A Schiefspiegler toolkit
http://atm.udjat.nl/articles/schief_kit.html
A Schiefspiegler toolkit By: Arjan te Marvelde In the 1958 Sky Publishing Publishing publication publication (Bull ( Bulletin etin A: G leani leanings ngs fo r ATMs), ATMs ), Anton Kutter presented a set of design guidelines for a two mirror type of tilted component telescope. This summary of the original article only covers the essential equations, necessary for deriving your own design. The strength of this 'Schiefspiegler 'Schiefspiegl er toolkit' toolkit' is is that it quickly provides ball-park dimensions and performance of a design. This design can be tuned afterwards with a ray-tracing optical design program. In contrast with the original article, the toolkit starts with the general schiefspiegler design and subsequently derives the anastigmatic and coma-free cases. The general design presented in the original is catadioptic, with a spherical concave primary, a spherical convex secondary mirror and a planconvex lens in the final lightcone of the system. The original also describes some more exotic variations, using a warped or toroidal secondary or a more complex corrector lens, but these are not discussed in this summary. The complete toolkit has been programmed into a spreadsheet, which is available from the download page of this website.
Overview of the Schiefspiegler. Parameters: F
Effective focal length
Γ
Fo cal plane inclination inclination
ƒ1
Primary mirror focal length
ƒ2
Secondary mirror focal length
y 1
Primary mirror lightcone lightcone radius
y 2
Secondary mirror lightcone radius
y' 1
Primary mirror radius
y' 2
Secondary mirror radius
φ1
Primary mirror inclination inclination
φ2
Secondary mirror inclination
∆
Secondary offset
ƒ3
Corrector focal length length
∆'
Primary Primary o ffset
y 3
Corrector lightcone radius
e
Mirror separation
y' 3
Corrector lens radius
p
Primary residual cone length
φ3
Corrector inclination
p'
Effective cone length
a1m Corrector to focus distance distance
γ
Variation-angle
s
Corrector t o seco ndary distance distance
ξ
Residual astigmatism [rad]
β
Residual coma [rad]
The drawing suggests that the normal on the secondary vertex is parallel to the direction of incoming light, but this is only roughly the case. Also, the secondary will in general be larger than the width of the lightcone. The primary diameter and the lightcone diameter at the primary are approximately the same.
General equations Before going into specific solutions of Schiefspieglers, the basic set of equations dictating the dimensions is given below. This set of equations can be used as a toolbox for calculating designs of this type of TCT.
Basic design parameters
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A Schiefspiegler toolkit
http://atm.udjat.nl/articles/schief_kit.html
The basic dimensions are taken from the set of equations that describe a Cassegrain system. Starting from given values for F , ƒ 1 and y 1 the following equations can be used to calculate the remaining values:
Magnification:
Primary residual cone length:
Mirror separation:
Effective cone length:
Secondary focal length:
Lightcone radius on the secondary:
Back focal length:
Focal ratio:
Secondary offset:
Primary inclination:
The parameter d represents the additional space required by larger secondary and tube diameter. Back focal length ( b) can be taken smaller when construction allows, this improves correction while conserving tubelength.
Residual astigmatism The equation for calculation of the residual astigmatism in the system consists of three parts, representing the three optical coponents of the system. For catoptic designs the third part (for the corrector lens) can be omitted (since ƒ3 is infinite).
where:
Residual coma As for residual astigmatism, the equation for calculation of the residual coma consists of three parts representing the three optical coponents of the system. Fo r catoptic designs the third part (fo r the corrector lens) can be o mitted (since ƒ3 is infinite).
where:
Position of corrector lens In case a correctorlens is used, the following formula determines its position:
where the differential effective cone length is given by:
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A Schiefspiegler toolkit
http://atm.udjat.nl/articles/schief_kit.html
and the differential system focal length is given by:
Note that the term ξ is the residual astigmatism of the t wo mirror system to be corrected! The parameters F m and p' m for the syst em meridional focal length and effective cone length, can be substituted with the system values F and p' or (better) derived with:
and
Anastigmatic design The anastigmatic design is optimized for zero astigmatism on the paraxial focus. Such anastigmatic designs can be constructed with apertures of up to 150mm. Larger apertures yield telescopes that are exceedingly long and impractical in their use. With the condition of z ero astigmatism (ξ =0) and omitting the term for the corrector lens, the following equation can derived from the equation of residual astigmatism:
When the focal lengths of both mirrors are equal, this equation simplifies to:
The primary offset parameter, essential for building the system, is given by:
Finally, the actual performance of the system is approximated with the formula for resiual coma (in radians), where the third term has been omitted:
The visible coma is approximately one third of this value. Some examples of anastigmatic designs, derived with these formulae (dimensions are in mm): 80mm F/20
110mm F/25
150mm F/20
150mm F/29
F
1600
2720
3000
4300
ƒ1
960
1620
1800
2550
80
110
150
150
1000
1620
1800
2720
40
55
70
70
e
540
915
1013
1443
p'
700
1185
1312
1867
∆
59
81
109
109
∆'
136
185
247
259
β
4.7"
2.5"
4.6"
1.7"
2y' 1 ƒ2 2y' 2
As can be seen, the paraxial residual coma decreases with decreasing focal ratio. That the focal ratio has to decrease with increasing primary diameter is due to the decreasing size of the airy disk. The visible coma is approximately 1/3 of what is calculated above, but even then the coma of the 150mm F/20 example is more than twice the theoretical resolution of a 150mm telescope. Off axis the values are worse, the magnitude of optical aberrations away from the optical axis should be examined with ray tracing methods. However a boundary approximation can be made when the angular radius of the desired field of view is added to/subtracted from the secondary inclination φ2.
