A Schiefspiegler toolkit

Arjan te Marvelde, Feb 2013

In a 1958 publication of Sky Publishing (Bulletin A: Gleanings for ATMs), Anton Kutter presented a set of design principles for a two-mirror type of tilted component telescope.
An updated version in PDF can be opened here.

This article sumarizes the original article and mainly covers the essential mathematics needed for deriving your own design. It contains is a lot of math in the beginning, but there are some worked out examples at the end of the article.
This 'Schiefspiegler toolkit' is an Excel sheet that it quickly provides ball-park dimensions and performance figures of a target design. Since the anastigmatic case is the most widely used variant a simplified version, only solving the anastigmatic design, can be found on the second sheet. This design then is best refined by means of a ray-tracing optical design program. There is a free version of OSLO, which goes up to 10 surfaces and has some limits on analysis tools which do not harm the amateur. An OSLO-LT lens file is provided that can be used as a starting point for further optimization of your Kutter design. You may also want to download WinSpot to evaluate the spot diagrams.

In contrast with the original publication, this article starts with the general equations for a Kutter schief and subsequently derives the anastigmatic and coma-free cases. The general design is of a catadioptric system, containing a spherical concave primary mirror, a spherical convex secondary mirror and a plan-convex lens in the final lightcone of the system. Kutters' article also describes some more exotic variations, using a warped or toroidal secondary or a more complex corrector lens, but these will not be discussed here.

Finally, some further design considerations are given, together with some optimized examples.


Original drawing in Kutters' article


Parameter definitions

The Kutter telescope design is ultimately based upon a Cassegrain layout. As shown, it can be considered a cut-out of a relatively large Cassegrain system, with only spherical mirrors. For this reason some of the equations are identical to those applying to the Cassegrain.

 
Cassegrain as basic system.

The detailed Kutter system layout presents some more variables used further down in the text. The drawing suggests that the normal vector on the secondary vertex is parallel to the direction of incoming light beam. In reality this is only roughly the case and the secondary tilt will depend on the type of system. Also, the calculations are based on local beamwidth of teh paraxial ray only. Taking into account the field of view, the secondary in reality needs to be somewhat larger than the width of the lightcone.


Parameters used in calculations.

F Effective focal length

Γ Focal plane inclination
1 Primary mirror focal length 2 Secondary mirror focal length
y1 Primary mirror lightcone radius y2 Secondary mirror lightcone radius
y'1 Primary mirror radius y'2 Secondary mirror radius
φ1 Primary mirror inclination

φ2 Secondary mirror inclination
Δ Secondary offset 3 Corrector focal length
Δ' Primary offset y3 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
γ Variation-angle s Corrector to secondary distance

ξ Residual astigmatism [rad] β Residual coma [rad]


General equations

Before going into specific solutions of Schiefspieglers, the basic set of equations dictating the dimensions is given. This set of equations will then be used as a toolbox for evaluation of specific designs of this type of tilted component telescope (TCT).

Basic design equations

The basic dimensions are taken from the set of equations that describe a Cassegrain system, since this is what a two-mirror schief essentially is. Starting from given values for F, 1 and y1 the following equations can be used to calculate the remaining parameters:

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 in the secondary offset 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. Kutter recommends approximately one 6th of e.
Note that the magnification A should be around 5/3.

Now that the system has been dimensioned, we will have a look at the remaining abberations in the focal plane. These equation will then be used in the strategies to minimize these abberations.

Residual astigmatism

The equation to calculate the residual astigmatism consists of three parts, representing the contributions of the three optical components in the system. For catoptic designs the third part, representing the corrector lens, can be omitted (since 3 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. Again, for catoptic designs the third part (for the corrector lens) can be omitted (since 3 is infinite).

where:

Position of corrector lens

In case a correctorlens is used, the following formula determines its position (refer to the reference design):

where the differential effective cone length is given by:

and the differential system focal length is given by:

The parameters Fm and p'm (the system meridional focal length and effective cone length) can simply be substituted with the system values F and p' or (better) derived with:

and

Image plane tilt

The image plane will inherently be tilted, or inclined to use Kutters' words. This tilt is roughly equal to the difference between φ2 and φ1. The exact formula to evaluate image tilt:

where r2 is the radius of curvature of the secondary mirror.

