2 Astigmatism in a misaligned telescope with an arbitrary pupil position

2.1 General formulation

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:

   

$$u_{{\rm pr}2} \left( \frac{s_{{\rm pr}1}}{f'_{1}} - 1 \right) \; u_{{\rm pr}1} \; - \; \alpha \; - \; \frac{h_{1}}{f'_{1}}$$ = (3)

$$-h\; \frac{s_{{\rm pr}1}}{d_{1} + s_{{\rm pr}2}}$$ h₁ = (4)

$$h \, - \delta \, - \, s_{{\rm pr}2}\alpha .$$ h₂ = (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

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

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

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

The coefficient of the astigmatic wavefront aberration of a two mirror telescope is then given by (Schroeder [1987], page 79)

  

$$c_{\rm ast}$$ = D₁ y₁² + D₂ y₂² (8)

$$\frac{n'_{i}}{16f_{i}^{'3}} \left( A_{0,i}u_{{\rm pr}i}^{2} \; + \; A_{1,i}u_{{\rm pr}i}h_{i}\; + \; A_{2,i}h_{i}^{2} \right)$$ D_i = (9)

with the expressions for A_0,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)

   

$$b_{{\rm s}i} s_{{\rm pr}i}^{2} \, + \, \left( 2f'_{i} - s_{{\rm pr}i} \right)^{2}$$ A_0,i = (10)

$$4f'_{i} \, - \, 2(1+b_{{\rm s}i})s_{{\rm pr}i}$$ A_1,i = (11)

$$1+b_{{\rm s}i}.$$ A_2,i = (12)

Introducing the expressions for A_0,i, A_1,i and A_2,iinto Eq. (9) one gets

 

$$\frac{n'_{i}}{16f_{i}^{'3}} \; \{ [b_{{\rm s}i} s_{{\rm pr}i}^{2} + (2f'_{i} - s_{{\rm pr}i})^{2}] \; u_{{\rm pr}i}^{2}$$ D_i =

$$+ \; [4f'_{i} -2(1+b_{{\rm s}i})s_{{\rm pr}i}] \; u_{{\rm pr}i} \; h_{i} \! + \! (1+b_{{\rm s}i}) \; h_{i}^{2} \}\cdot$$ (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

 

$$\frac{\delta}{d_{1}} \, (d_{1} + s_{{\rm pr}2})$$ h =

$$\frac{\delta}{d_{1}} \frac{f'_{1}s_{{\rm pr}1}}{s_{{\rm pr}1}-f'_{1}}.$$ = (14)

With this expression for h one gets for the Eqs. (3), (4) and (5)

   

$$u_{{\rm pr}2} \frac{s_{{\rm pr}1} - f'_{1}}{f'_{1}}u_{{\rm pr}1} \; - \; \alpha \; + \frac{\delta}{d_{1}}\frac{s_{{\rm pr}1}}{f'_{1}}$$ = (15)

$$-\frac{\delta}{d_{1}} \, s_{{\rm pr}1}$$ h₁ = (16)

$$\left( \frac{\delta}{d_{1}} - \alpha \right) \; \left( \frac{s_{{\rm pr}1}f'_{1}}{s_{{\rm pr}1} - f'_{1}} - d_{1} \right).$$ h₂ = (17)

Introducing these expressions into Eqs. (13) and (8) and using the relationships given by Wilson ([1996], Sect. 2.2.5.2)

     

$$\frac{f'_{2}}{f'_{1} - f'_{2} - d_{1}}$$ m₂ = (18)

$$m_{2}\, f'_{1}$$ f' = (19)

$$\frac{L}{m_{2}+1}$$ f'₂ = (20)

$$m_{2}\, (f'_{1} - d_{1})$$ L = (21)

$$\frac{L}{f'}\, y_{1}$$ y₂ = (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

   

$$c_{\rm ast} C_{0} u_{{\rm pr}1}^{2} + C_{1} u_{{\rm pr}1} + C_{2}$$ = (23)

$$\frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \; \bigg\{ \frac{f'}{L} \left(f'+d_{1} \right)$$ C₀ =

$$+ \; \frac{d_{1}^{2}}{L}\xi \; + \; \frac{s_{{\rm pr}1}}{f'} \left(f' + 2d_{1}\xi \right)$$

$$+ \; \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2} \left( -f'\zeta + L\xi\right) \bigg\}$$ (24)

$$\frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}$$ C₁ =

$$\bigg\{ ~ \alpha ~ \bigg[ (m_{2}+1)(f+L+d_{1}) - \frac{s_{{\rm pr}1}}{f'} L(m_{2}^{2}-1) \bigg]$$

