same system (Fig. 1) without gain are displayed in Fig. 2. These results confirmed that the QB states of a localized system play a role similar to the eigenmodes of the cavity of a conventional laser. The only difference is the complicated and system-dependent nature of the localized modes as opposed to the well-known modes of a conventional cavity. These results are in good agreement with the theoretical results described in Section 5, which show that inside systems in the localized regime, the single lasing modes just above threshold are close to the high-Q resonances of the passive system.

2.2. Diffusive Case

The decay rate observed corresponds to a quality factor of 30, to be compared with the value 104 found in the localized case. This result shows that a bad resonance in a leaky disordered system can nevertheless turn into a lasing mode in the presence of an active medium. This result is in stark contrast with the common belief that random lasing with resonant feedback involves the presence of resonances with high quality factors. It provides a consistent explanation for the experimental observation of random lasing with resonant feedback even far from the localized regime, without resorting to other scenarios such as those reviewed in Section 1 [19

19. V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002). [CrossRef] [PubMed]

, 20

20. V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004). [CrossRef]

, 21

21. S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004). [CrossRef] [PubMed]

, 22

22. S. Mujumdar, V. Türck, R. Torre, and D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007). [CrossRef]

].

The comparison of patterns between Fig. 3(a) and Fig. 3(b) shows that the lasing mode and the QB modes are close to each other inside the scattering system as confirmed by the evolution of the correlation function, which has been defined only inside the system. However, one also notices that outside the scattering medium the field distributions differ substantially. The free propagating field outside the scattering system in Fig. 3(b) reproduces the laser field distribution in Fig. 3(a) with significant distortions that are due to the enhancement of the amplitude towards the external boundaries of the total system. Hence, the comparison between both figures indicates that if the lasing modes and the QB modes are similar inside the scattering system, they differ noticeably outside. Moreover, a careful examination of the correlation function in Fig. 5(a) shows that it oscillates between two extremal values, which slowly depart from 1 and +1 well before the ultimate fast decay. This is in contrast with the long time behavior of the correlation function in the localized regime (not shown), which displays oscillations between 1 and +1 with a very good precision for time scales much longer than the time scale in Fig. 5(a). This result also indicates that inside the scattering system, the lasing mode is close to but not identical to a QB state.

3. Numerical Simulations: Time-Independent Models

Different models have been proposed in the frequency domain to solve the wave equation. In one dimension, it is possible to employ the transfer matrix method similar to that used in [43

43. X. Jiang and C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E 65, 025601(R) (2002). [CrossRef]

] for studying the lasing modes in an active layered random system. A direct comparison between TLMs and QB states of the corresponding passive random system is proposed in the first part of this section. In two dimensions, the multipole method has been used, which also provides a direct comparison of the QB states and the lasing modes of a 2D disordered open system. The comparison presented in the second part of this section has been carried out for refractive index of the scatterers ranging from nl=2.0 (localized regime) to nl=1.25 (diffusive regime). We alternatively used a different approach based on the finite element method to obtain the passive modes, which turned out to be much less computationally demanding in the weakly scattering regime. A brief description of both methods is provided in Appendices A, B.

3.1. One-Dimensional Random Lasers

Employing the transfer matrix method, similar to that used in [43

43. X. Jiang and C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E 65, 025601(R) (2002). [CrossRef]

], we study the lasing modes in a 1D random system and compare them with the QB states of the passive random system. The random system is composed of 161 layers. A dielectric material with index of refraction n1=1.05 separated by air gaps (n2=1) results in a spatially modulated index of refraction n(x). Outside the random medium n0=1. The system is randomized by specifying thicknesses for each layer as d1,2=d1,2(1+ηζ), where d1=100nm and d2=200nm are the average thicknesses of the layers, η=0.9 represents the degree of randomness, and ζ is a random number in (1,1). The length of the random structure L is normalized to L=24,100nm. Linear gain is simulated by appending an imaginary part to the dielectric function ϵ(x)=ϵ(x)+iϵ(x), where ϵ(x)=n2(x). This approximation is valid at or below threshold [49

49. X. Jiang, Q. Li, and C. M. Soukoulis, “Symmetry between absorption and amplification in disordered media,” Phys. Rev. B 59, R9007–R9010 (1999). [CrossRef]

]. The complex index of refraction is given by ñ(x)=ϵ(x)=n(x)+in, where n<0. We consider n to be constant everywhere within the random system. This yields a gain length lg=|1k|=1|n|k (k=2πλ is the vacuum frequency of a lasing mode), which is the same in the dielectric layers and the air gaps. The real part of the index of refraction is modified by the imaginary part as n(x)=n2(x)+n2.

We find the frequency k and threshold gain k of each lasing mode within the wavelength range 500 nm<λ<750nm. The results are shown in Fig. 6. Finding matching QB states for lasing modes with large thresholds (large |k|) is challenging because of large shifts of the solution locations [Fig. 6, region (c)]. However, there is a clear one-to-one correspondence with QB states for the lasing modes remaining [Fig. 6, regions (a) and (b)]. It is straightforward to find the matching QB states for these lasing modes and calculate their differences. The average percent difference between QB state frequencies and lasing mode frequencies in Fig. 6, region (a), is 0.013%, while it is 0.15% in Fig. 6, region (b). The average percent difference between QB state decay rates k0 and lasing thresholds k in Fig. 6, region (a), is 2.5% and in Fig. 6, region (b), is 21%.

The normalized intensities of the QB states IQB and lasing modes with linear gain ILG are also compared. Figure 7 shows representative pairs of modes from the three regions shown in Fig. 6. The spatially averaged relative difference between each pair of modes is calculated by
σd=|IQBILG|dAILGdA×100%.
(9)
For small thresholds [Fig. 7(a)] the difference between the lasing modes and the matching QB states is very small. The average percent difference between all pairs of modes in this region is σd=4.2%. For lasing modes with slightly larger thresholds [Fig. 7(b)] there are clear differences. Nevertheless, we may confidently match each lasing mode in this region with its corresponding QB state. The average percent difference between all pairs of modes in this region is σd=24%. As mentioned above, it is challenging to find matching pairs of lasing modes and QB states for large thresholds. Figure 7(c) compares the lasing mode with the largest threshold and the QB state with the largest decay rate [circled in Fig. 6, region (c)]. Though these two modes are fairly close to each other in terms of k, k0, and k, their intensity distributions are quite different. Indeed, there may be no correspondence between the two.

The deviation of the lasing modes from the QB states can be explained by the modification of the transfer matrix. In the passive system, k0 is constant, but ki=k0n(x) varies spatially. With the introduction of gain, k=kn becomes constant within the random system, and feedback due to the inhomogeneity of k is removed. However, introducing gain generates additional feedback inside the random system caused by the modification in the real part of the wave vector k=kn(x). Neglecting this effect results in some correspondence between lasing modes and QB states even at large thresholds [50

50. X. Wu, J. Andreasen, H. Cao, and A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007). [CrossRef]

]. Furthermore, since there is no gain outside the random system, k suddenly drops to zero at the system boundary. This discontinuity of k generates additional feedback for the lasing modes. In this weakly scattering system, the threshold gain is high. The large drop of k at the system boundary makes the additional feedback stronger.

3.2. Two-Dimensional Random Lasers

The same method is used to find the lasing modes (TLM) at threshold. It is necessary this time to find the poles of the scattering matrix in the 2D space (λ,ϵb) of real wavelengths (λ=0) and the imaginary component of the complex dielectric constant outside the scatterers where the gain is distributed. It can also be used to find the lasing modes when gain is localized inside of the scatterers. In this case the poles of the scattering matrix are searched in the space of real wavelengths (λ=0) and the imaginary part of the dielectric constant of cylinders ϵl.

