Wave transport in random media: The ballistic to diffusive transition

The character of wave transport through a strongly scattering medium, excited by a pulsed plane-wave source, is investigated as a function of sample thickness over the range from about one to 13 mean free paths. To examine the behavior theoretically, we perform a first-principles calculation of both the frequency correlation function of the transmitted field and the time-domain profile of the transmitted intensity. These quantities are investigated experimentally using an ultrasonic technique, which allows us to separate the ballistic and scattered components of the total transmitted field, and hence to measure the scattered component unambiguously in thin samples. For sample thicknesses greater than about four mean free paths, we find good agreement between our theory, the diffusion approximation, and our experimental data for both the frequency correlation function and the intensity time profile. In thinner samples, there are systematic differences between theory and experiment. To characterize the transition from ballistic to diffusive behavior in thin samples, we focus on the arrival time of the peak in the scattered component of the transmitted intensity; unexpectedly we find that the scattered peak arrival time exhibits an abrupt crossover between ballistic and diffusive behavior when the ratio of sample thickness to mean free path, $L/\mathcal{l},$ is approximately equal to 3. Excellent agreement is obtained between our theory and experiment for this crossover behavior over the entire range of sample thicknesses investigated.