Abstract

Performances of a far-ultraviolet (FUV) imaging spectrometer in an advanced design are presented with a toroidal uniform-line-space (TULS) grating. It provides high spatial resolution and spectral resolution for a broadband and a wide field of view. A particular analysis for the grating aberrations, including all the high-order coefficients neglected by previous existing designs, was generated for indicating their significance. The analysis indicates that these high-order off-axis aberrations would have a remarkable influence on the design results. The transcendental equations composed of these aberration coefficients do not have analytic solutions in algebra. To solve the problem, the past designs always do some simplified calculation which only suits a narrow field of view and waveband. Thus, the optimization of the genetic algorithm is introduced to propose reasonable ranges of optical parameters. Then ZEMAX software is used to obtain the final optical system from these ranges. By comparing different design results of the same example, our advanced TULS design performs better than conventional TULS design and spherical varied-line-space grating design, and as well as the toroidal varied-line-space design. It is demonstrated that aberrations are minimized when the TULS design is operated by our method. The advanced design is low-cost, easy to fabricate, and more suitable for FUV observations.

© 2011 Optical Society of America

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References

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  1. F. W. Schenkel, B. S. Ogorzalek, J. C. Larrabee, F. J. Leblanc, and R. E. Huffman, “Ultraviolet daytime auroral and ionospheric imaging,” Appl. Opt. 24, 3395–3405 (1985).
    [CrossRef] [PubMed]
  2. R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
    [CrossRef]
  3. L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
    [CrossRef]
  4. S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
    [CrossRef]
  5. D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
    [CrossRef]
  6. W. Werner, “Geometric optical aberration theory of diffraction gratings,” Appl. Opt. 6, 1691–1699 (1967).
    [CrossRef] [PubMed]
  7. H. G. Beutler, “Theory of the concave grating,” J. Opt. Soc. Am. 35, 311–350 (1945).
    [CrossRef]
  8. T. Namioka, “Theory of the concave grating. I,” J. Opt. Soc. Am. 49, 446–460 (1959).
    [CrossRef]
  9. M. C. E. Huber and G. Tondello, “Stigmatic performance of an EUV spectrograph with a single toroidal grating,” Appl. Opt. 18, 3948–3953 (1979).
    [CrossRef] [PubMed]
  10. D. M. Cotton, T. Cook, and S. Chakrabarti, “Single-element imaging spectrograph,” Appl. Opt. 33, 1958–1962 (1994).
    [CrossRef] [PubMed]
  11. T. Harada and T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3993 (1980).
    [CrossRef] [PubMed]
  12. T. Onaka, T. Miyata, H. Kataza, and Y. Okamoto, “Design for an aberration-corrected concave grating for a mid-infrared long-slit spectrometer,” Appl. Opt. 39, 1474–1479 (2000).
    [CrossRef]
  13. L. Poletto and R. J. Thomas, “Stigmatic spectrometers for extended sources: design with toroidal varied-line-space gratings,” Appl. Opt. 43, 2029–2038 (2004).
    [CrossRef] [PubMed]
  14. H. Haber, “Torus grating,” J. Opt. Soc. Am. 40, 153–165 (1950).
    [CrossRef]
  15. J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).
  16. K. De Jong, “The analysis and behavior of class of genetic adaptive systems,” Ph. D. dissertation, University of Michigan (1975).
  17. D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning reading (Addison-Wesley, 1989).
  18. W. R. Hunter, J. F. Osantowski, and G. Hass, “Reflectance of aluminum overcoated with MgF2 and LiF in the wavelength region from 1600 to 300 at various angles of incidence,” Appl. Opt. 10, 540–544 (1971).
    [CrossRef] [PubMed]
  19. T. Onaka, “Aberration-corrected concave grating for the mid-infrared spectrometer aboard the Infrared Telescope in Space,” Appl. Opt. 34, 659–666 (1995).
    [CrossRef] [PubMed]

2004 (2)

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

L. Poletto and R. J. Thomas, “Stigmatic spectrometers for extended sources: design with toroidal varied-line-space gratings,” Appl. Opt. 43, 2029–2038 (2004).
[CrossRef] [PubMed]

2002 (1)

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

2000 (1)

1997 (1)

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

1995 (1)

1994 (2)

D. M. Cotton, T. Cook, and S. Chakrabarti, “Single-element imaging spectrograph,” Appl. Opt. 33, 1958–1962 (1994).
[CrossRef] [PubMed]

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

1989 (1)

D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning reading (Addison-Wesley, 1989).