The multipole method is both accurate and efficient: the boundary conditions are analytically satisfied, thus providing enhanced convergence, particularly when the refractive index contrast is high. However, in the case of large systems the method can be slow (given that field expansions are global, rather than local) when it is necessary to locate all poles within a sizable wavelength range. Another extremely efficient time-independent numerical method based on the finite element method [60

60. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

] has been tested. This method is briefly described in Appendix B. We confirmed that the results obtained by both methods, the (purely numerical) finite element method and the (semi-analytic) multipole method were identical with a good precision.

3.2a. Localized Case

We first consider the localized case (nl=2.0) for which a complete comparison of the QB states and the lasing modes was possible with the time-dependent FDTD-based method (Subsection 2.1), thus providing a reference comparison for the multipole calculations. The lasing mode is found at a wavelength λ=446.335nm for a value of the imaginary part of the refractive index nl=1.967×104, representing the pumping threshold for this mode. The spatial distribution of its amplitude is shown in Fig. 8(b). The QB states of the passive system are calculated in the spectral vicinity of the lasing mode. The number of required multipoles was Nmax=4 (see Appendix A). Figure 8(a) shows the QB state that best resembles the lasing mode. Its wavelength and quality factor are, respectively, λ=446.339nm and Q=8047. The relative difference between the two modes is σd=0.05%. These calculations provide confirmation that the lasing modes and the QB states are the same inside the scattering region for high-Q-valued states.

3.2b. Diffusive Case

3.2c. Transition Case

It is both informative and interesting to follow the evolution of the lasing modes and the QB states spatial profile when the index of refraction is decreased progressively, allowing one to compare the QB states and the random lasing modes (TLM) systematically in a regime ranging from localized to diffusive. The QB state and lasing modes calculated for intermediate cylinder refractive indices nl=1.75 and nl=1.5 are displayed in Figs. 11, 12. We note that the highly spatially localized mode for nl=2 (Fig. 8) is replaced for nl=1.75 by a mode formed by two spatially localized peaks and several smaller peaks. For a refractive index of nl=1.5, the mode is still spatially localized, although in a larger area, but is now formed with a large number of overlapping peaks. A more systematic exploration of the nature of the lasing modes at the transition between localized states and extended resonances can be found in [29

29. C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009). [CrossRef]

]. There, a scenario for the transition has been proposed based on the existence of necklace states which form chains of localized peaks, resulting from the coupling between localized modes. The modes shown here support this scenario. It is important to note that the decreasing scattering and increasing leakage not only affect the degree of spatial extension of the mode but also the nature of the QB states. Indeed, it was shown in [29

29. C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009). [CrossRef]

] that, because of leakage, extended QB states have a nonvanishing imaginary part associated with a progressive component, in contrast to the purely stationary localized states. In Media 1, 2 , 3 , 4 we present animations of the time oscillation of the real part of the field R[Ψexp(iωt)] of the QB state and of the corresponding TLM for n=2 and n=1.25. The QB state is exponentially decaying in contrast to the lasing mode. The diffusive lasing mode clearly exhibit a progressive component, which does not exist in the localized lasing mode.

The values of wavelengths and quality factors of the QB states, lasing frequencies of the corresponding TLMs, and associated imaginary part of the refractive index are summarized in Table 1, together with the relative difference σd as defined in Eq. (9).

4. Threshold Lasing States versus Passive Cavity Resonances

Semiclassical laser theory treats classical electromagnetic fields coupled to quantized matter and yields the thresholds, frequencies and electric fields of the lasing modes, but not their linewidths or noise properties. To treat the spatial dependence of lasing modes, one must go beyond rate equation descriptions and use the coupled nonlinear Maxwell-Bloch (MB) equations for light coupled to homogeneously broadened two-level atoms or multilevel generalizations thereof. These equations will be presented in Section 5 below. While the MB description has been used since the inception of laser theory [61

61. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1982).

, 62

62. H. Haken, Light: Laser Dynamics (North-Holland, 1985), vol. 2.

], in almost all cases simplifications to these equations were made, most notably a neglect of the openness of the laser cavity. As random lasers are strongly open systems, it is necessary to treat this aspect of the problem correctly to obtain a good description of them.

Historically a first breakthrough in describing Fabry-Perot type lasers with open sides was the Fox-Li method [44

44. A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

, 45

45. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961). [CrossRef]

], which is an integral equation method of finding the passive cavity resonances of such a structure. It is widely assumed and stated that these resonances or QB states are the correct electromagnetic modes of a laser, at least at threshold. Often the nonlinear laser equations are studied with Hermitian cavity modes with phenomenological damping constants representing the cavity outcoupling loss obtained, e.g., from a Fox-Li calculation. It is worth noting that there are two kinds of cavity loss that occur in lasers; there is the outcoupling loss just mentioned and also the internal absorption of the cavity, which can be taken into account via the imaginary part of the passive cavity index of refraction. These are very different processes, as the former describes the usable coherent light energy emitted from the laser and the latter simply energy lost, usually as heat, in the laser cavity.

The QB states of an arbitrary passive cavity described by a linear dielectric function ϵc(x,ω) can be rigorously defined in terms of an electromagnetic scattering matrix S for the cavity. This matrix relates incoming waves at wave vector k (frequency ω=ck) to outgoing waves in all of the asymptotic scattering channels and can be calculated from the wave equation. Note that while we speak of the frequency of the incoming wave, in fact the S matrix is a time-independent quantity depending on the wave vector k. This is the wave vector outside the cavity; in random lasers we will be interested in spatially varying dielectric functions so that in the cavity there is no single wave vector of the field. For any laser, including the random laser, the cavity can be defined as simply the surface of last scattering, beyond which no backscattering occurs. The QB states are then the eigenvectors of the passive cavity S matrix with eigenvalue equal to infinity; i.e., one has outgoing waves with no incoming waves. Because this boundary condition is incompatible with current conservation, these eigenvectors have the complex wave vector k̃μ; these complex frequencies are the poles of the S matrix and their imaginary parts must always be negative to satisfy causality conditions. There is normally a countably infinite set of such QB states. Because of their complex wave vector, asymptotically the QB states vary as r(d1)2exp(+|Im[k̃μ]|r) and diverge at infinity, so they are not normalizable solutions of the time-independent wave equation. Therefore we see that QB states cannot represent the lasing modes of the cavity, even at threshold, as the lasing modes have a real frequency and wave vector outside the cavity with conserved photon flux.

When gain is added to the cavity the effect is to add another contribution to the dielectric function ϵg(x,ω), which in general has a real and imaginary part. The imaginary part of ϵg has an amplifying sign when the gain medium is inverted and depends on the pump strength; it compensates for the outcoupling loss as well as any cavity loss from the cavity dielectric function ϵc. The specific form of this function for the MB model will be given in Section 5 below. The TLMs are the solutions of the wave equation with ϵtotal(x)=ϵc(x)+ϵg(x) with only outgoing waves of real wave vector kμ [we neglect henceforth for simplicity the frequency dependence of ϵc(x)]. The kμ are the wave vectors of the TLMs with real lasing frequencies Ωμ=ckμ. These lasing wave vectors are clearly different from the complex k̃μ; moreover they are not equal to Re[k̃μ] as often supposed. This can be seen by the following continuity argument. Assume that ϵc(x) is purely real for simplicity, so that the S matrix is unitary and all of its poles are complex and lie in the negative half-plane. Turn on the pump, which we will call D0, anticipating our later notation, so that the inversion rises steadily from zero, continuously increasing the amplifying part of ϵg. The S matrix is no longer unitary, and its poles move continuously upward towards the real axis until each of them crosses the axis at a particular pump value, D0 (see Fig. 14); the place where each pole crosses is the real lasing frequency kμ for that particular TLM. Note that the poles do not move vertically to reach the real axis but always have some shift of the lasing frequency from the passive cavity frequency, mainly due to line-pulling towards the gain center. As the Q value of the cavity increases, the distance the poles need to move to reach the real axis decreases, so that the frequency shift from Re[k̃μ] can become very small, and the conventional picture becomes more correct. In general the poles of the S matrix are conserved quantities even in the presence of loss, so that the TLMs are in one-to-one correspondence with the QB states and thus are countably infinite, but for any cavity the pole that reaches the real axis first (i.e., at lowest pump D0) is the actual first lasing mode. At higher pump values the nonlinear effects of saturation and mode competition will affect the behavior; so only the lowest-threshold TLM describes an observable lasing mode for fixed pumping conditions, the first lasing mode at threshold. Which pole gets there first depends not only on the Q of the passive cavity resonance before gain is added, but also on the parameters of ϵg(x), which include the atomic transition frequency, the gain linewidth, and the pump conditions, as will be discussed below.

