1. Coma-corrected Cross-Dragone Design
Richard Hills
June 8th 2017
Simons Observatory Optics Workshop

2. Classical Cassegrain telescopes
Primary is a Paraboloid
Secondary is a Hyperboloid
Used for many optical telescopes until mid 1900’s inch last!
Still used for many radio telescopes, incl JCMT & ALMA.
Suffers from
Blockage, which causes diffraction and multiple reflections.
Coma aberration, which limits the field of view.
Optical properties described by equivalent paraboloid: same D as primary but f-ratio of final focus.
June 8th 2017
Simons Observatory Optics Workshop

3. Classical Cassegrain telescopes
Primary is a Paraboloid
Secondary is a Hyperboloid
Used for many optical telescopes until mid 1900’s inch last?
Still used for many radio telescopes, incl JCMT & ALMA.
Suffers from
Blockage, which causes diffraction and multiple reflections.
Coma aberration, which limits the field of view.
Optical properties described by equivalent paraboloid: same D as primary but f-ratio of final focus.
June 8th 2017
Simons Observatory Optics Workshop

4. Classical Cassegrain telescopes
Primary is a Paraboloid
Secondary is a Hyperboloid
Used for many optical telescopes until mid 1900’s inch last!
Still used for many radio telescopes, incl JCMT & ALMA.
Suffers from
Blockage, which causes diffraction and multiple reflections.
Aberrations, which limit the field of view.
Optical properties described by equivalent paraboloid: same D as primary but f-ratio of final focus.
June 8th 2017
Simons Observatory Optics Workshop

5. Classical Cassegrain telescopes
Primary is a Paraboloid
Secondary is a Hyperboloid
Used for many optical telescopes until mid 1900’s inch last!
Still used for many radio telescopes, incl JCMT & ALMA.
Suffers from
Blockage, which causes diffraction and multiple reflections.
Coma aberration, which limits the field of view.
Optical properties described by equivalent paraboloid: same D as primary but f-ratio of final focus.
June 8th 2017
Simons Observatory Optics Workshop

6. Simons Observatory Optics Workshop
Coma
For a parabolic mirror Coma is usually the largest aberration.
Spot size (in terms of angle) is proportional to field angle / (f-ratio)2.
June 8th 2017
Simons Observatory Optics Workshop

7. Coma in terms of wavefront
In terms of the position in the aperture (r , φ) the wavefront error, w is
w = α r 3 cosφ / 4f 2
where α is off-axis angle.
This arises because a parabola does not fulfil the Abbe sine condition relating r and θ.
We want w = α r = α f sinθ which means
that θ = sin-1(r/f) ≈ r/f + 1/6(r/f)3 …
For a parabola we have tan(θ/2) = r’/2f which
expands to θ ≈ r’/f - 1/12(r’/f)3 …
So to correct coma we need to introduce a radial shift in the the position of the ray in the aperture such that it ends up at r instead of at r’, with r = r’ + r’ 3/ 4f 2 . w r θ αf f
June 8th 2017
Simons Observatory Optics Workshop

8. Ritchey-Crétien telescopes
The optical astronomers have known for ~100 years that the way to do this is to add a fourth-order term to the primary and take it out again with the secondary.
This means that both mirrors are then hyperboloids although the deviations from the classical shapes are quite small.
Note that this is not a compromise. The images at the center of the field are still perfect to the extent that the mirrors are made to the correct shapes.
The only penalty is that the prime focus is no longer sharp.
June 8th 2017
Simons Observatory Optics Workshop

9. Off-axis Dual-Reflector Antennas
Radio and Radar people focussed on removing Blockage.
Going off-axis breaks the symmetry which distorts the beam and raises the cross-polarization level.
In 1970s it was realised that with two reflectors you can recover this by choosing the off-axis angles such that the illumination of the equivalent paraboloid is symmetrical.
Mizugutch-Dragone condition
tan(α/2) = m tan(β/2)
m = (e + 1) / (e – 1), e is eccentricity of the hyperboloid.
June 8th 2017
Simons Observatory Optics Workshop

