Universal and nonuniversal scaling of transmission in thin random layered media

The statistics of transmission through random 1D media are generally presumed to be universal and to depend only upon a single dimensionless parameter, the ratio of the sample length and the mean free path, s=L/l. For s much larger than unity, the probability distribution function of the logarithm of transmission, P(ln T) is Gaussian with average value -s and variance 2s. Here we show in numerical simulations and optical measurements that in random binary systems, and most prominently in systems for which s less than unity, the statistics of transmission are universal for transmission near an upper cutoff of unity and depend upon the character of the discrete disorder near a lower cutoff. The universal behavior of P(ln T) closely resembles a segment of a Gaussian and arises in random binary media with as few as three binary layers. Above the lower cutoff, but below the crossover to a universal expression, the shape of P(ln T) also depends upon the reflectivity of the interface between the layers. For a given value of s, P(ln T) evolves towards a universal distribution given by random matrix theory in the dense weak scattering limit as the numbers of layers increases. P(ln T) found in simulations is compared to results of random matrix calculations in the dense weak scattering limit but with an imposed minimum in transmission. Optical measurements in stacks of glass coverslips are compared to random matrix theory, and differences are ascribed to transverse disorder in the layers.