New technique to measure the cavity defects of Fabry-Perot interferometers

(Abridged): We define and test a new technique to accurately measure the cavity defects of air-spaced FPIs, including distortions due to the spectral tuning process typical of astronomical observations. We further develop a correction technique to maintain the shape of the cavity as constant as possible during the spectral scan. These are necessary steps to optimize the spectral transmission profile of a two-dimensional spectrograph using one or more FPIs. We devise a generalization of the techniques developed for the so-called phase-shifting interferometry to the case of FPIs. The technique is applicable to any FPI that can be tuned via changing the cavity spacing ($z$-axis), and can be used for any etalon regardless of the coating' reflectivity. The major strength of our method is the ability to fully characterize the cavity during a spectral scan, allowing for the determination of scan-dependent modifications of the plates. As a test, we have applied this technique to three 50 mm diameter interferometers, with cavity gaps ranging between 600 micron and 3 mm, coated for use in the visible range. We obtain accurate and reliable measures of the cavity defects of air-spaced FPIs, and of their evolution during the entire spectral scan. Our main, and unexpected, result is that the relative tilt between the two FPI plates varies significantly during the spectral scan, and can dominate the cavity defects; in particular, we observe that the tilt component at the extremes of the scan is sensibly larger than at the center of the scan. Exploiting the capability of the electronic controllers to set the reference plane at any given spectral step, we develop a correction technique that allows the minimization of the tilt during a complete spectral scan. The correction remains highly stable over long periods, well beyond the typical duration of astronomical observations.