10. Off-axis Dual Reflector Antennas
Radio and Radar people focussed on removing Blockage.
Going off-axis breaks the symmetry which distorts the beam and raises the cross-polarization level.
In 1970s it was realised that with two reflectors you can recover this by choosing the off-axis angles such that the illumination of the equivalent paraboloid is symmetrical.
Mizugutch-Dragone condition
tan(α/2) = m tan(β/2)
m = (e + 1) / (e – 1), e is eccentricity of the hyperboloid.
Ralph Graham, patent 1968
June 8th 2017
Simons Observatory Optics Workshop

11. Off-axis Dual Reflector Antennas
Radio and Radar people focussed on removing Blockage.
Going off-axis breaks the symmetry which distorts the beam and raises the cross-polarization level.
In 1970s it was realised that with two reflectors you can recover this by choosing the off-axis angles such that the illumination of the equivalent paraboloid is symmetrical.
Mizugutch-Dragone condition
tan(α/2) = m tan(β/2)
m = (e + 1) / (e – 1), e is eccentricity of the hyperboloid.
June 8th 2017
Simons Observatory Optics Workshop

12. Simons Observatory Optics Workshop
Dragone classic paper 1978
Generalized to multiple mirrors
Showed that the condition for restoring symmetry can be found by a simple geometric construction.
Alternative form of the requirement:
M tan p = (1 – M) tan i
where p and i are the angles of incidence and M is Lp / Ls .
Recommended the crossed design which was adopted by the microwave design community as the preferred form of “compact range” for antenna testing because of its uniform illumination and polarization purity.
Bell System Technical Journal, vol. 57, Sept. 1978, p
June 8th 2017
Simons Observatory Optics Workshop

13. Simons Observatory Optics Workshop
Dragone classic paper 1978
Generalized to multiple mirrors
Showed that the condition for restoring symmetry can be found by a simple geometric construction.
Alternative form of the requirement:
M tan p = (1 – M) tan i
where p and i are the angles of incidence and M is Lp / Ls .
Recommended the crossed design which was adopted by the microwave design community as the preferred form of “compact range” for antenna testing because of its uniform illumination and polarization purity.
Bell System Technical Journal, vol. 57, Sept. 1978, p
June 8th 2017
Simons Observatory Optics Workshop

14. Simons Observatory Optics Workshop
Dragone classic paper 1978
Generalized to multiple mirrors
Showed that the condition for restoring symmetry can be found by a simple geometric construction.
Alternative form of the requirement:
M tan p = (1 – M) tan i
where p and i are the angles of incidence and M is Lp / Ls .
Recommended the crossed design which was adopted by the microwave design community as the preferred form of “compact range” for antenna testing because of its uniform illumination and polarization purity. Lp Ls
Bell System Technical Journal, vol. 57, Sept. 1978, p
June 8th 2017
Simons Observatory Optics Workshop

15. Simons Observatory Optics Workshop
Dragone classic paper 1978
Generalized to multiple mirrors
Showed that the condition for restoring symmetry can be found by a simple geometric construction.
Alternative form of the requirement:
M tan p = (1 – M) tan i
where p and i are the angles of incidence and M is Lp / Ls .
Recommended the crossed design which was adopted by the microwave design community as the preferred form of “compact range” for antenna testing because of its uniform illumination and polarization purity.
Bell System Technical Journal, vol. 57, Sept. 1978, p
June 8th 2017
Simons Observatory Optics Workshop

16. A Ritchey-Crétien cross-Dragone design
As explained we need to move the positions of the rays in the aperture such that r = r’ + r’ 3 / 4f 2, i.e. Δr = r’ 3 / 4f 2.
Do this by putting an additional slope dz/dr into the secondary such that Δr = 2 dz/dr L where L is the distance between the mirrors.
We need to integrate this to find the shape of the required modification to the surface.
In the x-direction (perp to the picture) this is easy dz/dx = x 3 / (8 f 2 L) so z = x 4 / (32 f 2 L).
In the y-direction, along the mirror, it is a little more complicated because of a) the projection onto the elliptical shape and b) the separation L changes with y. y
L ≈ (1 + ky) L0
June 8th 2017
Simons Observatory Optics Workshop

17. A Ritchey-Crétien cross-Dragone design
Can still do the integral – gives terms in y4, y5, etc.
In fact we also want to put in terms in x2, y2 and y3 to keep the overall optical properties (f- ratio, etc) symmetry that same.
Result is that the required change in shape looks like this.
Note that these are expressed in polynominals, taking the actual centre of the mirror as the origin, not that of the conic.
The deviations from the original parabola are actually quite small in magnitude: about 1mm peak-to-peak for the f/2.5 case with 6m aperture shown here.
This shows the change in the shape of the primary.
Colour scale is from +0.1mm (red) to -0.6mm (blue).
Secondary is rather more shaped to -0.7mm.
June 8th 2017
Simons Observatory Optics Workshop

18. A Ritchey-Crétien cross-Dragone design
Must add matching deformations to the secondary in order to cancel the deformation we have introduced into the wavefront.
In reality these correction terms were found simply by asking Zemax to minimize the spot size at several off-axis positions and requiring that the on-axis focus also remains good and also limiting the deviation at the edge of the mirrors.
Primary
Secondary
Procedure is: 1) find a CD design with the right geometry using conics, ) make a version with polynomials that reproduces this, ) perturb the polynomials to get the improved off-axis performance.
June 8th 2017
Simons Observatory Optics Workshop

19. Simons Observatory Optics Workshop
It works as expected
Classic Design
Corrected
Spots are much smaller
The remaining aberrations are mainly astigmatism
That gives a wavefront error that scales as
α 2 r 2 cos2φ / f
i.e. quadratic with field offset instead of linear.
June 8th 2017
Simons Observatory Optics Workshop

20. Simons Observatory Optics Workshop
It works as expected
Spots are much smaller
The remaining aberrations are mainly astigmatism
That gives a wavefront error that scales as
α 2 r 2 cos2φ / f
i.e. quadratic with field offset angle α instead of linear.
June 8th 2017
Simons Observatory Optics Workshop

21. Simons Observatory Optics Workshop
It works as expected
Classic Design on left, Corrected on right
Plots of Strehl ratio at 150GHz.
Red ellipse is 80%
Contours are elliptical. Believe that this is intrinsic to off-axis design.
This is for an f/2.5 case with a 6m aperture.
Square are total of 8 degrees on a side, i.e. 64 square degrees
June 8th 2017
Simons Observatory Optics Workshop

22. Increase in FOV area Corrected over Classic versus f-ratio
In terms of area of FOV with Strehl > 0.8, the gain from the correction is really large at high frequencies.
At 350 microns already greater than factor of 5 at f/2.9, > x2 at 1mm.
The gain increases at shorter f-ratio.
Not likely we can use all of this at 350 microns any time soon!
Wavelength in mms.
June 8th 2017
Simons Observatory Optics Workshop

23. Simons Observatory Optics Workshop
Postscript
Steve Parschley pointed out a patent by Dragone which led me to further papers of his on this. It turns out that he had already worked all this out in I suspect that few people penetrated far enough into his paper to appreciate what he had done!
As far as I can see he only proposed adding the leading 4th-order term and didn’t add the higher- (and lower-) order terms needed because the distance between the mirrors changes between the top and bottom.
The only follow-up to this that I have been able to find so far is the analysis by Hanany and Marrone 2002, Appl Opts, 41, 4666, which only showed a fairly modest improvement. I think this was because the f-ratio was low ~f/1.3.
Dragone, C., IEEE AP-31, 764, 1983
June 8th 2017
Simons Observatory Optics Workshop

24. Simons Observatory Optics Workshop
Postscript 2
In another paper Dragone went further and pointed out that one can correct the Astigmatism as well as the Coma by choosing a particular combination of focal lengths.
He designed a “Unique” solution that does this, while still meeting the M-D condition, and says that this results in an antenna “having the widest possible field of view obtainable with two reflectors in a compact arrangement of short focal length”, although he does point out that the astigmatism correction only works in one direction – out of the plane of symmetry.
As far as I can see he didn’t actually make this statement quantitative. It turns out that, because the f-ratio is rather low (~f/1.45), the area of the DLFoV is not as large as we can get at ~f/2.5 with the crossed design, but it is still impressive. Using this geometry and optimising the shapes to remove coma, I get > 100sq deg with Strehl > 0.8 at 150GHZ for an aperture D of 1.4m.
Dragone, C., Elec Letts, 19, 1061, 1983
June 8th 2017
Simons Observatory Optics Workshop