Abstract

Closed-form solutions for centered imaging systems, corrected for any combination of third-order aberrations, have been derived. The theory is based on Seidel’s formulas for third-order aberrations, which have been reformulated so that they are expressed by mutually independent variables only. Chief importance is laid upon all-reflective systems. Special formulas for well-corrected two- and three-mirror systems are given.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 10, 1 (1905).
  2. K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 9, 1 (1905).
  3. M. H. Crétien, Rev. Opt. Theor. Instrum. 1, 49 (1922).
  4. H. Slevogt, Z. Instrumentenkd. 62, 312 (1942).
  5. H. Theissing and O. Zinke, Optik (Stuttg.) 3, 451 (1948).
  6. H. Köhler, Astron. Nachr. 278, 1 (1949).
    [Crossref]
  7. C. R. Burch, Proc. Phys. Soc. Lond. 59, 41 (1947).
    [Crossref]
  8. K. P. Norris, W. E. Seeds, and M. H. F. Wilkens, J. Opt. Soc. Am. 41, 456 (1951).
    [Crossref]
  9. B. Jurek, Optik (Stuttg.) 18, 413 (1961).
  10. P. Erdös, J. Opt. Soc. Am. 49, 877 (1959).
    [Crossref]
  11. O. N. Stavroudis, J. Opt. Soc. Am. 57, 741 (1967).
    [Crossref]
  12. S. Rosin, Appl. Opt. 7, 1483 (1968).
    [Crossref] [PubMed]
  13. M. Paul, Rev. Opt. Theor. Instrum. 14, 169 (1935).
  14. J. Picht, Optik (Stuttg.) 8, 129 (1951).
  15. J. L. Dessy, Publ. Astron. Soc. Pac. 75, 66 (1963).
    [Crossref]
  16. J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. 5, 2 (1969); IEEE Trans. Aerosp. Electron. Syst. 5, 261 (1969).
  17. D. Korsch, Appl. Opt. 11, 2986 (1972).
    [Crossref] [PubMed]

1972 (1)

1969 (1)

J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. 5, 2 (1969); IEEE Trans. Aerosp. Electron. Syst. 5, 261 (1969).

1968 (1)

1967 (1)

1963 (1)

J. L. Dessy, Publ. Astron. Soc. Pac. 75, 66 (1963).
[Crossref]

1961 (1)

B. Jurek, Optik (Stuttg.) 18, 413 (1961).

1959 (1)

1951 (2)

1949 (1)

H. Köhler, Astron. Nachr. 278, 1 (1949).
[Crossref]

1948 (1)

H. Theissing and O. Zinke, Optik (Stuttg.) 3, 451 (1948).

1947 (1)

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

1942 (1)

H. Slevogt, Z. Instrumentenkd. 62, 312 (1942).

1935 (1)

M. Paul, Rev. Opt. Theor. Instrum. 14, 169 (1935).

1922 (1)

M. H. Crétien, Rev. Opt. Theor. Instrum. 1, 49 (1922).

1905 (2)

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 10, 1 (1905).

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 9, 1 (1905).

Baker, J. G.

J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. 5, 2 (1969); IEEE Trans. Aerosp. Electron. Syst. 5, 261 (1969).

Burch, C. R.

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

Crétien, M. H.

M. H. Crétien, Rev. Opt. Theor. Instrum. 1, 49 (1922).

Dessy, J. L.

J. L. Dessy, Publ. Astron. Soc. Pac. 75, 66 (1963).
[Crossref]

Erdös, P.

Jurek, B.

B. Jurek, Optik (Stuttg.) 18, 413 (1961).

Köhler, H.

H. Köhler, Astron. Nachr. 278, 1 (1949).
[Crossref]

Korsch, D.

Norris, K. P.

Paul, M.

M. Paul, Rev. Opt. Theor. Instrum. 14, 169 (1935).

Picht, J.

J. Picht, Optik (Stuttg.) 8, 129 (1951).

Rosin, S.

Schwarzschild, K.

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 10, 1 (1905).

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 9, 1 (1905).

Seeds, W. E.

Slevogt, H.

H. Slevogt, Z. Instrumentenkd. 62, 312 (1942).

Stavroudis, O. N.