1985 (1)

1980 (1)

1979 (1)

1975 (2)

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).

K. De Jong, “The analysis and behavior of class of genetic adaptive systems,” Ph. D. dissertation, University of Michigan (1975).

1971 (1)

1967 (1)

1959 (1)

1950 (1)

1945 (1)

Beutler, H. G.

Chakrabarti, S.

Christensen, A. B.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

Cook, T.

Cotton, D. M.

De Jong, K.

K. De Jong, “The analysis and behavior of class of genetic adaptive systems,” Ph. D. dissertation, University of Michigan (1975).

DeMajistre, R.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

Dymond, K. F.

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Fritz, G. G.

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Goldberg, D. E.

D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning reading (Addison-Wesley, 1989).

Goldsten, J. O.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

Haber, H.

Habraken, S.

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

Harada, T.

Hass, G.

Holland, J. H.

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).

Huber, M. C. E.

Huffman, R. E.

Humm, D.

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Humm, D. C.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

Hunter, W. R.

Jamar, C.

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

Kataza, H.

Kil, H.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Kita, T.

Lampton, M.

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

Larrabee, J. C.

Leblanc, F. J.

McCoy, R. P.

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Meier, R. R.

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Mende, S. B.

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

Meng, C.-I.

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Meng, , C.-I.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

Miyata, T.

Morrison, D.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Namioka, T.

Ogorzalek, B. S.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

F. W. Schenkel, B. S. Ogorzalek, J. C. Larrabee, F. J. Leblanc, and R. E. Huffman, “Ultraviolet daytime auroral and ionospheric imaging,” Appl. Opt. 24, 3395–3405 (1985).
[CrossRef] [PubMed]

Okamoto, Y.

Onaka, T.

Osantowski, J. F.

Paxton, L.

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Paxton, L. J.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

Poletto, L.

Regeon, P. A.

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Rochus, P.

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

Schenkel, F. W.

Thomas, R. J.

Thonnard, S. E.

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Tondello, G.

Werner, W.

Wolven, B.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Zhang, Y.

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Appl. Opt. (9)

W. Werner, “Geometric optical aberration theory of diffraction gratings,” Appl. Opt. 6, 1691–1699 (1967).
[CrossRef] [PubMed]

W. R. Hunter, J. F. Osantowski, and G. Hass, “Reflectance of aluminum overcoated with MgF2 and LiF in the wavelength region from 1600 to 300 at various angles of incidence,” Appl. Opt. 10, 540–544 (1971).
[CrossRef] [PubMed]

M. C. E. Huber and G. Tondello, “Stigmatic performance of an EUV spectrograph with a single toroidal grating,” Appl. Opt. 18, 3948–3953 (1979).
[CrossRef] [PubMed]

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

F. W. Schenkel, B. S. Ogorzalek, J. C. Larrabee, F. J. Leblanc, and R. E. Huffman, “Ultraviolet daytime auroral and ionospheric imaging,” Appl. Opt. 24, 3395–3405 (1985).
[CrossRef] [PubMed]

T. Onaka, “Aberration-corrected concave grating for the mid-infrared spectrometer aboard the Infrared Telescope in Space,” Appl. Opt. 34, 659–666 (1995).
[CrossRef] [PubMed]

T. Onaka, T. Miyata, H. Kataza, and Y. Okamoto, “Design for an aberration-corrected concave grating for a mid-infrared long-slit spectrometer,” Appl. Opt. 39, 1474–1479 (2000).
[CrossRef]

D. M. Cotton, T. Cook, and S. Chakrabarti, “Single-element imaging spectrograph,” Appl. Opt. 33, 1958–1962 (1994).
[CrossRef] [PubMed]

