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2. Optical principles for spectrographs

2.1. Optical elements and geometry

The elements of a model for echelle spectrographs are mirrors, lenses, gratings, prisms, grisms. Here we will focus solely on the geometric aspects, i.e. the 2D-dispersion relation. Luminosity aspects brought about by the interference terms, e.g. the echelle blaze function, line spread functions, as well as geometrical vignetting or reflectivity and transmissivity of materials will be the subject of subsequent communications. The notations of Schroeder (1987) are followed for the definitions of angles and orientations (right-hand Cartesian frame).

A ray incident on an optical surface co-planar with the xy-plane has direction tex2html_wrap_inline1206 (see Fig. 1 (click here)) and projects onto the axes as

Conversely the angles tex2html_wrap_inline1206 are given by

Figure 1: Axes and angles notations

Since these angles are for incidences, their values are comprised in the interval tex2html_wrap_inline1210 to tex2html_wrap_inline1212. The orientation of the optical surface is described by the rotation angle tex2html_wrap_inline1214 about the z axis. The angles tex2html_wrap_inline1218 and tex2html_wrap_inline1220 denote rotations relative to x and y respectively. Referential changes are performed by applying the rotations in the following order:
and the transpose matrix provides the opposite change of the referential:
where each rotation matrix is of the form:

2.2. Diffraction by the echelle grating

The direction of incident rays on an echelle grating is defined by the two angles tex2html_wrap_inline1228 (Schroeder 1987). The general grating equation can be written as


where m is the order number, tex2html_wrap_inline1232 the wavelength, tex2html_wrap_inline1234 the groove separation in unit of length, n and n' the refractive indices before and after the grating, tex2html_wrap_inline1240 and tex2html_wrap_inline1242 the incidence angles, and tex2html_wrap_inline1244 and tex2html_wrap_inline1246 the direction of the diffracted rays.

For a reflection grating n' = -n and therefore tex2html_wrap_inline1250, hence

One can note that using the projection equations it is possible to write the two grating Eqs. (9) and (10) as:

This formulation allows to introduce the grating in the model in the form of a matrix. However, since no simple optical equation allows to derive z this term will need to be derived from the normalization relation:

2.3. Echelle relation

The echelle relation of a spectrograph derives from the echelle grating equation. For a constant tex2html_wrap_inline1240 and all rays corresponding to a constant product tex2html_wrap_inline1256 are diffracted in the direction tex2html_wrap_inline1244. In a simplistic model one could expect that the lines of constant value tex2html_wrap_inline1256 project as straight lines in the detector planes and for a perfect alignment of the detector as, say, columns on the detector. This assumption is however limited by aberrations and rotation of the detector.

Diffraction by the cross-disperser

The cross-disperser is either a grating, a prism or a grism of low spectral resolution. It displaces successive orders of the echelle grating vertically with respect to each other. It is normally rotated by an angle tex2html_wrap_inline1262 so that the dispersion equations of the cross-disperser needs to be applied after rotation to the referential of the cross-disperser. tex2html_wrap_inline1264 denotes the groove separation for this grating.

2.5. Planes, mirrors and grisms

The refraction at a plane is given by Snell-Descartes law

The projection of an incident beam is given by Eq. (1 (click here)) and the angle tex2html_wrap_inline1266 with the z axis is:

One can also express this relation using the direction cosine vector:

For n' = -n these are the equations for a mirror. This paper does not discuss grism based spectrographs although the above equations have been used to model such spectrographs. We indicate it here for the sake of completeness of the analytical framework. Grisms are represented by the association of a plane and a grating.

2.6. Lenses and curved mirrors with positive power

In the referential of the optical element the projections (x,y) on the focal plane at focal distance F are given by

Equations (4) and (5) allow to derive directly the values of tex2html_wrap_inline1276 and tex2html_wrap_inline1278 from the coordinates of the unit vector (x,y,z).

2.7. Projection onto detectors

Translation and rotation of the detector array are applied to the vector point (x,y) by

This rotation can be applied as a rotation matrix before entering the camera lens. This step is performed after the lens projection in order to determine the analytical form of the dispersion relation in the absence of detector rotation.

2.8. Aberrations and chromatism

In a complete optical train the above set of equations strictly apply only for on-axis rays and do not take into account field distortions, camera aberrations and wavelength dependencies of e.g. the focal lengths. In instruments like UVES (discussed below) these effects can account for discrepancies of several pixels at the detector.

Distortions are specific to the optical elements and layout, and are usually predictable and stable in time. As will be shown in Sect. 3 a model for a given instrument will typically match 99.5% using the physical description as developed above. For most applications it will not pay off to develop further the description by introducing off-axis optical equations. Instead the residuals can be corrected for by inserting at the proper location low order polynomial functions whose coefficients can for example be produced with the help of a ray tracing program.

In the following we will closely analyze the application of both, the exact (on axis) equations and a simplified analytical form respectively. Two notions of accuracy will be used. On the one hand the absolute accuracy of the model is limited by the optical effects not taken into account like e.g. aberrations and distortions. On the other hand the accuracy of the simplified analytical model will be limited by the numerical approximations performed to keep the analytical form simple (e.g. Taylor series expansions).

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