Transmission and reflection studies of periodic and random 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 ${L}_{c}^{0},$ 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 ${\ensuremath{\varepsilon}}^{\ensuremath{''}}$ of the dielectric function $\ensuremath{\varepsilon}.$ For the random system, T and R also show a nonmonotonic behavior versus L. For short systems $(L<{L}_{c})$ with gain $〈\mathrm{ln}T〉{=(l}_{g}^{\ensuremath{-}1}\ensuremath{-}{\ensuremath{\xi}}_{0}^{\ensuremath{-}1})L.$ For large systems $(L\ensuremath{\gg}{L}_{c})$ with gain $〈\mathrm{ln}T〉=\ensuremath{-}{(l}_{g}^{\ensuremath{-}1}+{\ensuremath{\xi}}_{0}^{\ensuremath{-}1})L.{L}_{c},{l}_{g},$ and ${\ensuremath{\xi}}_{0}$ are the critical, gain, and localization lengths, respectively. The dependence of the critical length ${L}_{c}$ on ${\ensuremath{\varepsilon}}^{\ensuremath{''}}$ 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.