Coma-free design By increasing φ2, as compared to the anastigmatic design, the coma can be cancelled completely. Obviously this will go at the cost of increased astigmatism. The condition of coma-free design is derived from the equation for residual coma:
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A Schiefspiegler toolkit
http://atm.udjat.nl/articles/schief_kit.html
The primary offset parameter, essential for building the system, is given by:
The residual astigmatism of this system is given by:
Astigmatism is more visible than coma, so for a telescope of equal dimensions preference should be given to the anastigmatic design. However the best overall images/Images, in a design using spherical mirrors and no corrector lens, will be obtained with φ2 increased somewhat in the direction of zero coma.
Catadioptic design The basis for the catadioptic design is also with φ2 somewhere between the anastigmatic and coma free boundary cases. The residual aberrations can then almost be eliminated by using an inclined plan-convex lens with the flat side facing the secondary mirror. According to A. Kutter, the proper value for φ2 is obtained by, starting with the coma-free case and decreasing ∆' to approximately 80%, so that the values of the paraxial residual aberrations ( ξ and β) are reduced to half of those in the boundary cases. The plan-convex lens that should be used has a focal length of approximately:
This value is not very c ritical, but w ill determine the inclination at which it sho uld be used. In a practical application t he inclination and po sition along the optical path and in sagittal direction should be adjustable, to be able to fine-tune the correction. The radius of the lightcone at the corrector lens is determined as follows:
Once all telescope dimensions are calculated, including the position of the corrector lens, the corrector inclination φ3 can be derived from the equations for residual astigmatism and coma, setting ξ and β equal to 0 . Finally, the required radius of the corrector lens follows from:
Example calculation So let's design an anastigmatic F/23 Kutter telescope, with an effective focal length of 3000mm. It's 32mm field of view corresponds with 0.6° (slightly larger than the moon): a field lens of 26mm diameter will give about half a degree. From the magnification factor value of 5/3 the primary focal length can be calculated: 1800mm. The secondary focal length can be taken identical, and its' diameter can be estimated to be roughly half of the primary diameter. This value is rounded up to allow for some constructional play (aiming at using a standard rainpipe...): 75mm. With the set of basic equations the design parameters are derived in the following order: Effective focal length (F)
3500 mm
Primary focal length (f 1)
2100 mm
Primary diameter (2y'1)
130 mm
Secondary focal length (f 2)
2100 mm
Secondary diameter (2y'2)
75 mm
Magnification (A)
1.667 X
Back Focal Length (b)
200 mm
Residual primary cone length (p)
863 mm
Mirror separation (e)
1238 mm
Secondary cone radius (y 2)
26.7 mm
Effective cone length (p')
1438 mm
Then, applying the equations for the anastigmatic design, the final parameters follow: Mirror offset (∆)
103 mm
Primary inclination ( φ1)
2.376 °
Secondary inclination (φ2)
5.792 °
Primary offset (∆')
249 mm
The paraxial residual coma β can now be calculated: 2.1". Between the extremities of the field of view in sagittal direction, residual coma ranges from 2.3" to 1.9", while the residual astigmatism ranges from 1.1 to -1.1". The visible coma will be approximately 0.7". For comparison, the airy disk for an aperture of 130mm and 550nm light has a diameter of about 2.1". After tuning the design in MODAS, with the following parameters:
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Surface#
Type
Height
Radius
Eccentricity
Thickness
Glass
Tilt-X
Tilt-Y
Tilt-Z
1:Obj
SPHRSURF
65
-4200
0
0
AIR
2.376
0
0
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A Schiefspiegler toolkit
http://atm.udjat.nl/articles/schief_kit.html
2:MST
SPHRSURF
35
-4200
0
-1239
AIR
-5.792
0
0
3:Img
FLATSURF
15
0
0
1459
AIR
0
0
0
The resulting spot diagram:
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