Abstract

We review the theory of spherical diffraction gratings with regard to their imaging properties in off-plane arrangements. Our study is restricted to gratings with equally spaced grooves, and it is focused on the quadrature configuration, where the incident and diffraction planes are orthogonal to each other. We identify regions of low astigmatism and propose some monochromator mounts.

© 2009 Optical Society of America

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References

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  1. W. Werner, “The geometric optical aberration theory of diffraction gratings,” Appl. Opt. 6, 1691-1699 (1967).
    [CrossRef] [PubMed]
  2. A. Danielsson and P. Lindblom, “Focusing conditions of the spherical concave grating. I,” Optik (Stuttgart) 41, 441-451 (1974).
  3. C. H. F. Velzel, “A general theory of the aberration of diffraction gratings and gratinglike optical instruments,” J. Opt. Soc. Am. 66, 346-353 (1976).
    [CrossRef]
  4. S. Morozumi, “Aberration theory of diffraction gratings,” Optik (Stuttgart) 53, 75-88 (1979).
  5. S. D. Brorson and H. A. Haus, “Diffraction gratings and geometrical optics,” J. Opt. Soc. Am. B 5, 247-248 (1988).
    [CrossRef]
  6. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), chap. 17.
  7. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), chap. 12.
  8. J. Dahlbacka and P. Lindblom, “Off-plane stigmatic imaging with spherical concave gratings: experimental verification,” Appl. Opt. 16, 288-289 (1977).
    [CrossRef] [PubMed]
  9. J. P. Schwenker, “Stigmatic monochromator design using spherical, normally ruled concave diffraction gratings,” Appl. Opt. 28, 3292-3293 (1989).
    [CrossRef] [PubMed]

1989 (1)

1988 (1)

1979 (1)

S. Morozumi, “Aberration theory of diffraction gratings,” Optik (Stuttgart) 53, 75-88 (1979).

1977 (1)

1976 (1)

1974 (1)

A. Danielsson and P. Lindblom, “Focusing conditions of the spherical concave grating. I,” Optik (Stuttgart) 41, 441-451 (1974).

1967 (1)

Brorson, S. D.

Dahlbacka, J.

Danielsson, A.

A. Danielsson and P. Lindblom, “Focusing conditions of the spherical concave grating. I,” Optik (Stuttgart) 41, 441-451 (1974).

Haus, H. A.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), chap. 17.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), chap. 17.

Lindblom, P.

J. Dahlbacka and P. Lindblom, “Off-plane stigmatic imaging with spherical concave gratings: experimental verification,” Appl. Opt. 16, 288-289 (1977).
[CrossRef] [PubMed]

A. Danielsson and P. Lindblom, “Focusing conditions of the spherical concave grating. I,” Optik (Stuttgart) 41, 441-451 (1974).

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), chap. 12.

Morozumi, S.

S. Morozumi, “Aberration theory of diffraction gratings,” Optik (Stuttgart) 53, 75-88 (1979).

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), chap. 12.

Schwenker, J. P.

Velzel, C. H. F.

Werner, W.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Optik (Stuttgart) (2)

S. Morozumi, “Aberration theory of diffraction gratings,” Optik (Stuttgart) 53, 75-88 (1979).

A. Danielsson and P. Lindblom, “Focusing conditions of the spherical concave grating. I,” Optik (Stuttgart) 41, 441-451 (1974).

Other (2)

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (Dover, 2000), chap. 17.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997), chap. 12.

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Figures (7)

Fig. 1
Fig. 1

Scheme of the spherical diffraction grating showing the notation used in the text.

Fig. 2
Fig. 2

Spot diagram obtained for a generalized Wadsworth mounting showing coma aberration for λ = 600 nm . In this plot, the grating grooves are along the vertical dimension and consequently the spectral direction is horizontal. The parameters of the simulation are given in the text. Units are millimeters.

Fig. 3
Fig. 3

Ray tracing in the Fresnel approximation for a diffraction grating in quadrature. (a) Rays that hit the grating along the plane X Z ; (b) rays that hit the grating in the Y Z plane.

Fig. 4
Fig. 4

Longitudinal astigmatism as a function of the object angular coordinate θ 0 and different values of α 0 (see the legend) for the configuration of quadrature. (a) r 0 = R cos θ 0 , (b) r 0 = R cos θ 0 , (c) r 0 = R ( cos θ 0 + 1 cos θ 0 ) 2 . Both θ 0 and α 0 are in degrees.

