A Pole-Based Approach to Composite Linear Optical Cavities with Internal Dielectric Reflectors

We develop a versatile description of Fabry–Pérot resonators comprising internal dielectric structures, based on transfer matrices and interpreted from a non-Hermitian one-pole self-energy viewpoint. For a two-mirror cavity containing an internal slab of arbitrary thickness and refractive index, we derive closed-form expressions for transmission and identify cavity poles as zeros of the reduced denominator in the complex-frequency plane. For a weak-reflector, we obtain leading-order expressions for the pole shifts of individual modes and show that the resulting mode pulling and linewidth change are, respectively, governed by the imaginary and real parts of a single complex quantity formed by the coherent sum of two mirror-side scattering paths. These expressions provide placement criteria for dispersive or dissipative operation and support practical workflows for extracting weak-reflector parameters from measured resonance traces, predicting slab-modified cavity spectra from the empty cavity calibration, and designing doubly resonant cavities with fine constraints on slab position. We extend the consideration to multiple reflectors, with emphasis on two-membrane and three-mirror geometries, relevant to coupled filter cavities and membrane-in-the-middle architectures. The coupled-pole parametrization relates these configurations, providing a compact framework for the analysis and design of composite Fabry–Pérot elements for precision filtering and quantum-noise shaping in advanced interferometric experiments.