Up: Analytical expressions for field

3 Recalculation of the coefficients B₀, B₁ and B₂ for the center of the adapter

3.1 Introduction

In the formulae given above the center of the field is defined by $u_{{\rm pr}1} = 0$ and corresponds to the direction of a star located on the M1 axis. In a decentered telescope the rays coming from such a star do not focus in the center of the adapter.

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 B₀, B₁ and B₂.

$\begin{figure}\resizebox{\hsize}{!}{\includegraphics{H1962F2.eps}} \end{figure}$

Figure 2: Pure lateral decenter of M2 by $\delta$ . C₂ is the center of curvature of M2

$\begin{figure}\resizebox{\hsize}{!}{\includegraphics{H1962F3.eps}} \end{figure}$

Figure 3: Pure rotation of M2 by $\alpha$ around its vertex. C₂ is the center of curvature of M2

3.2 Depointing of the telescope for a pure lateral decentering of M2

Depointing for a pure lateral decentering of M2
If M2 is laterally decentered by $\delta$ (see Fig. 2) the star centered in the adapter is producing an angle $\theta_{{\rm offset,d}}$ with the M1 axis:

$\begin{displaymath}\theta_{{\rm offset,d}} = -\frac{L}{f'_{2}f'} \, \delta. \end{displaymath}$ (72)

For the VLT at the Cassegrain focus the factor in front of $\delta$ is $-6.025 \ 10^{-5}/{\rm mm}$ .
Depointing for M2 tilted around its vertex
If M2 is tilted by an angle $\alpha$ around its pole (see Fig. 3), the star centered on the adapter is producing an angle $\theta_{{\rm offset,t}}$ with the M1 axis.

$\begin{displaymath}\theta_{{\rm offset,t}} = \frac{-2L}{f} \, \alpha . \end{displaymath}$ (73)

For the VLT at the Cassegrain focus the factor in front of $\alpha$ is -0.2743.
Depointing for M2 rotated around the coma-free point
When M2 is rotated around its coma-free point by an angle $\alpha$ we have a combination of pure tilt and lateral decentering with $\delta=-\alpha\, z$ . The resulting offset of the telescope is

$\displaystyle \theta_{{\rm offset,cfp}}$ = $\displaystyle \theta_{{\rm offset,t}} + \theta_{{\rm offset,d}}$ (74)

= $\displaystyle T\, \alpha$

with

$\begin{displaymath}T = \frac{-L}{f} \, \left(2-\frac{z}{f'_{2}}\right). \end{displaymath}$ (75)

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

3.3 New coefficients B₀, B₁ and B₂ with reference to the center of the adapter

To get the coefficients B₀, B₁ and B₂ with the reference of the field in the center of the adapter, we define new field angles as

$\begin{displaymath}\theta' = \theta - T\, \alpha. \end{displaymath}$

(76)

Replacing then $\theta$ by $\theta' + T\alpha$ in the Eqs. (1) and (2) leads to

$\displaystyle Z_{4}^{{\rm sys}}$	=	$\displaystyle B'_{0}\, (\theta_{x}^{'2} - \theta_{y}^{'2}) \, + \, B'_{1}(\theta'_{x}\alpha_{x} \, - \, \theta'_{y}\alpha_{y})$
		$\displaystyle + \, B'_{2}\, (\alpha_{x}^{2} - \alpha_{y}^{2})$	(77)
$\displaystyle Z_{5}^{{\rm sys}}$	=	$\displaystyle 2B'_{0}\, \theta'_{x}\theta'_{y} \, + \, B'_{1}(\theta'_{x}\alpha_{y} \, + \, \theta'_{y}\alpha_{x}) \, + \, 2B'_{2}\, \alpha_{x}\alpha_{y}.$	(78)

Equations (77) and (78) are similar to Eqs. (1) and (2), but with different coefficients B'₀, B'₁ and B'₂. The new parameters B'₀, B'₁ and B'₂ are given by

B'₀	=	B₀
B'₁	=	B₁ + 2B₀T
B'₂	=	B₂ + B₁T + B₀T².	(79)

For the VLT with the pupil at M1

B₀	=	$\;\, +71.788$	$\;\mu {\rm m}/{\rm deg}^{2}$	B'₀	=	$\;\, +71.788$	$\;\mu {\rm m}/{\rm deg}^{2}$
B₁	=	$\;\, -84.783$	$\;\mu {\rm m}/{\rm deg}^{2}$	B'₁	=	$\;\, -106.888$	$\;\mu {\rm m}/{\rm deg}^{2}$
B₂	=	$\;\, +0.0663$	$\;\mu {\rm m}/{\rm deg}^{2}$	B'₂	=	$\;\, +14.821$	$\;\mu {\rm m}/{\rm deg}^{2}$ .