Anastigmatic design

Now let's have a closer look at the anastigmatic design, which is optimized for zero astigmatism on the paraxial focus. Such anastigmatic designs can be constructed with apertures of up to 150mm. Larger apertures, without using a corrector lens, 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 further simplifies to:

The primary offset parameter, determining the system physical dimensions, is given by:

Finally, the actual performance of the system is approximated with the formula for resiual coma (in radians), where again the third term has been omitted:

The coma that will be actually visible 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

System F

1600

2720

3000

4300

Primary 1

960

1620

1800

2550

2y'1

80

110

150

150

Secondary 2

1000

1620

1800

2720

2y'2

40

55

70

70

e

540

915

1013

1443

Cone p'

700

1185

1312

1867

Δ

59

81

109

109

Δ'

136

185

247

259

Coma β

4.7"

2.5"

4.6"

1.7"

Airy disk

3.5"

2.5"

1.8"

1.8"

As can be seen, the paraxial residual coma β decreases with increasing (slower) focal ratio. At some point the coma equals the size of the airy disk. Assuming that the visible coma is approximately 1/3 of β as calculated above, the case of the 150mm F/20 example would have barely acceptable optical performance.

Off-axis the values are a bit worse, and also the effect of image plane tilt is not taken into account. The magnitude of optical aberrations away from the optical axis can be quickly estimated in the toolkit by varying the angle φ2 with a quarter of the FoV angle. Ultimately however, the performance should be examined with ray tracing methods.

Coma-free design

Starting with an anastigmatic design and then increasing φ2 the coma will be cancelled completely. Obviously this will go at the cost of increased astigmatism. The condition for the coma-free design is derived from the equation for residual coma:

The primary offset parameter, needed for building the system, is given by:

The residual astigmatism of this system is given by:

Astigmatism is more disturbing than coma, so for a two-mirror telescope of equal dimensions preference should be given to the anastigmatic design. However, the best overall performance in a design using spherical mirrors and no corrector lens will be obtained with φ2 increased slightly with respect to the anastigmat.

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 be almost eliminated by inserting an inclined plan-convex lens with the flat side facing the secondary mirror.
According to Kutter, the proper value for φ2 is obtained by starting with the coma-free case and decreasing Δ' to approximately 80% towards the anastigmatic case, so that the values of both 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 in meridional direction and the position along the optical path should be adjustable, to be able to fine-adjust 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:

Note that a corrector can also be made by means of a pair of off the shelf lenses. These lenses are chosen so that focal lengths cancel each other but the difference in tilt angle provides the desired correction. For such designs, refer to the final section of this article.


A calculated example

Now the toolbox has been equipped with sufficient math, let's design an anastigmatic F/27 Kutter telescope with an effective focal length of 3500mm and an aperture of 130mm. The 32mm field of view corresponds with 0.6° (slightly larger than the moon), and a field lens of 26mm diameter will give about half a degree.

From the magnification factor of 5/3 the target primary focal length can be calculated: 2100mm. The secondary focal length is taken identical and the diameter can be estimated to be roughly half of the primary diameter. This value is rounded up to allow for the field of view 70mm. When using a standard 80mm PVC pipe as a secondary tube the additional room (d) can for example be set at 5mm.
Finally, as a reasonable initial value for the back focal length, 200mm is chosen.

These values are inserted in the schief-kit spreadsheet, to yield a first-order unoptimized design:

Effective focal length (Feff) 3500 mm
Primary focal length (Fpri) 2100 mm
Primary diameter (Dpri) 130 mm
Secondary focal length (Fsec) 2100 mm
Secondary diameter (Dsec) 70 mm
Additional room secondary (Psec) 5 mm
Back Focal Length (b) 200 mm
Magnification (A) 1.667 X
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

The schief-kit will calculate the required mirror inclinations for anastigmatic and coma-free cases. Starting from these values the design can be further optimized in the second pane. The table below shows the result from changing the angle φ2 :

Anastigmatic Coma-free Catadioptic
Mirror offset (Δ) 111 mm 111 mm 111 mm
Primary inclination (φ1) 2.565 ° 2.565 ° 2.565 °
Secondary inclination (φ2) 6.256 ° 9.610 ° 8.000 °
Primary offset (Δ') 268 mm 407 mm 341 mm
Coma (β) 2.3" 0.0" 1.1"
Astigmatism (ξ) 0.0" -17.2" -8.1"

From this can be concluded that the astigmatism changes quite rapidly, and hence solutions without corrector lens are best taken anastigmatic. Another conclusion is that the secondary tilt is fairly critical, and should receive sufficient attention in construction (i.e. the value of Δ' and collimation means).
For illustration the starting point for a catadioptic solution is added, where coma and astigmatism are at about half of their range. Now a lens can be inserted and its position and tilt optimized to achieve lowest total abberation. This lens usually has a very long focus, in the order of 20-60m. Note: the toolkit excel already estimates better values for the catadioptic design.

When staying with the anastigmatic solution, the paraxial residual coma β is 2.3". The variation in φ2 required to estimate the range of coma and astigmatism is plus or minus half the field radius (i.e. +/-0.13°). The residual coma lies in the range [2.4"; 2.2"] and the astigmatism range is [+0.5"; -0.5"]. This should be compared with the airy disk diameter of 2.1".

The anastigmatic design is now loaded in OSLO-LT, with the following parameters:
Surface Type Height RoC Thickness Tilt-Y
1: Primary Sphere 65 -4200 0 2.565
2: Secondary Sphere 35 -4200 -1238 -6.256
3: Image Flat 13 infinite 1438 0

The resulting spot diagrams correspond fairly well with the analytically found performance values.



Design considerations

When fine-tuning a design with OSLO-LT, it is worthwhile to check the field at both sides of the optical axis (the multi-spot diagram by default only shows one side). One way to do this, is by using the slider wheel from the optimization menu. You can define a number of sliders for parameters indezed per defined surface. Two particularly useful types are TLA (tilt) and TH (thickness). The TLA defines meridional rotation about the Y-axis (which sticks out of the paper). The TH defines the distance to the next surface. In the slider wheel design select the multi-spot option, and the output will be image plane spot on the optical axis as well as maximum fieldangle on both sides. What you can see now is that a single-sided multi-spot diagram looking fine may actually be completely off on the other side of the optical axis.

There are a number of design choices to be made, and it is therefore interesting to analyze the effect of those choices on the performance of the system. Putting them in order:

  • Primary and secondary focal lengths
  • Secondary tilt
  • Corrector
  • Primary correction

Primary/Secondary focal length

When the Primary and secondary focal lengths are equal, the Petzval field curvature is zero. This is still the case when a corrector is inserted with two opposite lenses. For example this would be a planconcave and a planconvex of equal (but opposite) focal length.
The radius of the Petzval surface for this case is, assuming glass index of 1.5:

All radii are subtituted as their absolute (positive) value. The Petzval condition for a flat image plane is achieved when Rp is infinite.

The 200mm F/20 prescription given by Kutter has a secondary with a slightly longer focal length than the primary (2530mm vs 2400mm), but it also has a single long focus PCX lens corrector.
To meet the Petzval condition when the Rpcv is infinite the Rpcx of the PCX corrector lens can be easily calculated. In Kutters' 200mm F/20 example Rpri=4800 and Rsec=5060, so Rpcx is approximately 15000. This is precisely what Kutter prescribes for this system.

When a dual lens corrector is used where Rpcv=Rpcx, Rpri and Rsec should be chosen equal as well. If you need to resort to off the shelf lenses of differing focal length, the Petzval condition could in principle be met by changing the secondary focal length to match.
Unfortunately this will go at the cost of increased chromatic abberation. Bottom line: A flat image plane is a nice goal, but in practise the curvature will never be much in this type of system.

