## Abstract

Unlike conventional lasers, diffusive random lasers (DRLs) have no resonator to trap light and no high-Q resonances to support lasing. Because of this lack of sharp resonances, the DRL has presented a challenge to conventional laser theory. We present a theory able to treat the DRL rigorously and provide results on the lasing spectra, internal fields, and output intensities of DRLs. Typically DRLs are highly multimode lasers, emitting light at a number of wavelengths. We show that the modal interactions through the gain medium in such lasers are extremely strong and lead to a uniformly spaced frequency spectrum, in agreement with recent experimental observations.

Novel laser systems have emerged recently because of modern nanofabrication capabilities (*1*–*3*). The diffusive random laser (DRL), perhaps the most challenging of the new systems, consists of a random aggregate of particles that scatter light and have gain or are embedded in a background medium with gain (*2*, *4*–*8*). Whereas light scattering in such a random medium can give rise to Anderson-localized, high-Q resonances (*9*, *10*), in almost all experiments the localized regime is not reached, and the laser “cavity” has no isolated resonances in the absence of gain. Despite the lack of sharp resonances, the laser emission from the more recent DRLs (*2*, *5*, *6*) was observed to have the essential properties of conventional lasers: the appearance of coherent emission with line-narrowing above a series of thresholds and uncorrelated photon statistics far from threshold (*11*). Earlier work on random lasers (*4*, *7*) did not find isolated narrow lines and was interpreted as incoherent lasing, in which there was intensity feedback but not amplitude feedback. Later experiments (*2*) and recent numerical studies (*12*) indicated that the lasing involves coherent phase-sensitive feedback in at least some cases. Our work shows that standard coherent multimode lasing is possible even when the linear resonances are much broader than their spacing, raising the question of what determines the emission frequencies of DRLs because they are not determined by the position of passive cavity resonances. Furthermore, recent experiments on porous GaP DRLs have shown that the frequencies are rather uniformly spaced and stable from pulse to pulse, although the intensities vary substantially (*8*). We show that this is a result of strong nonlinear interactions between lasing modes combined with extreme leakiness, a regime particularly difficult to treat. In any multimode laser, the different modes compete with one another through the gain medium in a complex manner that depends on the spatial distribution of the electric field of each mode. This is particularly severe in the DRL, in which there are many spatially overlapping modes with similar (very short) lifetimes.

The finesse, *f*, of a resonator is defined as the ratio of the resonance spacing to the resonance width; standard laser theory only addresses lasers with high finesse (weakly open) resonators and cannot be applied to the DRL, which has finesse much less than unity. Hence, no analytic results have been derived relating to two-dimensional (2D) or 3D DRLs, and realistic numerical simulations have been limited because of the computational demands. We introduced a formulation of semiclassical laser theory in terms of biorthogonal modes, called constant flux (CF) states, which treats lasing media with any degree of outcoupling and includes the effects of nonlinear modal interactions to all orders (*13*, *14*). We present analytic and numerical results using this approach applied to a DRL.

The simplest model for a laser that captures all of the relevant spatial complexity uses the Maxwell-Bloch equations (*15*, *16*), which are three coupled nonlinear equations for the electric field, the polarization, and the inversion of the gain medium. For stationary multimode lasing, the modes predicted by these equations are determined by the time-independent self-consistent equation (*13*) (1) where the electric field is given by *e*(** x**,

*t*) = Σ

_{μ}Ψ

_{μ}(

**)e**

*x*^{–iΩμt}). In Eq. 1, the lasing frequencies Ω

_{μ}=

*ck*

_{μ}and the lasing mode functions Ψ

_{μ}(

**) are assumed to be unknown (henceforth we set the speed of light**

*x**c*= 1 and use the wave vector to denote frequency as well). In Eq. 1,

*k*

_{a}is the atomic frequency, γ

_{⊥}is the transverse relaxation rate, Γ

_{ν}= Γ(

*k*

_{ν}) is the gain profile evaluated at

*k*

_{ν},

*D*

_{0}(

**) =**

*x**D*

_{0}[1 +

*d*

_{0}(

**)] is the pump, which can vary in space, and ϵ(**

*x***) =**

*x**n*

^{2}(

**) is the dielectric function of the “resonator.” Electric field and pump strength are dimensionless, being measured in units and , where γ**

