- 3.1 Introduction
- 3.2 Depointing of the telescope for a pure lateral decentering of M2
- 3.3 New coefficients
*B*_{0},*B*_{1}and*B*_{2}with reference to the center of the adapter - 3.4 Effects of the correction of coma at the center of the adapter
- 3.5 Simulation for the ESO 3.6 m telescope

The center of the adapter is however always defined as being the center
of the field in a telescope and field measurements are made with
reference to this point. In this section we first calculate the offset
angles of the telescope corresponding to the misalignment of M2 and then
the effect of these offsets on the parameters *B*_{0}, *B*_{1} and
*B*_{2}.

*Depointing for a pure lateral decentering of M2*

If M2 is laterally decentered by (see Fig. 2) the star centered in the adapter is producing an angle with the M1 axis:

For the VLT at the Cassegrain focus the factor in front of is .*Depointing for M2 tilted around its vertex*

If M2 is tilted by an angle around its pole (see Fig. 3), the star centered on the adapter is producing an angle with the M1 axis.

For the VLT at the Cassegrain focus the factor in front of is -0.2743.*Depointing for M2 rotated around the coma-free point*

When M2 is rotated around its coma-free point by an angle we have a combination of pure tilt and lateral decentering with . The resulting offset of the telescope is

with

For the VLT at the Cassegrain focus one has*T*= -0.1539.

3.3 New coefficients

Replacing then by in the Eqs. (1) and (2) leads to

Equations (77) and (78) are similar to Eqs. (1) and (2), but with different coefficients

For the VLT with the pupil at M1

B_{0} |
= | B'_{0} |
= | |||||

B_{1} |
= | B'_{1} |
= | |||||

B_{2} |
= | B'_{2} |
= | . | ||||

B_{0} |
= | B'_{0} |
= | |||||

B_{1} |
= | B'_{1} |
= | |||||

B_{2} |
= | B'_{2} |
= | . | ||||

Equation (75) gives the relationship between the
misalignment angle and the corresponding field angle. The coefficient
of coma for this field angle can then be calculated from Eq.
(30). Finally, one needs the coefficient
of coma generated by a rotation around the center
of curvature of M2 by an angle .
This can be derived by
combining the expressions for coefficients of coma generated by a pure
lateral decentering by
and a rotation around the vertex of M2
by an angle
(see Eqs. (31) and
(32)). If
and
are related by
the total effect is a rotation around the
center of curvature of M2. One then obtains

Combining the three Eqs. (75), (30) and (80) gives the rotation angle of M2 which shifts the coma correction from the field center with to the center of the adapter.

For a classical telescope this reduces to the simple expression

Since the VLT is optically very close to a classical telescope and

The values of astigmatism calculated at different field positions by
Zemax were entered into our fitting software.
Two fittings were done, one with the set of parameters *B*_{0},
*B*_{1}, *B*_{2} for the original field center and one with the set
*B*'_{0}, *B*'_{1} and *B*'_{2} for the field center at the center of
the adapter. With the first
set we find
with a residual wavefront rms of
.
With the second set we find
,
very close to the input value,
with a small residual rms of
.

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