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

We report in this paper analytical and numerical results on the effect of amplification (due to non-Hermitian site potentials) on the transmission and reflection coefficients 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 system lengths before strongly oscillating with a maximum at a certain length. For larger lengths the transmission decays exponentially at 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 system 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 of the disorder and amplification giving rise to some non-monotonic behaviour in the peak of the transmission.