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Up: Analytical expressions for field


2 Astigmatism in a misaligned telescope with an arbitrary pupil position

2.1 General formulation

  \begin{figure}\includegraphics[width=10cm]{H1962F1.eps} \end{figure} Figure 1: Two mirror telescope with general position of M2 and general position of the stop

The general case which we study is shown in Fig.  1. The notation as well as the sign conventions for the angles and the distances are taken from Wilson ([1996]). The following list defines the parameters. i=1 stands for the primary mirror and i=2for the secondary mirror.

yi : semi-diameter of the mirror i
f'i : focal length of the mirror i
$s_{{\rm pr}i}$ : distance from the surface to the entrance
    pupil of mirror i
$b_{{\rm s}i}$ : aspheric constant of the mirror i
n'i : index of the exit medium of mirror i
$u_{{\rm pr}i}$ : angle of the incoming principle ray with
    the axis of the mirror i
hi : distance from the axis of the mirror i to
    the center of its entrance pupil
z : distance of the coma-free point from the
    surface of M2.

In this whole section the field center is defined as the optical axis of M1, that is $u_{{\rm pr}1} = 0$. The stop is located between the two mirrors at a distance $s_{{\rm pr}2}$ from M2 and decentered laterally by a value h from the M1 axis and h2 from the M2 axis. The entrance pupil is located at a distance $s_{{\rm pr}1}$behind M1 and decentered by h1 from the M1 axis. The vertex of the secondary mirror is decentered by a distance $\delta $ from the M1 axis and tilted by an angle $\alpha $ with respect to the M1 axis.

Initially we will only make the assumption that the lateral decenter $\delta $ and the rotation of M2 around its vertex by $\alpha $ are in the same plane.

After a bit of geometry following Fig. 1 we find:

$\displaystyle u_{{\rm pr}2}$ = $\displaystyle \left( \frac{s_{{\rm pr}1}}{f'_{1}} - 1 \right) \; u_{{\rm pr}1}
\; - \; \alpha \; - \; \frac{h_{1}}{f'_{1}}$ (3)
h1 = $\displaystyle -h\; \frac{s_{{\rm pr}1}}{d_{1} + s_{{\rm pr}2}}$ (4)
h2 = $\displaystyle h \, - \delta \, - \, s_{{\rm pr}2}\alpha .$ (5)

Knowing the distance $s_{{\rm pr}2}$ from M2 to the stop in a plane between M1 and M2 one can calculate $s_{{\rm pr}1}$ by

 \begin{displaymath}s_{{\rm pr}1} = \frac{s_{{\rm pr}2} + d_{1}}{s_{{\rm pr}2} + d_{1}- f'_{1}}\, f'_{1}.
\end{displaymath} (6)

Inversely, $s_{{\rm pr}2}$ is given by

 \begin{displaymath}s_{{\rm pr}2} = \frac{s_{{\rm pr}1}f'_{1}}{s_{{\rm pr}1} - f'_{1}} - d_{1}.
\end{displaymath} (7)

The coefficient of the astigmatic wavefront aberration of a two mirror telescope is then given by (Schroeder [1987], page 79)
$\displaystyle c_{\rm ast}$ = D1 y12 + D2 y22 (8)
Di = $\displaystyle \frac{n'_{i}}{16f_{i}^{'3}}
\left( A_{0,i}u_{{\rm pr}i}^{2}
\; + \; A_{1,i}u_{{\rm pr}i}h_{i}\; + \; A_{2,i}h_{i}^{2} \right)$ (9)

with the expressions for A0,i given by Schroeder ([1987]) in the Tables 5.6 and 5.9 (but adding the missing factor 1/2 in the expressions for astigmatism)
A0,i = $\displaystyle b_{{\rm s}i} s_{{\rm pr}i}^{2} \, + \,
\left( 2f'_{i} - s_{{\rm pr}i} \right)^{2}$ (10)
A1,i = $\displaystyle 4f'_{i} \, - \, 2(1+b_{{\rm s}i})s_{{\rm pr}i}$ (11)
A2,i = $\displaystyle 1+b_{{\rm s}i}.$ (12)

Introducing the expressions for A0,i, A1,i and A2,iinto Eq. (9) one gets
Di = $\displaystyle \frac{n'_{i}}{16f_{i}^{'3}} \;
\{ [b_{{\rm s}i} s_{{\rm pr}i}^{2} +
(2f'_{i} - s_{{\rm pr}i})^{2}] \; u_{{\rm pr}i}^{2}$  
    $\displaystyle + \; [4f'_{i} -2(1+b_{{\rm s}i})s_{{\rm pr}i}] \; u_{{\rm pr}i} \; h_{i}
\! + \! (1+b_{{\rm s}i}) \; h_{i}^{2} \}\cdot$ (13)

Our definition of the wavefront error differs by a factor -1 from the one given by Schroeder ([1987]), since we use the convention that a wavefront error is positive if the actual wavefront is in advance of the reference wavefront.

2.2 Explicit expressions for astigmatism in a schiefspiegler of general form

2.2.1 Telescope in focal form

In a two mirror telescope as shown in Fig. 1 the horizontal position of the stop may be anywhere to the left of the primary mirror. For the vertical position h of the stop we make the simplifying assumption that the center of the stop lies on the line connecting the vertices of the two mirrors. h is then given by
h = $\displaystyle \frac{\delta}{d_{1}} \, (d_{1} + s_{{\rm pr}2})$  
  = $\displaystyle \frac{\delta}{d_{1}} \frac{f'_{1}s_{{\rm pr}1}}{s_{{\rm pr}1}-f'_{1}}.$ (14)

With this expression for h one gets for the Eqs. (3), (4) and (5)
$\displaystyle u_{{\rm pr}2}$ = $\displaystyle \frac{s_{{\rm pr}1} - f'_{1}}{f'_{1}}u_{{\rm pr}1} \; - \; \alpha
\; + \frac{\delta}{d_{1}}\frac{s_{{\rm pr}1}}{f'_{1}}$ (15)
h1 = $\displaystyle -\frac{\delta}{d_{1}} \, s_{{\rm pr}1}$ (16)
h2 = $\displaystyle \left( \frac{\delta}{d_{1}} - \alpha \right) \;
\left( \frac{s_{{\rm pr}1}f'_{1}}{s_{{\rm pr}1} - f'_{1}} - d_{1} \right).$ (17)

Introducing these expressions into Eqs. (13) and (8) and using the relationships given by Wilson ([1996], Sect.
m2 = $\displaystyle \frac{f'_{2}}{f'_{1} - f'_{2} - d_{1}}$ (18)
f' = $\displaystyle m_{2}\, f'_{1}$ (19)
f'2 = $\displaystyle \frac{L}{m_{2}+1}$ (20)
L = $\displaystyle m_{2}\, (f'_{1} - d_{1})$ (21)
y2 = $\displaystyle \frac{L}{f'}\, y_{1}$ (22)

where m2 is the magnification of the secondary mirror, f' the focal length of the two mirror telescope and L is the distance from the secondary mirror to the focus of the telescope, one gets with n'1=-1 and n'2=1 for the coefficients of the astigmatic wavefront aberration
$\displaystyle c_{\rm ast}$ = $\displaystyle C_{0} u_{{\rm pr}1}^{2} + C_{1} u_{{\rm pr}1} + C_{2}$ (23)
C0 = $\displaystyle \frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \;
\bigg\{ \frac{f'}{L} \left(f'+d_{1} \right)$  
    $\displaystyle + \; \frac{d_{1}^{2}}{L}\xi
\; + \; \frac{s_{{\rm pr}1}}{f'} \left(f' + 2d_{1}\xi \right)$  
    $\displaystyle + \; \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2}
\left( -f'\zeta + L\xi\right) \bigg\}$ (24)
C1 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}$  
    $\displaystyle \bigg\{ ~ \alpha ~
\bigg[ (m_{2}+1)(f+L+d_{1})
- \frac{s_{{\rm pr}1}}{f'} L(m_{2}^{2}-1) \bigg]$  
    $\displaystyle + ~ \delta ~
\bigg[ (m_{2}+1)^{2} \,
\left( 1+\frac{d_{1}}{2L}(m_{2}+1)(1+b_{{\rm s}2}) \right)$  
    $\displaystyle + ~ \frac{s_{{\rm pr}1}}{f'} \frac{2}{d_{1}}
\bigg( d_{1}\xi + \frac{f'}{2}$  
    $\displaystyle ~~~- ~ \frac{d_{1}}{4}(m_{2}+1)^{2}
[m_{2}-1-(m_{2}+1)b_{{\rm s}2}] \bigg)$  
    $\displaystyle + ~ \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2}
\frac{2}{d_{1}}(-f'\zeta + L\xi) \bigg]
\bigg\}$ (25)
C2 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}$  
    $\displaystyle \bigg\{ ~ \alpha^{2} ~ L(m_{2}+1)$  
    $\displaystyle + ~ \alpha \delta ~
\bigg[ (m_{2}+1)^{2} ~ - ~
\frac{s_{{\rm pr}1}}{f'}\frac{L}{d_{1}}(m_{2}^{2}-1) \bigg]$  
    $\displaystyle + ~ \delta^{2} ~
\bigg[ \frac{1}{4L} (m_{2}+1)^{3} (1+b_{{\rm s}2})$  
    $\displaystyle - ~ \frac{s_{{\rm pr}1}}{f'}
\frac{1}{2d_{1}} (m_{2}+1)^{2}[m_{2}-1-(m_{2}+1)b_{{\rm s}2}]$  
    $\displaystyle + ~ \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2}
\frac{1}{d_{1}^{2}}(-f'\zeta + L\xi) \bigg]
\bigg\}$ (26)

$\displaystyle \zeta$ = $\displaystyle \frac{m_{2}^{3}}{4} (1 ~ + ~ b_{{\rm s}1})$ (27)
$\displaystyle \xi$ = $\displaystyle \frac{(m_{2}+1)^{3}}{4}
\left[ \left( \frac{m_{2}-1}{m_{2}+1} \right)^{2}
~ + ~ b_{{\rm s}2} \right].$ (28)

The expression for C0 is, apart from a factor needed for the conversion to Seidel coefficients, identical to the one given by Wilson ([1996], Sect.

If the stop is at the primary mirror, the parameters C0, C1and C2 should not depend on the asphericity $b_{{\rm s}1}$ of the primary mirror. This can be seen from the Eqs. (24), (25) and (26), which then depend only on the asphericity $b_{{\rm s}2}$ of the secondary mirror. Similarly, if the stop is at the secondary mirror, the parameters C0, C1 and C2 should not depend on the asphericity of the secondary mirror. This can easily be verified by introducing $s_{{\rm pr}2} = 0$ or, equivalently, from Eq. (6), $s_{{\rm pr}1}/f' = -d_{1}/L$ into the Eqs. (24), (25) and (26).

These expressions can be considerably simplified by using corresponding expressions for other aberration coefficients of two mirror telescopes (Wilson [1996], Sects. and 3.7.2): $c_{{\rm spher}}$ of spherical aberration, $c_{{\rm coma},{\rm cen}}$ of third order field coma of a centered system, and the coefficients of field independent third order coma generated by a lateral decenter by $\delta $( $c_{{\rm coma},\delta}$) and by a pure rotation of M2 around its vertex by $\alpha $( $c_{{\rm coma},\alpha}$).

$\displaystyle c_{{\rm spher}}$ = $\displaystyle \frac{1}{8} \,
\; \left( -f'\zeta + L\xi \right)$ (29)
$\displaystyle c_{{\rm coma},{\rm cen}}$ = $\displaystyle \frac{1}{2} \,
\; \bigg[-d_{1}\xi -\frac{f'}{2}$  
    $\displaystyle -\frac{s_{{\rm pr}1}}{f'}
\left( -f'\zeta + L\xi \right) \bigg]
\; u_{{\rm pr}1}$ (30)
$\displaystyle c_{{\rm coma},\delta}$ = $\displaystyle \frac{1}{4} \,
    $\displaystyle \bigg[ \frac{1}{2} (m_{2}+1)^{2} \;
\left[ m_{2} - 1 - (m_{2}+1)b_{{\rm s}2} \right]$  
    $\displaystyle + \frac{s_{{\rm pr}1}}{f'}\frac{2}{d_{1}}
(-f'\zeta+L\xi) \bigg] ~ \delta$ (31)
$\displaystyle c_{{\rm coma},\alpha}$ = $\displaystyle \frac{1}{4} \,
L(m_{2}^{2}-1) ~ \alpha.$ (32)

With the further definitions

 \begin{displaymath}c_{{\rm coma},{\rm cen}} = {c}_{{\rm coma},{\rm cen}}^{\ast} \; u_{{\rm pr}1}
\end{displaymath} (33)

C0,0 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2} \;
\left[ \frac{f'}{L} \left( f'+d_{1} \right)
\; + \; \frac{d_{1}^{2}}{L}\xi \right]$ (34)
C1,0 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}
\bigg[ ~ \alpha \, (m_{2}+1)(f+L+d_{1})$  
    $\displaystyle + ~ \delta \, (m_{2}+1)^{2} \,
\left( 1+\frac{d_{1}}{2L}(m_{2}+1)(1+b_{{\rm s}2}) \right)
\bigg]$ (35)
C2,0 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}
\bigg[ ~ \alpha^{2} ~ L(m_{2}+1)$  
    $\displaystyle + ~ \alpha \delta ~ (m_{2}+1)^{2}
\! + \! \delta^{2} ~ \frac{1}{4L} (m_{2}+1)^{3} (1+b_{{\rm s}2})
\bigg].$ (36)

Equations (24), (25) and (26) can be written as
C0 = $\displaystyle C_{0,0}
~ - ~ \frac{s_{{\rm pr}1}}{y_{1}} \; {c}_{{\rm coma},{\rm...
~ - ~ \left( \frac{s_{{\rm pr}1}}{y_{1}} \right)^{2}
2\, c_{{\rm spher}}$ (37)
C1 = C1,0  
    $\displaystyle - ~ \frac{s_{{\rm pr}1}}{y_{1}}
\left( \frac{\delta}{d_{1}}\, {c}...
...coma},{\rm cen}}^{\ast}
+ c_{{\rm coma},\alpha} + c_{{\rm coma},\delta} \right)$  
    $\displaystyle + ~ \left( \frac{s_{{\rm pr}1}}{y_{1}} \right)^{2}
4\, \frac{\delta}{d_{1}}\, c_{{\rm spher}}$ (38)
C2 = $\displaystyle C_{2,0}
~ - ~ \frac{s_{{\rm pr}1}}{y_{1}}\, \frac{\delta}{d_{1}} \;
\left( c_{{\rm coma},\alpha} + c_{{\rm coma},\delta} \right)$  
    $\displaystyle + ~ \left( \frac{s_{{\rm pr}1}}{y_{1}} \right)^{2}
6\, \left( \frac{\delta}{d_{1}} \right)^{2}\, c_{{\rm spher}}.$ (39)

These equations show a nice symmetry of the stop-shift terms. The linear terms are proportional to coefficients of coma and the quadratic terms are proportional to the coefficient of spherical aberration. The total coma in the linear coefficient of C1 is the sum of the coma for a centered system for a principal ray with the angle $\delta/d_{1}$, which, in the decentered system, is the angle of the principal ray connecting the vertices of the two mirrors, and the coma contributions from a pure decenter of M2 by $\delta $ and a pure rotation of M2 around its vertex by $\alpha $. The total coma in the linear coefficient of C2 contains only the two latter contributions.

