Abstract

Geometric theory of the grating is developed, with special emphasis on the relation between the properties of holographic gratings and those of mechanically ruled gratings. The theory is applicable to practically all types of holographic and mechanically ruled gratings, including aspheric gratings. General expressions are given for the effective grating constant, the grating equation, the focal curves, and the light-path function. Possible applications of the theory to the design of aberration-corrected gratings are also described.

© 1974 Optical Society of America

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  1. F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].
  2. T. Harada (private communication, 1973).
  3. Y. Sakayanagi, Sci. Light (Tokyo) 3, 79 (1954/55).
  4. H. Haber, J. Opt. Soc. Am. 40, 153 (1950).
    [Crossref]
  5. H. Greiner and E. Schäffer, Optik 16, 288 (1959); Optik 16, 350 (1959).
  6. I. V. Peisakhson and I. N. Tarnakin, Zh. Prikl. Spektrosk. 1, 289 (1964); Zh. Prikl. Spektrosk. 2, 218 (1965).
  7. S. C. Miller, , Upper Air Laboratory, University of Colorado (1950).
  8. T. Namioka, J. Opt. Soc. Am. 51, 4 (1961); J. Opt. Soc. Am. 51, 13 (1961).
    [Crossref]
  9. B. E. Woodgate, J. Opt. Soc. Am. 64, 654 (1974).
    [Crossref]
  10. Yu. P. Shchepetkin, Opt. Spektrosk. 4, 383 (1958); Opt. Spektrosk. 4, 513 (1958).
  11. E. Schönheit, Optik 23, 305 (1966).
  12. S. A. Strezhnev and A. L. Andreeva, Opt. Spektrosk. 28, 796 (1970) [Opt, Spectrosc. 28, 426 (1970)].
  13. R. P. Madden and D. L. Ederer, J. Opt. Soc. Am. 62, 722 (1972).
  14. A. Cornu, C. R. Acad. Sci. (Paris) 80, 645 (1875); C. R. Acad. Sci. (Paris) 116, 1215 (1893); C. R. Acad. Sci. (Paris) 116, 1421 (1893); C. R. Acad. Sci. (Paris) 117, 1032 (1893).
  15. B. Gale, Opt. Acta 13, 41 (1966).
    [Crossref]
  16. Y. Sakayanagi, Sci. Light (Tokyo) 16, 129 (1967).
  17. Y. Sakayanagi, Sci. Light (Tokyo) 3, 1 (1954/55).
  18. M. V. R. K. Murty, J. Opt. Soc. Am. 52, 768 (1962).
    [Crossref]
  19. M. Singh and K. Majumder, Sci. Light (Tokyo) 18, 57 (1969).
  20. J. D. Baumgardner, Ph.D. thesis (University of Rochester, 1970).
  21. J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.
  22. M. V. R. K. Murty and N. C. Das, J. Opt. Soc. Am. 61, 1001 (1971).
    [Crossref]
  23. G. Pieuchard (private communication, 1972).
  24. T. Namioka and H. Noda, in Proceedings of International Symposium for Synchrotron Radiation Users, edited by G. V. Marr and I. H. Munro (Daresbury Nuclear Physics Laboratory, Daresbury, 1973), p. 51.
  25. T. Namioka, H. Noda, and M. Seya, Sci. Light (Tokyo) 22, 77 (1973).
  26. M. Pouey, C. R. Acad. Sci. (Paris) B276, 531 (1973).
  27. W. T. Welford, Opt. Acta 10, 121 (1963).
    [Crossref]
  28. M. Pouey, Opt. Commun. 2, 339 (1970).
    [Crossref]
  29. Commercial classification by Jobin-Yvon.

