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. In this paper, we review numerical and theoretical studies aimed at clarifying the nature of the lasing modes in disordered scattering systems with gain. The last 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 will focus here on the near-threshold single-mode lasing regime where nonlinear effects associated with gain saturation and mode competition can be neglected. We will 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 (TLM) 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.

We study wave propagation in mixed, 1D disordered stacks of alternating right- and left-handed layers and reveal that the introduction of metamaterials substantially suppresses Anderson localization. At long wavelengths, the localization length in mixed stacks is orders of magnitude larger than for normal structures, proportional to the sixth power of the wavelength, in contrast to the usual quadratic wavelength dependence of normal systems. Suppression of localization is also exemplified in long-wavelength resonances which largely disappear when left-handed materials are introduced.

We study the propagation and localization of classical waves in one-dimensional disordered structures composed of alternating layers of left-and right-handed materials (mixed stacks) and compare them with structures composed of different layers of the same material (homogeneous stacks). The main characteristic of the transport is the transmission length, which coincides with the localization and ballistic lengths in the localized and ballistic regime respectively. For weakly scattering layers, we have developed an effective analytical approach and have calculated the transmission length within a wide region of the input parameters. This enables us to describe in a unified way the localized and ballistic regimes as well as the crossover between them. When both refractive index and layer thickness of a mixed stack are random, the transmission length in the long-wave range of the localized regime exhibits a quadratic power wavelength dependence with the different coefficients of proportionality for mixed and homogeneous stacks. Moreover, the transmission length of a mixed stack differs from reciprocal of the Lyapunov exponent of the corresponding infinite stack. In the ballistic regime, the transmission length of a mixed stack for a particular realization is always a strongly fluctuating quantity, while in a homogeneous stack, it fluctuates only in the near long-wave region. In the far long-wave part of ballistic region, the transmission length is practically non-random, and the stack becomes effectively uniform. The crossover region from the localization to the ballistic regime is relatively narrow for both M-and H-stacks. In mixed stacks with only refractive-index disorder, Anderson localization at long wavelengths is substantially suppressed, with the localization length growing with wavelength much faster than for homogeneous stacks. The crossover region becomes essentially wider and transmission resonances appear only in much longer stacks. The effects of absorption on the one-dimensional transport and localization have also been studied, both analytically and numerically. Specifically, it is shown that the crossover region is particularly sensitive to losses, so that even small absorption noticeably suppresses frequency dependent oscillations in the transmission length. All theoretical predictions are in an excellent agreement with the results of numerical simulations.