5. Self-Consistent Time-Independent Approach to Random Lasing

In Section 4 we gave a general argument based on the scattering matrix with the addition of gain to show that in general the QB states (passive cavity resonances) are never exactly the same as the TLMs, even inside the cavity. However the same argument indicated that inside a high-Q cavity the two sets of functions become very similar, since the poles of the S matrix are very close to the real axis and only a small amount of gain is required on order to move them to the real axis, which maps QB states onto TLMs. For localized states in the center of the sample the Q values should be exponentially large and, as found numerically, QBs and TLMs should be indistinguishable (again, inside the cavity; outside the QB states have an unphysical growth). As already noted, the set of TLMs defines only threshold modes; as soon as the first TLM has turned on, it will alter the gain medium for the other potential modes through spatial hole burning, and a nonlinear approach needs to be considered. Very recently such an approach has been developed that has the major advantage of being time independent and partially analytic, providing both ease of computation and greater physical insight. The approach, due to Türeci-Stone-Ge, is known as steady-state ab initio laser theory (SALT) [31

31. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006). [CrossRef]

, 34

34. L. Ge, R. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895 (2008). [CrossRef] [PubMed]

, 35

35. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]

]. It finds the stationary solutions of the MB semiclassical lasing equations in the multimode regime, for cavities of arbitrary complexity and openness, and to infinite order in the nonlinear interactions. As such it is ideal for treating diffusive or quasi-ballistic random lasers, which are extremely open and typically highly multimode even slightly above threshold. In this section we present the basic ideas with emphasis on TLMs, which are the focus of this review. The nonlinear theory has been reviewed in some detail elsewhere [35

35. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]

], and we just present a brief introduction to it here.

5.1. Maxwell-Bloch Threshold Lasing Modes

The resulting system of nonlinear coupled partial differential equations for the three fields E(x,t),P(x,t),D(x,t) are (c=1)
Ë+=1ϵc(x)2E+4πϵc(x)P̈+,
(10)
Ṗ+=(iωa+γ)P++g2iE+D,
(11)
Ḋ=γ(D0D)2i(E+(P+)*P+(E+)*).
(12)
Here g is the dipole matrix element of the atoms, and the units for the pump are chosen so that D0 is equal to the time-independent inversion of the atomic system in the absence of an electric field. This pump can be nonuniform: D0=D0(x) based on the experimental pump conditions, but we will not discuss that case here. The electric field, polarization, and inversion are real functions (E,P are vector functions in general, but we assume a geometry where they can be treated as scalars). In writing the equations above we have written these fields in the usual manner in terms of their positive and negative frequency components, E=E++E, P=P++P, and then made the rotating wave approximation in which the coupling of negative to positive components is neglected. There is no advantage in our treatment to making the standard slowly varying envelope approximation, and we do not make it.

5.2. Self-Consistent Steady-State Lasing Equations

The starting point of our formulation is to assume that there exists a steady-state multiperiodic solution of Eqs. (10, 11, 12) above; i.e., we try a solution of the form
E+(x,t)=μ=1NΨμ(x)eikμt,P+(x,t)=μ=1NPμ(x)eikμt.
(13)
Having taken c=1 we do not distinguish between frequency and wave vector. The functions Ψμ(x) are the unknown lasing modes, and the real numbers kμ are the unknown lasing frequencies; these functions and frequencies are not assumed to have any simple relationship to the QB states of the passive cavity and will be determined self-consistently. As the pump increases from zero the number of terms in the sum will vary, N=0,1,2,; at a series of thresholds each new mode will appear. The general nonlinear theory is based on a self-consistent equation that determines how many modes there are at a given pump and solves for these modes and their frequencies. However in this section we will discuss TLMs, and so we need only consider one term in the sum. Furthermore, at the first threshold the electric field is negligibly small, and so the inversion is equal to the external pump profile, assumed uniform in space, D(x,t)=D0. Assuming single-mode lasing, the equation for the polarization becomes
Pμ(x)=iD0g2Ψμ(x)(γi(kμka)).
(14)
Having found Pμ(x) in terms of Ψμ(x),D0, we substitute this result into the right-hand side of Maxwell’s equation along with Ψμ(x) for the electric field on the left-hand side. The result is
[2+ϵc(x)kμ2]Ψμ(x)=iD04πg2kμ2Ψμ(x)(γi(kμka)),
(15)
which can be written in the form
[2+(ϵc(x)+ϵg(x))kμ2]Ψμ(x)=0,
(16)
where ϵg(x) is the dielectric function of the gain medium, which only varies in space if the external pump or the gain atoms are nonuniform. Defining convenient units of the pump D0c=γ4πka2g2 and replacing D0D0D0c, we find that
ϵg(x)=D0ka2[γ(kμka)γ2+(kμka)2+iγ2γ2+(kμka)2].
(17)

Equation (16) shows that the TLMs are the solutions of the original Maxwell equation with the addition of a complex, pump- and frequency-dependent dielectric function that is uniform in space (for the assumed uniform pumping). The imaginary and the real parts of the gain dielectric function have the familiar symmetric and antisymmetric two-level resonance forms, respectively. The dependence on the atomic frequency ka encodes the usual atomic line-pulling effect. In the limit of a very broad gain curve (γ) the line-pulling effects can be neglected, and we find the simple result
ϵgiD0ka2,
(18)
i.e., a constant imaginary (amplifying) part of ϵg proportional to the pump strength. Such linear gain models have been studied before, although typically with a constant imaginary part of the index of refraction instead of a constant imaginary part of the dielectric function. Our results show that, in order to reproduce the TLMs of the MB equations, one needs to take
n(x)=ϵc(x)+ϵg(D0,kμka,γ)
(19)
so that the pump changes both the real and the imaginary parts of the index of refraction.

5.3. Solution for Threshold Lasing Modes and Constant-Flux States

The CF states satisfy the standard wave equation, Eq. (24), but with the non-Hermitian boundary condition already mentioned; hence their eigenvalues km2 are complex, with (it can be shown) a negative imaginary part, corresponding to amplification within the cavity. However, outside the cavity, by construction, they have the real wave vector kμ and a conserved photon flux. They are a complete basis set for each lasing frequency kμ, and hence they are a natural choice to represent the TLMs as well as the lasing modes above threshold. Hence we make the expansion
Ψμ(x)=m=1amμφmμ(x).
(26)
Substituting this expansion into Eq. (21), using biorthogonality, and truncating the expansion at N terms, leads to the eigenvalue problem
amμ=D0Λm(kμ)Ddxφ¯mμ*(x)pNapμφpμ(x)ϵc(x)D0pNTmp(0)apμ,
(27)
where Λm(k)iγ(k2ka2)[(γi(kka))(k2km2(k))].

