Abstract

The general two-mirror system used at finite conjugates is examined here. Relations for first-order geometric properties and third-order aberrations are given in terms of five design parameters: object distance, image distance, exit pupil size, and the two mirror magnifications. The conditions for aplanatic solutions are derived for conic mirrors. The curvatures of the astigmatic image surfaces are given, and the condition for anastigmatic solutions is derived. The relations are applied to infinite conjugate systems and spherical mirror systems as special cases.

© 1992 Optical Society of America

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References

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  1. R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), p. 331.
  2. R. G. Bingham, “A two-mirror focal length reducer and field corrector for prime foci,” in Optical and Infrared Telescopes for the 1990s, A. Hewitt, ed. (Kitt Peak National Observatory, Tucson, Arizona, 1980), pp. 965–974.
  3. A. Offner, “Null tests using compensators,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 439–458.
  4. S. Rosin, “Inverse Cassegrain systems,” Appl. Opt. 7, 1483–1497 (1968).
    [CrossRef] [PubMed]
  5. C. G. Wynne, “Two-mirror anastigmats,” J. Opt. Soc. Am. 59, 572–578 (1969).
    [CrossRef]
  6. A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Section B 70, 945–949 (1957).
    [CrossRef]
  7. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, England, 1986), pp. 130–153.
  8. O. E. Stavroudis, “Two-mirror systems with spherical reflecting surfaces,” J. Op. Soc. Am. 57, 741–748 (1967).
    [CrossRef]
  9. For example, see M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 32.
  10. C. R. Burch, “Reflecting microscopes,” Proc. Phys. Soc. London 59, 41–46 (1947).
    [CrossRef]

1969

1968

1967

O. E. Stavroudis, “Two-mirror systems with spherical reflecting surfaces,” J. Op. Soc. Am. 57, 741–748 (1967).
[CrossRef]

1957

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Section B 70, 945–949 (1957).
[CrossRef]

1947

C. R. Burch, “Reflecting microscopes,” Proc. Phys. Soc. London 59, 41–46 (1947).
[CrossRef]

Bingham, R. G.

R. G. Bingham, “A two-mirror focal length reducer and field corrector for prime foci,” in Optical and Infrared Telescopes for the 1990s, A. Hewitt, ed. (Kitt Peak National Observatory, Tucson, Arizona, 1980), pp. 965–974.

Burch, C. R.

C. R. Burch, “Reflecting microscopes,” Proc. Phys. Soc. London 59, 41–46 (1947).
[CrossRef]

Head, A. K.

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Section B 70, 945–949 (1957).
[CrossRef]

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), p. 331.

Offner, A.

A. Offner, “Null tests using compensators,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 439–458.

Rosin, S.

Spiegel, M. R.

For example, see M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 32.

Stavroudis, O. E.

O. E. Stavroudis, “Two-mirror systems with spherical reflecting surfaces,” J. Op. Soc. Am. 57, 741–748 (1967).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, England, 1986), pp. 130–153.

Wynne, C. G.

Appl. Opt.

J. Op. Soc. Am.

O. E. Stavroudis, “Two-mirror systems with spherical reflecting surfaces,” J. Op. Soc. Am. 57, 741–748 (1967).
[CrossRef]

J. Opt. Soc. Am.

Proc. Phys. Soc. London

C. R. Burch, “Reflecting microscopes,” Proc. Phys. Soc. London 59, 41–46 (1947).
[CrossRef]

Proc. Phys. Soc. London Section B

A. K. Head, “The two-mirror aplanat,” Proc. Phys. Soc. London Section B 70, 945–949 (1957).
[CrossRef]

Other

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, Bristol, England, 1986), pp. 130–153.

For example, see M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 32.

R. Kingslake, Lens Design Fundamentals (Academic, New York, 1978), p. 331.

R. G. Bingham, “A two-mirror focal length reducer and field corrector for prime foci,” in Optical and Infrared Telescopes for the 1990s, A. Hewitt, ed. (Kitt Peak National Observatory, Tucson, Arizona, 1980), pp. 965–974.

A. Offner, “Null tests using compensators,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 439–458.

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Figures (2)

Fig. 1
Fig. 1

Optical design parameters. All entities have a positive sign as shown.

Fig. 2
Fig. 2

Configurations with zero astigmatism: B/A = −5 (-■-); B/A = −1/5 (-♦-); B/A = 1/5 (-▲-); B/A = 1 (-□-); B/A = 5 (-⋄-).