Theissing, H.

H. Theissing and O. Zinke, Optik (Stuttg.) 3, 451 (1948).

Wilkens, M. H. F.

Zinke, O.

H. Theissing and O. Zinke, Optik (Stuttg.) 3, 451 (1948).

Appl. Opt. (2)

Astr. Mitt. Kgl. Sternw. Göttingen (2)

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 10, 1 (1905).

K. Schwarzschild, Astr. Mitt. Kgl. Sternw. Göttingen 9, 1 (1905).

Astron. Nachr. (1)

H. Köhler, Astron. Nachr. 278, 1 (1949).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

J. G. Baker, IEEE Trans. Aerosp. Electron. Syst. 5, 2 (1969); IEEE Trans. Aerosp. Electron. Syst. 5, 261 (1969).

J. Opt. Soc. Am. (3)

Optik (Stuttg.) (3)

B. Jurek, Optik (Stuttg.) 18, 413 (1961).

H. Theissing and O. Zinke, Optik (Stuttg.) 3, 451 (1948).

J. Picht, Optik (Stuttg.) 8, 129 (1951).

Proc. Phys. Soc. Lond. (1)

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

Publ. Astron. Soc. Pac. (1)

J. L. Dessy, Publ. Astron. Soc. Pac. 75, 66 (1963).
[Crossref]

Rev. Opt. Theor. Instrum. (2)

M. Paul, Rev. Opt. Theor. Instrum. 14, 169 (1935).

M. H. Crétien, Rev. Opt. Theor. Instrum. 1, 49 (1922).

Z. Instrumentenkd. (1)

H. Slevogt, Z. Instrumentenkd. 62, 312 (1942).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (88)

Equations on this page are rendered with MathJax. Learn more.