2.2.2 Telescope in afocal form

Equations (29) to (39) can be converted into equations valid for afocal telescopes, where both the total focal length f' and the position of the focus, which is linked to L, go to infinity. One can eliminate f' and L in favour of f'1, d1 and m2 with the Eqs. (19) and (21) and then let m2 go to infinity. This gives
$\displaystyle c_{{\rm spher},{\rm af}}$ = $\displaystyle \frac{1}{32} \,
\; \bigg[ -f'_{1}(1+b_{{\rm s}1})$  
    $\displaystyle + ~ (f'_{1} - d_{1})(1+b_{{\rm s}2}) \bigg]$ (40)
$\displaystyle c_{{\rm coma},{\rm cen},{\rm af}}$ = $\displaystyle - \bigg[\frac{1}{8} \,
\; d_{1}(1+b_{{\rm s}2}) ~$  
    $\displaystyle + ~ 4\frac{s_{{\rm pr}1}}{y_{1}} c_{{\rm spher},{\rm af}} \bigg]
\; u_{{\rm pr}1}$ (41)
$\displaystyle c_{{\rm coma},\delta,{\rm af}}$ = $\displaystyle \bigg[ \frac{1}{8} \,
\left(\frac{y_{1}}{f'_{1}}\right)^{3} (1 - b_{{\rm s}2})$  
    $\displaystyle + ~ \frac{s_{{\rm pr}1}}{y_{1}} \frac{4}{d_{1}}c_{{\rm spher},{\rm af}}
\bigg] ~ \delta$ (42)
$\displaystyle c_{{\rm coma},\alpha,{\rm af}}$ = $\displaystyle \frac{1}{4} \,
\, (f'_{1} - d_{1}) ~ \alpha$ (43)
$\displaystyle C_{0,0,{\rm af}}$ = $\displaystyle \frac{1}{16} \left( \frac{y_{1}}{f'_{1}} \right)^{2} \;
\frac{d_{1}^{2}}{f'_{1}-d_{1}}(1+b_{{\rm s}2})$ (44)
$\displaystyle C_{1,0,{\rm af}}$ = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'_{1}} \right)^{2}
\bigg[\alpha \, (2f'_{1} - d_{1}) ~$  
    $\displaystyle + ~ \delta \, \left( 1+\frac{d_{1}}{2(f'_{1} - d_{1})}
(1+b_{{\rm s}2}) \right)
\bigg]$ (45)
$\displaystyle C_{2,0,{\rm af}}$ = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'_{1}} \right)^{2}
\bigg[~ \alpha^{2} ~ (f'_{1} - d_{1})
+ ~ \alpha \delta ~$  
    $\displaystyle + ~ \delta^{2} ~ \frac{1+b_{{\rm s}2}}{4(f'_{1} - d_{1})}
\bigg].$ (46)

Equations (37), (38) and (39) for focal telescopes are then also valid for afocal telescopes if all expressions in these formulae are replaced by the corresponding expressions (40) to (46) for afocal telescopes.

2.3 Explicit expressions for astigmatism in a coma-free schiefspiegler

2.3.1 Telescope in focal form

A further simplification is possible if the telescope is corrected for coma at the center of the field, that is for $u_{{\rm pr}1} = 0$. The axes of M1 and M2 must then intersect at the coma-free point $P_{{\rm cfp}}$. The distance from the vertex of M2 to $P_{{\rm cfp}}$ is denoted by z. The lateral decenter $\delta $ and the misalignment angle $\alpha $ are then related by $\delta=-\alpha\, z$. z can be calculated from the requirement that the contributions to decentering coma from a pure lateral decenter $\delta $ and the simultaneous rotation $\alpha = -\delta/z$ cancel.

 \begin{displaymath}c_{{\rm coma},\delta} + c_{{\rm coma},\alpha} = 0.
\end{displaymath} (47)

This gives, using the Eqs. (31) and (32),
z = $\displaystyle 2L \; \frac{m_{2}-1}{m_{2}+1} \cdot$ (48)
    $\displaystyle \cdot \frac{1}{m_{2}-1-(m_{2}+1)b_{{\rm s}2}
- \frac{s_{{\rm pr}1}}{f'}\frac{4}{(m_{2}+1)^{2}d_{1}^{2}}
(-f'\zeta + L\xi)}\cdot$  

This equation shows that the position of the coma-free point depends on the stop position. If the stop is at the primary mirror or if the telescope is corrected for spherical aberration, z depends, for a given telescope geometry L and m2, only on the aspheric constant $b_{{\rm s}2}$ of the secondary mirror.

 \begin{displaymath}z = 2L\; \frac{m_{2}-1}{m_{2}+1} \;
\frac{1}{m_{2}-1-(m_{2}+1)b_{{\rm s}2}}\cdot
\end{displaymath} (49)

If the stop is at the secondary mirror z depends, for a given telescope geometry L and m2, only on the asphericity $b_{{\rm s}1}$ of the primary mirror.

 \begin{displaymath}z = 2L\; \frac{m_{2}^{2}-1}{2m_{2}(m_{2}^{2}-1)
- \frac{f'}{L}m_{2}^{3}(1+b_{{\rm s}1})}\cdot
\end{displaymath} (50)

The coefficient $c_{\rm ast}$ of third order astigmatism can then be expressed as

\begin{displaymath}c_{\rm ast} = B_{0} u_{{\rm pr}1}^{2} + B_{1} u_{{\rm pr}1}\alpha + B_{2}\alpha^{2}
\end{displaymath} (51)

B0 = C0 (52)
B1 = $\displaystyle -\frac{1}{4}\left(\frac{y_{1}}{f'}\right)^{2}
\; \frac{z}{d_{1}} \;
\bigg[ \frac{f'd_{1}}{L}\frac{(m_{2}+1)^{3}}{m_{2}-1}b_{{\rm s}2}$  
    $\displaystyle + ~ \frac{s_{{\rm pr}1}}{f'} \bigg( f' + 2d_{1}\xi$  
    $\displaystyle + 2\frac{2L+(m_{2}+1)d_{1}}{(m_{2}-1)L}(-f\zeta+L\xi)
\bigg) \bigg]$ (53)
B2 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}
\; \left( \frac{z...{d_{1}^{2}}{L}
\left( \frac{m_{2}+1}{m_{2}-1} \right)^{2} b_{{\rm s}2} \, \xi$  
    $\displaystyle + ~ \frac{s_{{\rm pr}1}}{f'} \frac{2d_{1}}{L}
\; \left( \frac{m_{2}+1}{m_{2}-1} \right)^{2}
b_{{\rm s}2} (-f\zeta+L\xi)$  
    $\displaystyle + ~ \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2}
    $\displaystyle ~~~~~~~~~ \left( \frac{4}{(m_{2}-1)^{2}(m_{2}+1)}
\frac{-f\zeta+L\xi}{L} - 1 \right) \bigg].$ (54)

Equations (52), (53) and (54) show that the parameters Bi are proportional to (z/d1)i. At a first glance they seem to be linear or quadratic equations in $s_{{\rm pr}1}/f'$. This is only the case for B0 since the distance z of the coma-free point appearing in B1 and B2 depends itself on the stop position, as can be seen from Eq. (49).

By introducing $z = -\delta/\alpha$ in the Eqs. (52), (53) and (54) the coefficient of third order astigmatism can be expressed as a polynomial in $u_{{\rm pr}1}$ and $\delta $, that is

 \begin{displaymath}c_{\rm ast} = B_{0} u_{{\rm pr}1}^{2}
+ B_{1}^{(\delta)} u_{{\rm pr}1}\delta
+ B_{2}^{(\delta)}\delta^{2}.
\end{displaymath} (55)

It is easy to see that now the parameter $B_{1}^{(\delta)}$is a linear function and the parameters B0 and $B_{2}^{(\delta)}$ are quadratic functions in $s_{{\rm pr}1}$.

Exactly as with C0, C1 and C2 the parameters B0, B1and B2 depend only on the asphericity $b_{{\rm s}2}$ of M2, if the stop is at the primary mirror, and only on the asphericity $b_{{\rm s}1}$ of M1, if the stop is at the secondary mirror.

Since the coupling between $\alpha $ and $\delta $ through z involves the expression for spherical aberration, the symmetry of the stop-shift terms has disappeared. But, the telescope will usually be corrected for spherical aberration. In this case all terms containing $-f'\zeta + L\xi$ vanish. Then, with the additional definitions

B0,0 = $\displaystyle \frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \;
\left[ \frac{f'}{L}\, (f'+d_{1})
+ \frac{d_{1}^{2}}{L}\xi \right]$ (56)
B1,0 = $\displaystyle -\frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \;
\frac{f'd_{1}}{L}\frac{(m_{2}+1)^{3}}{m_{2}-1}b_{{\rm s}2}$ (57)
B2,0 = $\displaystyle \frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \;
\left( \frac{m_{2}+1}{m_{2}-1} \right)^{2} b_{{\rm s}2} \, \xi .$ (58)

Equations (52), (53) and (54) reduce to
B0 = $\displaystyle B_{0,0} ~ -
~ \frac{s_{{\rm pr}1}}{y_{1}} c_{{\rm coma},{\rm cen}}^{\ast}$ (59)
B1 = $\displaystyle \frac{z}{d_{1}} \left( B_{1,0}
~ + ~ \frac{s_{{\rm pr}1}}{y_{1}} c_{{\rm coma},{\rm cen}}^{\ast} \right)$ (60)
B2 = $\displaystyle \left( \frac{z}{d_{1}} \right)^{2} B_{2,0}.$ (61)

The distance z from the secondary mirror to the coma-free point and the expression $c_{{\rm coma},{\rm cen}}^{\ast}$ are now no longer dependent on the stop position. Therefore, B0 and B1 are linear functions of the stop position and B2 is independent of the stop position.

2.3.2 Telescope in afocal form

The Eqs. (59), (60) and (61), all valid for focal telescopes corrected for spherical aberration, can be converted into equations valid for afocal telescopes, also corrected for spherical aberration, by using Eqs. (21) and (19) and letting m2 go to infinity. The distance from M2 to the coma-free point is then

 \begin{displaymath}z = 2\, \frac{f'_{1} - d_{1}}{1-b_{{\rm s}2}} .
\end{displaymath} (62)

For a classical or aplanatic afocal telescope the coma-free point is in the focus of M1. The astigmatism parameters are given by
B0 = $\displaystyle \frac{1}{4} \left(\frac{y_{1}}{f'_{1}}\right)^{2} \;
(1+b_{{\rm s...
...frac{d_{1}^{2}} {f'_{1}-d_{1}}
~ + ~ s_{{\rm pr}1}\frac{d_{1}}{2f'_{1}} \right]$ (63)
B1 = $\displaystyle -\frac{1}{4}\left(\frac{y_{1}}{f'_{1}}\right)^{2}
\; \frac{1}{1-b_{{\rm s}2}} \;
\bigg[ 2f'_{1}b_{{\rm s}2}$  
    $\displaystyle + ~ s_{{\rm pr}1}\frac{f'_{1}-d_{1}}{f'_{1}} (1+b_{{\rm s}2}) \bigg]$ (64)
B2 = $\displaystyle \frac{1}{4} \left( \frac{y_{1}}{f'_{1}} \right)^{2}
\; \frac{1+b_{{\rm s}2}}{(1-b_{{\rm s}2})^{2}}\, b_{{\rm s}2}
\; (f'_{1} - d_{1}).$ (65)