Equations (44)

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z ( ρ ) = ρ 2 2 R + ( 1 + κ ) ρ 4 8 R 3 ,
R 1 = 2 m 1 A 1 m 1 ,
R 2 = 2 B 1 m 2 ,
D = B + m 1 m 2 A m 2 .
k = y 2 y 1 = B m 1 m 2 A .
ϕ = m 1 2 m 2 A ( 1 m 2 ) B ( 1 m 1 ) m 1 m 2 A B ,
F / # i = B 2 y 2 .
x p = M 1 A ( B + m 1 m 2 A ) m 1 2 A B ( 1 m 1 ) ,
m p = y p y 2 = m 1 m 2 m 1 2 m 2 B A ( 1 m 1 ) ,
s I = 1 y 2 4 4 B 3 { m 1 m 2 4 ( A B ) ( 1 m 1 ) 3 [ κ 1 + ( m 1 + 1 m 1 1 ) 2 ] ( 1 m 2 ) 3 [ κ 2 + ( m 2 + 1 m 2 1 ) 2 ] } ,
s I I = y 2 3 h 4 B 3 × { 2 ( m 2 2 1 ) + m 2 2 ( 1 m 1 ) [ κ 1 ( 1 m 1 ) 2 ( 1 + A B m 1 m 2 ) + A B m 1 m 2 ( m 1 + 1 ) 2 + m 1 2 1 ] } ,
s I I I = y 2 2 h 2 4 A B 2 × { ( κ 1 + 1 ) ( A B m 1 m 2 + 1 ) 2 ( 1 m 1 ) 3 m 1 4 [ A B ( 1 2 m 1 m 2 + m 1 2 m 2 ) ( 1 m 1 ) ( A B m 1 m 2 ) 2 ] } ,
S I V = y 2 2 h 2 A B 2 [ A B ( 1 m 2 ) ( 1 m 1 ) m 1 + A k 0 ] ,
S V = y 2 h 3 4 A 2 B ( 1 + m 1 m 2 A B ) m 1 2 m 2 2 × { ( 1 + m 1 m 2 A B ) 2 [ ( 1 m 1 ) 3 ( κ 1 + 1 ) + 4 m 1 ( 1 m 1 ) ] + 6 ( 1 + m 1 m 2 A B ) ( m 1 2 1 ) + 8 ( 1 m 1 ) .
κ 1 = 1 ( 1 m 1 ) 2 ( 1 + A B m 1 m 2 ) × [ 2 ( m 2 2 1 ) m 2 2 ( 1 m 1 ) + A B m 1 m 2 ( m 1 + 1 ) 2 + m 1 2 1 ] .
s I I I = y 2 2 h 2 2 A B 2 [ A B m 1 2 m 2 2 2 m 2 + 1 m 2 + m 1 m 2 2 ( m 1 2 ) + 1 m 1 m 2 2 ] .
κ 2 = ( m 2 + 1 m 2 1 ) 2 + A B m 1 m 2 4 ( 1 m 1 1 m 2 ) 3 [ κ 1 + ( m 1 + 1 m 1 1 ) 2 ] .
k p = 1 B [ 1 m 2 B A ( 1 m 1 ) m 1 + B k 0 ] ,
k a = 1 2 B [ m 1 2 m 2 2 2 m 2 + 1 m 2 + B A m 1 m 2 2 ( m 1 2 ) + 1 m 1 m 2 2 ]
k t = k p + 3 k a ,
k s = k p + k a ,
k m = k p + 2 k a .
z ( h ) = h 2 2 k α , α = p , a , t , s , m .
k 0 = 1 B [ B A ( 1 m 1 ) m 1 + m 2 2 B k a 1 ] .
m 2 = ( m 2 + 1 ) ( m + B A ) 2 m ( B A + 1 ) .
A m 1 = B m 2 k ,
κ 1 = 2 k m 2 2 ( k 1 ) 1 ,
κ 2 = ( m 2 + 1 m 2 1 ) 2 m 2 3 k ( 1 m 2 ) 3 ( κ 1 + 1 ) .
m 2 = k 1 2 .
B m 2 = A k m 1
h B = m p A + x p h o
κ 1 = ( m 1 + 1 m 1 1 ) 2 ,
κ 2 = ( m 2 + 1 m 2 1 ) 2 .
m 2 3 + p m 2 2 + q m 2 + r = 0 ,
p = ( m 2 A B + 1 ) / ( m A B + 1 ) ,
q = ( m 3 A B + 1 ) / ( m A B + 1 ) ,
r = ( m 4 A B + 1 ) / ( m A B + 1 ) ,
m 2 = 5 + 1 2 ,
k = 5 + 3 5 1 .
m 2 = ( m + B A ) ( m 2 + 1 ) 2 m ( B k a + B A + 1 ) .
k 0 = 1 B [ B k p + B A ( m 2 m ) m + m 2 1 ] ,
m 4 + p m 3 + q m 2 + p m + ( p 2 ) 2 = 0 ,
p = 2 B A ,
q = 2 B 2 k a ( k 0 k p ) + 2 B ( B A + 1 ) ( k a + k 0 k p ) + ( B A ) 2 + 4 B A + 1 .

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