$$+ ~ \delta ~ \bigg[ (m_{2}+1)^{2} \, \left( 1+\frac{d_{1}}{2L}(m_{2}+1)(1+b_{{\rm s}2}) \right)$$

$$+ ~ \frac{s_{{\rm pr}1}}{f'} \frac{2}{d_{1}} \bigg( d_{1}\xi + \frac{f'}{2}$$

$$~~~- ~ \frac{d_{1}}{4}(m_{2}+1)^{2} [m_{2}-1-(m_{2}+1)b_{{\rm s}2}] \bigg)$$

$$+ ~ \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2} \frac{2}{d_{1}}(-f'\zeta + L\xi) \bigg] \bigg\}$$ (25)

$$\frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2}$$ C₂ =

$$\bigg\{ ~ \alpha^{2} ~ L(m_{2}+1)$$

$$+ ~ \alpha \delta ~ \bigg[ (m_{2}+1)^{2} ~ - ~ \frac{s_{{\rm pr}1}}{f'}\frac{L}{d_{1}}(m_{2}^{2}-1) \bigg]$$

$$+ ~ \delta^{2} ~ \bigg[ \frac{1}{4L} (m_{2}+1)^{3} (1+b_{{\rm s}2})$$

$$- ~ \frac{s_{{\rm pr}1}}{f'} \frac{1}{2d_{1}} (m_{2}+1)^{2}[m_{2}-1-(m_{2}+1)b_{{\rm s}2}]$$

$$+ ~ \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2} \frac{1}{d_{1}^{2}}(-f'\zeta + L\xi) \bigg] \bigg\}$$ (26)

where

  

$$\zeta \frac{m_{2}^{3}}{4} (1 ~ + ~ b_{{\rm s}1})$$ = (27)

$$\xi \frac{(m_{2}+1)^{3}}{4} \left[ \left( \frac{m_{2}-1}{m_{2}+1} \right)^{2} ~ + ~ b_{{\rm s}2} \right].$$ = (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. 3.2.4.2).

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. 3.2.4.2 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 $\alpha$( $c_{{\rm coma},\alpha}$).

    

$$c_{{\rm spher}} \frac{1}{8} \, \left(\frac{y_{1}}{f'}\right)^{4} \; \left( -f'\zeta + L\xi \right)$$ = (29)

$$c_{{\rm coma},{\rm cen}} \frac{1}{2} \, \left(\frac{y_{1}}{f'}\right)^{3} \; \bigg[-d_{1}\xi -\frac{f'}{2}$$ =

$$-\frac{s_{{\rm pr}1}}{f'} \left( -f'\zeta + L\xi \right) \bigg] \; u_{{\rm pr}1}$$ (30)

$$c_{{\rm coma},\delta} \frac{1}{4} \, \left(\frac{y_{1}}{f'}\right)^{3}$$ =

$$\bigg[ \frac{1}{2} (m_{2}+1)^{2} \; \left[ m_{2} - 1 - (m_{2}+1)b_{{\rm s}2} \right]$$

$$+ \frac{s_{{\rm pr}1}}{f'}\frac{2}{d_{1}} (-f'\zeta+L\xi) \bigg] ~ \delta$$ (31)

$$c_{{\rm coma},\alpha} \frac{1}{4} \, \left(\frac{y_{1}}{f'}\right)^{3} L(m_{2}^{2}-1) ~ \alpha.$$ = (32)

With the further definitions

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

   

$$\frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2} \; \left[ \frac{f'}{L} \left( f'+d_{1} \right) \; + \; \frac{d_{1}^{2}}{L}\xi \right]$$ C_0,0 = (34)

$$\frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2} \bigg[ ~ \alpha \, (m_{2}+1)(f+L+d_{1})$$ C_1,0 =

$$+ ~ \delta \, (m_{2}+1)^{2} \, \left( 1+\frac{d_{1}}{2L}(m_{2}+1)(1+b_{{\rm s}2}) \right) \bigg]$$ (35)

$$\frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2} \bigg[ ~ \alpha^{2} ~ L(m_{2}+1)$$ C_2,0 =

$$+ ~ \alpha \delta ~ (m_{2}+1)^{2} \! + \! \delta^{2} ~ \frac{1}{4L} (m_{2}+1)^{3} (1+b_{{\rm s}2}) \bigg].$$ (36)

