# Time-reversal invariant scaling of light propagation in one-dimensional non-Hermitian systems

## TL;DR

This paper demonstrates a universal scaling law in one-dimensional non-Hermitian optical systems that ensures identical transmission and reflection spectra between a medium and its time-reversed counterpart, highlighting a fundamental symmetry in light propagation.

## Contribution

It introduces a universal scaling approach that relates the optical properties of non-Hermitian media and their time-reversed systems, extending understanding of symmetry in light transmission.

## Key findings

- Universal scaling yields identical transmittance and reflectance in paired media.
- Scaled optical properties reduce to standard forms in time-reversal invariant systems.
- The approach clarifies the role of gain and loss landscapes in light propagation.

## Abstract

Light propagation through a normal medium is determined not only by the real part of the refractive index but also by its imaginary part, which represents optical gain and loss. Therefore, two media with different gain and loss landscapes can have very different transmission and reflection spectra, even when their real parts of the refractive index are identical. Here we show that while this observation is true for an arbitrary one-dimensional medium with refractive index $n(x)$ and its time-reversed partner with refractive index $n^*(x)$, there exists a universal scaling that gives identical transmittance and reflectance in these corresponding systems. Interestingly, these scaled transmittance and reflectance reduce to their standard, unscaled forms in a time-reversal invariant system, i.e., one without gain or loss.