Transmission and reflection studies of periodic and random systems with gain

The transmission ((T) and reflection (R) coefficients are studied in period ic systems and random systems with gain. For both the periodic electronic t ight-binding model and the periodic classical many-layered model, we obtain numerically and theoretically the dependence of T and R. The critical leng th of periodic system L-c(0), above which T decreases with the size of the system L while R approaches a constant value, is obtained to be inversely p roportional to the imaginary part epsilon'' of the dielectric function epsi lon. For the random system, T and R also show a nonmonotonic behavior versu s L. For short systems (L < L-c) with gain < In T >=(l(g)(-1) -xi(0)(-1))L. For large systems (L much greater than L-c,) with gain < In T >= -(l(g)(-1 )+xi(0)(-1))L. L-c, l(g), and xi(0) are the critical, gain, and localizatio n lengths, respectively. The dependence of the critical length L-c on epsil on'' and disorder strength W are also given. Finally, the probability distr ibution of the reflection R for random systems with gain is also examined. Some very interesting behaviors are observed. [S0163-182(99)02809-X].