*x*_{||}is the longitudinal relaxation rate and

*g*is the dipole matrix element of the gain medium. Each lasing mode Ψ

_{μ}depends nonlinearly on all of the other lasing modes through the denominator in Eq. 1, which represents the “spatial hole-burning” (

*15*) interaction with the other modes. Through this mechanism, modes that lase first tend to suppress lasing in other modes, particularly those with which they are correlated in space.

For simplicity we study a 2D DRL and take the gain medium to be a uniform disk of radius *R*, which contains randomly placed nanoparticles with constant index greater than unity. The light field in the cavity can be either transverse magnetic or transverse electric polarized perpendicular to the plane of the disk, leading to a scalar equation for its normal component. The integral in Eq. 1 is over the gain region, and the kernel *G*(** x**,

**;**

*x′**k*) is the Green function of the cavity wave equation with purely outgoing boundary conditions (

*13*). This represents the steady-state response of the passive cavity to a monochromatic dipole oscillating with frequency

*k*at

**. The nonhermitian boundary conditions on the Green function lead to a spectral representation**

*x′**G*(

**,**

*x***;**

*x′**k*) in terms of dual sets of biorthogonal functions φ

_{m}(

**,**

*x**k*) and φ̄

_{m}(

*,*

**x***k*), termed constant flux (CF) states, with complex eigenvalues,

*k*

_{m}(

*13*).

The CF states play the role of the linear cavity resonances within our theory and reduce to the quasi-bound (QB) states within the cavity to a good approximation for a high-Q resonator (*17*). Importantly, the CF states are complete within the cavity and generate a conserved photon flux outside the cavity, unlike the QB states (*13*); for the extremely leaky cavity of a DRL, the CF and QB states are significantly different but statistically similar (Fig. 1).

Because the CF basis is complete and conserves flux outside the gain region, it is an appropriate basis for representing arbitrary lasing modes Ψ_{μ}(** x**) of a DRL. To solve Eq. 1, we expand each mode in terms of CF states: . Substituting this expansion into 1 gives an equation for the complex vector of coefficients that completely determines Ψ

_{μ}: (2) The nonlinear operator is written out explicitly and discussed in (

*18*).

This formalism allows us to obtain analytic insight into the question of what determines the frequencies of the DRL. In single-mode lasers, each lasing frequency is a weighted mean of the real part of the cavity resonance frequency and the atomic frequency (*16*), which for a typical high-finesse system is very close to the cavity frequency with a small shift (“pull”) toward the atomic line. If we denote the “conventional” lasing frequency by , we find from Eq. 2 that for the DRL (3) where is a collective contribution due to all the other CF states, which has no analog in conventional lasers. In our parameter regime (*k*_{a}*R* ≈ 30), both the conventional and collective terms are important (although the conventional term is larger), and the lasing frequencies have no simple relationship to the cavity frequencies. The collective term is random in sign and does not always generate a pull toward line center (Fig. 1). We believe that at larger *k*_{a}*R* the collective term will dominate.

In Fig. 2, we plot the intensities associated with the lasing modes of Fig. 1 as a function of pump strength, *D*_{0}, measured by the length of the vector of CF coefficients, . The behavior is very different from conventional lasers, showing complex nonmonotonic and reentrant behavior in contrast to the linear increase found for uniform edge-emitting lasers (*14*). Analysis reveals that the complex behavior is due to the strong spatial hole-burning interactions in these systems. The Fig. 2 inset shows the lasing frequencies associated with the modes as a function of pump; of the eight lasing modes in the interval, there are six that form three pairs nearby in frequency, and their behavior is highly correlated. Evaluation of the overlap of the **a**^{μ} vector associated with each pair of modes confirms that not only their frequencies but also their decomposition into CF states are similar.