B = 1 2 1 k h i 4 { δ i r i 3 · n ¯ i + n i 2 · ( 1 r i 1 s i ) 2 · ( 1 n i 1 s i + 1 n i s i ) } ,
F = 1 2 1 k H i h i 3 { δ i r i 3 · n ¯ i + n i 2 · ( 1 r i 1 s i ) · ( 1 r i 1 t i ) · ( 1 n i 1 s i + 1 n i s i ) } ,
C = 1 2 1 k H i 2 h i 2 { δ i r i 3 · n ¯ i + n i 2 · ( 1 r i 1 t i ) 2 · ( 1 n i 1 s i + 1 n i s i ) } ,
E = 1 2 1 k H i 3 h i { δ i r i 3 · n ¯ i + n i 2 · ( 1 r i 1 t i ) 2 · ( 1 n i 1 s i + 1 n i s i + 1 n i 1 t i + 1 n i t i ) n i 2 · ( 1 r i 1 s i ) · ( 1 r i 1 t i ) · ( 1 n i 1 t i + 1 n i t i ) } ,
1 ρ = 1 k n ¯ i n i n i 1 · 1 r i ,
H 1 = t 1 n 0 , H i + 1 = H i · t i + 1 t i , h 1 = s 1 s 1 t 1 , h i + 1 = h i · s i + 1 s i .
z = 1 2 r i ( x 2 + y 2 ) + 1 8 r i 3 ( 1 + δ i ) ( x 2 + y 2 ) 2 .
m i = s i / s i , p i = t i / t i ,
r i n ¯ i = s i m i n i 1 + n i = t i p i n i 1 + n i ,
t i s i = p i n i 1 + n i m i n i 1 + n i ,
1 p i = t i s i · ( 1 m i + m i 1 n i ) n i 1 n i ,
s i + 1 s i = n i 1 n i + 1 · ( m i p i m i n i 1 + n i ) · ( m i + 1 n i + n i + 1 m i + 1 p i + 1 ) · m i + 1 · p i + 1 ,
s i = s 1 n 0 n 1 n i 1 n i · m 1 p 1 m 1 n 0 + n 1 · m i n i 1 + n i m i p i · j = 2 i m j p j ,
H i = f n k n 1 n 0 p 1 · m 1 p 1 m 1 n 0 + n 1 · p i n i 1 + n i n i 1 n i ( m i p i ) · j = i + 1 k m j 1 ,
h i = n 0 n i 1 n i · m i n i 1 + n i m i p i · j = 1 i p i .
p 1 = { n 1 n 0 for a refractive surface , 1 for a reflective surface .
f = s 1 · n 0 n k j = 2 k m j
s 1 j = 2 k m j .
B = 1 2 · n 0 4 p 1 3 n k 3 n 1 3 f · ( m 1 n 0 + n 1 m 1 p 1 ) 3 · i = 1 k ( b i δ i + b i ) ,
F = 1 2 · n 0 2 p 1 2 n k 2 n 1 2 f 2 · ( m 1 n 0 + n 1 m 1 p 1 ) 2 · i = 1 k ( g i δ i + g i ) ,
C = 1 2 · p 1 n k n 1 f · ( m 1 n 0 + n 1 m 1 p 1 ) · i = 1 k ( c i δ i + c i ) ,
E = 1 2 · 1 n 0 2 · i = 1 k ( e i δ i + e i ) ,
b i = ( m i n i 1 + n i ) 4 n ¯ i 2 n i 1 n i ( m i p i ) · j = 1 i p j · j = i + 1 k m j 3 ,
b i = ( m i n i 1 + n i n ¯ i ) 2 · ( m i n i + n i 1 ) · m i n i 1 + n i n ¯ 2 n i 1 2 · ( m i p i ) · j = 1 i p j · j = i + 1 k m j 3 ,
g i = ( p i n i 1 + n i ) · ( m i n i 1 + n i ) 3 n ¯ i 2 n i 1 n i · ( m i p i ) · j = i + 1 k m j 2 ,
g i = ( p i n i 1 + n i n ¯ i ) · ( m i n i 1 + n i n ¯ i ) · ( m i n i + n i 1 ) · m i n i 1 + n i n ¯ i 2 n i 1 2 · ( m i p i ) · j = i + 1 k m j 2 ,
