Abstract

Under certain circumstances, a system of two mirrors can be corrected to eliminate Seidel spherical aberration, coma, and astigmatism. Schwarzschild investigated the case of such a system with a flat field and an object at infinity. This paper considers the full range of solutions for different magnifications and field curvatures, with positive or negative powers and two real, one real and one virtual, and two virtual conjugates. If aspherical terms of higher than the fourth order in aperture are used, highly corrected systems of large relative aperture, covering extended field sizes, are possible. A numerical example is given of an f/0.7 system covering ±0.10 radians.

© 1969 Optical Society of America

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  1. K. Schwarzschild’s work on two-mirror systems is contained in the second of three papers, “Untersuchung zur geometrischen Optik I, II and III,” published in 1905 in the Abhandlungen der Königl. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse. Neue Folge. Band IV. No. 2, and reprinted the same year in the Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen, Part 2.
  2. H. Chrétien, Rev. Opt. 1, 49 (1922).
  3. E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955), p. 277.
  4. C. R. Burch, Proc. Phys. Soc. (London) 59, 41 (1947).
    [Crossref]
  5. P. Erdos, J. Opt. Soc. Am. 49, 877 (1959).
    [Crossref]
  6. S. Rosin, Appl. Opt. 7, 1483 (1968).
    [Crossref] [PubMed]
  7. H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).
  8. C. G. Wynne, Proc. Phys. Soc. (London) 73, 777 (1959).
    [Crossref]

1968 (1)

1959 (2)

C. G. Wynne, Proc. Phys. Soc. (London) 73, 777 (1959).
[Crossref]

P. Erdos, J. Opt. Soc. Am. 49, 877 (1959).
[Crossref]

1947 (1)

C. R. Burch, Proc. Phys. Soc. (London) 59, 41 (1947).
[Crossref]

1922 (1)

H. Chrétien, Rev. Opt. 1, 49 (1922).

Brueggemann, H. P.

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).

Burch, C. R.

C. R. Burch, Proc. Phys. Soc. (London) 59, 41 (1947).
[Crossref]

Chrétien, H.

H. Chrétien, Rev. Opt. 1, 49 (1922).

Erdos, P.

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955), p. 277.

Rosin, S.

Schwarzschild’s, K.

K. Schwarzschild’s work on two-mirror systems is contained in the second of three papers, “Untersuchung zur geometrischen Optik I, II and III,” published in 1905 in the Abhandlungen der Königl. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse. Neue Folge. Band IV. No. 2, and reprinted the same year in the Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen, Part 2.

Wynne, C. G.

C. G. Wynne, Proc. Phys. Soc. (London) 73, 777 (1959).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Proc. Phys. Soc. (London) (2)

C. G. Wynne, Proc. Phys. Soc. (London) 73, 777 (1959).
[Crossref]

C. R. Burch, Proc. Phys. Soc. (London) 59, 41 (1947).
[Crossref]

Rev. Opt. (1)

H. Chrétien, Rev. Opt. 1, 49 (1922).

Other (3)

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955), p. 277.

K. Schwarzschild’s work on two-mirror systems is contained in the second of three papers, “Untersuchung zur geometrischen Optik I, II and III,” published in 1905 in the Abhandlungen der Königl. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse. Neue Folge. Band IV. No. 2, and reprinted the same year in the Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen, Part 2.

H. P. Brueggemann, Conic Mirrors (Focal Press, London, 1968).

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Figures (8)

Fig. 1
Fig. 1

Curves showing the values of M and p for which two-mirror anastigmats exist, for values of dk of −1, −2, −3, ∞, +1, and +2. The shaded regions indicate solutions with 100% axial obscuration.

Fig. 2
Fig. 2

Layout of two-mirror anastigmats with positive power k, M = 0.2, for different values of p.

Fig. 3
Fig. 3

Layout of two-mirror anastigmats with positive power k, M = 0, for different values of p.

Fig. 4
Fig. 4

Layout of two-mirror anastigmats with positive power k, M = −0.2, for different values of p.

Fig. 5
Fig. 5

Layout of two-mirror anastigmats with positive power k, M = −0.5, for different values of p.

Fig. 6
Fig. 6

Layout of two-mirror anastigmats with negative power k, M = 0.5, p = −1.5, and M = −0.5, p = 3.5.

Fig. 7
Fig. 7

f/0.7 flat-field two-mirror anastigmat (M = 0) showing ray paths on axis (full lines) and at a field angle of 0.10 rad (dotted lines).

