Anderson localization in a partially random Bragg grating and a conserved area theorem

We investigate the gradual emergence of the disorder-related phenomena in intermediate regimes between a deterministic periodic Bragg grating and a fully random grating and highlight two critical properties of partially disordered Bragg gratings. First, the integral of the logarithm of the transmittance over the reciprocal wavevector space is a conserved quantity. Therefore, adding disorder merely redistributes the transmittance over the reciprocal space. Second, for any amount of disorder, the average transmittance decays exponentially with the number of grating layers in the simple form of $\exp(-\eta N)$ for sufficiently large $N$, where $\eta$ is a constant and $N$ is the number of layers. Conversely, the simple exponential decay form does not hold for small $N$ except for a highly disordered system. Implications of these findings are demonstrated.