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1 Introduction

A general theory of low order field aberrations of decentered optical systems has been given by Shack & Thompson ([1980]). In particular it has been shown that the general field dependence of third order astigmatism can be described by a binodal pattern, known as ovals of Cassini. Only for special cases such as a centered system do the two nodes coincide and the field dependence reduces to the well known rotationally symmetric pattern with a quadratic dependence on the distance to the field center. These general geometric properties have been used by McLeod ([1996]), starting from equations by Schroeder ([1987]), for the alignment of an aplanatic two mirror telescope. We shall recall this method briefly.

In a Cassegrain telescope the absence of decentering coma in the center of the field does not imply that the optical axes of the primary (M1) and the secondary mirrors (M2) coincide. The axes of the primary and the secondary mirrors must intersect at the coma free point, but the axis of M2 may still form an angle $\alpha $ with respect to the axis of M1. For this case McLeod showed that the components Z4and Z5 of third order astigmatism of a two mirror telescope with the stop at the primary mirror for a field angle $\theta$ with components $\theta_{x}$ and $\theta_{y}$are given by

$\displaystyle Z_{4}^{{\rm sys}}$ = $\displaystyle B_{0}\, (\theta_{x}^{2} - \theta_{y}^{2})
\, + \, B_{1}(\theta_{x}\alpha_{x}
\, - \, \theta_{y}\alpha_{y})$  
    $\displaystyle + \, B_{2}\, (\alpha_{x}^{2} - \alpha_{y}^{2})$ (1)
$\displaystyle Z_{5}^{{\rm sys}}$ = $\displaystyle 2B_{0}\, \theta_{x}\theta_{y}
\, + \, B_{1}(\theta_{x}\alpha_{y}
\, + \, \theta_{y}\alpha_{x})
\, + \, 2B_{2}\, \alpha_{x}\alpha_{y}.$ (2)

B0 is the coefficient of field astigmatism for a centered telescope, B1 and B2 only appear in decentered systems. Numerical values for B0, B1 and B2 were obtained by using general formulae for field astigmatism of individual mirrors and adding the effects of the two mirrors. The values for $\alpha_{x}$ and $\alpha_{y}$ could then be obtained from measurements of $Z_{4}^{{\rm sys}}$ and $Z_{5}^{{\rm sys}}$ in the field of the telescope.

For a centered two mirror telescope Wilson ([1996]) has derived expressions for the low order field aberrations including third order astigmatism (B0) showing the dependence on fundamental design parameters and optical properties of the total telescope and on the position of the stop along the optical axis. This gives more physical insight into the characteristics of field aberrations of two mirror telescopes. Similarly we derive, for decentered two mirror telescopes, explicit expressions for third order astigmatism, i.e. for the parameters B1 and B2, and discuss the field dependence of astigmatism for fundamental types of two mirror telescopes.

In a centered optical system the field center, projected towards the sky, is the direction parallel to the optical axis of M1. The formulae for the parameters B0, B1 and B2 are initially derived for this reference system. However, in a decentered system the image of an object in the field center is not in the mechanical center of the adapter, where the instruments are located. For all measurements in the telescope, the practical field center is the center of the adapter. The evaluation of the measurements of field astigmatism has to take this difference of the origins of the reference systems into account. The structure of the Eqs. (1) and (2) will remain the same, but the parameters B0, B1 and B2 will change and $\theta$ will denote the field angle with respect to the center of the adapter.

One case of practical interest is the collimation at the Cassegrain focus of the VLT where the stop is at the secondary mirror and the telescope is only corrected for spherical aberration. We apply our formulae to measurements of astigmatism in the field of the Cassegrain focus to determine the misalignment angles $\alpha_{x}$ and $\alpha_{y}$. In addition, the possibility to change the misalignment angles by well defined values by rotating the primary mirror around its vertex allows a measurement of the accuracy of this collimation method.

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