Completing the interaction-suppression theory of multimode random lasers in the Anderson localized regime: open-system mode statistics, full-vector 3D Maxwell–Bloch dynamics, gain saturation, and the crossover to interacting lasing

In Anderson-localized random lasers, Stano and Jacquod (Nature Photonics 7, 66, 2013) predicted that strong disorder suppresses interactions between simultaneously lasing modes: each lasing mode is a single, pump-independent, Anderson-localized mode of the passive cavity, with modal interactions turned on only by absorption or delocalization. The authors explicitly framed this result as resting on three idealizations — constant-flux-mode statistics borrowed from closed non-absorbing systems, scalar one- and two-dimensional stationary theory where a full-vector Maxwell–Bloch analysis is required in three dimensions, and unbounded inversion density — and proposed monitoring the pump dependence of the lasing frequencies as an experimental diagnostic of Anderson localization. The open problem is to establish, with these idealizations removed, the actual domain of validity of the interaction-suppression criteria, and to test experimentally, through the proposed pump-frequency diagnostic, the crossover from non-interacting localized lasing to interacting delocalized lasing. The corresponding questions remain open: the open-system wavefunction-statistics assumption is unproven, the three-dimensional vector criterion is unconfirmed, the impact of bounded inversion on the onset of interactions is unquantified, and the proposed pump-frequency diagnostic has not been executed.

Background

A random laser is a laser in which the optical feedback that traps light comes not from mirrors or a designed cavity but from multiple scattering in a disordered gain medium, such as a semiconductor powder or a dye-doped suspension of scatterers. Because the light is multiply scattered, wave interference dominates transport, and in a sufficiently strongly scattering medium interference can bring transport to a halt: this is Anderson localization, where modes of the medium decay exponentially with a characteristic localization length $\xi$ . Early numerical and experimental work (Vanneste and Sebbah, Phys. Rev. A 79, 041802(R) (2009); Türeci, Ge, Rotter and Stone, Science 320, 643 (2008)) established two contrasting regimes: in weakly scattering, diffusive random lasers the lasing modes are extended and interact strongly (mode switching, frequency repulsion), while in strongly scattering media long-lived localized modes can lase individually.

The quantitative theory used to describe multimode lasing in such systems is the steady-state ab initio laser theory (SALT) of Türeci, Stone and Collier (Phys. Rev. A 74, 043822 (2006)), a semiclassical stationary solution of the Maxwell–Bloch equations (MBE) — the coupled equations for the classical electromagnetic field and the quantum two-level polarization and population inversion of the gain medium. SALT expands the lasing field in constant-flux (CF) states, which are scattering eigenstates of the open, passive (unpumped) cavity with purely outgoing boundary conditions; unlike the resonances (quasimodes) of the open system, CF states form a complete, well-behaved basis at fixed real wavenumber. The gain enters through a threshold matrix, and the lasing modes interact through spatial hole burning — the local depletion of the population inversion by the lasing fields themselves, which couples all simultaneously lasing modes.

Stano and Jacquod (Nature Photonics 7, 66–71 (2013)) analyzed this multimode theory in the Anderson-localized regime and showed that localization suppresses modal interactions: the corrections to a single-pole approximation (one lasing mode dominated by a single CF component) are exponentially small in $L/\xi$ , where $L$ is the system size, because exponentially localized modes separated by more than a localization length have exponentially small matrix-element products. They concluded that the criteria for the absence of interactions are $\xi \ll L$ in one dimension and $\xi \ll L$ together with $k_\mu \ell \lesssim L/\xi$ in two dimensions — the latter bound derived when $\xi$ is not exponentially larger than the wavelength $2\pi/k_\mu$ — where $k_\mu$ is the lasing wavenumber and $\ell$ the transport mean free path, and they expected the same criterion in three dimensions. Their one-dimensional numerics confirmed that lasing modes keep their frequency, position and spatial extent (measured by inverse-participation-ratio-type quantities, e.g. the participation volume $\int |\Psi|^2\,dx / \int |\Psi|^4\,dx$ , the inverse of the inverse participation ratio, which shrinks as a mode localizes) unchanged as the pump is raised by orders of magnitude and many mode thresholds are crossed, while adding absorption or reducing the disorder (delocalized modes) turns interactions back on, with mode switching and pump-dependent frequencies.

The authors themselves, however, stated the limits of this analysis. First, their estimates assume that CF states have the same wavefunction correlations as the localized modes of a corresponding closed, non-absorbing system — an assumption they caution is no longer true in the weakly scattering, delocalized regime and whose range of validity in open geometries was not established. Second, the two- and three-dimensional criteria were derived for scalar fields, and they recalled that an analysis of the MBE for full vector fields is required in three dimensions. Third, the numerics imposed no upper bound on the population inversion density, and the authors noted that incorporating a finite maximal density would lead to mode saturation but would also limit the number of lasing modes and thus constrain the investigation of mode interactions, leaving the effect of physical gain saturation (a finite maximal inversion) on mode interactions open. Fourth, they proposed as an experimental diagnostic that monitoring how little lasing frequencies shift with pump strength could detect Anderson localization — an observation that does not require spatially resolving the lasing field. In addition, the theory is stationary and semiclassical; in the Supplementary Information the authors noted that for quasi-degenerate hybridized (Mott) state pairs the static inversion approximation made to derive the SALT equations may break down, so that further investigations are necessary to determine the relevance of such states in random lasing in the Anderson localized regime.

Problem Statement

The problem is to determine, with the idealizations of the 2013 analysis removed, under what physical conditions Anderson localization suppresses interactions between simultaneously lasing modes in multimode random lasers, and to test experimentally, through the proposed pump-frequency diagnostic, the crossover from non-interacting localized lasing to interacting delocalized lasing. Its components jointly complete one theory — each removes one acknowledged idealization from the same single-pole (non-interacting) analysis of multimode lasing, and together they fix the domain of validity of the Stano–Jacquod prediction that lasing modes in strongly scattering random lasers are pump-independent Anderson-localized passive-cavity modes: (1) establish, analytically or with controlled numerics, the range of validity of treating the open cavity's constant-flux states (the scattering eigenstates of the passive, open cavity with purely outgoing boundary conditions, which form the expansion basis of the steady-state laser theory) as having the wavefunction correlations of localized modes of closed, non-absorbing systems — in particular where and how this fails in weakly scattering or strongly open geometries — and correspondingly bound the corrections to the non-interacting single-pole approximation; (2) confirm or revise the dimensional interaction-suppression criteria ( $\xi \ll L$ in one dimension; $\xi \ll L$ together with $k_\mu \ell \lesssim L/\xi$ in two dimensions, a bound derived when $\xi$ is not exponentially larger than the wavelength $2\pi/k_\mu$ ; the same expected in three dimensions; here $\xi$ is the localization length, $L$ the linear system size, $k_\mu$ the lasing wavenumber and $\ell$ the transport mean free path): the one- and two-dimensional criteria by any controlled argument, and the three-dimensional expectation with a full-vector three-dimensional Maxwell–Bloch treatment of random lasers in the localized regime; (3) quantify how a finite maximal inversion density (gain saturation) limits the number of simultaneously lasing modes and how it changes their interactions; and (4) execute the proposed experimental diagnostic of monitoring how much lasing frequencies shift with pump strength, whose invariance would signal Anderson localization without spatially resolving the field, and thereby probe the crossover from non-interacting localized lasing to interacting delocalized lasing. All four components are jointly required: resolving a subset advances only that component and leaves the task open, so only a result covering all four resolves the task; for component (4), a quantitative design of the pump-frequency diagnostic together with a supporting numerical (Maxwell–Bloch or SALT) demonstration counts as completing that component, and a laboratory measurement is not required.

References

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