One sees that the TLMs in the CF basis are determined by the condition that an eigenvalue of the matrix D0T(0)(kμ) is equal to unity. Since the matrix T(0)(kμ) is independent of D0, it is natural to focus on this object, which we call the threshold matrix. It is a complex matrix with no special symmetries, implying that its eigenvalues λμ are all complex for a general value of kμ. If the real control parameter D0 (the pump) is set equal to 1|λμ|, then the matrix D0T(0)(kμ) will have an eigenvalue of modulus unity, but not a real eigenvalue equal to unity as required, and no solution for the TLMs exists for this choice of kμ. It is the phase condition that λμ (kμ) must be real that determines the allowed lasing frequencies. In practice one orders the λμ in decreasing modulus based on an initial approximation to the lasing frequency, kμ, and then tunes kμ slowly until each eigenvalue flows through the real axis [which is guaranteed by the dominant k dependence contained in the factor Λm(k)]. Normally the eigenvalues do not switch order during this flow, and the largest eigenvalue λμ will determine the lowest threshold TLM, with threshold D0(μ)=1λμ(kμ), where kμ is the frequency that makes the largest eigenvalue T(0)(kμ) real. The eigenvector corresponding to λμ gives the coefficients for the CF expansion of the TLM of the first mode Ψμ(x). TLMs with higher thresholds can be found by imposing the reality condition on smaller eigenvalues of T(0)(kμ). This approach has been described in detail elsewhere [33

33. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643 (2008). [CrossRef] [PubMed]

, 35

35. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]

], and provides a much more efficient method for finding TLMs than solving the self-consistent differential equation, Eq. (16).

We immediately see from Eqs. (25, 27) that for an arbitrarily shaped cavity of uniform dielectric constant ϵc the matrix T(0)(kμ) is diagonal owing to the biorthogonality of the CF states. Thus each TLM is a single CF state, corresponding to one of the kμ that satisfies the reality condition. In this case the expansion of Ψμ(x) consists of just one term, and the threshold lasing equation is equivalent to Eq. (24) with appropriate relabeling. When ϵc varies in space, as for random lasers, the threshold matrix is not diagonal, and there can in principle be many CF states contributing to one TLM [63

63. Recently it was shown that one can define a variant of the CF states, termed the threshold constant flux (TCF) states, for which one TCF state is the TLM. Above threshold additional TCF states are needed to describe the lasing modes, but in general fewer than for the CF states defined here. The above threshold theory using these TCF states is almost identical to the theory described in [33, 35]. Details are in L. Ge, Y. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” arXiv.org, arXiv:1008.0628v1 (submitted to Phys. Rev. A).

]. However, since φm(x),φ¯p(x) are uncorrelated fluctuating functions of space, it turns out that the threshold matrix in random lasers is approximately diagonal and the threshold modes are dominated by one, pseudorandom CF state determined by solving Eq. (24) for the appropriate random dielectric function ϵc(x). This is shown in Fig. 15. In summary, the theory leading to the threshold equation (27) gives an efficient time-independent method for finding the TLMs of random lasers in any disorder regime. In general these TLMs are very close to a single CF state determined by Eq. (24) at the lasing frequency kμ. With this new method TLMs of random lasers can be found for complex 2D and even 3D geometries. In Figs. 16, 17 we compare TLMs, CF states and QB states for the 2D random laser model used in [33

33. H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643 (2008). [CrossRef] [PubMed]

], illustrating the agreement of TLMs with CF states even for weak scattering, while a significant deviation from the closest QB state is found.

5.4. Nonlinear Steady-State ab Initio Laser Theory

The key to generalizing SALT to the multimode nonlinear regime is to return to the fundamental MB equations and go beyond the assumption that the inversion D(x,t) is equal to the constant threshold pump D0. Once lasing modes have turned on, their spatially varying electric fields cause varying degrees of stimulated emission from the gain atoms and hence tend to reduce the inversion D from the pump value D0 in a manner that varies in space and in principle in time. However it has been shown that if γγ, then the time dependence of the inversion is weak, and although D varies in space, it is a good approximation to take D(x,t)=D(x). This stationary inversion approximation has been used in laser theory for many years, going back to Haken [62

62. H. Haken, Light: Laser Dynamics (North-Holland, 1985), vol. 2.

], but has not been incorporated into an ab initio method such as SALT. We will not review the details of the derivation of the nonlinear multimode theory of Türeci-Stone-Ge, which have been given elsewhere [31

31. H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006). [CrossRef]

, 35

35. H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]

]. Instead we just state that the net effect of the nonlinear interactions within the stationary inversion approximation is just to replace the uniform inversion, as follows,
D0D01+νΓ(kν)(|Ψν(x)|2),
(28)
in all of the equations of the theory of the TLMs. Here ν labels all above-threshold modes and Γ(kν) is a Lorentzian centered at the lasing frequency of mode ν with width γ. If we make this substitution into Eq. (21), we arrive at the fundamental integral equation of SALT:
Ψμ(x)=iD0γγi(kμka)kμ2ka2DdxG(x,x;kμ)Ψμ(x)ϵc(x)(1+νΓν|Ψν(x)|2).
(29)

The new tool of SALT allows one to study random lasers with full nonlinear interactions in 2D and even in 3D. The elimination of time dependence in this theory makes larger and more complex cavities computationally tractable. The theory also provides a new language based on CF states to describe the lasing modes. Now detailed statistical studies as well as comparisons to statistical models based on random matrix theory, disordered media theory, and wave chaos theory are needed. Such studies are in progress.

6. Conclusion

A decade of theoretical study of random lasers has clarified the nature of the lasing modes in disordered systems with multiple scattering and gain. Most important, it has been established that high-Q passive cavity modes such as those created by Anderson localization or by rare fluctuations of various kinds are not necessary in order to have self-organized laser oscillation at a frequency distinct from the atomic transition frequency (gain center). In addition, this study has emphasized a point of general importance in laser theory, that TLMs are not identical to the QB states (resonances) of the passive cavity. This point is demonstrated by a number of numerical calculations presented above and also can be understood from the realization that the QB states are eigenvectors of the unitary S matrix of the cavity without gain, but at complex frequency, whereas the TLMs are eigenvectors of the nonunitary S matrix of the cavity with gain and with real frequency. The difference between these eigenvectors (within the cavity), which is large in the weak scattering limit, becomes small in the diffusive regime as the Q of the cavity increases and is negligible, e.g., for Anderson localized modes and for high-Q modes of conventional cavities. The new basis set of constant flux (CF) states provides a better approximation for finding the TLMs of random lasers and coincides with the exact lasing modes of uniform index cavities. Further statistical and analytical study is necessary to characterize the properties of random lasers in the different regimes, weak scattering, diffusive, and localized, and to understand the effects of nonlinear interactions.

Appendix A: Multipole Method

This appendix details the principle of the multipole method as used in this paper and its implementation. Although we describe here the method for 2D systems, it can be also applied to 3D structures.

We consider a random collection of Nc nonoverlapping cylinders with arbitrary complex dielectric constant ϵl=ϵl+iϵl=nl2 and arbitrary radii al located in a uniform medium with complex dielectric constant ϵb=ϵb+iϵb=nb2 (Fig. 19), where nl=nl+inl and nb=nb+inb are the refractive indices of the cylinders and the background. The complex dielectric permittivities of the cylinders and the background can be arbitrary and may be frequency dependent.

In two dimensions, the solution of the electromagnetic field problem decouples into two fundamental polarizations, in each of which the field may be characterized by a single field component: V(r)=Ez (for TM polarization) and V(r)=Hz (for TE polarization). In the coordinate system that is used, the z axis is aligned with the cylinder axes.

The field component V satisfies the Helmholtz equation
2V(r)+k2n2(r)V(r)=0.
(A1)
For TM polarization, both V(r) and its normal derivative νV(r) are continuous across all boundaries, while for TE polarization the corresponding boundary conditions are the continuity of V(r) and its weighted normal derivative νVn2(r). Here, n(r) denotes the refractive index of the relative medium and ν is an unit outward normal vector.

