Localization and absorption of waves in a weakly dissipative disordered medium

The effect of a small imaginary part ${\epsilon }_2$ to the dielectric constant on the propagation of waves in a disordered medium near the Anderson localization transition is considered. The n→0 replica-field representation of the averaged Green’s function leads to a nonlinear σ model with a symmetry-breaking perturbation proportional to ${\epsilon }_2$. In d=2+ε, the renormalized energy absorption coefficient is shown to increase anomalously with frequency ω near the mobility edge ${\omega }^{\mathrm{*}}$ as α∼(${\omega }^{\mathrm{*}}$-ω${)}^{\mathrm{-}\left(d\mathrm{-}2\right)\nu /2\mathrm{}}$, ν=1/ε. It is shown that the wavelength ${\lambda }^{\mathrm{*}}$ below which localization occurs is related to the elastic mean free path l by (l/${\lambda }^{\mathrm{*}}$ ${)}^{d\mathrm{-}1}$∼1/ε (d>2). This may occur near the limit of attainable disorder from a quenched random array of small dielectric or metallic spheres.