Abstract

The “plate-diagram” method of quantifying and manipulating the Seidel aberrations of an optical system has been used to develop a procedure that has successfully determined the complete solution set of three-mirror anastigmats in which two surfaces are left strictly spherical. The procedure also readily identified solutions in which the Petzval sum is zero, and four distinct families of flat-field three-mirror anastigmats with two mirrors strictly spherical have thus been found. The success of the method is strong support for the argument that algebraic approaches to optical design can yield results distinctly superior to currently favored optimization-based design methods, at least for some types of optical systems.

© 2002 Optical Society of America

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References

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  1. H. L. Aldis, “On the construction of photographic objectives,” Photograp. J. 24, 291–299 (1900).
  2. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102, 159–165 (1942).
  3. C. R. Burch, “On aspheric anastigmatic systems,” Proc. Phys. Soc. 55, 433–444 (1943).
    [CrossRef]
  4. C. R. Burch, “Application of the plate diagram to reflecting telescope design,” Opt. Act 26, 493–504 (1979).
    [CrossRef]
  5. E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, UK, 1955), pp. 229–259.
  6. M. Paul, “Systèmes correcteurs pour réflecteurs astronomiques,” Rev. d’Opt. 14, 169–202 (1935).
  7. N. J. Rumsey, “Pairs of spherical mirrors as field correctors for paraboloid mirrors,” Proc. Astron. Soc. Aust. 2, 22–23 (1971).
  8. N. J. Rumsey, “Pairs of spherical mirrors as prime focus correctors for the Anglo-Australian Telescope,” Proc. Astron. Soc. Aust. 2, 126–127 (1972).
  9. A. Rakich, “A complete survey of three-mirror anastigmatic reflecting telescope systems with one aspheric surface,” M.Sc. thesis (University of Canterbury, Canterbury, New Zealand, 2001).
  10. A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1953).
  11. J. G. Baker, “On improving the effectiveness of large telescopes,” Trans. Aerosp. Electron. Syst. 2, 261–272 (1969).
    [CrossRef]
  12. L. G. Cook, “Wide field of view three-mirror anastigmat (TMA) employing spherical secondary and tertiary mirrors,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londoño, R. E. Fischer, eds., Proc SPIE766, 158–162 (1987).

1979 (1)

C. R. Burch, “Application of the plate diagram to reflecting telescope design,” Opt. Act 26, 493–504 (1979).
[CrossRef]

1972 (1)

N. J. Rumsey, “Pairs of spherical mirrors as prime focus correctors for the Anglo-Australian Telescope,” Proc. Astron. Soc. Aust. 2, 126–127 (1972).

1971 (1)

N. J. Rumsey, “Pairs of spherical mirrors as field correctors for paraboloid mirrors,” Proc. Astron. Soc. Aust. 2, 22–23 (1971).

1969 (1)

J. G. Baker, “On improving the effectiveness of large telescopes,” Trans. Aerosp. Electron. Syst. 2, 261–272 (1969).
[CrossRef]

1943 (1)

C. R. Burch, “On aspheric anastigmatic systems,” Proc. Phys. Soc. 55, 433–444 (1943).
[CrossRef]

1942 (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102, 159–165 (1942).

1935 (1)

M. Paul, “Systèmes correcteurs pour réflecteurs astronomiques,” Rev. d’Opt. 14, 169–202 (1935).

1900 (1)

H. L. Aldis, “On the construction of photographic objectives,” Photograp. J. 24, 291–299 (1900).

Aldis, H. L.

H. L. Aldis, “On the construction of photographic objectives,” Photograp. J. 24, 291–299 (1900).

Baker, J. G.

J. G. Baker, “On improving the effectiveness of large telescopes,” Trans. Aerosp. Electron. Syst. 2, 261–272 (1969).
[CrossRef]

Burch, C. R.

C. R. Burch, “Application of the plate diagram to reflecting telescope design,” Opt. Act 26, 493–504 (1979).
[CrossRef]

C. R. Burch, “On aspheric anastigmatic systems,” Proc. Phys. Soc. 55, 433–444 (1943).
[CrossRef]

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102, 159–165 (1942).

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1953).

Cook, L. G.

L. G. Cook, “Wide field of view three-mirror anastigmat (TMA) employing spherical secondary and tertiary mirrors,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londoño, R. E. Fischer, eds., Proc SPIE766, 158–162 (1987).

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, UK, 1955), pp. 229–259.

Paul, M.

M. Paul, “Systèmes correcteurs pour réflecteurs astronomiques,” Rev. d’Opt. 14, 169–202 (1935).

Rakich, A.

A. Rakich, “A complete survey of three-mirror anastigmatic reflecting telescope systems with one aspheric surface,” M.Sc. thesis (University of Canterbury, Canterbury, New Zealand, 2001).

Rumsey, N. J.

N. J. Rumsey, “Pairs of spherical mirrors as prime focus correctors for the Anglo-Australian Telescope,” Proc. Astron. Soc. Aust. 2, 126–127 (1972).

N. J. Rumsey, “Pairs of spherical mirrors as field correctors for paraboloid mirrors,” Proc. Astron. Soc. Aust. 2, 22–23 (1971).

Mon. Not. R. Astron. Soc. (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102, 159–165 (1942).

Opt. Act (1)

C. R. Burch, “Application of the plate diagram to reflecting telescope design,” Opt. Act 26, 493–504 (1979).
[CrossRef]

Photograp. J. (1)

H. L. Aldis, “On the construction of photographic objectives,” Photograp. J. 24, 291–299 (1900).

Proc. Astron. Soc. Aust. (2)

N. J. Rumsey, “Pairs of spherical mirrors as field correctors for paraboloid mirrors,” Proc. Astron. Soc. Aust. 2, 22–23 (1971).