In the vicinity of the lth cylinder, we may represent the exterior field in the background medium (refractive index nb) in local coordinates as rl=(rl,θl)=rcl, where cl represents the center of the cylinder, and we write
V(r)=m=[AmlJm(knbrl)+BmlHm(1)(knbrl)]eimθl.
(A2)
This local expansion is valid only in an annulus extending from the surface of the cylinder l to the surface of the nearest adjacent cylinder.

The global field expansion (also referred to as a Wijngaard expansion), which is valid everywhere in the background matrix, comprises only outgoing cylindrical harmonic terms:
V(r)=q=1Ncm=BmqHm(1)(k|rcq|)eimarg(rcq).
(A3)

Correspondingly, the field inside any cylinder l is written in an interior expansion:
V(r)=m=CmlJm(knl|rcl|)eimarg(rcl).
(A4)

Then, applying Graf’s addition theorem [53

53. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Complex Media 13, 9–25 (2003). [CrossRef]

] to the terms on the right-hand side of Eq. (A3) (see Fig. 19), we may express the global field expansion in terms of the local coordinate system for the lth cylinder. Equating this with the local expansion (A2), we deduce the field identity (also known as the Rayleigh identity):
Aml=q=1,qlNcp=HmplqBpq,
(A5)
where
Hmplq=Hmp(1)(kclq)ei(mp)θlq.
(A6)
Here, (clq,θlq) are the polar coordinates of the vector clq=cqcl, the position of cylinder q relative to cylinder l.

This is the first connection between the standing wave ({Aml}) and outgoing ({Bml}) multipole coefficients, one which follows solely from the system geometry. Equation (A5) indicates that the local field in the vicinity of cylinder l is due to sources on all other cylinders (ql), the contributions of which to the multipole term of order mp at cylinder l are given by Hmplq.

The second relation between the {Aml} and {Bml} multipole coefficients is obtained from the field continuity equations (i.e., the boundary conditions) at the interface of cylinder l and the local exterior (A2) and interior field (A4) expansions. From these, we obtain
Bml=RmlAml,
(A7)
Cml=TmlAml,
(A8)
where the interface reflection and transmission coefficients, for both Ez and Hz polarization, are given by
Rml=ξnlJm(knlal)Jm(knbal)nbJm(knlal)Jm(knbal)ξnlJm(knlal)Hm(1)(knbal)nbJm(knlal)Hm(1)(knbal),
(A9)
Tml=2i(πkaL)ξnlJm(nlkal)Hm(1)(knbal)nbJm(knlal)Hm(1)(knbal),
(A10)
in which ξ=1 for TM polarization and ξ=nb2(r)nl2(r) for TE polarization.

To derive a simple closed form expression for the solution of the problem, we use partitioned matrix notation, introducing vectors Al=[Aml] and Bl=[Bml] and expressing Eq. (A5) in the form
Al=qHlqBq,
(A11)
where Al and Bl denote vectors of multipole coefficients for cylinder l. The matrix H is block partitioned according to Hlq=[Hmplq] for lq (A6), and Hll=[0], each block of which is a matrix of Toeplitz form. Correspondingly, the matrix forms of Eqs. (A7, A8) are
B=RA,
(A12)
C=TA,
(A13)
where R=diagRl is a block diagonal matrix of diagonal matrices Rl=diagRml, and with corresponding definitions applying for the transmission matrices.

Formal system (A15) is of infinite dimension and so must be truncated to generate a computational solution, the accuracy of which is governed by the number of retained multipole coefficients Nm=2Nmax+1, where Nmax is the truncation order of the multipole series; i.e., only the terms corresponding to the cylindrical harmonics of order n=Nmax,,Nmax are retained.

Appendix B: Finite Element Method

We have also used the finite element method [60

60. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

], implemented in a commercial software (Comsol), to solve wave equation (A1) and calculate the complex eigenvalues and eigenfunctions of the passive modes of the systems that were calculated by the multipole method. The method suitably applies for modeling passive or active modes in a cavity, which is surrounded by perfectly matched layers [64

64. Ü. Pekel and R. Mittra, “A finite-element-method frequency-domain application of the perfectly matched layer (PML) concept,” Microwave Opt. Technol. Lett. 9(3) 117–122 (2007). [CrossRef]

] to simulate open boundaries. It is possible to obtain all the leaky modes, even the resonances characterized by a very small quality factor (as small as 5), in a reasonable computation time with a commercial PC, provided the size of the geometry is smaller than hundred times the wavelength. This is in contrast with the other methods described in this paper, which require much heavier computation.

One of the most important steps of the finite element method is the creation of the mesh that describes the system. Figure 20 shows a close up of a typical mesh calculated for the 2D random system of Fig. 2. The maximum size of elements must be smaller than seven times the wavelength [65

65. P. Ambre, “Modélisation et caractérisation des fibres microstructurées air/silice pour application aux télécommunications optiques,” Ph.D. thèses (Université de Limoges, 2003).

].

Acknowledgments

P. Sebbah thanks the French National Research Agency which supports this work under grant ANR-08-BLAN-0302-01, the PACA region, and the CG06. A. A. Asatryan acknowledges the support from the Australian Research Council under its Centres of Excellence and Discovery Grants programs. A. D. Stone acknowledges support from the National Science Foundation under grant DMR-0908437. H. E. Türeci acknowledges support from the Swiss NSF under grant PP00P2-123519/1.