1974 (1)

1973 (2)

T. Namioka, H. Noda, and M. Seya, Sci. Light (Tokyo) 22, 77 (1973).

M. Pouey, C. R. Acad. Sci. (Paris) B276, 531 (1973).

1972 (1)

R. P. Madden and D. L. Ederer, J. Opt. Soc. Am. 62, 722 (1972).

1971 (1)

1970 (3)

S. A. Strezhnev and A. L. Andreeva, Opt. Spektrosk. 28, 796 (1970) [Opt, Spectrosc. 28, 426 (1970)].

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].

M. Pouey, Opt. Commun. 2, 339 (1970).
[Crossref]

1969 (1)

M. Singh and K. Majumder, Sci. Light (Tokyo) 18, 57 (1969).

1967 (1)

Y. Sakayanagi, Sci. Light (Tokyo) 16, 129 (1967).

1966 (2)

B. Gale, Opt. Acta 13, 41 (1966).
[Crossref]

E. Schönheit, Optik 23, 305 (1966).

1964 (1)

I. V. Peisakhson and I. N. Tarnakin, Zh. Prikl. Spektrosk. 1, 289 (1964); Zh. Prikl. Spektrosk. 2, 218 (1965).

1963 (1)

W. T. Welford, Opt. Acta 10, 121 (1963).
[Crossref]

1962 (1)

1961 (1)

1959 (1)

H. Greiner and E. Schäffer, Optik 16, 288 (1959); Optik 16, 350 (1959).

1958 (1)

Yu. P. Shchepetkin, Opt. Spektrosk. 4, 383 (1958); Opt. Spektrosk. 4, 513 (1958).

1950 (1)

1875 (1)

A. Cornu, C. R. Acad. Sci. (Paris) 80, 645 (1875); C. R. Acad. Sci. (Paris) 116, 1215 (1893); C. R. Acad. Sci. (Paris) 116, 1421 (1893); C. R. Acad. Sci. (Paris) 117, 1032 (1893).

Andreeva, A. L.

S. A. Strezhnev and A. L. Andreeva, Opt. Spektrosk. 28, 796 (1970) [Opt, Spectrosc. 28, 426 (1970)].

Baumgardner, J. D.

J. D. Baumgardner, Ph.D. thesis (University of Rochester, 1970).

Cordelle, J.

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Cornu, A.

A. Cornu, C. R. Acad. Sci. (Paris) 80, 645 (1875); C. R. Acad. Sci. (Paris) 116, 1215 (1893); C. R. Acad. Sci. (Paris) 116, 1421 (1893); C. R. Acad. Sci. (Paris) 117, 1032 (1893).

Das, N. C.

Ederer, D. L.

R. P. Madden and D. L. Ederer, J. Opt. Soc. Am. 62, 722 (1972).

Flamand, J.

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Gale, B.

B. Gale, Opt. Acta 13, 41 (1966).
[Crossref]

Gerasimov, F. M.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].

Greiner, H.

H. Greiner and E. Schäffer, Optik 16, 288 (1959); Optik 16, 350 (1959).

Haber, H.

Harada, T.

T. Harada (private communication, 1973).

Koshelev, B. V.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].

Labeyrie, A.

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Madden, R. P.

R. P. Madden and D. L. Ederer, J. Opt. Soc. Am. 62, 722 (1972).

Majumder, K.

M. Singh and K. Majumder, Sci. Light (Tokyo) 18, 57 (1969).

Miller, S. C.

S. C. Miller, , Upper Air Laboratory, University of Colorado (1950).

Murty, M. V. R. K.

Namioka, T.

T. Namioka, H. Noda, and M. Seya, Sci. Light (Tokyo) 22, 77 (1973).

T. Namioka, J. Opt. Soc. Am. 51, 4 (1961); J. Opt. Soc. Am. 51, 13 (1961).
[Crossref]

T. Namioka and H. Noda, in Proceedings of International Symposium for Synchrotron Radiation Users, edited by G. V. Marr and I. H. Munro (Daresbury Nuclear Physics Laboratory, Daresbury, 1973), p. 51.

Noda, H.