c i = ( p i n i 1 + n i ) 2 · ( m i n i 1 + n i ) 2 n ¯ i 2 n i 1 n i ( m i p i ) · j = 1 i p j 1 · j = i + 1 k m j ,
c i = ( p i n i 1 + n i n ¯ i ) 2 · ( m i n i + n i 1 ) · m i n i 1 + n i n ¯ i 2 n i 1 2 · ( m i p i ) · j = 1 i p j 1 · j = i + 1 k m j ,
e i = ( p i n i 1 + n i ) 3 · m i n i 1 + n i n ¯ i 2 n i 1 n i · ( m i p i ) · j = 1 i p j 2 ,
e i = { ( p i n i 1 + n i n ¯ i ) 2 · ( m i n i + n i m i n i 1 + n i · ( p i n i 1 + n i ) + ( p i n i + n i 1 ) ) ( p i n i 1 + n i ) · ( p i n i + n i 1 ) · ( p i n i 1 + n i n ¯ i ) · m i n i 1 + n i n ¯ i m i n i 1 + n i } · m i n i 1 + n i n ¯ i 2 n i 1 2 · ( m i p i ) · j = 1 i p j 2 .
1 ρ = 1 f · 1 n k n 1 · m 1 n 0 + n 1 m 1 p 1 · i = 1 k ( m i p i ) · j = 2 i p j 1 · j = i + 1 k m j .
i = 1 k b i δ i = b 0 ,
i = 1 k g i δ i = g 0 ,
i = 1 k c i δ i = c 0 ,
i = 1 k e i δ i = e 0 ,
i = 1 k b i = b 0 , i = 1 k g i = g 0 , i = 1 k c i = c 0 , i = 1 k e i = e 0 .
δ 1 = | b 0 b 2 g 0 g 2 | | b 1 b 2 g 1 g 2 | , δ 2 = | b 1 b 0 g 1 g 0 | | b 1 b 2 g 1 g 2 | ;
δ 1 = | b 0 b 2 b 3 g 0 g 2 g 3 c 0 c 2 c 3 | | b 1 b 2 b 3 g 1 g 2 g 3 c 1 c 2 c 3 | , δ 2 = | b 1 b 0 b 3 g 1 g 0 g 3 c 1 c 0 c 3 | | b 1 b 2 b 3 g 1 g 2 g 3 c 1 c 2 c 3 | , δ 3 = | b 1 b 2 b 0 g 1 g 2 g 0 c 1 c 2 c 0 | | b 1 b 2 b 3 g 1 g 2 g 3 c 1 c 2 c 3 | ;
δ 1 = | b 0 b 2 b 3 b 4 g 0 g 2 g 3 g 4 c 0 c 2 c 3 c 4 e 0 e 2 e 3 e 4 | | b 1 b 2 b 3 b 4 g 1 g 2 g 3 g 4 c 1 c 2 c 3 c 4 e 1 e 2 e 3 e 4 | , δ 2 = | b 1 b 0 b 3 b 4 g 1 g 0 g 3 g 4 c 1 c 0 c 3 c 4 e 1 e 0 e 3 e 4 | | b 1 b 2 b 3 b 4 g 1 g 2 g 3 g 4 c 1 c 2 c 3 c 4 e 1 e 2 e 3 e 4 | δ 3 = | b 1 b 2 b 0 b 4 g 1 g 2 g 0 g 4 c 1 c 2 c 0 c 4 e 1 e 2 e 0 e 4 | | b 1 b 2 b 3 b 4 g 1 g 2 g 3 g 4 c 1 c 2 c 3 c 4 e 1 e 2 e 3 e 4 | , δ 4 = | b 1 b 2 b 3 b 0 g 1 g 2 g 3 g 0 c 1 c 2 c 3 c 0 e 1 e 2 e 3 e 0 | | b 1 b 2 b 3 b 4 g 1 g 2 g 3 g 4 c 1 c 2 c 3 c 4 e 1 e 2 e 3 e 4 |
o = j = 1 k ( m i p i ) · j = 2 i p j 1 · j = i + 1 k m j .
r i = f i · n ¯ i / n i ,
s i = f i · ( m i n i 1 + n i ) / n i ,
d i = s i + s i + 1 / m i + 1 ,
f i = f · n 1 n k m 1 p 1 m 1 n 0 + n 1 · j = 2 i p j · j = i + 1 k m j 1 n i 1 ( m i p i )
b 0 = 1 4 { ( m 1 1 ) 2 ( m 1 + 1 ) 2 ( m 1 p 1 ) · p 1 m 2 3 + ( m 2 1 ) 2 ( m 2 + 1 ) 2 ( m 2 p 2 ) · p 1 p 2 } ,
b 1 = ( m 1 + 1 ) 4 4 ( m 1 p 1 ) · p 1 m 2 3 ,
b 2 = ( m 2 + 1 ) 4 4 ( m 2 p 2 ) · p 1 p 2 ,