Tables and Figures

Table 1. QB State Valuesa   for Four Index Values n of Scatterers

table-icon
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Fig. 1 Example of a random realization of 896 circular scatterers contained in a square box of size L=5μm and optical index n=1. The radius and the optical index of the scatterers are, respectively, r=60nm and n=2. The total system of size 9μm is bounded by perfectly matched layers (not shown) to simulate an open system.
Fig. 2 (a) Spatial distribution of the amplitude of a lasing mode in the localized regime (n=2) and (b) that of the corresponding QB states of the same random system without gain. The squares delimit the scattering medium. The amplitude rather than the intensity is represented for a better display of the small values of the field.
Fig. 3 (a) Spatial distribution of the amplitude of a lasing mode in the diffusive regime. (b) Spatial distribution of the field amplitude after the pump has been stopped and the polarization term has been set to zero. The spatial distribution of scatterers is the collection shown in Fig. 1, but here the optical index of the scatterers is n=1.25 instead of n=2 in Fig. 2
Fig. 4 Short-time behavior over a few cycles of the correlation function, CE(t0,t), for (a) a localized lasing mode as in Fig. 2 and for (b) a diffusive lasing mode as in Fig. 3. The periodic square function in (a) is typical of a standing wave, while the sine-like function in (b) is characteristic of a traveling wave [28].
Fig. 5 Correlation function (solid curves) and energy decay (dashed curves) versus time of (a) the lasing mode when the pump is turned off and (b) an arbitrary field distribution at the frequency of the lasing mode.
Fig. 6 The frequencies k of QB states (crosses) and lasing modes with linear gain (open diamonds) together with the decay rates k0 of QB states and the lasing thresholds k of lasing modes. The horizontal dashed lines separate three different regions of behavior: (a) lasing modes are easily matched to QB states, (b) clear differences appear but matching lasing modes to QB states is still possible, (c) lasing modes have shifted so much it is difficult to match them to QB states. The QB state with the largest decay rate and the lasing mode with the largest threshold are circled, though they may not be a matching pair.
Fig. 7 Spatial intensity distributions of QB states IQB(x) (red solid lines) and lasing modes ILG(x) (black dashed lines) from each of the three regions in Fig. 6. Representative samples were chosen for each case. (a) The lasing mode intensity is nearly identical to the QB state intensity with σd=1.7%. (b) A clear difference appears between the lasing mode and the QB state, with σd=21.8%, but they are still similar. (c) The lasing mode with the largest threshold and QB state with the largest decay rate are compared, with σd=198%. Though these two modes are fairly close to each other [circled in Fig. 6 region (c)], their intensity distributions are quite different.
Fig. 8 (a) Intensity |E|2 of the localized QB state (Media 1) and (b) corresponding lasing mode (Media 2) calculated by using a multipole method for a 2D disordered scattering system of the kind shown in Fig. 1 with the refractive index of the cylinders nl=2.0.
Fig. 9 (a) Intensity |E|2 of the diffusive QB state (Media 3) and (b) the lasing mode (Media 4) calculated by using the multipole method for the same random configuration as in Fig. 8 but with the refractive index of the cylinders of nl=1.25.
Fig. 10 Intensity |E|2 of the diffusive QB state (blue dashed curve) and lasing mode (red solid curve) for x=2.75 and nl=1.25.
Fig. 11 (a) Intensity |E|2 of a QB state and (b) a lasing mode calculated by using multipole method for the same random configuration as above but with the refractive index of the cylinders nl=1.75.
Fig. 12 Same as in Fig. 11 but for nl=1.5.
Fig. 13 Intensity |E|2 of QB state (blue dashed curves) and lasing mode (red solid curves) at x=2.75 for (a) nl=1.75, (b) n=1.5, (c) n=1.25.
Fig. 14 Shift of the poles of the S matrix in the complex plane onto the real axis to form TLMs when the imaginary part of the dielectric function ϵϵc+ϵg varies for a simple 1D edge-emitting cavity laser [34]. The cavity is a region of length L and uniform index (a)nc=1.5, (b) n=1.05 (ϵc=2.25,1.0025) terminated in vacuum at both ends. The calculations are based on the MB model discussed in Section 5, with parameters kaL=39 and γ=2. (a) nc=1.5; squares of different colors represent Im[ϵg]=0,0.032,0.064,0.096,0.128; (b) nc=1.05; squares of different colors represent Im[ϵg]=0,0.04,0.08,0.12,0.16. Note the increase in the frequency shift in the complex plane for the leakier cavity. The center of the gain curve is at kL=39, which determines the visible line-pulling effect.
Fig. 15 Typical values of the threshold matrix elements T(0) in a 2D random laser schematized in the inset of Fig. 18 below, using sixteen CF states. The off-diagonal elements are one to two orders of magnitude smaller than the diagonal ones.
Fig. 16 (a) False color plot of one TLM in a 2D random laser modeled as an aggregate of subwavelength particles of index of refraction n=1.2 and radius r=R30 against a background index n=1 imbedded in a uniform disk of gain material of radius R [see inset, panel (d)]. The frequency of the lasing mode is kR=59.9432, which is pulled from (b) the real part of the dominating CF state kmR=59.87660.8593i towards the transition frequency kaR=60. The spatial profile of the TLM and CF state agree very well, whereas (c) the corresponding QB state k̃mR=59.86020.8660i differs from that of the TLM and the CF state noticeably, as can be seen in (d), where we plot the internal intensity along the θ=200° direction [white line in (a)].
Fig. 17 (a) False color plot of one TLM in a 2D random laser similar to that in Fig. 16 but with particles of radius r=R60, corresponding to weaker scattering [see inset, panel (d)]. The frequency of the lasing mode is kR=29.9959, which is very close to (b) the CF state kmR=30.00581.3219i but shifted from (c) the corresponding QB state k̃mR=29.88131.3790i. (d) Internal intensity of the three states in the θ=π direction [white line in (a)]; because of weaker scattering the QB state now differs substantially from the CF and TLM, which still agree quite well with each other.
Fig. 18 (a) CF (dots) and QB (crosses) frequencies in a 2D random laser modeled as an aggregate of subwavelength particles of index of refraction n=1.2 against a background index n=1 imbedded in a uniform disk of gain material (see inset). The two sets of complex frequencies are statistically similar but differ substantially. The solid curve shows the gain curve Γ(k) with γ=1. (b) Lasing frequencies of the same random system well above threshold (colored lines). Colored circles denote the CF state dominating the correspondingly colored modes at threshold.
Fig. 19 Geometry and local coordinate systems.
Fig. 20 Close up of a typical mesh created by Comsol to describe the 2D random system of Fig. 2.
1.

D. S. Wiersma, “The smallest random laser,” Nature 406, 132–135 (2000). [CrossRef] [PubMed]

2.

H. Cao, “Random lasers with coherent feedback,” in Optical Properties of Nanostructured Random MediaV. M. Shalaev, ed., Vol. 82 of Topics in Applied Physics (Springer-Verlag, 2002), pp. 303–330 [CrossRef]

3.

H. Cao, “Lasing in random media,” Waves Random Complex Media 13, R1–R39 (2003). [CrossRef]

4.

H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, and R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003). [CrossRef]

5.

D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008). [CrossRef]

6.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958). [CrossRef]

7.

C. Gouedart, D. Husson, C. Sauteret, F. Auzel, and A. Migus, “Generation of spatially incoherent short pulses in laser pumped neodymium stoichiometric crystal powders,” J. Opt. Soc. Am. B 10, 2358–2363 (1993). [CrossRef]

8.

N. M. Lawandy, R. M. Balachandra, A. S. L. Gomes, and E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994). [CrossRef]

9.

W. L. Sha, C. H. Liu, and R. R. Alfano, “Spectral and temporal measurements of laser action of Rhodamine 640 dye in strongly scattering media,” Opt. Lett. 19, 1922–1924 (1994). [CrossRef] [PubMed]

10.

H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, and R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998). [CrossRef]

11.

H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999). [CrossRef]

12.

H. Cao, Y. G. Zhao, H. C. Ong, and R. P. H. Chang, “Far-field characteristics of random lasers,” Phys. Rev. B 59, 15107–15111 (1999). [CrossRef]

13.

S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, and R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999). [CrossRef]

14.

H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, and R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000). [CrossRef] [PubMed]

15.

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, and C. Q. Cao, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001). [CrossRef] [PubMed]

16.

A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009). [CrossRef]

17.

V. Milner and A. Z. Genack, “Photon localization laser: low-threshold lasing in a random amplifying layered medium via wave localization,” Phys. Rev. Lett. 94, 073901 (2005). [CrossRef] [PubMed]

18.

K. L. van der Molen, R. W. Tjerkstra, A. P. Mosk, and A. Lagendijk, “Spatial extent of random laser modes,” Phys. Rev. Lett. 98, 143901 (2007). [CrossRef] [PubMed]

19.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Random resonators and prelocalized modes in disordered dielectric films,” Phys. Rev. Lett. 89, 016802 (2002). [CrossRef] [PubMed]

20.

V. M. Apalkov, M. E. Raikh, and B. Shapiro, “Almost localized photon modes in continuous and discrete models of disordered media,” J. Opt. Soc. Am. B 21, 132–140 (2004). [CrossRef]

21.

S. Mujumdar, M. Ricci, R. Torre, and D. S. Wiersma, “Amplified extended modes in random lasers,” Phys. Rev. Lett. 93, 053903 (2004). [CrossRef] [PubMed]

22.

S. Mujumdar, V. Türck, R. Torre, and D. S. Wiersma, “Chaotic behavior of a random laser with static disorder,” Phys. Rev. A 76, 033807 (2007). [CrossRef]

23.

A. A. Chabanov, Z. Q. Zhang, and A. Z. Genack, “Breakdown of diffusion in dynamics of extended waves in mesoscopic media,” Phys. Rev. Lett. 90, 203903 (2003). [CrossRef] [PubMed]

24.

J. Fallert, R. J. B. Dietz, J. Sartor, D. Schneider, C. Klingshirn, and H. Kalt, “Co-existence of strongly and weakly localized random laser modes,” Nat. Photonics 3, 279–282 (2009). [CrossRef]

25.

X. Jiang and C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000). [CrossRef] [PubMed]

26.

