On Prism Cross-Dispersers -- Modelling Échelle Spectrograms

Abstract: In this paper, we elaborate on correctly predicting \'Echelle spectrograms by employing the fully three-dimensional representation of Snell's law to model the effects of prisms as cross-dispersers in \'Echelle spectrographs. We find that it is not sufficient to simply apply the frequently used trigonometric prism dispersion equation to describe recorded spectra. This vector equation approach is not limited to a single dispersive element when modelling multi-prism cross-disperser configurations. Our results help to understand the main levers in an \'Echelle spectrograph as well as contribute to auto-calibration algorithms for minimizing calibration efforts in daily operation.