Fig. 8
Fig. 8

Spot diagrams for the f/0.7 flat-field two-mirror anastigmat, on axis (a) and at angles from the axis of 0.07 rad (b) and 0.10 rad (c). The circle represents an image spread of 0.2 mrad.

Tables (1)

Tables Icon

Table I Flat-field anastigmat mirror pair, focal length = 1.00, M = 0, f/0.7, field angle ±0.1 rad.

Equations (26)

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S 1 I = - 2 h 1 2 i 1 2 c 1 ,             S 2 I = 2 h 2 2 i 2 2 c 2 ,
S II = S I ( E + 1 / h i ) ,             S III = S I ( E + 1 / h i ) 2 ,
S II # = S I # E ,             S III # = S I # E 2 ,
E 1 = 0 ,             E 2 = - d / h 1 h 2 ,
S 1 II = - 2 h 1 i 1 c 1 S 1 III = - 2 c 1 S 1 II # = 0 S 1 III # = 0 S 2 II = 2 h 2 i 2 c 2 [ 1 - ( d i 2 / h 1 ) ] S 2 III = 2 c 2 [ 1 - ( d i 2 / h 1 ) ] 2 . }
S 2 II # = 2 h 1 i 1 c 1 - 2 h 2 i 2 c 2 [ 1 - ( d i 2 / h 1 ) ] ,
S 2 I # = - 2 S 1 # II h 1 h 2 / d ;             S 2 III # = - S 2 II # d / h 1 h 2 .
d = ( c 2 - c 1 ) h 1 h 2 h 1 i 1 c 1 - h 2 i 2 c 2 ,
S 2 I # = - 2 c 1 c 2 - c 1 ( h 1 2 i 1 2 c 1 - h 2 2 i 2 2 c 2 ) ;
S 1 I # = 2 c 2 c 2 - c 1 ( h 1 2 i 1 2 c 1 - h 2 2 i 2 2 c 2 ) .
P = 2 ( c 1 - c 2 ) .
S 1 I # = 16 a 1 h 1 4 ,
k = 2 ( c 2 - c 1 - 2 d c 1 c 2 ) .
i 1 = 1 2 ( u 2 - u 1 ) i 2 = 1 2 ( u 2 - u 2 ) h 1 c 1 = 1 2 ( u 1 + u 2 ) h 2 c 2 = 1 2 ( u 2 + u 2 ) }
u 1 = h k M / ( 1 - M ) ,             u 2 = h k / ( 1 - M ) .
u 2 = h k 2 ( 1 - M ) × [ p ( 1 + M ) ± { ( 1 + M ) 2 ( 1 + p ) 2 + ( 1 - M ) 2 } 1 2 ] .
h 1 = h [ 1 + k M ( 1 + p ) 2 c 1 ( 1 - M ) ] ,
c 1 = k 4 [ p ± { ( 1 + M ) 2 ( 1 + p ) 2 + ( 1 - M ) 2 } 1 2 1 - M ] c 2 = k 4 [ - p ± { ( 1 + M ) 2 ( 1 + p ) 2 + ( 1 - M ) 2 } 1 2 ( 1 - M ) ] . }
d k = - 2 ( 1 - M ) 2 2 M p + 1 + M 2 .
h 1 / h 2 = - h 2 * / h 1 * .
d k = - 4 / ( 1 - p ) c 1 / k = 1 4 ( p ± 1 ) c 2 / k = 1 4 ( - p ± 1 ) }
d = 1 / c 1 - 1 / c 2 ,
d k = - 2 ( 1 - M ) 2 / ( 1 + M ) 2 c 1 / k = 1 4 ( 1 ± { 5 M 2 + 6 M + 1 } 1 2 ( 1 - M ) ) c 2 / k = 1 4 ( - 1 ± { 5 M 2 + 6 M + 1 } 1 2 ( 1 - M ) ) . }
p = M - 1 1 + 3 M ;             p = 1 - M 3 + M ;             M = - 1.
z = 0.34207 × 10 - 1 ρ 4 + 0.21631 × 10 - 2 × ρ 6 + 0.29599 × 10 - 1 ρ 8 - 0.75682 × 10 - 2 ρ 10
z = 0.10446 × 10 - 2 ρ 4 + 0.26630 × 10 - 3 × ρ 6 - 0.28766 × 10 - 4 ρ 8 + 0.15894 × 10 - 4 ρ 10