C. Vanneste and P. Sebbah, “Selective excitation of localized modes in active random media,” Phys. Rev. Lett. 87, 183903 (2001). [CrossRef]

27.

P. Sebbah and C. Vanneste, “Random laser in the localized regime,” Phys. Rev. B 66, 144202 (2002). [CrossRef]

28.

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98, 143902 (2007). [CrossRef] [PubMed]

29.

C. Vanneste and P. Sebbah, “Complexity of two-dimensional quasimodes at the transition from weak scattering to Anderson localization,” Phys. Rev. A 79, 041802(R) (2009). [CrossRef]

30.

X. Wu, W. Fang, A. Yamilov, A. A. Chabanov, A. A. Asatryan, L. C. Botten, and H. Cao, “Random lasing in weakly scattering systems,” Phys. Rev. A 74, 053812 (2006). [CrossRef]

31.

H. E. Türeci, A. D. Stone, and B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006). [CrossRef]

32.

H. E. Türeci, A. D. Stone, and L. Ge, “Theory of the spatial structure of nonlinear lasing modes,” Phys. Rev. A 76, 013813 (2007). [CrossRef]

33.

H. E. Türeci, L. Ge, S. Rotter, and A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643 (2008). [CrossRef] [PubMed]

34.

L. Ge, R. Tandy, A. D. Stone, and H. E. Türeci, “Quantitative verification of ab initio self-consistent laser theory,” Opt. Express 16, 16895 (2008). [CrossRef] [PubMed]

35.

H. E. Türeci, A. D. Stone, L. Ge, S. Rotter, and R. J. Tandy, “Ab initio self-consistent laser theory and random lasers,” Nonlinearity 22, C1–C18 (2009). [CrossRef]

36.

P. Sebbah, D. Sornette, and C. Vanneste, “Anomalous diffusion in two-dimensional Anderson-localization dynamics,” Phys. Rev. B 48, 12506–12510 (1993). [CrossRef]

37.

F. A. Pinheiro, M. Rusek, A. Orlowski, and B. A. van Tiggelen, “Probing Anderson localization of light via decay rate statistic,” Phys. Rev. E 69, 026605 (2004). [CrossRef] .

38.

The idea of considering localized modes for random lasing can already be found in P. Sebbah, D. Sornette, and C. Vanneste, “Wave automaton for wave propagation in the time domain,” Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of OSA Proceedings Series (Optical Society of America, 1994), p. 68.

39.

P. Sebbah, “A new approach for the study of wave propagation and localization,” Ph.D. thesis (Université de Nice—Sophia Antipolis, 1993).

40.

A. E. Siegman, Lasers (University Science Books, 1986).

41.

A. Taflove, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 1995).

42.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1995). [CrossRef]

43.

X. Jiang and C. M. Soukoulis, “Localized random lasing modes and a path for observing localization,” Phys. Rev. E 65, 025601(R) (2002). [CrossRef]

44.

A. G. Fox and T. Li, “Resonant modes in an optical maser,” Proc. IRE 48, 1904–1905 (1960).

45.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961). [CrossRef]

46.

S. M. Dutra and G. Nienhuis, “Quantized mode of a leaky cavity,” Phys. Rev. A 62, 063805 (2000). [CrossRef]

47.

J. C. J. Paasschens, T. Sh. Misirpashaev, and C. W. J. Beenakker, “Localization of light: dual symmetry between absorption and amplification,” Phys. Rev. B 54, 11887–11890 (1996). [CrossRef]

48.

A. A. Asatryan, N. A. Nicorovici, L. C. Botten, M. C. de Sterke, P. A. Robinson, and R. C. McPhedran, “Electromagnetic localization in dispersive stratified media with random loss and gain,” Phys. Rev. B 57, 13535–13549 (1998). [CrossRef]

49.

X. Jiang, Q. Li, and C. M. Soukoulis, “Symmetry between absorption and amplification in disordered media,” Phys. Rev. B 59, R9007–R9010 (1999). [CrossRef]

50.

X. Wu, J. Andreasen, H. Cao, and A. Yamilov, “Effect of local pumping on random laser modes in one dimension,” J. Opt. Soc. Am. B 24, A26–A33 (2007). [CrossRef]

51.

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994). [CrossRef]

52.

E. Centeno and D. Felbacq, “Characterization of defect modes in finite bidimensional photonic crystals,” J. Opt. Soc. Am. A 16, 2705–2712 (1999). [CrossRef]

53.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, “Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals,” Waves Random Complex Media 13, 9–25 (2003). [CrossRef]

54.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. Martijn de Sterke, and N. A. Nicorovici, “Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,” Phys. Rev. E 63, 046612 (2001). [CrossRef]

55.

A. A. Asatryan, L. C. Botten, N. A. Nicorovici, R. C. McPhedran, and C. Martijn de Sterke, “Frequency shift of sources embedded in finite two-dimensional photonic clusters,” Waves Random Complex Media 16, 151–165 (2006). [CrossRef]

56.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of optical wave-guides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994). [CrossRef]

57.

T. P. White, B. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers I: formulation,” J. Opt. Soc. Am. B 10, 2322–2330 (2002). [CrossRef]

58.

B. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers II: implementation and results,” J. Opt. Soc. Am. B 10, 2331–2340 (2002). [CrossRef]

59.

D. P. Fussell, R. C. McPhedran, C. Martijn de Sterke, and A. A. Asatryan, “Three-dimensional local density of states in a finite two-dimensional photonic crystal composed of cylinders,” Phys. Rev. E 67, 045601(R) (2003). [CrossRef]

60.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

61.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1982).

62.

H. Haken, Light: Laser Dynamics (North-Holland, 1985), vol. 2.

63.

Recently it was shown that one can define a variant of the CF states, termed the threshold constant flux (TCF) states, for which one TCF state is the TLM. Above threshold additional TCF states are needed to describe the lasing modes, but in general fewer than for the CF states defined here. The above threshold theory using these TCF states is almost identical to the theory described in [33, 35]. Details are in L. Ge, Y. Chong, and A. D. Stone, “Steady-state ab initio laser theory: generalizations and analytic results,” arXiv.org, arXiv:1008.0628v1 (submitted to Phys. Rev. A).

64.

Ü. Pekel and R. Mittra, “A finite-element-method frequency-domain application of the perfectly matched layer (PML) concept,” Microwave Opt. Technol. Lett. 9(3) 117–122 (2007). [CrossRef]

65.

P. Ambre, “Modélisation et caractérisation des fibres microstructurées air/silice pour application aux télécommunications optiques,” Ph.D. thèses (Université de Limoges, 2003).

aop-3-1-88-i001 Jonathan Andreasen studied computational physics at Illinois State University and received his Ph.D. degree in physics and astronomy from Northwestern University, Evanston, Illinois, in 2009. He is a Research Affiliate in the Department of Applied Physics, Yale University, New Haven, Connecticut. Since 2010, he has been a Chateaubriand postdoctoral fellow in the Laboratoire de Physique de la Matière Condensée, Université de Nice—Sophia Antipolis, France. His research interests are computational electrodynamics, nonlinear dynamics and chaos, laser excitation, microcavity lasers, collective excitations, and applications of computer vision and genetic algorithms to optics.

aop-3-1-88-i002 Ara Asatryan graduated with a master degree, first class honors from Yerevan State University in 1983, Armenia. His Ph.D. studies were done at the General Physics Institute of the Soviet Union Academy of Sciences, Moscow, supervised by Prof. Yu. A. Kravtsov. He got his Ph.D. in radiophysics including quantum electronics from the Radiophysics Institute of the Armenian Academy of Sciences in 1988. He started his employment as a lecturer at Yerevan State University in 1988 and become a reader in 1992. In 1994 he migrated to Australia and worked as a Research Associate at Macquarie University in 1995—1997 and at Sydney University in 1998–2001. Since 2002 he has worked at the University of Technology Sydney as a Research Associate and became a Senior Research Associate in 2004. His main area of research is in wave propagation in disordered media, Anderson localization, photonic crystals, and metamaterials. Some of his major results are the characterization of the effects of disorder in the propagation of waves in random photonic crystals, construction of the Green’s function for finite photonic clusters and calculation of their local density of states, characterization of the effects of metamaterials on the Anderson localization, and calculation of the modes of random lasers. He is an Author of 70 refereed papers, four book chapters, and more than 100 conference presentations. He is the Senior member of the Optical Society of America and a member of the Australian Optical Society. He is a regular referee for Physical Review Letters, Physical Review A, B, and E, Optics Letters, Optics Express, and other major journals.

