Abstract

We describe the aberrations of diffraction gratings without any restrictions on the symmetry of the substrate, the curvature of the grooves, or the position of the entrance slit. The theory can also be applied to holographic gratings, holographic lenses, synthetic holograms, zone plates, and other diffracting instruments. The description comprises second-order focusing conditions, including the generalization of the Rowland circle and the conditions for astigmatism, length, slope, and curvature of the focal lines, comatic and fourth-order aberrations. In particular we show the relations between the aberrations of second and higher orders. The theory is applied in this paper to astigmatic mountings of the Rowland type, and to mountings of the Wadsworth and Eagle type that we call anastigmatic. We give special attention to the conditions for freedom of coma of anastigmatic mountings.

© 1976 Optical Society of America

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Corrections

C. H. F. Velzel, "Errata," J. Opt. Soc. Am. 67, 1695-1695 (1977)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-67-12-1695

References

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  1. W. Werner, Appl. Opt. 6, 1691 (1967).
    [Crossref] [PubMed]
  2. W. Werner, thesis (Delft, 1970) (unpublished).
  3. H. Noda, T. Namioka, and M. Seya, J. Opt. Soc. Am. 64, 1031 (1974).
    [Crossref]
  4. F. Zernike, in Festschrift, Pieter Zeeman (Martinus Nijhoff, The Hague, 1935), p. 323.
  5. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [Crossref]
  6. T. Namioka, J. Opt. Soc. Am. 49, 446 (1959).
    [Crossref]
  7. G. Stroke, Handbuch der Physik XXIX (Springer, Berlin, 1967), p. 447.
  8. W. T. Welford, Progress in Optics, Vol. IV (North-Holland, Amsterdam, 1965).
  9. G. S. Monk, Light, Principles and Experiments, 1st ed. (McGraw-Hill, New York, 1937), p. 424.
  10. A. Danielson and P. Lindblom, Optik 41, 441 (1974).
  11. A. Danielson and P. Lindblom, Optik 41, 465 (1975).
  12. M. Pouey, J. Opt. Soc. Am. 64, 1616 (1974).
    [Crossref]

1975 (1)

A. Danielson and P. Lindblom, Optik 41, 465 (1975).

1974 (3)

1967 (1)

1959 (1)

1945 (1)

Beutler, H. G.

Danielson, A.

A. Danielson and P. Lindblom, Optik 41, 465 (1975).

A. Danielson and P. Lindblom, Optik 41, 441 (1974).

Lindblom, P.

A. Danielson and P. Lindblom, Optik 41, 465 (1975).

A. Danielson and P. Lindblom, Optik 41, 441 (1974).

Monk, G. S.

G. S. Monk, Light, Principles and Experiments, 1st ed. (McGraw-Hill, New York, 1937), p. 424.

Namioka, T.

Noda, H.

Pouey, M.

Seya, M.

Stroke, G.

G. Stroke, Handbuch der Physik XXIX (Springer, Berlin, 1967), p. 447.

Welford, W. T.

W. T. Welford, Progress in Optics, Vol. IV (North-Holland, Amsterdam, 1965).

Werner, W.

W. Werner, Appl. Opt. 6, 1691 (1967).
[Crossref] [PubMed]

W. Werner, thesis (Delft, 1970) (unpublished).

Zernike, F.

F. Zernike, in Festschrift, Pieter Zeeman (Martinus Nijhoff, The Hague, 1935), p. 323.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Optik (2)

A. Danielson and P. Lindblom, Optik 41, 441 (1974).

A. Danielson and P. Lindblom, Optik 41, 465 (1975).

Other (5)

W. Werner, thesis (Delft, 1970) (unpublished).

F. Zernike, in Festschrift, Pieter Zeeman (Martinus Nijhoff, The Hague, 1935), p. 323.

G. Stroke, Handbuch der Physik XXIX (Springer, Berlin, 1967), p. 447.

W. T. Welford, Progress in Optics, Vol. IV (North-Holland, Amsterdam, 1965).

G. S. Monk, Light, Principles and Experiments, 1st ed. (McGraw-Hill, New York, 1937), p. 424.

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Figures (5)

FIG. 1
FIG. 1

Coordinate system for a ray path APB. Object and image points are given in the XYZ coordinate system. The grating surface is given in the ξ, η, ζ coordinate system; ξ and η are referred to as pupil coordinates.

