Hydrodynamic and field-theoretic approaches of light localization in open media

Many complex systems exhibit hydrodynamic (or macroscopic) behavior at large scales characterized by few variables such as the particle number density, temperature and pressure obeying a set of hydrodynamic (or macroscopic) equations. Does the hydrodynamic description exist also for waves in complex open media? This is a long-standing fundamental problem in studies on wave localization. Practically, if it does exist, owing to its simplicity macroscopic equations can be mastered far more easily than sophisticated microscopic theories of wave localization especially for experimentalists. The purposes of the present paper are two-fold. On the one hand, it is devoted to a review of substantial recent progress in this subject. We show that in random open media the wave energy density obeys a highly unconventional macroscopic diffusion equation at scales much larger than the elastic mean free path. The diffusion coefficient is inhomogeneous in space; most strikingly, as a function of the distance to the interface, it displays novel single parameter scaling which captures the impact of rare high-transmission states that dominate long-time transport of localized waves. We review aspects of this novel macroscopic diffusive phenomenon. On the other hand, it is devoted to a review of the supersymmetric field theory of light localization in open media. In particular, we review its application in establishing a microscopic theory of the aforementioned unconventional diffusive phenomenon.