aop-3-1-88-i003 Hui Cao received the Ph.D. degree in applied physics from Stanford University, Stanford, California, in 1997. Currently, she is a Professor in the Department of Applied Physics and Physics, Yale University, New Haven, Connecticut. She has coauthored one book, three book chapters, three review articles and 130 journal papers. Dr. Cao received the David and Lucille Packard Fellow in 1999, the Alfred P. Sloan Fellow in 2000, the National Science Foundation (NSF) CAREER Award in 2001, the Friedrich Wilhelm Bessel Research Award of the Alexander von Humboldt Foundation in 2005, and the APS Maria Goeppert-Mayer Award in 2006. She is a fellow of the Optical Society of America (OSA) and the American Physical Society (APS).

aop-3-1-88-i004 Li Ge received the B.S. degree in Physics from Peking University, China, in 2004. He then came to the United States and received his Ph.D. degree in Physics from Yale University in 2010. His doctoral research focused on self-consistent ab initio laser theory (SALT) under the supervision of Prof. A. Douglas Stone. Currently he is working with Prof. Hakan Türeci as a postdoctoral associate in the Department of Electrical Engineering at Princeton University.

aop-3-1-88-i005 Laurent Labonté was born in Périgueux, France, on May 15, 1979. He received the Ph.D. degree in Electronics and Photonics from the University of Limoges (France), where he studied theoretically and experimentally the characteristics of the propagation in microstructure fiber. Since 2006, he has been a lecturer at the Laboratoire de Physique de la Matière Condensée, and his research interests have been in the design of original optic fibers. The applications of this research are in classical and quantum telecommunication. On the other hand, more generally, he is interested in studying the propagation of light waves in a guided structure.

aop-3-1-88-i006 Patrick Sebbah is a researcher of the CNRS (Centre Nationale de la Recherche Scientifique) since 1994, presently at the Laboratoire de Physique de la Matiere Condensee, University of Nice—Sophia Antipolis, France. He received an education in general physics and laser physics. He graduated from the Ecole Nationale Superieure des Telecommunications de Paris (1988) and from the University Paris XI—Orsay (1989). He received his Ph.D. degree in physics from the University of Nice—Sophia Antipolis in 1993 for his dissertation on wave propagation and localization. He did his postdoctoral work at the Physics Department of Queens College, CUNY, New York, USA, in the group of Prof. A. Z. Genack. His research interests are light/acoustics wave/microwave propagation in complex media, multiple scattering and localization, random lasing, and nonlinear scattering.

aop-3-1-88-i008 Hakan E. Türeci studied Physics at Bilkent University and received his Ph.D. from Yale University in 2003 for his dissertation on mesoscopic optics. He did his postdoctoral work in the Condensed Matter Theory group at Yale University and in the Institute for Quantum Electronics at ETH Zurich. In 2009, he was appointed as SNF Professor for Mesoscopic Quantum Optics at ETH. Since 2010, he has been an Assistant Professor in the Department of Electrical Engineering at Princeton University. His research interests span a wide range of theoretical problems in quantum optics and condensed matter physics.

aop-3-1-88-i009 Christian Vanneste studied Physics at Orsay University and received his Ph.D. from Nice University in 1982 for his dissertation on the Josephson effect. After a postdoctoral stay at IBM Thomas J. Watson Research Center, he joined CNRS (Centre National de la Recherche Scientifique) in the Laboratoire de Physique de la Matière Condensée, University of Nice—Sophia Antipolis. His current research interests are wave propagation in complex media and random lasers.

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 21, 2010
Revised Manuscript: August 4, 2010
Manuscript Accepted: August 6, 2010
Published: October 1, 2010

Virtual Issues
(2011) Advances in Optics and Photonics

Citation
J. Andreasen, A. A. Asatryan, L. C. Botten, M. A. Byrne, H. Cao, L. Ge, L. Labonté, P. Sebbah, A. D. Stone, H. E. Türeci, and C. Vanneste, "Modes of random lasers," Adv. Opt. Photon. 3, 88-127 (2011)
http://www.opticsinfobase.org/aop/abstract.cfm?URI=aop-3-1-88


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References

  1. D. S. Wiersma, “The smallest random laser,” Nature 406, 132–135 (2000). [CrossRef] [PubMed]
  2. H. Cao, “Random lasers with coherent feedback,” in Optical Properties of Nanostructured Random MediaV. M. Shalaev, ed., Vol. 82 of Topics in Applied Physics (Springer-Verlag, 2002), pp. 303–330 [CrossRef]
  3. H. Cao, “Lasing in random media,” Waves Random Complex Media 13, R1–R39 (2003). [CrossRef]
  4. H. Cao, J. Y. Xu, Y. Ling, A. L. Burin, E. W. Seeling, X. Liu, R. P. H. Chang, “Random lasers with coherent feedback,” IEEE J. Sel. Top. Quantum Electron. 9, 111–118 (2003). [CrossRef]
  5. D. S. Wiersma, “The physics and applications of random lasers,” Nat. Phys. 4, 359–367 (2008). [CrossRef]
  6. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492 (1958). [CrossRef]
  7. C. Gouedart, D. Husson, C. Sauteret, F. Auzel, A. Migus, “Generation of spatially incoherent short pulses in laser pumped neodymium stoichiometric crystal powders,” J. Opt. Soc. Am. B 10, 2358–2363 (1993). [CrossRef]
  8. N. M. Lawandy, R. M. Balachandra, A. S. L. Gomes, E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994). [CrossRef]
  9. W. L. Sha, C. H. Liu, R. R. Alfano, “Spectral and temporal measurements of laser action of Rhodamine 640 dye in strongly scattering media,” Opt. Lett. 19, 1922–1924 (1994). [CrossRef] [PubMed]
  10. H. Cao, Y. G. Zhao, H. C. Ong, S. T. Ho, J. Y. Dai, J. Y. Wu, R. P. H. Chang, “Ultraviolet lasing in resonators formed by scattering in semiconductor polycristalline films,” Appl. Phys. Lett. 73, 3656–3658 (1998). [CrossRef]
  11. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, R. P. H. Chang, “Random laser action in semiconductor powder,” Phys. Rev. Lett. 82, 2278–2281 (1999). [CrossRef]
  12. H. Cao, Y. G. Zhao, H. C. Ong, R. P. H. Chang, “Far-field characteristics of random lasers,” Phys. Rev. B 59, 15107–15111 (1999). [CrossRef]
  13. S. V. Frolov, Z. V. Vardeny, K. Yoshino, A. Zakhidov, R. H. Baughman, “Stimulated emission in high-gain organic media,” Phys. Rev. B 59, R5284–R5287 (1999). [CrossRef]
  14. H. Cao, J. Y. Xu, D. Z. Zhang, S.-H. Chang, S. T. Ho, E. W. Seelig, X. Liu, R. P. H. Chang, “Spatial confinement of laser light in active random media,” Phys. Rev. Lett. 84, 5584–5587 (2000). [CrossRef] [PubMed]
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  16. A. Lagendijk, B. van Tiggelen, D. S. Wiersma, “Fifty years of Anderson localization,” Phys. Today 62(8), 24–29 (2009). [CrossRef]
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