Equations 1 and 2 imply that modes with similar **a**^{μ} vectors and similar frequencies will compete strongly because this leads to a hole-burning denominator that is spatially correlated with the numerator. However, it is not obvious that frequency quasi-degeneracy should be associated with spatial correlation for the DRL. For random lasers with Anderson localization (*9*, *10*), the CF states would typically be spatially disjoint, the *T*_{mn}(*k*) operator (compare with Eq. S3) would be approximately diagonal, and there would be no such spatial correlation. In additional calculations not shown, we do find that for larger index nanoparticles, which begin to localize the CF states, the modal interactions are reduced. But for the DRL, *T*_{mn}(*k*) is not diagonal, and frequency degeneracy would require an eigenvalue degeneracy in this complex pseudo-random matrix [see discussion in (*18*)]. Instead, there is eigenvalue repulsion in the complex plane and strong mixing of eigenvectors, resulting in large spatial overlap of quasidegenerate lasing modes and strong hole-burning interaction. This interaction, in the absence of some special symmetry, tends to suppress one of the two modes, leading to well-spaced lasing frequencies as found by (*8*). Hence, the rigid lasing-frequency spectrum could distinguish the DRL from an Anderson-localized laser.

This strong interaction of nearly degenerate modes is reflected in a very large increase in the lasing threshold of the second mode of each pair (Fig. 2 caption) (*18*). These interaction effects are strongly nonlinear and hence highly sensitive to statistical fluctuations. To illustrate this, in Fig. 3 we contrast the intensity behavior of Fig. 2, for which the pump was uniform in space [*d*_{0}(** x**)= 0], with a case for which we have added to the uniform pump a random white noise term

*d*

_{0}(

**) of standard deviation ±30% (normalized to the same total power). For this nonuniform pump, the third uniform mode (green) now turns on first. It is thus able to suppress the seventh uniform mode (purple) over the entire range of pump powers and acquires an intensity almost a factor of 3 greater at the same average pump power. The intensities of all the interacting pairs show similar high sensitivity to pump profile, whereas their frequencies remain relatively stable (Fig. 3 inset). Exactly such behavior was observed in shot-to-shot spectra of DRLs in experiments (**

*x**8*).

Lastly, we consider the spatial variation of the electric field in DRLs (Fig. 4). The false-color representation of the multimode electric field in the laser has a striking property: It is consistently brighter at the edge of the disk than at its center, even though the gain is uniform and there are no special high-Q modes localized near the edge. To demonstrate that this effect is not a statistical fluctuation associated with this particular disorder configuration, we have averaged the behavior of the entire basis set of CF states over disorder configurations. The result is a nonrandom average growth of intensity toward the boundary. The origin of this effect is known from earlier work on distributed feedback lasers with weak reflectivity (*19*); if the single-pass loss is large, then the single-pass gain must also be large in order to lase, leading to a visible nonuniformity of the lasing mode, with growth in the direction of the loss boundary (on average the radial direction for the DRL). Because the DRL has fractional finesse (which is not achievable in a 1D geometry), this effect is much larger in these systems and should be observable. This effect means that the electric field fluctuations in DRLs will differ substantially from the random matrix/quantum chaos fluctuations of linear cavity modes (*20*), first because each mode is a superposition of pseudo-random CF states and second because these CF states themselves are not uniform on average.

The coexistence of gain, nonlinear interactions, and overlapping resonances (fractional finesse) makes the DRL a more complex and richer system than the widely studied linear wave-chaotic systems. It remains to be seen whether concepts from random matrix theory and semiclassical quantum mechanics (quantum chaos) will prove fruitful in this context. The theory presented here is ab initio in the sense that it generates all properties of the lasing states from knowledge of the dielectric function of the host medium and basic parameters of the gain medium; it should be applicable to any novel lasercavity system.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/320/5876/643/DC1

Materials and Methods

Figs. S1 to S3

References and Notes