FIG. 2
FIG. 2

Diagram of a Rowland-type mounting. The entrance and exit slits A and B lie on a circle with diameter r that touches the spherical grating surface in 0.

FIG. 3
FIG. 3

Aberration curve of a Rowland-type mounting with i′ = 0, i = 1 4 π, numerical aperture ρ / s = 1 4. The drawn curve shows the effect of astigmatism and coma; the interrupted curve gives the effect of fourth-order aberration in addition.

FIG. 4
FIG. 4

Focal curve of an off-plane anastigmatic mounting (off-plane Eagle mounting).

FIG. 5
FIG. 5

Aberration curve of an off-plane anastigmatic mounting due to elliptical coma and fourth-order aberrations. The pupil azimuth is given alongside the curve. The widths Δ(3)λ/λ0 and Δ(4)λ/λ0 discussed in the text are also shown in the figure. The numerical aperture was taken as 1 4 and the off-plane angle i as 1 4 π.

Equations (71)

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ζ ( ξ , η ) = Z 20 ξ 2 + Z 11 ξ η + Z 02 η 2 + + Z k 1 ξ k η l + + Z 04 η 4 .
F = A P + P B + G ( ξ , η ) ,
G ( ξ , η ) = m ξ λ / p
G ( ξ , η ) = m Λ { C P ± D P } ,
F ξ = 0 ,             F η = 0 ;
F = F 00 + F 10 ξ + F 01 η + + F k l ξ k η l + F 04 η 4 .
F ξ = α 00 + α 10 ξ + α 01 η + α k l ξ k η l + α 03 η 3 , F η = β 00 + β 10 ξ + β 01 η + β k l ξ k η l + β 03 η 3 ,
α k l = ( k + 1 ) F k + 1 , l ,             β k l = ( l + 1 ) F k , l + 1 .
α 00 = 0 ,             β 00 = 0.
A P = [ ( x - ξ ) 2 + ( y - η ) 2 + ( z - ζ ) 2 ] 1 / 2 , P B = [ ( x - ξ ) 2 + ( y - η ) 2 + ( z - ζ ) 2 ] 1 / 2 ,
α 00 = - x / s - x / s + m λ / p , β 00 = - y / s - y / s .
M + M = 0 ,             L + L = m λ / p .
- L - L + m λ / p + α 10 ξ + α 01 η = 0 , - M - M + β 10 ξ + β 01 η = 0.
L 2 = α 10 ξ + α 01 η ,             M 2 = β 10 ξ + β 01 η .
α 10 = 1 - L 2 s - 2 N Z 20 + 1 - L 1 2 s - 2 N 1 Z 20 , β 01 = 1 - M 2 s - 2 N Z 02 + 1 - M 1 2 s - 2 N 1 Z 02 , α 01 = β 10 = - L M s - N Z 11 - L 1 M 1 s - N 1 Z 11 ,
