On Fabry-P\'erot etalon-based instruments. IV. Analytical formulation of telecentric etalons

Fabry-P\'erot etalons illuminated with collimated beams have been characterized analytically in detail since their invention. Meanwhile, most of the features of etalons located in telecentric planes have been studied only numerically, despite the wide use of this configuration in astrophysical instrumentation over decades. In this work we present analytical expressions for the transmitted electric field and its derivatives that are valid for etalons placed in slow telecentric beams, like the ones commonly employed in solar instruments. We use the derivatives to infer the sensitivity of the electric field to variations in the optical thickness for different reflectivities and apertures of the incident beam and we compare them to the collimated case. This allows us to estimate the wavefront degradation produced by roughness errors on the surfaces of the Fabry-P\'erot and to establish the maximum allowed RMS value of the cavity irregularities across the footprint of the incident beam on the etalon that ensures diffraction-limited performance. We also evaluate the wavefront degradation intrinsic to these mounts, which is produced only by the finite aperture of the beam and that must be added to the one produced by defects. Finally, we discuss the differences in performance of telecentric and collimated etalon-based instruments and we generalize our formulation to anisotropic etalons.Comment: 10 pages, 7 figure