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A white light illuminated etalon is a valuable resource for spectrographcalibration in radial velocity exoplanet detection, and other astronomicalapplications. These etalons benefit from low drift (${\sim} 10^{-11}$/day) andwell-characterized stability of their mode structure. However, measurements ofseveral etalon systems across bandwidths greater than $500$ nm indicate thatthe modes exhibit complex, wavelength-dependent drift. Surprisingly, modes indifferent regions of the spectrum were found to drift in different directions,implying that the optical length of the etalon is getting both longer andshorter at the same time. In this paper, we provide a solution to this puzzlingobservation. With Fresnel analysis and the transfer matrix method, we model thereflective phase of the multi-layer dielectric mirrors in the etalon used as acalibrator for the Habitable Zone Planet Finder (HPF). We use this phase tocalculate the etalon mode positions and are able to reproduce the observedoscillatory chromatic drift of the etalon's mode spectrum across 800-1300 nm.Despite the complexity of the mirror structure, our modeling indicates that thegradual relaxation of the outermost layers of the etalon mirrors is thedominant source of the observed behavior. We also model the effect oftemperature, incident angle alignment variations, and manufacturing tolerances,and show that they are likely not causes of the chromatic mode frequencyshifts. Our work highlights techniques that can be employed in the design ofbroad bandwidth mirrors for future etalons to make them more useful for thehighest precision astronomical spectroscopy.

A resolution to the mystery of an etalon that is optically expanding and contracting at the same time