( 1 - L 2 s - 2 N Z 20 + 1 - L 1 2 s - 2 N 1 Z 20 ) × ( 1 - M 2 s - 2 N Z 02 + 1 - M 1 2 s - 2 N Z 02 ) = ( L M s + N Z 11 + L 1 M 1 s + N 1 Z 11 ) 2 .
( 1 - L 2 ) / s - 2 N Z 20 + ( 1 - L 1 2 ) / s 1 - 2 N 1 Z 20 = 0 ,
1 / s - 2 N Z 02 + 1 / s 2 - 2 N 1 Z 02 = 0.
( 1 - L 2 s - 2 N Z 20 ) ( 1 - M 2 s - 2 N Z 02 ) = ( L M s + N Z 11 ) 2 , ( 1 - L 2 s - 2 N Z 20 ) = c 2 ( 1 - M 2 s - 2 N Z 02 ) ,
η / ξ = tan ϑ 1 , 2 = - ( α 10 / α 01 ) 1 , 2 ,
L 2 = ρ sin ( ϑ - ϑ 1 ) ( α 10 2 + α 01 2 ) 1 / 2 , M 2 = ρ sin ( ϑ - ϑ 1 ) ( β 10 2 + β 01 2 ) 1 / 2 ,
x = L 1 s 1 + Δ ( 1 ) x = L 1 s 1 + L 2 s 1 , y = M 1 s 1 + Δ ( 1 ) y = M 1 s 1 + M 2 s 1 , z = N 1 s 1 + Δ ( 1 ) z ,
Δ ( 1 ) x L 1 + Δ ( 1 ) y M 1 + Δ ( 1 ) z N 1 = 0.
- Δ ( 1 ) y = Δ ( 1 ) x cot ϑ 1 .
Δ ( 1 ) x Δ ( 2 ) x + Δ ( 1 ) y Δ ( 2 ) y + Δ ( 1 ) z Δ ( 2 ) z = 0 ,
( 1 - L 2 ) / s - 2 N Z 20 + ( 1 - L 1 2 ) / s - 2 N 1 Z 20 = 0 ,
( 1 - M 2 ) / s - 2 N Z 02 + ( 1 - M 1 2 ) / s - 2 N 1 Z 02 = 0 ,
L M / s + N Z 11 + L 1 M 1 / s + N 1 Z 11 = 0.
2 Z 20 M ( 1 - M 2 ) = 2 Z 02 M ( 1 - L 1 L ) + Z 11 ) ( 1 - M 2 ) ( L 1 - L ) .
Z 20 ( 1 - M 2 ) = Z 02 ( 1 - L 1 L )
1 s = 2 L 1 ( N + N 1 ) Z 02 ( 1 - M 2 ) ( L + L 1 ) ,
1 s = 1 s L L 1 .
( 1 + N 2 ) / s + ( 1 + N 1 2 ) / s = 2 ( N + N 1 ) ( Z 20 + Z 02 ) .
α k 1 = A k 1 + A k 1 ,             β k 1 = B k 1 + B k 1 ,
L 3 = α 20 ξ 2 + α 11 ξ η + α 02 η 2 , M 3 = β 20 ξ 2 + β 11 ξ η + β 02 η 2 ,
A 20 = 3 2 L ( 1 - L 2 ) s 2 - 3 L N s Z 20 - 3 N Z 30 , A 11 = 2 B 20 = M ( 1 - 3 L 2 ) s 2 - 2 M N s Z 20 - 2 L N s Z 11 - 2 N Z 21 , 2 A 02 = B 11 = L ( 1 - 3 M 2 ) s 2 - 2 L N s Z 02 - 2 M N s Z 11 - 2 N Z 12 , B 02 = 3 2 M ( 1 - M 2 ) s 2 - 3 M N s Z 02 - 3 N Z 03 .
A 20 = 3 2 ( L / s ) A 10 - 3 N Z 30 , A 11 = 2 B 20 = ( M / s ) A 10 + 2 ( L / s ) A 01 - 2 N Z 21 , 2 A 02 = B 11 = ( L / s ) B 01 + 2 ( M / s ) B 10 - 2 N Z 12 , B 02 = 3 2 ( M / s ) B 01 - 3 N Z 03 ,
A 10 = ( 1 - L 2 ) / s - 2 N Z 20 , B 01 = ( 1 - M 2 ) / s - 2 N Z 02 ,             A 01 = B 10 = - L m / s - N Z 11 ,
