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:
and collimation by 
about Y
about X
about X
, and skip two rotations about Y)
about X
.
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:
