Transmission and reflection studies of periodic and random systems with gain

TL;DR

This paper investigates how transmission and reflection coefficients behave in periodic and random systems with gain, revealing critical lengths, nonmonotonic behaviors, and probability distributions through numerical and theoretical analysis.

Contribution

It provides a comprehensive analysis of transmission and reflection in systems with gain, including critical length dependence and probability distribution insights, which are novel contributions.

Findings

Critical length inversely proportional to the imaginary part of dielectric function.

Nonmonotonic behavior of T and R in random systems with gain.

Distinct exponential behaviors of ln T for short and large systems with gain.

Abstract

The transmission (T) and reflection (R) coefficients are studied in periodic systems and random systems with gain. For both the periodic electronic tight-binding model and the periodic classical many-layered model, we obtain numerically and theoretically the dependence of T and R. The critical length of periodic system L[sub c][sup 0], above which T decreases with the size of the system L while R approaches a constant value, is obtained to be inversely proportional to the imaginary part cursive-epsilon (double-prime) of the dielectric function cursive-epsilon . For the random system, T and R also show a nonmonotonic behavior versus L. For short systems (L < Lc) with gain (left-angle)ln T(right-angle) = (l[sub g][sup -1] - xi [sub 0][sup -1])L. For large systems (L(very-much-greater-than)Lc) with gain (left-angle)ln T(right-angle) = - (l[sub g][sup -1] + xi [sub 0][sup -1])L. Lc, lg, and xi 0 are the critical, gain, and localization lengths, respectively. The dependence of the critical length Lc on cursive-epsilon (double-prime) and disorder strength W are also given. Finally, the probability distribution of the reflection R for random systems with gain is also examined. Some very interesting behaviors are observed.