N. J. Rumsey, “Pairs of spherical mirrors as prime focus correctors for the Anglo-Australian Telescope,” Proc. Astron. Soc. Aust. 2, 126–127 (1972).

Proc. Phys. Soc. (1)

C. R. Burch, “On aspheric anastigmatic systems,” Proc. Phys. Soc. 55, 433–444 (1943).
[CrossRef]

Rev. d’Opt. (1)

M. Paul, “Systèmes correcteurs pour réflecteurs astronomiques,” Rev. d’Opt. 14, 169–202 (1935).

Trans. Aerosp. Electron. Syst. (1)

J. G. Baker, “On improving the effectiveness of large telescopes,” Trans. Aerosp. Electron. Syst. 2, 261–272 (1969).
[CrossRef]

Other (4)

L. G. Cook, “Wide field of view three-mirror anastigmat (TMA) employing spherical secondary and tertiary mirrors,” in Recent Trends in Optical Systems Design: Computer Lens Design Workshop, C. Londoño, R. E. Fischer, eds., Proc SPIE766, 158–162 (1987).

E. H. Linfoot, Recent Advances in Optics (Clarendon, Oxford, UK, 1955), pp. 229–259.

A. Rakich, “A complete survey of three-mirror anastigmatic reflecting telescope systems with one aspheric surface,” M.Sc. thesis (University of Canterbury, Canterbury, New Zealand, 2001).

A. E. Conrady, Applied Optics and Optical Design (Dover, New York, 1953).

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Figures (10)

Fig. 1
Fig. 1

Upper part, Paul three-mirror system, with paraboloid primary and spherical secondary and tertiary mirrors; lower part, corresponding plate diagram. Note that x4 has been made zero by placing the aperture stop on the primary mirror.

Fig. 2
Fig. 2

Ray reflected at a convex surface. All quantities shown here are positive. Note that P in this case is the length of the perpendicular from the center of curvature of the mirror to the incident ray. All other quantities follow the usual paraxial conventions as given by Conrady.10

Fig. 3
Fig. 3

Solution set for three-mirror anastigmats obtained by using Eq. (23) with the primary mirror aspherized. In this, and in the following figures representing solution sets, white points represent solutions with positive Petzval curvature, black points represent systems with negative Petzval curvature, and gray points represent coordinates for which no physically realizable anastigmats exist. Note that flat-field “Paul–Rumsey” systems can be found along a curve defined by where a white region abuts the black region in this graph.

Fig. 4
Fig. 4

Solution set for three-mirror anastigmats obtained by using Eq. (24) with the primary mirror aspherized. In this case the secondary mirror is concave (c2 is negative) for all solutions. No flat-field solutions exist in this set.

Fig. 5
Fig. 5

Solution set for three-mirror anastigmats obtained by using Eq. (25) with the primary mirror aspherized. A flat-field curve can be discerned along the left-hand edge of the central strip of solutions.

Fig. 6
Fig. 6

Solution set for three-mirror anastigmats obtained by using Eq. (23) with the secondary mirror aspherized, representing systems that have not previously been described in the literature. This graph corresponds closely to that shown in Fig. 1. Here there are two flat-field curves.

Fig. 7
Fig. 7

Solution set for three-mirror anastigmats obtained by using Eq. (24) with the secondary mirror aspherized. No flat-field solutions exist in this set.

Fig. 8
Fig. 8

Solution set for three-mirror anastigmats obtained by using Eq. (25) with the secondary mirror aspherized. No flat-field solutions exist in this set.

Fig. 9
Fig. 9

Solution set for three-mirror anastigmats obtained by using the method described in Section 4. Here the primary and secondary mirrors are spherical and the tertiary is aspherized. No flat-field solutions exist in this set.

Fig. 10
Fig. 10

Paul–Rumsey system. Here a configuration has been selected in which the primary mirror and the tertiary mirror could be made on the same substrate.

Tables (2)

Tables Icon

Table 1 Constructional Parameters for an Example of a 200-mm-Aperture, f/2.7 Paul–Rumsey System

Tables Icon

Table 2 Constructional Parameters for an Example of a 200-mm-Aperture System of the Type Described by Cook a

Equations (31)

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Coma:xnWn,
Astigmatism:xn2Wn.
Sphericalaberration:W1+W2+W3
+W4=0,
Coma:x1W1+x2W2+x3W3
+x4W4=0,
Astigmatism:x12W1+x22W2+x32W3
+x42W4=0.
Coma:x1W1+x2W2+x3W3=0,
Astigmatism:x12W1+x22W2+x32W3=0.
W1=-0.25N1c13y14
=-0.25×(-1)×0.53×0.24=0.000050,
x3W3=-0.00010-x2W2,
x32W3=-0.00020-x22W2.
W2=-0.25N2c2i22y22.
x3=x32W3x3W3=-0.00020-x22W2-0.00010-x2W2,
W3=-0.00010-x2W2x3.
W3=-0.25N3c3P32(u3+c3P3)2
-0.25N3c3P32(u3+c3P3)2-W3=0.
a0=-4W3P34,
a1=u32P32,
a2=2 u3P3,
Y=3a1-a22,
Z=-27a0+9a1a2-a23,
Q=4Y3+Z2,
SolutionA:=-a23-23Y3Z+Q3+Z+Q3323,
SolutionB:=-a23+(1-j3)Y34(Z+Q)3-(1+j3)Z+Q3623,
SolutionC:=-a23+(1+j3)Y34(Z+Q)3-(1-j3)Z+Q3623.
ν1=-(W1+W2+W3).
k1=4ν1c13y14.
k2=4ν2c23y24.

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