Fig. 5
Fig. 5

Locus of object and image position for anastigmatic imaging in quadrature. The dashed curve is part of a circumference of radius R.

Fig. 6
Fig. 6

Spot diagrams of the image point for the anastigmatic (left) and approximate anastigmatic (right) mounting. (a) and (e) λ = 200 nm , (b) and (f) λ = 500 nm , (c) and (g) λ = 800 nm . In (d) and (h) we show the image for λ = 500 nm as in (b) and (f), but we include two close wavelengths: λ = 499 nm , λ = 501 nm . Units are millimeters.

Fig. 7
Fig. 7

Same as Fig. 6 but r 0 = R cos θ 0 (left) and r 0 = R cos θ 0 (right).

Equations (45)

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U ( x 1 , y 1 , z 1 ) = K S R ( x , y ) exp [ i 2 π λ ( A P + P B ) ] d x d y ,
R ( x , y ) = m = c m exp ( i 2 π m x d ) .
U ( x 1 , y 1 , z 1 ) = m = c m S exp [ i 2 π λ ( A P + P B + m λ x d ) ] d x d y ,
U m ( x 1 , y 1 , z 1 ) = c m S exp [ i 2 π λ ( A P + P B + m λ x d ) ] d x d y .
F = ( A P + P B + m λ x d ) = r 0 + r 1 + F x x + F y y + 1 2 ( F x x x 2 + 2 F x y x y + F y y y 2 ) + W ( x , y ) ,
x 0 r 0 + x 1 r 1 = m λ d ,
y 0 r 0 + y 1 r 1 = 0 ,
sin θ 0 cos α 0 + sin θ 1 cos α 1 = m λ d ,
sin θ 0 sin α 0 + sin θ 1 sin α 1 = 0.
U m ( x 1 , y 1 , z 1 ) = c m S exp [ i π λ ( F x x x 2 + 2 F x y x y + F y y y 2 ) ] d x d y .
tan 2 ψ = 2 F x y F x x F y y .
K G = | F x x F x y F x y F y y | ,
F x y 2 = F x x F y y .
( 1 + cos 2 θ 0 r 0 + 1 + cos 2 θ 1 r 1 2 cos θ 0 + cos θ 1 R ) 2 = ( sin 2 θ 0 r 0 sin 2 θ 1 r 1 ) 2 + 4 sin 2 θ 0 sin 2 θ 1 r 0 r 1 cos 2 ( α 1 α 0 ) ,
( 1 + cos 2 θ 0 r 0 + 1 + cos 2 θ 1 r 1 2 cos θ 0 + cos θ 1 R ) 2 = ( sin 2 θ 0 r 0 + sin 2 θ 1 r 1 ) 2 4 sin 2 θ 0 sin 2 θ 1 r 0 r 1 sin 2 ( α 1 α 0 ) .
( cos 2 θ 0 r 0 + cos 2 θ 1 r 1 cos θ 0 + cos θ 1 R ) ( 1 r 0 + 1 r 1 cos θ 0 + cos θ 1 R ) = 0.
1 r 0 + 1 r 1 = 2 R .
r 0 sin 2 θ 1 = r 1 sin 2 θ 0 = R 1 cos 2 θ 0 cos 2 θ 1 cos θ 0 + cos θ 1 .
sin θ 1 = sin θ 0 tan | α 0 | ,
sin θ 0 = m λ d cos α 0 .
sin 2 θ 0 + sin 2 θ 1 = ( m λ d ) 2 ,
r 1 r 0 = tan 2 α 0 .
θ 1 = θ 0 ,
r 1 = r 0 = R 2 ( cos θ 0 + 1 cos θ 0 ) .
r 0 R 1 cos θ 0 cos θ 1 2 ( 1 cos θ 1 ) ,
r 1 R 1 cos θ 0 cos θ 1 sin 2 θ 0 .