Equations (24), (25) and (26) can be written as

   

$$C_{0,0} ~ - ~ \frac{s_{{\rm pr}1}}{y_{1}} \; {c}_{{\rm coma},{\rm... ...{\ast} ~ - ~ \left( \frac{s_{{\rm pr}1}}{y_{1}} \right)^{2} 2\, c_{{\rm spher}}$$ C₀ = (37)

C₁ = C_1,0

$$- ~ \frac{s_{{\rm pr}1}}{y_{1}} \left( \frac{\delta}{d_{1}}\, {c}... ...coma},{\rm cen}}^{\ast} + c_{{\rm coma},\alpha} + c_{{\rm coma},\delta} \right)$$

$$+ ~ \left( \frac{s_{{\rm pr}1}}{y_{1}} \right)^{2} 4\, \frac{\delta}{d_{1}}\, c_{{\rm spher}}$$ (38)

$$C_{2,0} ~ - ~ \frac{s_{{\rm pr}1}}{y_{1}}\, \frac{\delta}{d_{1}} \; \left( c_{{\rm coma},\alpha} + c_{{\rm coma},\delta} \right)$$ C₂ =

$$+ ~ \left( \frac{s_{{\rm pr}1}}{y_{1}} \right)^{2} 6\, \left( \frac{\delta}{d_{1}} \right)^{2}\, c_{{\rm spher}}.$$ (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

       

$$c_{{\rm spher},{\rm af}} \frac{1}{32} \, \left(\frac{y_{1}}{f'_{1}}\right)^{4} \; \bigg[ -f'_{1}(1+b_{{\rm s}1})$$ =

$$+ ~ (f'_{1} - d_{1})(1+b_{{\rm s}2}) \bigg]$$ (40)

$$c_{{\rm coma},{\rm cen},{\rm af}} - \bigg[\frac{1}{8} \, \left(\frac{y_{1}}{f'_{1}}\right)^{3} \; d_{1}(1+b_{{\rm s}2}) ~$$ =

$$+ ~ 4\frac{s_{{\rm pr}1}}{y_{1}} c_{{\rm spher},{\rm af}} \bigg] \; u_{{\rm pr}1}$$ (41)

$$c_{{\rm coma},\delta,{\rm af}} \bigg[ \frac{1}{8} \, \left(\frac{y_{1}}{f'_{1}}\right)^{3} (1 - b_{{\rm s}2})$$ =

$$+ ~ \frac{s_{{\rm pr}1}}{y_{1}} \frac{4}{d_{1}}c_{{\rm spher},{\rm af}} \bigg] ~ \delta$$ (42)

$$c_{{\rm coma},\alpha,{\rm af}} \frac{1}{4} \, \left(\frac{y_{1}}{f'_{1}}\right)^{3} \, (f'_{1} - d_{1}) ~ \alpha$$ = (43)

$$C_{0,0,{\rm af}} \frac{1}{16} \left( \frac{y_{1}}{f'_{1}} \right)^{2} \; \frac{d_{1}^{2}}{f'_{1}-d_{1}}(1+b_{{\rm s}2})$$ = (44)

$$C_{1,0,{\rm af}} \frac{1}{4} \left( \frac{y_{1}}{f'_{1}} \right)^{2} \bigg[\alpha \, (2f'_{1} - d_{1}) ~$$ =

$$+ ~ \delta \, \left( 1+\frac{d_{1}}{2(f'_{1} - d_{1})} (1+b_{{\rm s}2}) \right) \bigg]$$ (45)

$$C_{2,0,{\rm af}} \frac{1}{4} \left( \frac{y_{1}}{f'_{1}} \right)^{2} \bigg[~ \alpha^{2} ~ (f'_{1} - d_{1}) + ~ \alpha \delta ~$$ =

$$+ ~ \delta^{2} ~ \frac{1+b_{{\rm s}2}}{4(f'_{1} - d_{1})} \bigg].$$ (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.

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

This gives, using the Eqs. (31) and (32),

 

$$2L \; \frac{m_{2}-1}{m_{2}+1} \cdot$$ z = (48)

$$\cdot \frac{1}{m_{2}-1-(m_{2}+1)b_{{\rm s}2} - \frac{s_{{\rm pr}1}}{f'}\frac{4}{(m_{2}+1)^{2}d_{1}^{2}} (-f'\zeta + L\xi)}\cdot$$

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.

$$z = 2L\; \frac{m_{2}-1}{m_{2}+1} \; \frac{1}{m_{2}-1-(m_{2}+1)b_{{\rm s}2}}\cdot$$ (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.