For the VLT with the pupil at M2

B₀	=	$\;\, +86.614$	$\;\mu {\rm m}/{\rm deg}^{2}$	B'₀	=	$\;\, +86.614$	$\;\mu {\rm m}/{\rm deg}^{2}$
B₁	=	$\;\, -87.167$	$\;\mu {\rm m}/{\rm deg}^{2}$	B'₁	=	$\;\, -113.837$	$\;\mu {\rm m}/{\rm deg}^{2}$
B₂	=	$\;\, +0.0663$	$\;\mu {\rm m}/{\rm deg}^{2}$	B'₂	=	$\;\, +15.539$	$\;\mu {\rm m}/{\rm deg}^{2}$ .

In both cases the new coefficients B'₁ are approximately 30% larger than the original coefficients B₁.

3.4 Effects of the correction of coma at the center of the adapter

The formulae in Sect. 2.3 have been derived under the assumption that the telescope was a coma-free schiefspiegler. This means that the telecope is corrected for coma for the field center with $u_{{\rm pr}1} = 0$ . But, in reality the coma correction will ensure that the telescope is free of coma at the center of the adapter. The formulae derived above can therefore, strictly speaking, not be applied to non-aplanatic telescopes. But, a short calculation will show that, in practice, the inaccuracies introduced by this effect are negligible.

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 $c_{{\rm coma},{\rm coc}}$ of coma generated by a rotation around the center of curvature of M2 by an angle $\varphi$ . This can be derived by combining the expressions for coefficients of coma generated by a pure lateral decentering by $\delta$ and a rotation around the vertex of M2 by an angle $u_{{\rm pr}2}$ (see Eqs. (31) and (32)). If $\delta$ and $u_{{\rm pr}2}$ are related by $\delta = -2f'_{2}u_{{\rm pr}2}$ the total effect is a rotation around the center of curvature of M2. One then obtains

$\begin{displaymath}c_{{\rm coma},{\rm coc}} = \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{3} (m_{2}-1)\frac{f'L}{z}\; T \; \varphi. \end{displaymath}$

(80)

Combining the three Eqs. (75), (30) and (80) gives the rotation angle $\varphi$ of M2 which shifts the coma correction from the field center with $u_{{\rm pr}1} = 0$ to the center of the adapter.

$\displaystyle \varphi$	=	$\displaystyle -\; \frac{(2d_{1}\xi + f') \, z} {(m_{2}-1)f'L} \; \alpha$
	=	$\displaystyle - \; \frac{2\; \alpha}{(m_{2}+1)[m_{2}-1-(m_{2}+1)b_{{\rm s}2}]} \; \frac{2d_{1}\xi + f'}{f'}\cdot$	(81)

For a classical telescope this reduces to the simple expression

$\begin{displaymath}\varphi_{\rm class} = -\frac{1}{m_{2}(m_{2}-1)}\alpha \cdot \end{displaymath}$

(82)

Since the VLT is optically very close to a classical telescope and m₂ = -7.556, $\varphi$ is approximately fifty times smaller than $\alpha$ . This shows that the difference between a schiefspiegler with coma corrected for the center of the field and one with coma corrected for the center of the adapter is negligible. The formulae for field astigmatism derived above for a schiefspiegler free of coma at the center of the field can therefore also be used for a schiefspiegler with coma corrected for the center of the adapter.

3.5 Simulation for the ESO 3.6 m telescope

As an example we checked our formula with a simulation done with Zemax for the 3.6 m telescope on La Silla. We simulated M2 tilted around the coma-free point by an angle $\alpha = 0.21136^{\circ}$ , which corresponds to an unusually large decenter of $\delta=10\, {\rm mm}$ .

$\begin{figure}\includegraphics[width=10cm]{H1962F4.eps} \end{figure}$

Figure 4: Change of $\alpha$ due to a rotation of M1 around its vertex

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 B₀, B₁, B₂ 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 $\alpha = 0.3867^{\circ}$ with a residual wavefront rms of $697.2 \, {\rm nm}$ . With the second set we find $\alpha = 0.21041^{\circ}$ , very close to the input value, with a small residual rms of $10.2 \, {\rm nm}$ .