Abstract

A grazing-incidence spectrograph is designed by use of the flat-field image-focusing property of a spherical varied-line-space grating. Optimum grating parameters for mechanical ruling are selected by application of genetic algorithms. Two gratings, one for 2–5-nm and the other for 5–20-nm spectral regions, are designed, and their fabrication tolerances are analyzed.

© 1999 Optical Society of America

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References

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  1. E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), p. 455.
  2. J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
    [CrossRef]
  3. T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3993 (1980).
    [CrossRef] [PubMed]
  4. C. Palmer, “Absolute astigmatism correction for flat field spectrographs,” Appl. Opt. 28, 1605–1607 (1989).
    [CrossRef] [PubMed]
  5. T. Harada, S. Moriyama, T. Kita, “Mechanically ruled stigmatic concave grating,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 175–179 (1975).
  6. T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
    [CrossRef] [PubMed]
  7. T. Kita, T. Harada, “Use of aberration-corrected concave gratings in optical dumultiplexers,” Appl. Opt. 22, 819–825 (1983).
    [CrossRef]
  8. T. Kita, T. Harada, N. Nakano, H. Kuroda, “Mechanically ruled aberration corrected concave grating for a flat-field grazing incidence spectrograph,” Appl. Opt. 22, 512–513 (1983).
    [CrossRef] [PubMed]
  9. N. Nakano, H. Kuroda, T. Kita, T. Harada, “Development of a flat-field grazing-incidence XUV spectrometer and its application in picosecond XUV spectroscopy,” Appl. Opt. 23, 2386–2392 (1984).
    [CrossRef] [PubMed]
  10. G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
    [CrossRef]
  11. M. C. Hettrick, S. Bowyer, R. F. Malina, C. Martin, S. Mrowka, “Extreme Ultraviolet Explorer spectrometer,” Appl. Opt. 24, 1737–1756 (1985).
    [CrossRef] [PubMed]
  12. T. Harada, T. Kita, S. Bowyer, M. Hurwitz, “Design of spherical varied line-space gratings for a high resolution EUV spectrometer,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 2–10 (1991).
    [CrossRef]
  13. T. Harada, H. Sakuma, K. Takahashi, T. Watanabe, H. Hara, T. Kita, “Design of a high-resolution extreme-ultraviolet imaging spectrometer with aberration-corrected concave gratings,” Appl. Opt. 37, 6803–6810 (1998).
    [CrossRef]
  14. H. Noda, T. Namioka, M. Seya, “Design of holographic concave grating for Seya–Namioka monochromators,” J. Opt. Soc. Am. 64, 1043–1048 (1974).
    [CrossRef]
  15. T. Takahashi, T. Katayama, “Automatic design of holographic gratings for Seya–Namioka monochromators,” J. Opt. Soc. Am. 68, 1254–1256 (1978).
    [CrossRef]
  16. W. R. McKinney, C. Palmer, “Numerical design method for aberration-reduced concave grating spectrometers,” Appl. Opt. 26, 3108–3118 (1987).
    [CrossRef] [PubMed]
  17. M. Koike, T. Yamazaki, Y. Harada, “Design of holographic gratings recorded with aspheric wave-front recording optics for soft x-ray flat field spectrographs,” J. Electron. Spectrosc. Relat. Phenom. (to be published).
  18. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  19. A. Osyczka, S. Kudnda, “A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm,” Struct. Optim. 10, 94–99 (1995).
    [CrossRef]
  20. A. Osyczka, S. Kundu, “A modified distance method for multicriteria optimization using genetic algorithms,” Computers Ind. Eng. 30, 871–882 (1996).
    [CrossRef]
  21. S. Kundu, A. Osyczka, “The effect of genetic algorithm selection mechanisms on multicriteria optimization using distance method,” presented at the Fifth International Conference on Intelligent Systems, Reno, Nevada, 1996.
  22. K. Siegbahn, N. Kholine, G. Golikov, “A high resolution and large transmission electron spectrometer,” Nucl. Instrum. Methods A 384, 563–574 (1997).
    [CrossRef]

1998 (1)

1997 (1)

K. Siegbahn, N. Kholine, G. Golikov, “A high resolution and large transmission electron spectrometer,” Nucl. Instrum. Methods A 384, 563–574 (1997).
[CrossRef]

1996 (1)

A. Osyczka, S. Kundu, “A modified distance method for multicriteria optimization using genetic algorithms,” Computers Ind. Eng. 30, 871–882 (1996).
[CrossRef]

1995 (1)

A. Osyczka, S. Kudnda, “A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm,” Struct. Optim. 10, 94–99 (1995).
[CrossRef]

1992 (1)

1989 (1)

1987 (1)

1985 (1)

1984 (1)

1983 (2)

1980 (1)

1978 (1)

1975 (1)

T. Harada, S. Moriyama, T. Kita, “Mechanically ruled stigmatic concave grating,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 175–179 (1975).

1974 (1)

Bowyer, S.

M. C. Hettrick, S. Bowyer, R. F. Malina, C. Martin, S. Mrowka, “Extreme Ultraviolet Explorer spectrometer,” Appl. Opt. 24, 1737–1756 (1985).
[CrossRef] [PubMed]

T. Harada, T. Kita, S. Bowyer, M. Hurwitz, “Design of spherical varied line-space gratings for a high resolution EUV spectrometer,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 2–10 (1991).
[CrossRef]

Damerell, A. R.

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

Flamand, J.

J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
[CrossRef]

Garvey, T.

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Golikov, G.

K. Siegbahn, N. Kholine, G. Golikov, “A high resolution and large transmission electron spectrometer,” Nucl. Instrum. Methods A 384, 563–574 (1997).
[CrossRef]

Hara, H.

Harada, T.

T. Harada, H. Sakuma, K. Takahashi, T. Watanabe, H. Hara, T. Kita, “Design of a high-resolution extreme-ultraviolet imaging spectrometer with aberration-corrected concave gratings,” Appl. Opt. 37, 6803–6810 (1998).
[CrossRef]

T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
[CrossRef] [PubMed]

N. Nakano, H. Kuroda, T. Kita, T. Harada, “Development of a flat-field grazing-incidence XUV spectrometer and its application in picosecond XUV spectroscopy,” Appl. Opt. 23, 2386–2392 (1984).
[CrossRef] [PubMed]

T. Kita, T. Harada, N. Nakano, H. Kuroda, “Mechanically ruled aberration corrected concave grating for a flat-field grazing incidence spectrograph,” Appl. Opt. 22, 512–513 (1983).
[CrossRef] [PubMed]

T. Kita, T. Harada, “Use of aberration-corrected concave gratings in optical dumultiplexers,” Appl. Opt. 22, 819–825 (1983).
[CrossRef]

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

T. Harada, S. Moriyama, T. Kita, “Mechanically ruled stigmatic concave grating,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 175–179 (1975).

T. Harada, T. Kita, S. Bowyer, M. Hurwitz, “Design of spherical varied line-space gratings for a high resolution EUV spectrometer,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 2–10 (1991).
[CrossRef]

Harada, Y.

M. Koike, T. Yamazaki, Y. Harada, “Design of holographic gratings recorded with aspheric wave-front recording optics for soft x-ray flat field spectrographs,” J. Electron. Spectrosc. Relat. Phenom. (to be published).

Hettrick, M. C.

Hurwitz, M.

T. Harada, T. Kita, S. Bowyer, M. Hurwitz, “Design of spherical varied line-space gratings for a high resolution EUV spectrometer,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 2–10 (1991).
[CrossRef]

Katayama, T.

Kholine, N.

K. Siegbahn, N. Kholine, G. Golikov, “A high resolution and large transmission electron spectrometer,” Nucl. Instrum. Methods A 384, 563–574 (1997).
[CrossRef]

Kiehn, G. P.

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

Kita, T.

T. Harada, H. Sakuma, K. Takahashi, T. Watanabe, H. Hara, T. Kita, “Design of a high-resolution extreme-ultraviolet imaging spectrometer with aberration-corrected concave gratings,” Appl. Opt. 37, 6803–6810 (1998).
[CrossRef]

T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
[CrossRef] [PubMed]

N. Nakano, H. Kuroda, T. Kita, T. Harada, “Development of a flat-field grazing-incidence XUV spectrometer and its application in picosecond XUV spectroscopy,” Appl. Opt. 23, 2386–2392 (1984).
[CrossRef] [PubMed]

T. Kita, T. Harada, N. Nakano, H. Kuroda, “Mechanically ruled aberration corrected concave grating for a flat-field grazing incidence spectrograph,” Appl. Opt. 22, 512–513 (1983).
[CrossRef] [PubMed]

T. Kita, T. Harada, “Use of aberration-corrected concave gratings in optical dumultiplexers,” Appl. Opt. 22, 819–825 (1983).
[CrossRef]

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

T. Harada, S. Moriyama, T. Kita, “Mechanically ruled stigmatic concave grating,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 175–179 (1975).

T. Harada, T. Kita, S. Bowyer, M. Hurwitz, “Design of spherical varied line-space gratings for a high resolution EUV spectrometer,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 2–10 (1991).
[CrossRef]

Koike, M.

M. Koike, T. Yamazaki, Y. Harada, “Design of holographic gratings recorded with aspheric wave-front recording optics for soft x-ray flat field spectrographs,” J. Electron. Spectrosc. Relat. Phenom. (to be published).

Kudnda, S.

A. Osyczka, S. Kudnda, “A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm,” Struct. Optim. 10, 94–99 (1995).
[CrossRef]

Kundu, S.

A. Osyczka, S. Kundu, “A modified distance method for multicriteria optimization using genetic algorithms,” Computers Ind. Eng. 30, 871–882 (1996).
[CrossRef]

S. Kundu, A. Osyczka, “The effect of genetic algorithm selection mechanisms on multicriteria optimization using distance method,” presented at the Fifth International Conference on Intelligent Systems, Reno, Nevada, 1996.

Kuroda, H.

Laud, P.

J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
[CrossRef]

Lerner, J.

J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
[CrossRef]

Loewen, E. G.

E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), p. 455.

Malina, R. F.

Martin, C.

McKinney, W. R.

Moriyama, S.

T. Harada, S. Moriyama, T. Kita, “Mechanically ruled stigmatic concave grating,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 175–179 (1975).

Mrowka, S.

Nakano, N.

Namioka, T.

Noda, H.

Osyczka, A.

A. Osyczka, S. Kundu, “A modified distance method for multicriteria optimization using genetic algorithms,” Computers Ind. Eng. 30, 871–882 (1996).
[CrossRef]

A. Osyczka, S. Kudnda, “A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm,” Struct. Optim. 10, 94–99 (1995).
[CrossRef]

S. Kundu, A. Osyczka, “The effect of genetic algorithm selection mechanisms on multicriteria optimization using distance method,” presented at the Fifth International Conference on Intelligent Systems, Reno, Nevada, 1996.

Palmer, C.

Passereau, G.

J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
[CrossRef]

Popov, E.

E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), p. 455.

Sakuma, H.

Seya, M.

Siegbahn, K.

K. Siegbahn, N. Kholine, G. Golikov, “A high resolution and large transmission electron spectrometer,” Nucl. Instrum. Methods A 384, 563–574 (1997).
[CrossRef]

Smith, R. A.

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

Takahashi, K.

Takahashi, T.

Thevenson, A.

J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
[CrossRef]

Watanabe, T.

West, J.

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

Willi, O.

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

Yamazaki, T.

M. Koike, T. Yamazaki, Y. Harada, “Design of holographic gratings recorded with aspheric wave-front recording optics for soft x-ray flat field spectrographs,” J. Electron. Spectrosc. Relat. Phenom. (to be published).

Appl. Opt. (9)

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

C. Palmer, “Absolute astigmatism correction for flat field spectrographs,” Appl. Opt. 28, 1605–1607 (1989).
[CrossRef] [PubMed]

T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
[CrossRef] [PubMed]

T. Kita, T. Harada, “Use of aberration-corrected concave gratings in optical dumultiplexers,” Appl. Opt. 22, 819–825 (1983).
[CrossRef]

T. Kita, T. Harada, N. Nakano, H. Kuroda, “Mechanically ruled aberration corrected concave grating for a flat-field grazing incidence spectrograph,” Appl. Opt. 22, 512–513 (1983).
[CrossRef] [PubMed]

N. Nakano, H. Kuroda, T. Kita, T. Harada, “Development of a flat-field grazing-incidence XUV spectrometer and its application in picosecond XUV spectroscopy,” Appl. Opt. 23, 2386–2392 (1984).
[CrossRef] [PubMed]

M. C. Hettrick, S. Bowyer, R. F. Malina, C. Martin, S. Mrowka, “Extreme Ultraviolet Explorer spectrometer,” Appl. Opt. 24, 1737–1756 (1985).
[CrossRef] [PubMed]

T. Harada, H. Sakuma, K. Takahashi, T. Watanabe, H. Hara, T. Kita, “Design of a high-resolution extreme-ultraviolet imaging spectrometer with aberration-corrected concave gratings,” Appl. Opt. 37, 6803–6810 (1998).
[CrossRef]

W. R. McKinney, C. Palmer, “Numerical design method for aberration-reduced concave grating spectrometers,” Appl. Opt. 26, 3108–3118 (1987).
[CrossRef] [PubMed]

Computers Ind. Eng. (1)

A. Osyczka, S. Kundu, “A modified distance method for multicriteria optimization using genetic algorithms,” Computers Ind. Eng. 30, 871–882 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

Jpn. J. Appl. Phys. (1)

T. Harada, S. Moriyama, T. Kita, “Mechanically ruled stigmatic concave grating,” Jpn. J. Appl. Phys. 14, Suppl. 14-1, 175–179 (1975).

Nucl. Instrum. Methods A (1)

K. Siegbahn, N. Kholine, G. Golikov, “A high resolution and large transmission electron spectrometer,” Nucl. Instrum. Methods A 384, 563–574 (1997).
[CrossRef]

Struct. Optim. (1)

A. Osyczka, S. Kudnda, “A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm,” Struct. Optim. 10, 94–99 (1995).
[CrossRef]

Other (7)

E. G. Loewen, E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997), p. 455.

J. Lerner, J. Flamand, P. Laud, G. Passereau, A. Thevenson, “Diffraction gratings ruled and holographic—a review,” in Periodic Structures, Gratings, Moire Patterns, and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 82–88 (1980).
[CrossRef]

G. P. Kiehn, T. Garvey, R. A. Smith, O. Willi, A. R. Damerell, J. West, “Absolute calibration of an XUV time resolving spectrograph,” in X Rays from Laser Plasmas, M. C. Richardson ed., Proc. SPIE831, 150–153 (1988).
[CrossRef]

T. Harada, T. Kita, S. Bowyer, M. Hurwitz, “Design of spherical varied line-space gratings for a high resolution EUV spectrometer,” in International Conference on the Application and Theory of Periodic Structures, J. M. Lerner, W. R. McKinney, eds., Proc. SPIE1545, 2–10 (1991).
[CrossRef]

S. Kundu, A. Osyczka, “The effect of genetic algorithm selection mechanisms on multicriteria optimization using distance method,” presented at the Fifth International Conference on Intelligent Systems, Reno, Nevada, 1996.

M. Koike, T. Yamazaki, Y. Harada, “Design of holographic gratings recorded with aspheric wave-front recording optics for soft x-ray flat field spectrographs,” J. Electron. Spectrosc. Relat. Phenom. (to be published).

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the optical layout.

Fig. 2
Fig. 2

Flowchart of the design optimization process with genetic algorithms.

Fig. 3
Fig. 3

Focal curve of grating I for a grazing-incidence spectrograph. The grating parameters are σ0 = 1/2400 mm, R = 57.68 m, and b 2 = -95.97. The mounting parameters are α = 89° and r = 564 mm.

Fig. 4
Fig. 4

Ray-traced spectral images of 2-, 3-, 4-, and 5-nm spectra obtained with grating I without correction of higher-order aberration. The grating parameters are σ0 = 1/2400 mm, R = 57.68 m, b 2 = -95.97, b 3 = 0, and b 4 = 0. The mounting parameters are α = 89°, r = 564 mm, and r′ = -563.2/sin β.

Fig. 5
Fig. 5

Ray-traced spectral images of 2-, 3-, 4-, and 5-nm spectra obtained with grating I with correction of higher-order aberration. The grating parameters are σ0 = 1/2400 mm, R = 57.68 m, b 2 = -95.97, b 3 = 9.492 × 103, and b 4 = -9.78 × 105. The mounting parameters are the same as for Fig. 4.

Fig. 6
Fig. 6

Focal curve of grating II for a grazing-incidence spectrograph. The grating parameters are σ0 = 1/1200 mm, R = 13.45 m, and b 2 = -19.855. The mounting parameters are α = 87° and r = 564 mm.

Fig. 7
Fig. 7

Ray-traced spectral images of 5-, 10-, 15-, and 20-nm spectra obtained with grating II. The grating parameters are σ0 = 1/1200 mm, R = 13.45 m, b 2 = -19.855, b 3 = 4.35 × 102, and b 4 = -1.12 × 104. The mounting parameters are α = 87°, r = 564 mm, and r′ = -563.2/sin β.

Fig. 8
Fig. 8

Ray-traced spectral images of 2-, 3-, 4-, and 5-nm spectra obtained with grating I with radius error after focal correction: (a) +1% radius error, (b) -1% radius error. Other specifications of the grating and the mounting are the same as for Fig. 5.

Fig. 9
Fig. 9

Ray-traced spectral images of 5-, 10-, 15-, and 20-nm spectra obtained with grating II with radius error after focal correction: (a) +1% radius error, (b) -1% radius error. Other specifications of the grating and the mounting are the same as for Fig. 7.

Equations (13)

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

F=AP+PB+nmλ,
F=r+r+wF10+w2F20+l2F02+w3F30+wl2F12+w4F40+w2l2F22+l4F04+Tw5,
σ=σ01+2b2R w+3b3R2 w2+4b4R3 w3+,
Fij=Cij+mλσ0 Mij,
C10=-sin α-sin β,
M10=1,
C20=12cos2 αr-cos αR+12cos2 βr-cos βR,
M20=b2/R,
C30=12sin αrcos2 αr-cos αR+12sin βr×cos2 βr-cos βR,
M30=b3/R2,
C40=184 sin2 αr2cos2 αr-cos αR-1r×cos2 αr-cos αR2+1R21r-cos αR+184 sin2 βr2cos2 βr-cos βR-1r×cos2 βr-cos βR2+1R21r-cos βR,
M40=b4/R3,
r=rR cos2 βrcos α+cos β-2sin α+sin βb2-R cos2 α.

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