Formation of optimal-order necklace modes in one-dimensional random photonic superlattices

We study the appearance of resonantly coupled optical modes, optical necklaces, in Anderson-localized one-dimensional random systems with resonant layers. It has been stated that the increasingly low probability of their appearance in thick samples should reduce the theoretically predicted number of coupled resonances, leading to the formation of optimal-order necklaces (OONs). Here we examine through transmission phase studies how the optimal order m*, the number of resonances in a necklace, shifts gradually toward higher orders with increasing sample size. We present a model for OON formation in random samples with resonant layers and derive an empirical formula that predicts m*. We discuss the situation when in a sample length L the number of resonances degenerate in energy exceeds the optimal one, and show how the extra resonances are pushed out to the miniband edges of the necklace, reducing the order of the latter always by multiples of two.