Localization of classical waves in a random medium: A self-consistent theory

We study localization of classical waves in a model of point scatterers, idealizing a random arrangement of dielectric spheres (ε=1+Δε) of volume Vs and mean spacing a in a matrix (ε=1). At distances much-gt a energy transport is diffusive. A self-consistent equation for the frequency-dependent diffusion coefficient is obtained and evaluated in the approximation where noncritical quantities are calculated in the coherent potential approximation. The velocity of energy transport and the phase velocity are renormalized in a similar way, even for finite-size scatterers. We find localization for d=3 dimensions in a frequency window centered at ω congruent 2π/a, and for values of the average change in the dielectric constant Δ bar ε=(Vsa-3)Δε exceeding ∼1.7