T. Namioka, H. Noda, and M. Seya, Sci. Light (Tokyo) 22, 77 (1973).

T. Namioka and H. Noda, in Proceedings of International Symposium for Synchrotron Radiation Users, edited by G. V. Marr and I. H. Munro (Daresbury Nuclear Physics Laboratory, Daresbury, 1973), p. 51.

Peisakhson, I. V.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].

I. V. Peisakhson and I. N. Tarnakin, Zh. Prikl. Spektrosk. 1, 289 (1964); Zh. Prikl. Spektrosk. 2, 218 (1965).

Pieuchard, G.

G. Pieuchard (private communication, 1972).

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Pouey, M.

M. Pouey, C. R. Acad. Sci. (Paris) B276, 531 (1973).

M. Pouey, Opt. Commun. 2, 339 (1970).
[Crossref]

Sakayanagi, Y.

Y. Sakayanagi, Sci. Light (Tokyo) 16, 129 (1967).

Y. Sakayanagi, Sci. Light (Tokyo) 3, 1 (1954/55).

Y. Sakayanagi, Sci. Light (Tokyo) 3, 79 (1954/55).

Schäffer, E.

H. Greiner and E. Schäffer, Optik 16, 288 (1959); Optik 16, 350 (1959).

Schönheit, E.

E. Schönheit, Optik 23, 305 (1966).

Seya, M.

T. Namioka, H. Noda, and M. Seya, Sci. Light (Tokyo) 22, 77 (1973).

Shchepetkin, Yu. P.

Yu. P. Shchepetkin, Opt. Spektrosk. 4, 383 (1958); Opt. Spektrosk. 4, 513 (1958).

Singh, M.

M. Singh and K. Majumder, Sci. Light (Tokyo) 18, 57 (1969).

Strezhnev, S. A.

S. A. Strezhnev and A. L. Andreeva, Opt. Spektrosk. 28, 796 (1970) [Opt, Spectrosc. 28, 426 (1970)].

Tarnakin, I. N.

I. V. Peisakhson and I. N. Tarnakin, Zh. Prikl. Spektrosk. 1, 289 (1964); Zh. Prikl. Spektrosk. 2, 218 (1965).

Welford, W. T.

W. T. Welford, Opt. Acta 10, 121 (1963).
[Crossref]

Woodgate, B. E.

Yakovlev, E. A.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].

C. R. Acad. Sci. (Paris) (2)

A. Cornu, C. R. Acad. Sci. (Paris) 80, 645 (1875); C. R. Acad. Sci. (Paris) 116, 1215 (1893); C. R. Acad. Sci. (Paris) 116, 1421 (1893); C. R. Acad. Sci. (Paris) 117, 1032 (1893).

M. Pouey, C. R. Acad. Sci. (Paris) B276, 531 (1973).

J. Opt. Soc. Am. (6)

Opt. Acta (2)

B. Gale, Opt. Acta 13, 41 (1966).
[Crossref]

W. T. Welford, Opt. Acta 10, 121 (1963).
[Crossref]

Opt. Commun. (1)

M. Pouey, Opt. Commun. 2, 339 (1970).
[Crossref]

Opt. Spektrosk. (3)

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Koshelev, Opt. Spektrosk. 28, 790 (1970) [Opt. Spectrosc. 28, 423 (1970)].

S. A. Strezhnev and A. L. Andreeva, Opt. Spektrosk. 28, 796 (1970) [Opt, Spectrosc. 28, 426 (1970)].

Yu. P. Shchepetkin, Opt. Spektrosk. 4, 383 (1958); Opt. Spektrosk. 4, 513 (1958).

