Imaginary Potential as a Counter of Delay Time for Wave Reflection from a 1D Random Potential

TL;DR

This paper establishes a direct relationship between delay time distribution and reflection coefficient in 1D random potentials, using an imaginary potential to count delay times, and extends the analysis to amplifying media.

Contribution

It introduces a novel method linking delay time to reflection coefficient via an imaginary potential, providing new insights into wave reflection in disordered media.

Findings

Delay time distribution matches previous results with infinite moments.

In amplifying media, all moments of delay time are finite.

The method offers a natural way to quantify delay times in random systems.

Abstract

We show that the delay time distribution for wave reflection from a one-dimensional random potential is related directly to that of the reflection coefficient, derived with an arbitrarily small but uniform imaginary part added to the random potential. Physically, the reflection coefficient, being exponential in the time dwelt in the presence of the imaginary part, provides a natural counter for it. The delay time distribution then follows straightforwardly from our earlier results for the reflection coefficient, and coincides with the distribution obtained recently by Texier and Comtet [C.Texier and A. Comtet, Phys.Rev.Lett. {\bf 82}, 4220 (1999)],with all moments infinite. Delay time distribution for a random amplifying medium is then derived . In this case, however, all moments work out to be finite.