Amplification and disorder effects in a Kronig-Penney chain of active potentials

Abstract

We report in this paper the analytical and numerical results on the effect of amplification on the transmission and reflection coefficient of a periodic one-dimensional Kronig-Penney lattice. A qualitative agreement is found with the tight-binding model where the transmission and reflection increase for small lengths before strongly oscillating with a maximum at a certain length. For larger lengths the transmission decays exponentially with the same rate as in the growing region while the reflection saturates at a high value. However, the maximum transmission (and reflection) moves to larger lengths and diverges in the limit of vanishing amplification instead of going to unity. In very large samples, it is anticipated that the presence of disorder and the associated length scale will limit this uninhibited growth in amplification. Also, there are other interesting competitive effects between disorder and localization giving rise to some nonmonotonic behavior in the peak of transmission.

Nonlinear interaction effect on the phase distribution in one-dimensional disordered lattices

“…This gives rise to a delocalization effect, in agreement with our previous results on the effect of NL on the transmissive properties of these systems [11]. However, an extensive characterization of the transmissive properties of 1D systems by their PD should be completed in order to understand some unexpected behaviours in the transmission, such as the multistability, the chaotic behaviour for NL systems and the asymptotic exponential decay of transmission for amplifying systems [21]. Some of these interesting topics will be the subject of some forthcoming investigations.…”

Transmission and reflection in a perfectly amplifying and absorbing medium

“…¿From the above expression we find that the positive value of N max is obtained only when η < 0 i.e. the peak of T occurs in an amplifying medium [6,7,9,18]. On the other hand T odd goes to infinity as N → N max .…”

“…For proper understanding of the effect of randomness in active media it is important to study the perfect absorbing as well as amplifying system. But much less attention has been paid in this direction [9,24,25].…”

“…In Ref. [9] a perfectly amplifying medium has been studied in the Kronig Penny model. The authors obtain exact expressions for transmission coefficient and the cross-over length.…”

“…However, we feel that the work on perfect amplifying/absorbing medium is not complete. Some interesting results which may help to understand the properties of active media in the presence of randomness seem to have been overlooked in previous studies [9,24,25]. So, here we perform the analytical study of perfect amplifying as well as absorbing media using Eq.…”