Optik (2)

E. Schönheit, Optik 23, 305 (1966).

H. Greiner and E. Schäffer, Optik 16, 288 (1959); Optik 16, 350 (1959).

Sci. Light (Tokyo) (5)

Y. Sakayanagi, Sci. Light (Tokyo) 3, 79 (1954/55).

Y. Sakayanagi, Sci. Light (Tokyo) 16, 129 (1967).

Y. Sakayanagi, Sci. Light (Tokyo) 3, 1 (1954/55).

T. Namioka, H. Noda, and M. Seya, Sci. Light (Tokyo) 22, 77 (1973).

M. Singh and K. Majumder, Sci. Light (Tokyo) 18, 57 (1969).

Zh. Prikl. Spektrosk. (1)

I. V. Peisakhson and I. N. Tarnakin, Zh. Prikl. Spektrosk. 1, 289 (1964); Zh. Prikl. Spektrosk. 2, 218 (1965).

Other (7)

S. C. Miller, , Upper Air Laboratory, University of Colorado (1950).

T. Harada (private communication, 1973).

J. D. Baumgardner, Ph.D. thesis (University of Rochester, 1970).

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

G. Pieuchard (private communication, 1972).

T. Namioka and H. Noda, in Proceedings of International Symposium for Synchrotron Radiation Users, edited by G. V. Marr and I. H. Munro (Daresbury Nuclear Physics Laboratory, Daresbury, 1973), p. 51.

Commercial classification by Jobin-Yvon.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the optical system.

Fig. 2
Fig. 2

Recording system for a holographic stigmatic concave grating with variable spacing and straight grooves.

Fig. 3
Fig. 3

Recording system for a holographic stigmatic concave grating with variable spacing and curved grooves.

Equations (74)

