Non-linear supersymmetric σ-model for diffusive scattering of classical waves with resonance enhancement

We derive a non-linear σ-model for the transport of light (classical waves) through a disordered medium. We compare this extension of the model with the well-established non-linear σ-model for the transport of electrons (Schrodinger waves) and display similarities of and differences between both cases. Motivated by experimental work (M. van Albada et al., Phys. Rev. Lett., 66 (1991) 3132), we then generalize the non-linear σ-model further to include resonance scattering. We find that the form of the effective action is unchanged but that a parameter of the effective action, the mean level density, is modified in a manner which correctly accounts for the data.