Reflection of waves from slowly decaying complex permittivity profiles

Wave propagation through rapidly but continuously varying media is surprisingly subtle, and in a pair of recent papers [Horsley, J. Opt. 18, 044016 (2016)2040-897810.1088/2040-8978/18/4/044016; Longhi, Eur. Phys. Lett. 112, 64001 (2015)EULEEJ0295-507510.1209/0295-5075/112/64001] it was found that planar media with a spatially varying permittivity ϵ(x) obeying the spatial Kramers-Kronig relations do not reflect waves incident from one side, however rapid the changes in ϵ(x). Within this large class of media there are some examples where the dissipation or gain is not asymptotically negligible and it has been pointed out [Longhi, Eur. Phys. Lett. 112, 64001 (2015)EULEEJ0295-507510.1209/0295-5075/112/64001] that it is impossible to define meaningful reflection and transmission coefficients in such cases. Here we show - using an exactly soluble example - that despite the lack of any meaningful reflection and transmission coefficients, these profiles are still reflectionless from one side in the sense that the profile generates no counterpropagating wave for incidence from one side. This finding is demonstrated through examining the propagation of pulses through the profile, where from one side we find that no second reflected pulse is generated, while from the other there is. We conclude with a discussion of the effect of truncating these infinitely extended profiles, illustrating how the reflectionless behavior may be retained over a wide range of incident angles