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x = i = 0 j = 0 a i j y i z j ,             a 00 = a 10 = 0 ,             j = even .
n λ 0 = [ C P - D P ] - [ C O - D O ] ,
C P = [ ( x C - ξ ) 2 + ( y C - w ) 2 + ( z C - l ) 2 ] 1 2 , D P = [ ( x D - ξ ) 2 + ( y D - w ) 2 + ( z D - l ) 2 ] 1 2 , C O = ( r C 2 + z C 2 ) 1 2 ,             D O = ( r D 2 + z D 2 ) 1 2 ,
ξ = i = 0 j = 0 a i j w i l j ,             a 00 = a 10 = 0 ,             j = even .
σ ( n , l ) w ( n + 1 , l ) - w ( n , l ) .
σ [ w / n ] w = l = 0 .
λ 0 = w n [ ( x D - ξ D P - x C - ξ C P ) ξ w + y D - w D P - y C - w C P ] .
σ = λ 0 { [ 1 + ( z D / r D ) 2 ] - 1 2 sin δ - [ 1 + ( z C / r C ) 2 ] - 1 2 sin γ } - 1 .
sin δ > [ 1 + ( z C / r C ) 2 ] - 1 2 [ 1 + ( z D / r D ) 2 ] 1 2 sin γ .
σ = λ 0 [ 1 + ( z D / r D ) 2 ] 1 2 ( sin δ - sin γ ) - 1 ,             δ > γ .
F = A P + P B + n m λ ,
A P = [ ( x - ξ ) 2 + ( y - w ) 2 + ( z - l ) 2 ] 1 2 , P B = [ ( x - ξ ) 2 + ( y - w ) 2 + ( z - l ) 2 ] 1 2 ,
F = A P + P B + ( m λ / λ 0 ) × { [ C P - D P ] - [ C O - D O ] } .
F = F 000 + w F 100 + l F 011 + 1 2 w 2 F 200 + 1 2 l 2 F 020 + 1 2 w 3 F 300 + 1 2 w l 2 F 120 + w l F 111 + 1 8 w 4 F 400 + 1 4 w 2 l 2 F 220 + 1 8 l 4 F 040 + 1 4 w 2 F 202 + 1 4 l 2 F 022 + 1 2 l 3 F 031 + 1 2 w 2 l F 211 + ,
F i j k = M i j k + ( m λ / λ 0 ) H i j k ,
M 000 = r [ 1 + ( z / r ) 2 ] 1 2 + r [ 1 + ( z / r ) 2 ] 1 2 ,
H 000 = r C [ 1 + ( z C / r C ) 2 ] 1 2 - r D [ 1 + ( z D / r D ) 2 ] 1 2 ,
M 100 = - [ 1 + ( z / r ) 2 ] - 1 2 sin α - [ 1 + ( z / r ) 2 ] - 1 2 sin β ,
H 100 = - [ 1 + ( z C / r C ) 2 ] - 1 2 sin γ + [ 1 + ( z D / r D ) 2 ] - 1 2 sin δ ,
M 011 = - ( z / r ) [ 1 + ( z / r ) 2 ] - 1 2 - ( z / r ) [ 1 + ( z / r ) 2 ] - 1 2 ,
H 011 = - ( z C / r C ) [ 1 + ( z C / r C ) 2 ] - 1 2 + ( z D / r D ) [ 1 + ( z D / r D ) 2 ] - 1 2 ,
M 200 = ( cos 2 α / r ) + ( cos 2 β / r ) - 2 a 20 ( cos α + cos β ) ,
H 200 = ( cos 2 γ / r C ) - ( cos 2 δ / r D ) - 2 a 20 ( cos γ - cos δ ) ,
M 020 = ( 1 / r ) + ( 1 / r ) - 2 a 02 ( cos α + cos β ) ,
H 020 = ( 1 / r C ) - ( 1 / r D ) - 2 a 02 ( cos γ - cos δ ) ,
M 300 = [ T ( r , α ) / r ] sin α + [ T ( r , β ) / r ] sin β - 2 a 30 ( cos α + cos β ) ,
H 300 = [ T ( r C , γ ) / r C ] sin γ - [ T ( r D , δ ) / r D ] sin δ - 2 a 30 ( cos γ - cos δ ) ,
M 120 = [ S ( r , α ) / r ] sin α + [ S ( r , β ) / r ] sin β - 2 a 12 ( cos α + cos β ) ,
H 120 = [ S ( r C , γ ) / r C ] sin γ - [ S ( r D , δ ) / r D ] sin δ - 2 a 12 ( cos γ - cos δ ) ,
