Abstract

A merit function that is closely correlated with the rms spread of an infinite number of ray-traced spots by means of analytic spot-diagram formulas is critically evaluated in comparison with exact ray tracing. The analytic spot-diagram formulas are found to generate spot diagrams that are almost indistinguishable from ray-traced ones. The rms spread of ray-traced spots about the mean shows varying degrees of statistical dispersion depending on the number of rays traced, and it approaches the value given by the merit function as the number of rays is increased. All the results clearly show that the merit function behaves as defined and provides enough accuracy and a sufficiently short computing time for the design of highly sophisticated grating instruments.

© 1994 Optical Society of America

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References

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  1. T. Namioka, “Theory of the concave grating. III. Seya-Namioka monochromator,” J. Opt. Soc. Am. 49, 951–961 (1959).
    [CrossRef]
  2. H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
    [CrossRef]
  3. H. Noda, T. Namioka, M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromators,” J. Opt. Soc. Am. 64, 1043–1048 (1974).
    [CrossRef]
  4. T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
    [CrossRef]
  5. T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3993 (1980).
    [CrossRef] [PubMed]
  6. R. Grange, M. Laget, “Holographic diffraction gratings generated by aberrated wave fronts: application to a high-resolution far-ultraviolet spectrograph,” Appl. Opt. 30, 3598–3603 (1991).
    [CrossRef] [PubMed]
  7. A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
    [CrossRef]
  8. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [CrossRef] [PubMed]
  9. M. P. Chrisp, “X-ray spectrograph design,” Appl. Opt. 22, 1519–1529 (1983).
    [CrossRef] [PubMed]
  10. T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
    [CrossRef]
  11. K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
    [CrossRef] [PubMed]
  12. W. R. McKinney, C. Palmer, “Numerical design method for aberration-reduced concave grating spectrometers,” Appl. Opt. 26, 3108–3118 (1987).
    [CrossRef] [PubMed]
  13. R. Grange, “Aberration-reduced holographic gratings for Rowland circle spectrographs,” Appl. Opt. 31, 3744–3749 (1992).
    [CrossRef] [PubMed]
  14. T. Namioka, S. Morozumi, “Design of constant-deviation monochromators,” Nucl. Instrum. Methods 177, 141–146 (1980).
    [CrossRef]
  15. T. Namioka, “A merit function for the design of toroidal holographic gratings,” Bull. Res. Inst. Sci. Meas. Tohoku Univ. 29, 65–75 (1980).
  16. T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 139, 219–227 (1992).
    [CrossRef]
  17. In Ref. 13 the merit function Φ is defined by Φ = ΣwiΣ(δy + fδz)2 instead of Φ = ΣwiΣ[(δy)2 + f(δz)2]. As δy and δz include signs, δy + fδz does not have any physical significance.
  18. W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

1992 (2)

R. Grange, “Aberration-reduced holographic gratings for Rowland circle spectrographs,” Appl. Opt. 31, 3744–3749 (1992).
[CrossRef] [PubMed]

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 139, 219–227 (1992).
[CrossRef]

1991 (1)

1989 (1)

A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
[CrossRef]

1987 (1)

1986 (1)

1983 (3)

M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
[CrossRef] [PubMed]

M. P. Chrisp, “X-ray spectrograph design,” Appl. Opt. 22, 1519–1529 (1983).
[CrossRef] [PubMed]

T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
[CrossRef]

1980 (3)

T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3993 (1980).
[CrossRef] [PubMed]

T. Namioka, S. Morozumi, “Design of constant-deviation monochromators,” Nucl. Instrum. Methods 177, 141–146 (1980).
[CrossRef]

T. Namioka, “A merit function for the design of toroidal holographic gratings,” Bull. Res. Inst. Sci. Meas. Tohoku Univ. 29, 65–75 (1980).

1976 (1)

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

1974 (2)

1959 (1)

Attwood, D.

W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

Beguiristain, R.

W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

Chrisp, M. P.

Goto, K.

K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
[CrossRef] [PubMed]

T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
[CrossRef]

Grange, R.

Harada, T.

Ito, K.

Katayama, T.

K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
[CrossRef] [PubMed]

T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
[CrossRef]

Kita, T.

Koike, M.

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 139, 219–227 (1992).
[CrossRef]

K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
[CrossRef] [PubMed]

W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

Laget, M.