( cos 2 θ 0 r 0 + 1 r 1 cos θ 0 + cos θ 1 R ) ( 1 r 0 + cos 2 θ 1 r 1 cos θ 0 + cos θ 1 R ) = sin 2 θ 0 sin 2 θ 1 r 0 r 1 cos 2 ( α 1 α 0 ) ,
cos 2 θ 0 r 0 + 1 r 1 = 1 R ( cos θ 0 + cos θ 1 ) ,
1 r 0 + cos 2 θ 1 r 1 = 1 R ( cos θ 0 + cos θ 1 ) .
r 1 x = R cos θ 1 ,
r 1 y = R cos θ 0 cos 2 θ 1 cos θ 0 cos θ 1 sin 2 θ 0 ,
Δ r 1 = r 1 y r 1 x = R sin 2 θ 0 sin 2 θ 1 cos θ 0 cos θ 1 cos θ 1 ( cos θ 0 cos θ 1 sin 2 θ 0 ) .
r 1 x = R cos θ 0 sin 2 θ 0 + cos θ 1 ,
r 1 y = R cos θ 1 ,
Δ r 1 = r 1 y r 1 x = R sin 2 θ 0 cos θ 0 cos θ 1 sin 2 θ 1 cos θ 1 + cos θ 0 sin 2 θ 0 .
F = ( A P + P B + m λ x d ) = ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 + ( x x 1 ) 2 + ( y y 1 ) 2 + ( z z 1 ) 2 + m λ x d .
F x = x 0 r 0 x 1 r 1 + m λ d = sin θ 0 cos α 0 sin θ 1 cos α 1 + m λ d .
F y = y 0 r 0 y 1 r 1 = sin θ 0 sin α 0 sin θ 1 sin α 1 .
F x x = 1 r 0 ( 1 z 0 R x 0 2 r 0 2 ) + 1 r 1 ( 1 z 1 R x 1 2 r 1 2 ) = 1 sin 2 θ 0 cos 2 α 0 r 0 + 1 sin 2 θ 1 cos 2 α 1 r 1 cos θ 0 + cos θ 1 R .
F y y = 1 r 0 ( 1 z 0 R y 0 2 r 0 2 ) + 1 r 1 ( 1 z 1 R y 1 2 r 1 2 ) = 1 sin 2 θ 0 sin 2 α 0 r 0 + 1 sin 2 θ 1 sin 2 α 1 r 1 cos θ 0 + cos θ 1 R .
F x y = x 0 y 0 r 0 2 x 1 y 1 r 1 2 = sin 2 θ 0 r 0 sin α 0 cos α 0 sin 2 θ 1 r 1 sin α 1 cos α 1 .
F x x x = 3 x 0 3 r 0 5 + 3 x 0 ( 1 z 0 R ) r 0 3 3 x 1 2 r 1 5 + 3 x 1 ( 1 z 1 R ) r 1 3 = 3 sin θ 0 cos α 0 r 0 2 ( 1 sin 2 θ 0 cos 2 α 0 r 0 cos θ 0 R ) + 3 sin θ 1 cos α 1 r 1 2 ( 1 sin 2 θ 1 cos 2 α 1 r 1 cos θ 1 R ) .
F x x y = 3 x 0 2 y 0 r 0 5 + y 0 ( 1 z 0 R ) r 0 3 3 x 1 2 y 1 r 1 5 + 3 y 1 ( 1 z 1 R ) r 1 3 = sin θ 0 sin α 0 r 0 2 ( 1 3 sin 2 θ 0 cos 2 α 0 r 0 cos θ 0 R ) + sin θ 1 sin α 1 r 1 2 ( 1 3 sin 2 θ 1 cos 2 α 1 r 1 cos θ 1 R ) .
F x y y = 3 x 0 y 0 2 r 0 5 + x 0 ( 1 z 0 R ) r 0 3 3 x 1 y 1 2 r 1 5 + 3 x 1 ( 1 z 1 R ) r 1 3 = sin θ 0 cos α 0 r 0 2 ( 1 3 sin 2 θ 0 sin 2 α 0 r 0 cos θ 0 R ) + sin θ 1 cos α 1 r 1 2 ( 1 3 sin 2 θ 1 sin 2 α 1 r 1 cos θ 1 R ) .
F y y y = 3 y 0 3 r 0 5 + 3 y 0 ( 1 z 0 R ) r 0 3 3 y 1 2 r 1 5 + 3 y 1 ( 1 z 1 R ) r 1 3 = 3 sin θ 0 sin α 0 r 0 2 ( 1 sin 2 θ 0 sin 2 α 0 r 0 cos θ 0 R ) + 3 sin θ 1 sin α 1 r 1 2 ( 1 sin 2 θ 1 sin 2 α 1 r 1 cos θ 1 R ) .

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