g 0 = 1 4 { ( p 1 1 ) ( m 1 1 ) ( m 1 + 1 ) 2 ( m 1 p 1 ) · m 2 2 + ( p 2 1 ) ( m 2 1 ) ( m 2 + 1 ) 2 ( m 2 p 2 ) } ,
g 1 = ( p 1 + 1 ) ( m 1 + 1 ) 3 4 ( m 1 p 1 ) ,
g 2 = ( p 2 + 1 ) ( m 2 + 1 ) 3 4 ( m 2 p 2 ) ,
( m 1 p 1 ) · m 2 + ( m 2 p 2 ) 1 p 2 = 0 .
r 1 = 2 f m 2 ( m 1 + 1 ) ,
r 2 = 2 f p 2 m 1 + 1 · m 1 p 1 m 2 p 2 ,
d 1 = f m 2 ( 1 + p 2 · m 2 + 1 m 1 + 1 · m 1 p 1 m 2 p 2 ) ,
s 2 = f p 2 · m 2 + 1 m 1 + 1 · m 1 p 1 m 2 p 2 .
b 1 δ 1 + b 2 δ 2 = b 0
p 2 = m 2 1 + ( 1 / ) ( m 2 + 1 ) .
δ 1 = 1 + 2 m 2 2 · 1 + ,
δ 2 = 2 m 2 ( 1 + ) ( m 2 + 1 ) 3 ( m 2 1 m 2 + 1 ) 2 ,
r 1 = 2 f m 2 ,
r 2 = 2 f m 2 + 1 ,
d 1 = f m 2 ( 1 + ) ,
s 2 = f .
m 2 ( + 1 ) + 1 = 0
m 2 = 1 1 + .
2 + ( 2 + d 1 s 2 ) + 1 = 0 ,
< 1 2 ( 3 5 ) 0.382 .
m 2 3 ( δ 1 + 1 ) + ( m 2 + 1 ) 3 [ δ 2 + ( m 2 1 m 2 + 1 ) 2 ] = 0 .
b 0 = 1 4 { ( m 1 2 1 ) 2 ( m 1 p 1 ) · p 1 m 2 3 m 3 3 + ( m 2 3 1 ) 2 ( m 2 p 2 ) · p 1 p 2 m 3 3 + ( m 3 2 1 ) 2 ( m 3 p 3 ) · p 1 p 2 p 3 } ,
b 1 = 1 4 · ( m 1 + 1 ) 4 ( m 1 p 1 ) · p 1 m 2 3 m 3 3 ,
b 2 = 1 4 · ( m 2 + 1 ) 4 ( m 2 p 2 ) · p 1 p 2 m 3 3 ,
b 3 = 1 4 · ( m 3 + 1 ) 4 ( m 3 p 3 ) · p 1 p 2 p 3 ,
g 0 = 1 4 { ( p 1 1 ) ( m 1 1 ) · ( m 1 + 1 ) 2 m 1 p 1 · m 2 2 m 3 3 + ( p 2 1 ) ( m 2 1 ) · ( m 2 + 1 ) 2 ( m 2 p 2 ) · m 3 2 + ( p 3 1 ) ( m 3 1 ) ( m 3 + 1 ) 2 ( m 3 p 3 ) } ,
g 1 = 1 4 ( p 1 + 1 ) ( m 1 + 1 ) 3 ( m 1 p 1 ) · m 2 2 m 3 2 ,
g 2 = 1 4 ( p 2 + 1 ) ( m 2 + 1 ) 3 ( m 2 p 2 ) · m 3 2 ,
g 3 = 1 4 ( p 3 + 1 ) ( m 3 + 1 ) ( m 3 p 3 ) ,
c 0 = 1 4 { ( p 1 1 ) 2 ( m 1 + 1 ) 2 ( m 1 p 1 ) · m 2 m 3 p 1 + ( p 2 1 ) 2 ( m 2 + 1 ) 2 ( m 2 p 2 ) · m 3 p 1 p 2 + ( p 3 1 ) 2 p 1 p 2 p 3 · ( m 3 + 1 ) 2 ( m 3 p 3 ) } ,
c 1 = 1 4 ( p 1 + 1 ) 2 ( m 1 + 1 ) 2 ( m 1 p 1 ) · m 2 m 3 p 1 ,
c 2 = 1 4 ( p 2 + 1 ) 2 ( m 2 + 1 ) 2 ( m 2 p 2 ) · m 3 p 1 p 2 ,
c 3 = 1 4 ( p 3 + 1 ) 2 ( m 3 + 1 ) 2 ( m 3 p 3 ) · 1 p 1 p 2 p 3 .
( m 1 p 1 ) p 2 m 2 m 3 + ( m 2 p 2 ) m 3 + ( m 3 p 3 ) 1 p 3 = 0
p 3 = m 3 1 ( m 1 p 1 ) p 2 m 2 m 3 ( m 2 p 2 ) m 3 .
r 1 = 2 f m 2 m 3 · 1 m 1 + 1 ,
r 2 = 2 f p 2 m 3 ( m 1 + 1 ) · m 1 p 1 m 2 p 2 ,
r 3 = 2 f · m 1 p 1 m 1 + 1 · p 2 p 3 m 3 p 3 ,
d 1 = f m 2 m 3 ( 1 + p 2 · m 2 + 1 m 1 + 1 · m 1 p 1 m 2 p 2 ) ,
d 2 = f p 2 m 3 · m 1 p 1 m 1 + 1 · ( m 2 + 1 m 2 p 2 + p 3 · m 3 + 1 m 3 p 3 ) ,
s 3 = f p 2 p 3 · m 1 p 1 m 1 + 1 · m 3 + 1 m 3 p 3 ,