For a Mersenne telescope with $b_{{\rm s}1} = b_{{\rm s}2} = -1$ one obtains immediately B0 = B2 = 0. This gives therefore pure linear astigmatism which is proportional to the misalignment angle $\alpha $with the center of the pattern at the center of the field. This is a nice example of the general statement by Shack & Thompson ([1980]) that a system which is free of astigmatism in the centered configuration will show either linear or constant astigmatism in the decentered configuration.

2.4 Conclusions for specific types of telescopes and stop positions

2.5 Specific stop positions at coma-free schiefspieglers

For a general coma-free schiefspiegler the following conclusions can be drawn for the stop positions at the primary and secondary mirrors.

2.6 Specific types of coma-free schiefspieglers corrected for spherical aberration

From the general formulae given above more specific conclusions can be drawn if the telescopes is corrected for spherical aberration. In all cases we will also discuss the results in the limit of large magnifications, defined here as $m_{2} \rightarrow \infty$ together with a finite distance L from the secondary mirror to the focus. The semi-diameter y2 of M2, its radius of curvature 2f'2 and the difference between f'1 and d1will go to zero. This is different from the case of an afocal telescope, where L goes to infinity, while y2, 2f'2 and f'1-d1 remain finite. For the limit case with finite L we get from Eqs. (56) to (58) the following expressions for the astigmatism parameters:

 \begin{displaymath}\begin{array}{@{}l l c c}
& & {\rm Stop \; at \; M1}
& \;\...
...ow & \displaystyle{0} &
\;\;\; \displaystyle{0}.
\end{array} \end{displaymath} (66)

We now discuss a few telescope types.

2.7 Numerical examples for the VLT

In this sections we give numerical examples for the Cassegrain focus of the VLT telescope. The optical parameters used for these calculations are summarized in the following table.

2f'1 -28804.832 $\; {\rm mm}$ 2f'2 -4553.561 $\; {\rm mm}$
$b_{{\rm s}1}$ -0.996962 $b_{{\rm s}2}$ -1.66926
y1 4057.50 $\; {\rm mm}$ y2 556.55 $\; {\rm mm}$
d1 -12426.946 $\; {\rm mm}$    
z -1997.995 $\; {\rm mm}$    
L 14926.950 $\; {\rm mm}$    
m2 -7.556    

If the pupil was located at the primary mirror, we would have $s_{{\rm pr}1} = 0$. This would give the following numerical values for the parameters B0, B1 and B2.

B0 = $\displaystyle +71.788 \;\mu {\rm m}/{\rm deg}^{2}$  
B1 = $\displaystyle -84.784 \;\mu {\rm m}/{\rm deg}^{2}$  
B2 = $\displaystyle +0.06630 \;\mu {\rm m}/{\rm deg}^{2}.$ (70)

As has been discussed before, the value of B2 is effectively negligible compared with the values of B0 and B1 and the ratio B1/B0 is approximately equal to -1.

In reality the pupil is located at the secondary mirror. Then $s_{{\rm pr}2} = 0$ and from Eq. (6) one gets $s_{{\rm pr}1} = 90600.23 \, {\rm mm}$. This gives the following numerical values for the parameters:

B0 = $\displaystyle +86.614 \;\mu {\rm m}/{\rm deg}^{2}$  
B1 = $\displaystyle -87.167 \;\mu {\rm m}/{\rm deg}^{2}$  
B2 = $\displaystyle +0.06630 \;\mu {\rm m}/{\rm deg}^{2}.$ (71)

Since the spherical aberration is zero the value of B2 is, as can be seen from Eq. (54), identical to the one for the stop at the primary mirror. But, there is a significant difference between the values of B0 for the two stop positions.

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