Analytical approximation for photonic array modes in 1D photonic crystal superlattices.

We present a comprehensive analytical approximation for array modes (both the modal fields and their associated propagation constants) for 1D photonic crystal superlattices (i.e., large periodic arrays of repeated sequences of different coupled waveguides/lasers). In this class, a regular periodicity of a photonic lattice is supplemented with the additional periodicity of a larger scale. Our approximation is a vectorial approach, accounting for the TE and TM polarizations. It can be applied to both the low- and high-contrast photonic devices. We used the model of standing waves for analytical evaluation of envelopes of array modes in a photonic superlattice. Combination of the model of standing waves with the coupled-mode formalism for infinite photonic superlattices allows evaluation of propagation constants of the array modes. Both the evaluations require only a fraction of a second for computation. Still, the results, acquired with the analytical approximation, are very close to those of well-established approaches. Furthermore, for the first time, analytical expressions for the modal fields and propagation constants become available.