Maser, J.

W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

McKinney, W. R.

Meyer-Ilse, W.

W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

Morioka, Y.

Morozumi, S.

T. Namioka, S. Morozumi, “Design of constant-deviation monochromators,” Nucl. Instrum. Methods 177, 141–146 (1980).
[CrossRef]

Namioka, T.

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 139, 219–227 (1992).
[CrossRef]

K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
[CrossRef] [PubMed]

T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
[CrossRef]

T. Namioka, “A merit function for the design of toroidal holographic gratings,” Bull. Res. Inst. Sci. Meas. Tohoku Univ. 29, 65–75 (1980).

T. Namioka, S. Morozumi, “Design of constant-deviation monochromators,” Nucl. Instrum. Methods 177, 141–146 (1980).
[CrossRef]

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

H. Noda, T. Namioka, M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromators,” J. Opt. Soc. Am. 64, 1043–1048 (1974).
[CrossRef]

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[CrossRef]

T. Namioka, “Theory of the concave grating. III. Seya-Namioka monochromator,” J. Opt. Soc. Am. 49, 951–961 (1959).
[CrossRef]

Noda, H.

K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
[CrossRef] [PubMed]

T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
[CrossRef]

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

H. Noda, T. Namioka, M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromators,” J. Opt. Soc. Am. 64, 1043–1048 (1974).
[CrossRef]

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[CrossRef]

Palmer, C.

Sasaki, T.

Seya, M.

Takahashi, A.

A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
[CrossRef]

Appl. Opt. (7)

Bull. Res. Inst. Sci. Meas. Tohoku Univ. (1)

T. Namioka, “A merit function for the design of toroidal holographic gratings,” Bull. Res. Inst. Sci. Meas. Tohoku Univ. 29, 65–75 (1980).

J. Mod. Opt. (1)

A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
[CrossRef]

J. Opt. Soc. Am. (3)

Jpn. J. Appl. Phys. (1)

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

Nucl. Instrum. Methods (2)

T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods 208, 215–222 (1983).
[CrossRef]

T. Namioka, S. Morozumi, “Design of constant-deviation monochromators,” Nucl. Instrum. Methods 177, 141–146 (1980).
[CrossRef]

Nucl. Instrum. Methods A (1)

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 139, 219–227 (1992).
[CrossRef]

Other (2)

In Ref. 13 the merit function Φ is defined by Φ = ΣwiΣ(δy + fδz)2 instead of Φ = ΣwiΣ[(δy)2 + f(δz)2]. As δy and δz include signs, δy + fδz does not have any physical significance.

W. Meyer-Ilse, M. Koike, R. Beguiristain, J. Maser, D. Attwood, “X-Ray Microscopy Resource Center at the Advanced Light Source,” in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, Proc. Soc. Photo-Opt. Instrum. Eng.1741, 112–115 (1992).

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the optical system.

Fig. 2
Fig. 2

Spot diagrams and line profiles constructed for the model monochromator equipped with Grule. The results are (a) computed from Eqs. (1) and (2) and (b) obtained by exact ray tracing. Each spot diagram is constructed with 1000 randomly generated rays.

Fig. 3
Fig. 3

Same as Fig. 2, except that the model monochromator is equipped with Gholo. The results shown in (a) and (b) are computed and obtained in the same way as those shown in Figs. 2(a) and 2(b), respectively.

Fig. 4
Fig. 4

Statistical dispersion of the qY*’s and qZ*’s obtained by tracing ten different sets of N randomly generated rays through the model monochromator equipped with Grule. The number of rays used is N = 10, 50, 100, 500, 1000, and 2000. The values of qY and qZ are calculated from Eqs. (6) and (7).

Fig. 5
Fig. 5

Same as Fig. 4, except that the model monochromator is equipped with Gholo.

