A Schiefspiegler toolkit

By: Arjan te Marvelde

In the 1958 Sky Publishing publication (Bulletin A: Gleanings for 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 toolkit' 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:

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

The basic dimensions are taken from the set of equations that describe a Cassegrain system. Starting from given values for F, ƒ₁ and y₁ 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. For catoptic designs the third part (for the corrector lens) can be omitted (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:

and the differential system focal length is given by:

Note that the term ξ is the residual astigmatism of the two mirror system to be corrected! The parameters F_m and p'_m for the system 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 zero 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):

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:

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 Δ' 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 critical, but will determine the inclination at which it should be used. In a practical application the inclination and position 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 (f1)	2100 mm
Primary diameter (2y'1)	130 mm
Secondary focal length (f2)	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 (y2)	26.7 mm
Effective cone length (p')	1438 mm

Then, applying the equations for the anastigmatic design, the final parameters follow:

Mirror offset (Δ)	103 mm
Primary inclination (φ1)	2.376 °
Secondary inclination (φ2)	5.792 °
Primary offset (Δ')	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:

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
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: