Locally finite free space as limiting case of PT-symmetric medium

TL;DR

The paper proves that a finite layered PT-symmetric system behaves like free space in the limit of infinite layers, with perfect transmission and zero reflection for all wave vectors.

Contribution

It explicitly demonstrates that a PT-symmetric layered medium approaches free space behavior as the number of layers tends to infinity.

Findings

Transfer matrix approaches identity as N→∞

Transmission coefficient approaches unity

Reflection amplitude approaches zero

Abstract

We explicitly prove that the transfer matrix of a finite layered $PT$-symmetric system of fix length $L$ consisting of $N$ units of the potential system `$+iV$' and `$-iV$' of equal thickness becomes a unit matrix in the limit $N \rightarrow \infty$. This result is true for waves of arbitrary wave vector $k$. This shows that in this limit, the transmission coefficient is always unity while the reflection amplitude is zero for all waves traversing this length $L$. Therefore, a free space of finite length $L$ can be represented as a $PT$-symmetric medium.