Phase of the Transmission Coefficient of Waves in One-dimensional Random Media

We study the behavior of the phase of the transmission coefficient t in random media by using the invariant imbedding method of wave propagation. We calculate the disorder average of t/t* for waves propagating in one-dimensional random media with uncorrelated Gaussian disorder in a numerically exact manner. We find that the analytical formula derived by Mello et al. [Phys. Rev. Lett. 67, 342 (1991)] is valid only after a major modification and only in the limit where the so-called random phase approximation is valid. We find that the disorder average of t/t* converges to a finite complex number in the large-size limit. We discuss the implications of our results for the probability distribution of the phase of the transmission coefficient.