Tables (2)

Tables Icon

Table 1 Comparison of the Spot Positions Computed from Eqs. (1) and (2) with Those Determined by Exact Ray Tracing through the Model Monochromator Equipped with Gholo

Tables Icon

Table 2 Statistical Dispersion of the qY*’s and qZ*’s Obtained by Tracing Ten Different Sets of N Randomly Generated Rays through the Model Monochromator Equipped with Grule

Equations (15)

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Y = w f 100 + w 2 f 200 + l 2 f 020 + l z f 011 + z 2 f 002 + w 3 f 300 + w l 2 f 120 + w l z f 111 + w z 2 f 102 + O ( w 4 / R 3 ) ,
Z = z g 001 + l g 010 + w l g 110 + w z g 101 + w 2 l g 210 + l 3 g 030 + l 2 z g 021 + l z 2 g 012 + w 2 z g 201 + O ( w 4 / R 3 ) ,
Q ( λ i ) = 1 W L H W / 2 W / 2 L / 2 L / 2 H / 2 H / 2 ( Y Y ¯ ) 2 d w d l d z + μ W L H W / 2 W / 2 L / 2 L / 2 H / 2 H / 2 Z 2 d w d l d z ,
Y ¯ = 1 W L H W / 2 W / 2 L / 2 L / 2 H / 2 H / 2 Y d w d l d z .
Q ( λ i ) = q Y 2 ( λ i ) + μ [ q Z 2 ( λ i ) ] ,
q Y 2 ( λ i ) = 1 12 W 2 f 100 2 + 1 360 [ W 4 ( 2 f 200 2 + 9 f 100 f 300 ) + 2 L 4 f 020 2 + 2 H 4 f 002 2 ] + 1 144 [ 2 W 2 f 100 ( L 2 f 120 + H 2 f 102 ) + L 2 H 2 f 011 2 ] + 1 960 W 2 [ 2 W 2 f 300 ( L 2 f 120 + H 2 f 102 ) + L 4 f 120 2 + H 4 f 102 2 ] + 1 448 W 6 f 300 2 + 1 1728 W 2 L 2 H 2 ( f 111 2 + 2 f 120 f 120 ) ,
q Z 2 ( λ i ) = 1 2 ( L 2 g 010 2 + H 2 g 001 2 ) + 1 40 L 4 g 010 g 030 + 1 144 { W 2 L 2 ( g 110 2 + 2 g 010 g 210 ) + H 2 [ W 2 ( g 101 2 + 2 g 001 g 201 ) + 2 L 2 ( g 010 g 012 + g 001 g 021 ) ] } + 1 960 [ W 2 L 2 g 210 ( W 2 g 210 + 2 L 2 g 030 ) + L 2 H 2 g 012 ( 2 L 2 g 030 + H 2 g 012 ) + H 2 ( W 4 g 201 2 + L 4 g 021 2 ) ] + 1 448 L 6 g 030 2 + 1 864 W 2 L 2 H 2 ( g 210 g 012 + g 021 g 201 ) .
Q = i ( λ i ) Q ( λ i ) ,
σ n = σ 0 + 2 a n + 6 b n 2 + 4 c n 3 ,
n σ = w + Γ ( 1 2 n 20 w 2 + 1 2 n 02 l 2 + 1 2 n 30 w 3 + 1 2 n 12 w l 2 + 1 8 n 40 w 4 + 1 4 n 22 w 2 l 2 + 1 8 n 04 l 4 + ) ,
Γ = 1 for the ruled grating ,
Γ = σ / λ 0 for the holographic grating .
σ 0 = ( 1 300 ) × 10 3 + a b mm , 2 a = 2 . 514256 × 10 9 mm , 6 b = 1 . 280905 × 10 14 mm , 4 c = 1 . 050669 × 10 20 mm , n 20 = 2 . 262842 × 10 4 mm 1 , n 30 = 2 . 817677 × 10 7 mm 2 , n 40 = 7 . 498703 × 10 10 mm 3 , n 02 = n 12 = n 22 = n 04 = n i j ( i + j 5 ) = 0 .
λ 0 = 457 . 9 nm , r C = 21008 . 188 mm , r D = 9513 . 894 mm , γ = 61 . 727481 , δ = 48 . 016256 ,
n 20 = 3 . 121372 × 10 5 mm 1 , n 02 = 5 . 237068 × 10 5 mm 1 , n 30 = 2 . 374130 × 10 9 mm 2 , n 12 = 5 . 364042 × 10 9 mm 2 , n 40 = 6 . 765285 × 10 13 mm 3 , n 22 = 7 . 074779 × 10 13 mm 3 , n 04 = 7 . 098286 × 10 13 mm 3 , n i j ( i + j 5 ) = 0 .

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