L 3 = ρ 2 ( α 20 + α 02 2 + α 20 - α 02 2 cos 2 ϑ + α 11 2 sin 2 ϑ ) , M 3 = ρ 2 ( α 11 + 2 β 02 4 + α 11 - 2 β 02 4 cos 2 ϑ + α 02 sin 2 ϑ ) .
α 20 - α 11 tan ϑ 1 + α 02 tan 2 ϑ 1 = 0.
L 4 = α 30 ξ 3 + α 21 ξ 2 η + α 12 ξ η 2 + α 03 η 3 , M 4 = β 30 ξ 3 + β 21 ξ 2 η + β 12 ξ η 2 + β 03 η 3 .
A 30 = - 1 2 1 - 6 L 2 - 15 L 4 s 3 + 2 1 - 3 L 2 s 2 N Z 20 - 4 L N s Z 30 + 2 1 - N 2 s Z 20 2 - 4 N Z 40 , A 21 = 3 B 30 = 3 2 3 L M - 5 L 3 M s 3 + 3 2 1 - 3 L 2 s 2 N Z 11 - 9 L M s 2 N Z 20 - 3 L s N Z 21 + 3 1 - N 2 s Z 11 Z 20 - 3 N Z 31 , A 12 = B 21 = - 1 2 1 - 3 L 2 - 3 M 2 - 15 L 2 M 2 s 3 + 1 - 3 L 2 s 2 N Z 20 + 1 - 3 M 2 s 2 N Z 02 - 6 L M s 2 N Z 11 - 2 L s N Z 12 - 2 M s N Z 22 + 2 1 - N 2 s Z 20 Z 02 + 1 - N 2 s Z 11 2 - 2 N Z 22 , 3 A 03 = B 12 = 3 2 3 L M - 5 L M 3 s 3 + 3 2 1 - 3 M 2 s 2 N Z 11 - 9 L M s 2 N Z 02 - 3 M s N Z 12 + 3 1 - N 2 s Z 11 Z 02 - 3 N Z 13 , B 03 = - 1 2 1 - 6 M 2 - 15 M 4 s 3 + 2 1 - 3 M 2 s 2 N Z 02 - 4 M s N Z 03 + 2 1 - N 2 s Z 02 2 - 4 N Z 04 .
A 30 = 4 3 L s A 20 - 1 2 s A 10 2 + 2 s Z 20 2 - 4 N Z 40 , A 21 = 3 B 30 = 3 2 L s A 11 + M s A 20 - 3 2 1 s A 10 A 01 + 3 1 s Z 11 Z 20 - 3 N Z 31 , A 12 = B 21 = M s A 11 + L s B 11 - 1 s A 10 B 01 - 1 s A 01 B 10 + 2 s Z 02 Z 20 + 1 s Z 11 2 - 2 N Z 22 , 3 A 03 = B 12 = 3 2 M s B 11 + L s B 02 - 3 2 1 s B 01 B 10 + 3 1 s Z 11 Z 02 - 3 N Z 13 , B 03 = 4 3 M s A 02 - 1 2 s B 01 2 + 2 s Z 02 2 - 4 N Z 04 .
B 21 = A 12 = A 30 = B 03 = 2 3 Z 20 2 - 4 N Z 40 , A 21 = B 30 = A 03 = B 12 = 0.
L 4 = ρ 3 [ ( 3 4 α 30 + 1 2 α 12 ) cos ϑ + ( 3 4 α 03 + 1 4 α 21 ) sin ϑ + ( 1 4 α 30 - 1 4 α 12 ) cos 3 ϑ + ( - 1 4 α 03 + 1 4 α 21 ) sin 3 ϑ ] M 4 = ρ 3 [ ( 1 4 α 21 + 3 4 α 03 ) cos ϑ + ( 3 4 β 03 + 1 4 α 12 ) sin ϑ + ( 1 12 α 21 - 3 4 α 03 ) cos 3 ϑ + ( - 1 4 β 03 + 1 4 α 12 ) sin 3 ϑ ] .
L 2 = ( 1 - L 2 s - 2 N Z 20 + 1 - L 2 s - 2 N Z 20 ) ξ , - ( L M s + N Z 11 + L M s + N Z 11 ) η , M 2 = ( 1 - M 2 s - 2 N Z 02 + 1 - M 2 s - 2 N Z 02 ) ξ , - ( L M s + N Z 11 + L M s + N Z 11 ) η ,