L. Poletto and R. J. Thomas, “Stigmatic spectrometers for extended sources: design with toroidal varied-line-space gratings,” Appl. Opt. 43, 2029–2038 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

Opt. Eng. (1)

R. P. McCoy, K. F. Dymond, G. G. Fritz, S. E. Thonnard, R. R. Meier, and P. A. Regeon, “Special sensor ultraviolet limb imager: an ionospheric and neutral density profiler for the Defense Meteorological Satellite Program satellites,” Opt. Eng. 33, 423–429 (1994).
[CrossRef]

Proc. SPIE (3)

L. J. Paxton, A. B. Christensen, D. Morrison, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, D. C. Humm, J. O. Goldsten, R. DeMajistre, and C.-I. Meng, “GUVI: a hyperspectral imager for geospace,” Proc. SPIE 5660, 228–239 (2004).
[CrossRef]

S. Habraken, C. Jamar, P. Rochus, S. B. Mende, and M. Lampton, “Optical design of the FUV spectrographic imager for the IMAGE Mission,” Proc. SPIE 3114, 544–553 (1997).
[CrossRef]

D. Morrison, L. Paxton, D. Humm, B. Wolven, H. Kil, Y. Zhang, B. S. Ogorzalek, and C.-I. Meng, “On-orbit calibration of the special sensor ultraviolet scanning imager (SSUSI)—a far-UV imaging spectrograph on DMSP F16,” Proc. SPIE 4485, 328–337 (2002).
[CrossRef]

Other (3)

J. H. Holland, Adaptation in Natural and Artificial Systems (University of Michigan Press, 1975).

K. De Jong, “The analysis and behavior of class of genetic adaptive systems,” Ph. D. dissertation, University of Michigan (1975).

D. E. Goldberg, Genetic algorithms in search, optimization, and machine learning reading (Addison-Wesley, 1989).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the grating.

Fig. 2
Fig. 2

Relationships between different aberration coefficients F Z 2 , F 22 , F 12 , F 04 , F 40 , and w and l.

Fig. 3
Fig. 3

Genetic algorithm flowchart.

Fig. 4
Fig. 4

Layout of imaging spectrometer optical system.

Fig. 5
Fig. 5

MTF of telescope.

Fig. 6
Fig. 6

Convergence history of aberrations.

Fig. 7
Fig. 7

RMS spots radii versus wavelength of different designs.

Fig. 8
Fig. 8

MTF of the design system under central and marginal wavelengths.

Fig. 9
Fig. 9

Relation curve of detector’s spectral resolution and wavelength.

Tables (5)

Tables Icon

Table 1 Terms with Their Associate Aberration

Tables Icon

Table 2 Parameters of Imaging Spectrometer

Tables Icon

Table 3 Parameters of Telescope

Tables Icon

Table 4 Optimum Ranges of Parameters by Genetic Algorithm

Tables Icon

Table 5 Parameters of Optical Structure

Equations (22)