Secondary tilt

The primary tilt angle φ1 is determined by the location of the secondary. Smaller primary tilt means larger primary to secondary separation, and hence also the image plane moves inward. Usually, for construction reasons, the image plane is located behind the primary (i.e. b>0).
The abberations caused by the primary tilt are compensated by varying the angle of the secondary, φ2. An optimum angle can be found that leaves the smallest on-axis spot after correction. This angle is found about halfway the anastigmatic and coma-free boundary cases. Where exactly the optimum is depends on the corrector, this can correct astigmatism and coma only to a certain extent. When in the OSLO-LT model the optimum correction is approached for a certain φ2, the spots above and under optical axis should be about equal in size. If not, φ2 must be adjusted until this is the case, after which the corrector again is optimized. This is in general also the configuration where the smallest on-axis spot will be achieved.

Larger secondary tilt results in larger image plane tilt, which is approximately given by φ21. This is a bad thing, because the off-axis spotsizes will appear as if inside and outside focus respectively. Larger image plane tilt obviously leads to larger deviation. For an image tilt of the apparent defocus is about 10% of the distance from the optical axis, i.e. 1mm for every 10mm off axis.
As stated, the required secondary tilt is in turn determined by the primary tilt. The primary tilt is determined by mirror focal ratio and mechanical constraints. The focal ratio cannot be decreased indefinitely, since the length of the system will grow beyond manageable. For optical performance the primary tilt should not be much larger than though, which consequently sets a limit to the aperture of Kutter systems.

Corrector

The corrector can compensate the residual coma and astigmatism for a certain combination of φ1 and φ2. Several types of corrector have been proposed, a single long-focus PCX lens, a set of meniscus lenses or a combination of PCV and PCX lenses. The choice here will be between the use of stock components or to specially make what is needed. Since the corrector is a critical element for larger aperture systems, it is probably best to start the design optimization from here.

Stock PCV and PCX lenses of sufficient diameter are obtainable up to about 1000mm focal length, for example from Melles Griot or Ross optical. Anti reflection coating is strictly not needed to prevent ghost images, because both lenses are used at an angle.

Correction of primary

The primary can be given a bit of parabolization in order to minimize the on-axis spot size. Kutter recommends a value of -0.55 for his 200mm F20 system. This enhanced on-axis behaviour however goes at the cost of increased off-axis coma.
In contrast, an all spherical system can deliver a zero-tilt image plane when the corrector is placed on the right location. This goes at the cost of an enlarged spot size, but this is almost uniform over the field of view. A field-wide Strehl value of more than 90% can be achieved this way.


Optimized examples

TBC

Below follows an OSLO-LT prescription of a 200mm F20 catadioptric Kutter design. You can cut and paste this in a .txt file save it and give it a .len extension.

// OSLO 6.1 64187 0 53853
LEN NEW "Kutter 200F20" 4074.4 10
EBR 100.0
ANG 0.2
DES "Udjat"
UNI 1.0
// SRF 0
AIR
TH 1.0e+20
AP 3.4906726816e+17
NXT // SRF 1
AIR
TH 1500.0
APN 1
AY1 A -110.0
AY2 A 110.0
AX1 A -110.0
AX2 A 110.0
ATP A 1
AAC A 4
NXT // SRF 2
RFH
TCE 0.0
RD -4800.0
TH -1365.0
AP 100.0
AST
CC -0.55
DT 1
TLA -3.336
BEN
NXT // SRF 3
RFH
TCE 236.0
RD -5060.0
TH 1200.0
DT 1
TLA 9.7
BEN
NXT // SRF 4
AIR
NXT // SRF 5
GLA N-BK7
TH 3.0
AP 25.0
DT 1
GC -1
TLA 0.85
NXT // SRF 6
AIR
RD 518.7000000000001
AP 25.0
NXT // SRF 7
GLA N-BK7
RD 518.7000000000001
TH 2.6
AP 25.0
DT 1
GC -3
DCZ 10.0
TLA 11.15
NXT // SRF 8
AIR
AP 25.0
NXT // SRF 9
AIR
DT 1
GC -5
NXT // SRF 10
AIR
TH 549.6502356402747
DT 1
TLA -8.881784197e-16
CBK 1
WV 0.58756 0.48613 0.65627
WW 1.0 1.0 1.0
END 10
DLFS 1
DLAP 3
DLRS 3
DLAS On
SDSA On



Disclaimer: There are no known bugs in the schief-kit. However, it is recommended to have a second opinion by means of another tool (like WinSpot or OSLO-LT), as shown in the worked out examples.