General Formulation for the Calibration and Characterization of   Narrow-gap Etalons: the OSIRIS/GTC Tunable Filters Case
  Tunable filters are a powerful way of implementing narrow-band imaging mode over wide wavelength ranges, without the need of purchasing a large number of narrow-band filters covering all strong emission or absorption lines at any redshift. However, one of its main features is a wavelength variation across the field of view, sometimes termed the phase effect. In this work, an anomalous phase effect is reported and characterized for the OSIRIS instrument at the 10.4m Gran Telescopio Canarias. The transmitted wavelength across the field of view of the instrument depends, not only on the distance to the optical centre, but on wavelength. This effect is calibrated for the red tunable filter of OSIRIS by measuring both normal-incidence light at laboratory and spectral lamps at the telescope at non-normal incidence.   This effect can be explained by taking into account the inner coatings of the etalon. In a high spectral resolution etalon, the gap between plates is much larger than the thickness of the inner reflective coatings. In the case of a tunable filter, like that in OSIRIS, the coatings thickness could be of the order of the cavity, which changes drastically the effective gap of the etalon. We show that by including thick and dispersive coatings into the interference equations, the observed anomalous phase effect can be perfectly reproduced. In fact, we find that, for the OSIRIS red TF, a two-coatings model fits the data with a rms of 0.5\AA\ at all wavelengths and incidence angles. This is a general physical model that can be applied to other tunable-filter instruments. 
J.J. Gonzalez, J. Cepa, J.I. Gonzalez-Serrano, and M. Sanchez-Portal
astro-ph.IM