L 3 , 2 = [ ( 2 L 1 s + 2 L 1 N 1 Z 20 ) L 2 + 2 M 1 N 1 Z 20 M 2 ] ξ + [ ( M 1 s + L 1 N 1 Z 11 ) L 2 + ( L 1 s + M 1 N 1 Z 11 ) M 2 ] η , M 3 , 2 = [ ( 2 M 1 s + 2 M 1 N 1 Z 02 ) M 2 + 2 L 1 N 1 Z 02 · L 2 ] η + [ ( L 1 s + M 1 N 1 Z 11 ) M 2 + ( M 1 s + L 1 N 1 Z 11 ) L 2 ] ξ .
Z 20 = Z 02 = 1 / 2 r , 2 Z 40 = Z 22 = 2 Z 04 = 1 / 4 r 3 ;
cos 2 i / s - cos i / r + cos 2 i / s 1 - cos i / r = 0 , 1 / s - cos i / r + 1 / s 2 - cos i / r = 0 ,
L 2 = 0 ,             M 2 = ( η / r ) ( sin i tan i + sin i tan i ) ( η / r ) P 2 .
L 3 = ( η 2 / r 2 ) ( P 3 + Q 3 ) ,             M 3 = ( ξ η / r 2 ) ( 2 P 3 + Q 3 ) ,
R = P 2 2 s 1 / 2 ( P 3 + Q 3 ) .
A 30 = A 12 = B 21 = B 03 = 1 2 P 2
L 4 = ( ξ 2 / r 3 ) P 4 + ( ξ η 2 / r 3 ) Q 4 ,             M 4 = ( ξ 2 η / r 3 ) R 4 + ( η 3 / r 3 ) S 4 ,
P 4 = 1 2 P 2 , Q 4 = 1 2 P 2 + tan i ( 5 P 3 + 4 Q 3 ) - 3 tan 2 i P 2 , R 4 = 1 2 P 2 + tan i ( 2 P 3 + Q 3 ) - 3 2 tan 2 i P 2 , S 4 = 1 2 P 2 + tan i ( P 3 + Q 3 ) + P 2 2 / cos i .
m Δ λ / p = ( ρ 3 / r 3 ) P 2 = ( ρ 3 / r 3 ) sin i tan i .
α 20 = 0 , α 11 = 2 β 20 = M ( 1 / s + 1 / s ) A 10 , 2 α 02 = β 11 = 2 M ( 1 / s + 1 / s ) B 10 , β 02 = 3 2 M ( 1 / s + 1 / s ) B 01 ,
L L 1 = L ( m λ / p - L ) = 0.
α 30 = α 12 = β 21 = β 03 = 4 ( 1 + N ) ( L 2 Z 20 3 - Z 40 ) , α 21 = β 30 = β 12 = α 03 = 0.
β 01 = ( λ - λ 0 / λ 0 ) 2 L 2 / s .
α 20 = 3 2 [ ( λ - λ 0 ) / λ 0 ] L N ( N + 1 ) 1 / r 2 , α 11 = 2 β 20 = 0 , 2 α 02 = β 11 = [ ( λ - λ 0 ) / λ 0 ] L ( N + 1 2 ) ( N + 1 ) 1 / r 2 , β 02 = 0 ,
m Δ λ / p = ρ 2 1 2 ( α 20 + α 02 ) = ( Δ λ / λ 0 ) L .
n = 2 ( λ - λ 0 ) / Δ λ
n Ω = 32 π ( N + 1 ) / ( 5 N + 1 ) .
L 1 = L ,             1 / s = 1 / s ,             ( 1 - L 2 ) / s = 2 N Z 20 .
M = ± L .
s = 1 Z 20 1 + cos 2 i 4 cos i .
2 α 02 = β 11 = - 4 L 3 / s 2
Δ ( 3 ) λ / λ 0 = 1 2 L 2 b 2 / s 2 ,
β 03 = α 30 = ( 4 / s ) Z 20 2 - 8 N Z 40 , α 12 = β 21 = - 6 L 4 / s 3 + ( 4 / s ) Z 20 2 - 4 N Z 22 .
Δ ( 4 ) λ / λ 0 = 2 ( α 30 + α 12 ) b 3 ( 1 / L )