Up: Analytical expressions for field
Subsections
2 Astigmatism in a misaligned telescope with an arbitrary pupil
position

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.
y_{i} 
: 
semidiameter of the mirror i 
f'_{i} 
: 
focal length of the mirror i 

: 
distance from the surface to the entrance 


pupil of
mirror i 

: 
aspheric constant of the mirror i 
n'_{i} 
: 
index of the exit medium of mirror i 

: 
angle of the incoming principle ray with 


the axis of the mirror i 
h_{i} 
: 
distance from the axis of the mirror i to 


the center of its entrance pupil 
z 
: 
distance of the comafree point from the 


surface of M2. 
In this whole section the field center is
defined as the optical axis of M1, that is
.
The stop is
located between the two mirrors at a distance
from M2 and
decentered laterally by a value h from the M1 axis and h_{2} from
the M2 axis. The entrance pupil is located at a distance
behind M1 and decentered by h_{1} from the M1 axis. The vertex of
the secondary
mirror is decentered by a distance
from the M1 axis and
tilted by an
angle
with respect to the M1 axis.
Initially we will only make the assumption that the lateral decenter
and the rotation of M2 around its vertex by
are
in the same plane.
After a bit of geometry following Fig. 1 we find:

= 

(3) 
h_{1} 
= 

(4) 
h_{2} 
= 

(5) 
Knowing the distance
from M2 to the stop in a plane between
M1 and M2 one can calculate
by

(6) 
Inversely,
is given by

(7) 
The coefficient of the astigmatic wavefront
aberration of a two mirror telescope is then given by
(Schroeder [1987], page 79)

= 
D_{1} y_{1}^{2} + D_{2} y_{2}^{2} 
(8) 
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)
A_{0,i} 
= 

(10) 
A_{1,i} 
= 

(11) 
A_{2,i} 
= 

(12) 
Introducing the expressions for A_{0,i}, A_{1,i} and A_{2,i}into Eq. (9) one gets
D_{i} 
= 





(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.
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 
= 



= 

(14) 
With this expression for h one gets for the Eqs.
(3), (4) and (5)

= 

(15) 
h_{1} 
= 

(16) 
h_{2} 
= 

(17) 
Introducing these expressions into Eqs. (13) and
(8) and using the
relationships given by Wilson ([1996], Sect. 2.2.5.2)
m_{2} 
= 

(18) 
f' 
= 

(19) 
f'_{2} 
= 

(20) 
L 
= 

(21) 
y_{2} 
= 

(22) 
where m_{2} 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

= 

(23) 
C_{0} 
= 









(24) 
C_{1} 
= 





















(25) 
C_{2} 
= 





















(26) 
where
The expression for C_{0} 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 C_{0}, C_{1}and C_{2} should not depend on the asphericity
of the
primary mirror. This can be seen from the Eqs. (24),
(25) and (26), which then depend only on the
asphericity
of the secondary mirror. Similarly, if the stop
is at the secondary mirror, the parameters C_{0}, C_{1} and
C_{2} should not depend on the asphericity of the secondary
mirror. This can easily be verified by introducing
or,
equivalently, from
Eq. (6),
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):
of
spherical aberration,
of third order field coma of a
centered system, and the coefficients of field independent
third order coma generated by a lateral decenter by (
)
and by
a pure rotation of M2 around its vertex by (
).

= 

(29) 

= 





(30) 

= 









(31) 

= 

(32) 
With the further definitions

(33) 
C_{0,0} 
= 

(34) 
C_{1,0} 
= 





(35) 
C_{2,0} 
= 





(36) 
Equations (24), (25) and (26) can
be written as
C_{0} 
= 

(37) 
C_{1} 
= 
C_{1,0} 








(38) 
C_{2} 
= 





(39) 
These equations show a nice symmetry of the stopshift 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 C_{1} is
the sum of the coma for a centered system for a principal ray with the
angle
,
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
and a pure
rotation of M2 around its vertex by .
The total
coma in the linear coefficient of C_{2} contains only the two latter
contributions.
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}, d_{1} and m_{2} with the Eqs. (19)
and (21) and then let m_{2} go to infinity.
This gives

= 





(40) 

= 





(41) 

= 





(42) 

= 

(43) 

= 

(44) 

= 





(45) 

= 





(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 comafree
schiefspiegler
A further simplification is possible if the telescope is
corrected for coma at the center of the field, that is for
.
The axes of M1 and M2 must then intersect at the
comafree point
.
The distance from the vertex of M2 to
is denoted by z. The lateral decenter
and the misalignment
angle
are then related by
.
z can be calculated from the requirement that the contributions to
decentering coma from a pure lateral decenter
and the
simultaneous rotation
cancel.

(47) 
This gives, using the Eqs. (31) and
(32),
z 
= 

(48) 




This equation shows that the position of the comafree 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 m_{2}, only on the aspheric
constant
of the secondary mirror.

(49) 
If the stop is at the secondary mirror z depends, for a given
telescope geometry L and m_{2}, only on the
asphericity
of the primary mirror.

