The rapid evolution of detector and instrument technology has provided astronomers with the capabilities to acquire large amounts of high signal-to-noise, multi-dimensional observational data. In order to exploit optimally these data, the entire chain of the observation process from instrument configuration control through calibration, analysis and archival has to be tailored towards very high standards. In contemporary data calibration and analysis very little has been done so far to relate the optical layout and its engineering parameters with the performance on scientific targets and calibration sources. Even less use has been made of the physical principles underlying the characteristics of a given instrument in predicting the performance to a large degree of accuracy (cf. Rosa 1995).
A number of developments make it more practical today to efficiently use optical models in the post-observational process (see Ballester 1996). First of all instrument configuration control, which is self evident for space missions, is now being enforced more strictly at ground based instruments. Second the development of service observing modes at major facilities like the Very Large Telescope (VLT) of the European Southern Observatory (ESO) necessitates the development of data quality control procedures. And third, the need to process large amounts of data in a homogeneous manner calls for robust automatic calibration techniques (Ballester 1994). First principle based analysis techniques have been applied for example in the algorithm to correct for the grating scatter in the HST Faint Object Spectrograph (FOS) (Rosa 1994), which was implemented in major data analysis environments (Bushouse et al. 1995). A model based solution to first order spectrograph dispersion relations has been used by Dahlem & Rosa (1997) to assess the uncertainties inherent in low order polynomial fits for wavelength calibrations for the FOS. Another interesting case is the HST PSF model (e.g. Hasan & Burrows 1994), which shows the relation between using a model of the optics to retrieve parameters for the refurbishing mission and its use to generate PSFs for deconvolution.
One of the most demanding cases of data calibration and analysis are
2D echelle spectra. Traditionally, they require complex data reduction
procedures to cope simultaneously with both, the geometrical
distortion of the raw data introduced by order curvature and line
tilt, and with the spread of the signal across the tilted lines and
between successive orders respectively (cf. Hensberge & Verschueren
1990). Un-supervised wavelength calibration for these instruments can
only be achieved by reducing to a minimum the information needed to
start the calibration process. This requires the most efficient use of
the a priori knowledge from the optical properties of the
instrument under consideration. For instance the echelle relation is commonly employed to start the calibration process, and can in the simplest cases be determined with only 2 interactive line identifications. However, the validity of the simple echelle relation is limited to those cases where the detector rotation and optical aberrations are negligible, as pointed out for instance by Hall et al. (1994). Another approach is to use encoder values of the grating angles fed into analytic formulae for the dispersion relation, presented by Goodrich & Veilleux (1988) for an in-plane spectrograph. However, a more general solution capable of handling off-plane situations has not been developed so far.
We study in this article the principles governing the determination of an accurate form of the echelle relation for off-plane echelle spectrographs, and analyze the applications of such models for the calibration of echelle spectrographs. In a first step we derive the characteristic equations of echelle spectrographs from first principles (Sect. 2 and Appendix A). This formalism is then confronted with the results of ray tracing analysis for the demanding case of an off-plane echelle spectrograph (Sect. 3). In Sect. 4 the application of models to the task of instrument configuration control is discussed and verified on actual data from an instrument that has been in service for a decade. In Sect. 5 and Appendix B methods derived from such optical models are introduced that allow one to reduce to a minimum the information needed to start the wavelength calibration process. Section 6 will be used to discuss possible applications of such analytical models for the calibration of astronomical instruments, advanced data analysis, and observation planning.