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

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.