(50) 
The coefficient
of third order astigmatism can then be
expressed as

(51) 
with
B_{0} 
= 
C_{0} 
(52) 
B_{1} 
= 









(53) 
B_{2} 
= 













(54) 
Equations (52), (53) and (54) show that
the parameters B_{i} are proportional to
(z/d_{1})^{i}.
At a first glance they seem to be linear or quadratic equations in
.
This is
only the case for B_{0} since the distance z of the
comafree point appearing in B_{1} and B_{2} depends
itself on the stop position, as can be seen from Eq.
(49).
By introducing
in the Eqs. (52),
(53) and (54)
the coefficient of third order astigmatism can be expressed as a
polynomial in
and ,
that is

(55) 
It is easy to see that now the parameter
is a linear function and the parameters B_{0} and
are quadratic functions in
.
Exactly as with C_{0}, C_{1} and C_{2} the parameters B_{0}, B_{1}and B_{2} depend only on the
asphericity
of M2, if the stop is at the primary mirror,
and only on the asphericity
of M1, if the stop is at the
secondary mirror.
Since the coupling between
and
through z involves
the expression for spherical aberration, the symmetry of the
stopshift terms has disappeared. But,
the telescope will usually be corrected for spherical aberration.
In this case all terms containing
vanish.
Then, with the additional definitions
B_{0,0} 
= 

(56) 
B_{1,0} 
= 

(57) 
B_{2,0} 
= 

(58) 
Equations (52), (53) and
(54) reduce to
B_{0} 
= 

(59) 
B_{1} 
= 

(60) 
B_{2} 
= 

(61) 
The distance z from the secondary mirror to the comafree point and
the expression
are
now no longer dependent on the stop position. Therefore, B_{0} and
B_{1} are linear functions of the stop position and B_{2} is
independent of the stop position.
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 m_{2} go to infinity.
The distance from M2 to the comafree point is then

(62) 
For a classical or aplanatic afocal telescope the comafree point is in the focus of M1.
The astigmatism
parameters are given by
B_{0} 
= 

(63) 
B_{1} 
= 





(64) 
B_{2} 
= 

(65) 
For a Mersenne telescope with
one obtains
immediately
B_{0} = B_{2} = 0. This gives therefore pure linear
astigmatism which is proportional to the misalignment angle 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.
For a general comafree 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 (
)
the Eqs. (53) and
(54) show that one has, as for a centered system,
B_{1}=B_{2}=0.
Because of z=2f'_{2} the
comafree 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 ,
one has B_{2}=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
in Eq.
(54)
one can show that B_{2} is proportional to .
Therefore, for a
comafree schiefspiegler with a parabolic primary mirror one node is
at the position of the image corresponding to the field center.
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
together with a
finite distance L from the secondary mirror
to the focus. The semidiameter y_{2} of M2, its radius
of curvature 2f'_{2} and the difference between f'_{1} and d_{1}will go to zero. This is different from the
case of an afocal telescope, where L goes to infinity, while
y_{2}, 2f'_{2} and
f'_{1}d_{1} remain finite. For the limit case
with finite L we get from Eqs. (56) to (58) the
following expressions for the astigmatism parameters:

(66) 
We now discuss a few telescope types.
 Classical telescope
The primary mirror is parabolic and .
Therefore B_{2} 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
.
For the limit of large magnifications one has
and therefore
from Eq. (66) independently of the stop position:
Since the distance L between the vertex of
M2 and the focus will generally be similar to the
focal length of M1 the ratio
B_{1}/B_{0} will be close to 1.
 RitcheyChretien telescope
All three parameters are independent of the stop position, since,
in addition to spherical aberration, coma is zero as well.
Since modern RitcheyChretien telescopes with large
values of m_{2} are very close to classical telescopes, the
conclusions drawn for classical telescopes will be approximately
valid. That is, B_{2} will be much smaller than
B_{0} and B_{1}, a point also discussed by Wilson
([1999],
Sect. 2.2.1), and the ratio
B_{1}/B_{0} will be close to 1.
 Telescope with a spherical secondary mirror (DallKirkham)
Because of
the parameter B_{2} is
zero and B_{1} is only proportional to the coma of the
telescope. One node is therefore at the image position
corresponding to the field center with
.
If, in
addition, the stop is at the primary mirror, B_{1} 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):

(68) 
Whereas B_{1} converges to values independent of m_{2},
B_{0} is proportional to m_{2}. 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
B_{0} of quadratic astigmatism vanishes. For the VLT with the
Nasmyth focus the stop would be
in front of the
secondary mirror. The coefficient of linear astigmatism would be
small,
,
but the coefficient of
field coma would be very large,
.
 Telescope with a spherical primary mirror
For large magnifications we have in the limit
and then from Eq. (66):

(69) 
B_{1} converges to values independent of m_{2}. The
limit values of B_{0} are by a factor
m_{2}f'_{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.
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

2f'_{2} 
4553.561


0.996962 

1.66926 
y_{1} 
4057.50

y_{2} 
556.55

d_{1} 
12426.946



z 
1997.995



L 
14926.950



m_{2} 
7.556 


If the pupil was located at the primary mirror, we would have
.
This would give the following numerical values for the
parameters B_{0}, B_{1} and B_{2}.
B_{0} 
= 


B_{1} 
= 


B_{2} 
= 

(70) 
As has been discussed before, the value of B_{2} is effectively
negligible compared with the values of B_{0} and B_{1} and the
ratio
B_{1}/B_{0} is approximately equal to 1.
In reality the pupil is located at the secondary mirror. Then
and from Eq. (6) one gets
.
This gives the following numerical values for
the parameters:
B_{0} 
= 


B_{1} 
= 


B_{2} 
= 

(71) 
Since the spherical aberration is zero the value of B_{2} 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 B_{0} for the two stop
positions.
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