Extended dispersion relation in finite-length, one-dimensional, periodic photonic band-gap structures

We show that the general dispersion relation of a finite-length, one-dimensional, periodic photonic band-gap structure can be extended, by using the approach of the effective index of refraction and the Bloch vector, to yield an explicit relation between the real and imaginary components of an optical field emerging from such a structure. In consequence, by means of this Kramers-Krönig-like relation, the recovery of the effective index can be obtained with a good approximation directly from transmissivity without involving any numerical integration algorithm. The expression for a two-layer, step-index-profile quarter-wave stack, derived in the case of TE polarisation, is described in details. Some numerical results are given and discussed. The performance of the established relation as used for index recovery is evaluated.