  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.
