In this section we use the general framework derived above to model the UVES spectrograph and compare the results with those from a Code V ray tracing analysis.

The UV Visual Echelle Spectrograph UVES (D'Odorico 1997), to see first light in 1999 on the VLT, is a two-arm cross-dispersed echelle spectrograph covering the wavelength range (blue) and (red), with the possibility to use dichroics. The nominal resolution is 40 000 for a 1 slit, the maximum resolution that can be attained with a narrow slit or image slicer is 120 000 in the red and 90 000 in the blue with 2-pixel sampling. The dioptric cameras offer fields with a diameter of 43.5 mm (blue) and 87 mm (red), to be recorded by baseline CCD detectors of pixels of size in the blue arm and 2 or 4 such devices in the red arm. The instrument components are placed inside a passive enclosure which provides thermal isolation from the environment. The control and CCD electronics are located in temperature controlled cabinets outside the enclosure. All functions (filters, ADC etc.) are permanently on-board and remotely selectable without manual intervention. A continuous flow of liquid N2 coolant for the CCD's is supplied by an external vessel with an autonomy of at least 14 days. These measures are expected to lead to high stability and repeatability of calibrations both over short and extended periods of time.

UVES was selected as a case study because the off-plane design is introducing a pronounced slit curvature, and therefore offers a demanding case for any modeling approach. Elaborate ray tracing results are available from the optical design phase. Moreover, since the design of this instrument is tailored towards ensuring a high degree of stability (see above), it makes it a perfect candidate for model based calibration and data analysis approaches.

For our study we use the optical design parameters of the red arm configuration of UVES with the 316 grooves/mm cross-disperser. This configuration is described by

- an echelle reflection grating (31.6 grooves/mm). The beam enters at degrees and degrees. The output axis is oriented at degrees, degrees.
- an a-focal system composed of two collimators of focal
length
*F*_{1}=*F*_{2}= 2.0 meters. The system introduces field curvature in the direction of the dispersion of the echelle grating. In our model, field curvature is approximated by a polynomial. - a reflecting cross-disperser (316 grooves/mm). The difference between incident and diffracted directions is degrees. At the selected central wavelength () the beam enters at degrees and is diffracted at degrees.
- a camera introducing lateral chromatism and field distortion. In the extended model these are represented by bivariate polynomials, values at grid points being obtained from the Code V analysis.

A detailed presentation of the steps and equations involved in the UVES model is given in Appendix A.

Without the polynomial correction for aberrations the analytical model reproduces the positional information of the Code V ray tracing analysis with an rms error of 8 pixels of the detector, i.e. an error of less than 0.5%. Locally the genuine analytical model is much more accurate as regards curvatures, tilts and distances (see Sect. 3.2.2 (click here)).

Including a polynomial correction for the aberrations and field distortions, the comparison with ray tracing shows a precision of the analytical model of better than 1 pixel anywhere in the field. Regarding line tilt, the results are in full agreement with Code V.

Since the UVES mode considered has all vital elements of a genuine echelle spectrograph and in addition the pronounced line curvature from off-plane configurations we conclude that the formalism developed in the previous Section is indeed capable of modeling the geometric aspects with the accuracy necessary for applications in the domain of calibration and data analysis.

One important aspect of our simulation is to show in detail the curvature of the slit images. We recall that this is made possible only by the rigorous off-plane formulation. The simulation of orders 63, 76 and 98 is shown in Fig. 2 (click here).

**Figure 2:** Slit curvature in the red arm. The curvature is shown on a
scale exaggerated by a factor 10 along the *x*-axis and 2 along the
*y*-axis. Positions on the detector are in mm for lines taken at the
limits of the free spectral range in 3 different orders (order 98
is at the top of the graph, 76 in the middle, and 63 at the bottom)

It is interesting to note that one of the first practical applications of this analytical model was to determine the optimal slit rotation angle minimizing the slit curvature. Further simulation showed that the curvature can be minimized by rotating the slit by an angle of 5.95 degrees. In this case, for a pixel size of 15 m, the slit image extends across 200 pixels while the curvature reaches a maximum value of only 1 pixel at the extremes.

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