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

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

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

with

   

B₀ = C₀ (52)

$$-\frac{1}{4}\left(\frac{y_{1}}{f'}\right)^{2} \; \frac{z}{d_{1}} \; \bigg[ \frac{f'd_{1}}{L}\frac{(m_{2}+1)^{3}}{m_{2}-1}b_{{\rm s}2}$$ B₁ =

$$+ ~ \frac{s_{{\rm pr}1}}{f'} \bigg( f' + 2d_{1}\xi$$

$$+ 2\frac{2L+(m_{2}+1)d_{1}}{(m_{2}-1)L}(-f\zeta+L\xi) \bigg) \bigg]$$ (53)

$$\frac{1}{4} \left( \frac{y_{1}}{f'} \right)^{2} \; \left( \frac{z... ...ac{d_{1}^{2}}{L} \left( \frac{m_{2}+1}{m_{2}-1} \right)^{2} b_{{\rm s}2} \, \xi$$ B₂ =

$$+ ~ \frac{s_{{\rm pr}1}}{f'} \frac{2d_{1}}{L} \; \left( \frac{m_{2}+1}{m_{2}-1} \right)^{2} b_{{\rm s}2} (-f\zeta+L\xi)$$

$$+ ~ \left( \frac{s_{{\rm pr}1}}{f'} \right)^{2} (-f\zeta+L\xi)$$

$$~~~~~~~~~ \left( \frac{4}{(m_{2}-1)^{2}(m_{2}+1)} \frac{-f\zeta+L\xi}{L} - 1 \right) \bigg].$$ (54)

Equations (52), (53) and (54) show that the parameters B_i 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

$$c_{\rm ast} = B_{0} u_{{\rm pr}1}^{2} + B_{1}^{(\delta)} u_{{\rm pr}1}\delta + B_{2}^{(\delta)}\delta^{2}.$$ (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

   

$$\frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \; \left[ \frac{f'}{L}\, (f'+d_{1}) + \frac{d_{1}^{2}}{L}\xi \right]$$ B_0,0 = (56)

$$-\frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \; \frac{f'd_{1}}{L}\frac{(m_{2}+1)^{3}}{m_{2}-1}b_{{\rm s}2}$$ B_1,0 = (57)

$$\frac{1}{4} \left(\frac{y_{1}}{f'}\right)^{2} \; \frac{d_{1}^{2}}{L} \left( \frac{m_{2}+1}{m_{2}-1} \right)^{2} b_{{\rm s}2} \, \xi .$$ B_2,0 = (58)

Equations (52), (53) and (54) reduce to

   

$$B_{0,0} ~ - ~ \frac{s_{{\rm pr}1}}{y_{1}} c_{{\rm coma},{\rm cen}}^{\ast}$$ B₀ = (59)

$$\frac{z}{d_{1}} \left( B_{1,0} ~ + ~ \frac{s_{{\rm pr}1}}{y_{1}} c_{{\rm coma},{\rm cen}}^{\ast} \right)$$ B₁ = (60)

$$\left( \frac{z}{d_{1}} \right)^{2} B_{2,0}.$$ B₂ = (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

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

For a classical or aplanatic afocal telescope the coma-free point is in the focus of M1. The astigmatism parameters are given by

   

$$\frac{1}{4} \left(\frac{y_{1}}{f'_{1}}\right)^{2} \; (1+b_{{\rm s... ...frac{d_{1}^{2}} {f'_{1}-d_{1}} ~ + ~ s_{{\rm pr}1}\frac{d_{1}}{2f'_{1}} \right]$$ B₀ = (63)

$$-\frac{1}{4}\left(\frac{y_{1}}{f'_{1}}\right)^{2} \; \frac{1}{1-b_{{\rm s}2}} \; \bigg[ 2f'_{1}b_{{\rm s}2}$$ B₁ =

$$+ ~ s_{{\rm pr}1}\frac{f'_{1}-d_{1}}{f'_{1}} (1+b_{{\rm s}2}) \bigg]$$ (64)

$$\frac{1}{4} \left( \frac{y_{1}}{f'_{1}} \right)^{2} \; \frac{1+b_{{\rm s}2}}{(1-b_{{\rm s}2})^{2}}\, b_{{\rm s}2} \; (f'_{1} - d_{1}).$$ B₂ = (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.

• Stop at the primary mirror
  For a spherical secondary ( $b_{{\rm s}2} = 0$) the Eqs. (53) and (54) show that one has, as for a centered system, B₁=B₂=0. Because of z=2f'₂ the coma-free point is at the center of curvature of M2. Since a rotation of a spherical mirror around its center of curvature does not change the optical characteristics of the telescope, the astigmatism pattern has to be the same as the one of the centered system.
  If $\xi=0$, one has B₂=0, that is one node is at the position of the image corresponding to the field center.
• Stop at the secondary mirror
  If one uses the substitution $s_{{\rm pr}1}/f = -d_{1}/L$ in Eq. (54) one can show that B₂ is proportional to $\zeta$. Therefore, for a coma-free schiefspiegler with a parabolic primary mirror one node is at the position of the image corresponding to the field center.

