This appendix describes the model used for the CASPEC and UVES spectrographs. The optical design of the two instruments is similar enough to justify the use of the same description for both instruments. The analytical model taking into account the 3-dimensional grating equations involves the following successive steps:

- Slit rotation by an angle and collimation by
- Diffraction by the echelle grating
- Rotation by about
*Y* - Rotation by about
*X* - Rotation by -90 degrees about
*Z* - Diffraction by the cross-disperser
- Rotation by 90 degrees about
*Z* - Rotation by about
*X* - (We assume , and skip two rotations about
*Y*) - Rotation by about
*X* - Projection onto the detector plane
- Offset in position
- Rotation of the detector by an angle .

The *x*-axis of our coordinate system is perpendicular to the grooves of
the echelle grating, the *z*-axis normal to the surface. Nomenclature is
as in Sect. 2 of the main article. In addition, *s* is the position
on the slit, the focal length of the collimator,
the angle of rotation of the slit.

For practical purposes we construct a 4-dim vector using the wavelength
and the projections *x*, *y*, *z*.
The initial ray has the directions and projections

The list of matrices for the above sequence of operations is given
below, where after each pass through a grating the *z*-component of the
vector is determined from the normalization relation

One can at this stage either determine the complete expression of and

or make the approximation and obtain

Finally one applies the rotation of the detector:

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