M 111 = - ( z / r 2 ) sin α - ( z / r 2 ) sin β ,
H 111 = - ( z C / r C 2 ) sin γ + ( z D / r D 2 ) sin δ ,
M 400 = [ 4 T ( r , α ) / r 2 ] sin 2 α - [ T 2 ( r , α ) / r ] + [ 4 T ( r , β ) / r 2 ] sin 2 β - [ T 2 ( r , β ) / r ] + 4 a 20 2 [ ( 1 / r ) + ( 1 / r ) ] - 8 a 30 [ ( sin α cos α / r ) + ( sin β cos β / r ) ] - 8 a 40 ( cos α + cos β ) ,
H 400 = [ 4 T ( r C , γ ) / r C 2 ] sin 2 γ - [ T 2 ( r C , γ ) / r C ] - [ 4 T ( r D , δ ) / r D 2 ] sin 2 δ + [ T 2 ( r D , δ ) / r D ] + 4 a 20 2 [ ( 1 / r C ) - ( 1 / r D ) ] - 8 a 30 [ ( sin γ cos γ / r C ) - ( sin δ cos δ / r D ) ] - 8 a 40 ( cos γ - cos δ ) ,
M 220 = [ 2 S ( r , α ) / r 2 ] sin 2 α + [ 2 S ( r , β ) / r 2 ] sin 2 β - [ T ( r , α ) S ( r , α ) / r ] - [ T ( r , β ) S ( r , β ) / r ] + 4 a 20 a 02 [ ( 1 / r ) + ( 1 / r ) ] - 4 a 22 ( cos α + cos β ) - 4 a 12 [ ( sin α cos α / r ) + ( sin β cos β / r ) ] ,
H 220 = [ 2 S ( r C , γ ) / r C 2 ] sin 2 γ - [ 2 S ( r D , δ ) / r D 2 ] sin 2 δ - [ T ( r C , γ ) S ( r C , γ ) / r C ] + [ T ( r D , δ ) S ( r D , δ ) / r D ] + 4 a 20 a 02 [ ( 1 / r C ) - ( 1 / r D ) ] - 4 a 22 ( cos γ - cos δ ) - a 12 [ ( sin γ cos γ / r C ) - ( sin δ cos δ / r D ) ] ,
M 040 = 4 a 02 2 [ ( 1 / r ) + ( 1 / r ) ] - 8 a 04 ( cos α + cos β ) - [ S 2 ( r , α ) / r ] - [ S 2 ( r , β ) / r ] ,
H 040 = 4 a 02 2 [ ( 1 / r C ) - ( 1 / r D ) ] - 8 a 04 ( cos γ - cos δ ) - [ S 2 ( r C , γ ) / r C ] + [ S 2 ( r D , δ ) / r D ] ,
M 202 = ( z / r ) 2 [ ( 2 sin 2 α / r ) - T ( r , α ) ] + ( z / r ) 2 [ ( 2 sin 2 β / r ) - T ( r , β ) ] ,
H 202 = ( z C / r C ) 2 [ ( 2 sin 2 γ / r C ) - T ( r C , γ ) ] - ( z D / r D ) 2 [ ( 2 sin 2 δ / r D ) - T ( r D , δ ) ] ,
M 022 = - ( z / r ) 2 [ ( 2 / r ) + S ( r , α ) ] - ( z / r ) 2 [ ( 2 / r ) + S ( r , β ) ] ,
H 022 = - ( z C / r C ) 2 [ ( 2 / r C ) + S ( r C , γ ) ] + ( z D / r D ) 2 [ ( 2 / r D ) + S ( r D , δ ) ] ,
M 031 = ( z / r 2 ) S ( r , α ) + ( z / r 2 ) S ( r , β ) ,
H 031 = ( z C / r C 2 ) S ( r C , γ ) - ( z D / r D 2 ) S ( r D , δ ) ,
M 211 = ( z / r 2 ) [ T ( r , α ) - ( 2 sin 2 α / r ) ] + ( z / r 2 ) [ T ( r , β ) - ( 2 sin 2 β / r ) ] ,
H 211 = ( z C / r C 2 ) [ T ( r C , γ ) - ( 2 sin 2 γ / r C ) ] - ( z D / r D 2 ) [ T ( r D , δ ) - ( 2 sin 2 δ / r D ) ] ,
T ( r , α ) = ( cos 2 α / r ) - 2 a 20 cos α ,
S ( r , α ) = ( 1 / r ) - 2 a 02 cos α .
F / w = 0             and             F / l = 0 ,