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:

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

We now discuss a few telescope types.

• Classical telescope
  The primary mirror is parabolic and $\xi=0$. Therefore B₂ vanishes. The consequence of this is that one of the two nodes is exactly at the image position for an object on the axis of the primary mirror, that is with $u_{{\rm pr}1} = 0$. For the limit of large magnifications one has $b_{{\rm s}2} \rightarrow -1$ and therefore from Eq. (66) independently of the stop position:
  

  $$\rightarrow \frac{1}{4} \frac{y_{1}^{2}}{L}$$ B₀

  $$\rightarrow \frac{1}{4} \frac{y_{1}^{2}}{f'_{1}}$$ B₁

  $$\frac{B_{1}}{B_{0}} \rightarrow \frac{L}{f'_{1}}\cdot$$ (67)

  
  
  Since the distance L between the vertex of M2 and the focus will generally be similar to the focal length of M1 the ratio B₁/B₀ will be close to -1.
• Ritchey-Chretien telescope
  All three parameters are independent of the stop position, since, in addition to spherical aberration, coma is zero as well. Since modern Ritchey-Chretien telescopes with large values of m₂ are very close to classical telescopes, the conclusions drawn for classical telescopes will be approximately valid. That is, B₂ will be much smaller than B₀ and B₁, a point also discussed by Wilson ([1999], Sect. 2.2.1), and the ratio B₁/B₀ will be close to -1.
• Telescope with a spherical secondary mirror (Dall-Kirkham)
  Because of $b_{{\rm s}2} = 0$ the parameter B₂ is zero and B₁ is only proportional to the coma of the telescope. One node is therefore at the image position corresponding to the field center with $u_{{\rm pr}1} = 0$. If, in addition, the stop is at the primary mirror, B₁ is also zero. In that case the two nodes coincide and one has pure quadratic astigmatism with the center on the axis of the primary mirror.
  For the limit of large magnifications one has from Eq. (66):
  

  $$\begin{array}{@{}l l c c} & & {\rm Stop \; at \; M1} & {\r... ...0} & \;\;\; \displaystyle{ -4\frac{L}{f'} }\cdot \end{array}$$ (68)

  
  
  Whereas B₁ converges to values independent of m₂, B₀ is proportional to m₂. The binodal nature of the field astigmatism diminishes therefore linearly with increasing magnification of the telescope. For a certain stop position between the primary and the secondary mirror the coefficient B₀ of quadratic astigmatism vanishes. For the VLT with the Nasmyth focus the stop would be $3.9\, {\rm m}$ in front of the secondary mirror. The coefficient of linear astigmatism would be small, $B_{1} \approx 22\, \mu {\rm m}/{\rm deg}$, but the coefficient of field coma would be very large, $c^{\ast}_{{\rm coma},{\rm cen}} \approx 660\, \mu {\rm m}/{\rm deg}$.
• Telescope with a spherical primary mirror
  For large magnifications we have in the limit $m_{2} \rightarrow \infty$ $b_{{\rm s}2} \rightarrow m_{2}f'_{1}/L$ and then from Eq. (66):
  

  $$\begin{array}{@{}l l c c} & & {\rm Stop \; at \; M1} & {\r... ...isplaystyle{ -4\left( \frac{L}{f'} \right)^{2} } \end{array}$$ (69)

  
  
  B₁ converges to values independent of m₂. The limit values of B₀ are by a factor m₂f'₁/L larger than the ones for a telescopes with a spherical secondary. The binodal nature of the field astigmatism diminishes therefore quadratically with increasing magnification of the telescope.

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.

 

$$+71.788 \;\mu {\rm m}/{\rm deg}^{2}$$ B₀ =

$$-84.784 \;\mu {\rm m}/{\rm deg}^{2}$$ B₁ =

$$+0.06630 \;\mu {\rm m}/{\rm deg}^{2}.$$ B₂ = (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:

 

$$+86.614 \;\mu {\rm m}/{\rm deg}^{2}$$ B₀ =

$$-87.167 \;\mu {\rm m}/{\rm deg}^{2}$$ B₁ =

$$+0.06630 \;\mu {\rm m}/{\rm deg}^{2}.$$ B₂ = (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.