We derive, starting from Maxwell`s equations, a set of models that describe integrated Fabry--P`{e}rot resonators. Our work focuses on structures in which the mirrors are implemented through periodic variations of the refractive index, as commonly found in photonic crystal and Bragg-type configurations. Such systems exhibit rich optical behavior due to the interplay between confinement and periodicity.

Our methodology is based on two complementary approaches. On one hand, we employ a modal expansion of photonic waveguides, which provides an intuitive and computationally efficient framework to describe guided modes and their interactions. On the other hand, we use the plane wave expansion method, which is particularly well suited to capture the effects of periodic media and accurately describe band structure properties.

The goal of our models is to incorporate the complex dynamics arising from the underlying periodic crystal, going beyond simplified descriptions of resonator behavior. By doing so, we aim to account for effects such as dispersion engineering, mode coupling, and bandgap-induced confinement. This comprehensive framework opens the door to the exploration of new and potentially rich physical phenomena in integrated photonic resonators, with possible applications in optical communications, sensing, and quantum photonics.