Focusing through random media: Eigenchannel participation number and intensity correlation

Using random matrix calculations, we show that the contrast between maximally focused intensity through random media and the background of the transmitted speckle pattern for diffusive waves is mu(N) = 1 + N-eff, where N-eff is the eigenchannel participation number for the transmission matrix. For diffusive waves, N-eff is the inverse of the degree of intensity correlation kappa. The profile of the focused beam relative to the ensemble average intensity is expressed in terms of the square of the normalized spatial field correlation function F(Delta r) and kappa. These results are demonstrated in microwave experiments and provide the parameters for optimal focusing and the limits of imaging.