[ 1 + ( z / r ) 2 ] - 1 2 sin α + [ 1 + ( z 0 / r 0 ) 2 ] - 1 2 sin β 0 = ( m λ / λ 0 ) { [ 1 + ( z D / r D ) 2 ] - 1 2 sin δ - [ 1 + ( z C / r C ) 2 ] - 1 2 sin γ } = m λ / σ
( z / r ) [ 1 + ( z / r ) 2 ] - 1 2 + ( z 0 / r 0 ) [ 1 + ( z 0 / r 0 ) 2 ] - 1 2 = ( m λ / λ 0 ) { ( z D / r D ) [ 1 + ( z D / r D ) 2 ] - 1 2 - ( z C / r C ) [ 1 + ( z C / r C ) 2 ] - 1 2 }
( cos 2 α / r ) + ( cos 2 β / r ) - 2 a 20 ( cos α + cos β ) + ( m λ / λ 0 ) [ ( cos 2 γ / r C ) - ( cos 2 δ / r D ) - 2 a 20 ( cos γ - cos δ ) ] = 0
( 1 / r ) + ( 1 / r ) - 2 a 02 ( cos α + cos β ) + ( m λ / λ 0 ) [ ( 1 / r C ) - ( 1 / r D ) - 2 a 02 ( cos γ - cos δ ) ] = 0.
I i j k λ 1 λ 2 F i j k 2 d λ = minimum ,             i + j + k 2.
A i j k = ( σ / λ 0 ) H i j k ,             i + j + k 2.
[ 1 + ( z D / r D ) 2 ] - 1 2 sin δ - [ 1 + ( z C / r C ) 2 ] - 1 2 sin γ = λ 0 / σ
H i j k = ( λ 0 / σ ) A i j k
x C = R ,             y C = R tan γ ,             z C = 0 , x D = R ,             y D = z D = 0 ,             δ = 0.
γ = - sin - 1 ( λ 0 / σ ) < 0.
w ( n , l ) = - ( n λ 0 / sin γ ) - [ ( n λ 0 ) 2 / 2 R tan γ ] = n σ - n 2 σ 2 ( sin 2 γ / 4 R ) .
σ ( n , l ) = σ [ 1 - σ ( n + 1 2 ) ( sin 2 γ / 2 R ) ] = σ { 1 + ( λ 0 / 2 R ) [ 1 - ( λ 0 / σ ) 2 ] 1 2 } + ( n σ λ 0 / R ) [ 1 - ( λ 0 / σ ) 2 ] 1 2 .
d n = σ 0 + n · Δ σ ,
σ 0 = σ { 1 + ( λ * / 2 R ) [ 1 - ( λ * / σ ) 2 ] 1 2 } ,
Δ σ = ( σ λ * / R ) [ 1 - ( λ * / σ ) 2 ] 1 2 .
w ( n , l ) = { 1 - [ 1 - ( sin 2 γ / 2 R ) σ ] n } / ( sin 2 γ / 2 R ) n σ - n ( n - 1 ) σ 2 ( sin 2 γ / 4 R )
σ ( n , l ) = σ [ 1 - ( sin 2 γ / 2 R ) w ( n , l ) ] = σ [ 1 - σ ( sin 2 γ / 2 R ) ] n σ [ 1 - n σ ( sin 2 γ / 2 R ) ] .
x C = R cos 2 γ ,             y C = R sin γ cos γ ,             z C = 0 , x D = R ,             y D = z D = δ = 0.
w ( n , l ) = - ( n λ 0 / sin γ ) - [ ( n λ 0 ) 2 / 2 R tan γ ] ,
σ ( n , l ) = - ( λ 0 / sin γ ) [ 1 + ( λ 0 cos γ / 2 R ) ] - ( n λ 0 2 / R tan γ ) .
R sin γ = w ( n 0 , 0 ) = - ( n 0 λ 0 / sin γ ) - [ ( n 0 λ 0 ) 2 / 2 R tan γ ] .
sin γ = tan γ .
n 0 λ 0 = R [ cos γ - ( 1 + sin 2 γ ) 1 2 ] .
n = n - n 0 .
d n = σ ( n , 0 ) sec γ = - ( λ 0 / sin γ ) [ 1 + ( λ 0 / 2 R cos γ ) ] - ( n λ 0 2 / R sin γ cos γ ) = σ { 1 + ( λ * / 2 R ) [ 1 - ( λ * / σ ) 2 ] - 1 2 } + ( n σ λ * / R ) [ 1 - ( λ * / σ ) 2 ] - 1 2 .
w ( n , l ) = ( cot γ / 2 R ) [ 2 R 2 sin 2 γ - 2 R n λ 0 cos γ - ( n λ 0 ) 2 ] + ( 1 / 2 R ) { 4 R 2 ( R 2 - l 2 ) sin 2 γ - [ n λ 0 ( 2 R cos γ + n λ 0 ) - 2 R 2 sin 2 γ ] 2 } 1 2 .