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

u = w 2 2 R + l 2 2 ρ + w 4 8 R 3 + l 4 8 ρ 3 + w 2 l 2 4 R 2 ρ + ... .
F = < A P > + < A P > = < A O > + < A O > w d m λ = F 00 + F 10 + F 20 + F 30 + F 40 + F 02 + F 12 + F 22 + F 04 ++ F Z 0 + F Z 1 + F Z 2 + O ( w 5 ) + O ( l 5 ) + O ( z 3 ) + ...
{ F 00 = r + r F Z 0 = 1 2 ( z 2 r + z 2 r ) ,
{ F 10 = w ( m λ d sin i sin θ ) F Z 1 = w 2 ( z 2 sin i r 2 + z 2 sin θ r 2 ) l ( z r + z r ) ,
{ F 40 = w 4 8 [ 4 sin 2 i r 2 ( cos 2 i r cos i R ) 1 r ( cos 2 i r cos i R ) 2 + 1 R 2 ( 1 r cos i R ) + 4 sin 2 θ r 2 ( cos 2 θ r cos θ R ) 1 r ( cos 2 θ r cos θ R ) 2 + 1 R 2 ( 1 r cos θ R ) ] F 22 = w 2 l 2 4 [ 2 sin 2 i r 2 ( 1 r cos i ρ ) 1 r ( cos 2 i r cos i R ) ( 1 r cos i ρ ) + 1 R ρ ( 1 r cos i R ) + 2 sin 2 θ r 2 ( 1 r cos θ ρ ) 1 r ( cos 2 θ r cos θ R ) ( 1 r cos θ ρ ) + 1 R ρ ( 1 r cos θ R ) ] F 04 = l 4 8 [ 1 ρ 2 ( 1 r cos i ρ ) 1 r ( 1 r cos i ρ ) 2 + 1 ρ 2 ( 1 r cos θ ρ ) 1 r ( 1 r cos θ ρ ) 2 ] ,
{ F 30 = w 3 2 [ sin i r ( cos 2 i r cos i R ) + sin θ r ( cos 2 θ r cos θ R ) ] F 12 = w l 2 2 [ sin i r ( 1 r cos i ρ ) + sin θ r ( 1 r cos θ ρ ) ] ,
{ F 02 = l 2 2 ( 1 r cos i ρ + 1 r cos θ ρ ) F Z 2 = w 2 2 ( z 2 sin 2 i r 3 + z 2 sin 2 θ r 3 ) w l ( z sin i r 2 + z sin θ r 2 ) l 2 4 ( z 2 r 3 + z 2 r 3 ) z 2 8 [ 2 w 2 ( cos 2 i r 3 cos i R r 2 ) + l 2 ( 4 r 3 2 cos i ρ r 2 ) ] z 2 8 [ 2 w 2 ( cos 2 θ r 3 cos θ R r 2 ) + l 2 ( 4 r 3 2 cos θ ρ r 2 ) ] F 20 = w 2 2 ( cos 2 i r cos i R + cos 2 θ r cos θ R ) .
Minimize   W = i = 1 , j = 2 F i j ,
C = i = 1 m c i ,
if   g i 0     then     c i = g i ,
if   g i < 0     then     c i = 0.
Φ = W [ 1 + P · C ] ,
D i = ( Φ max + Φ min ) Φ i ,
D c = D i ( D i / n ) ,
145 mm < ρ < 155 mm , 3 ° < i < 8 ° ,
148 mm < L S G = r / cos α < 152 mm , 148 mm < L G I = r / cos α < 152 mm ,
{ cos 2 i / L S G + cos 2 θ / L G I cos i / R = 0 1 / L S G + 1 / L G I cos i / ρ cos θ / ρ = 0 ρ = R cos i cos θ .
{ L S G cos α = R = L G I cos β b 2 = σ 0 R 2 m λ [ cos i R cos 2 i L S G + cos θ R cos 2 θ L G I ] b 3 = σ 0 R 2 m λ [ sin i L S G ( cos i R cos 2 i L S G ) + sin θ L G I ( cos θ R cos 2 θ L G I ) ] b 4 = σ 0 R 3 8 m λ [ 4 sin 2 i L S G 2 ( cos i R cos 2 i L S G ) + 1 L S G ( cos i R cos 2 i L S G ) 2 + 1 R 2 ( cos i R 1 L S G ) + 4 sin 2 θ L G I 2 ( cos θ R cos 2 θ L G I ) + 1 L G I ( cos θ R cos 2 θ L G I ) 2 + 1 R 2 ( cos θ R 1 L G I ) ] .
{ L S G cos α = R = L G I cos β ρ = L S G ( cos i + cos θ ) M 1 + M σ 1 = 1 m K 2 ( cos i + cos θ ) K 1 ( cos i + cos θ ) λ 1 ( cos i + cos θ ) λ 2 ( cos i + cos θ ) σ 2 = 3 2 m λ [ sin i cos i L S G ( cos i L S G 1 R ) + sin θ cos θ L S G ( cos θ L S G 1 R ) ] σ 3 = 1 2 m λ [ 4 sin 2 i cos i L S G 2 ( cos i L S G 1 R ) cos 2 i L S G ( cos i L S G 1 R ) 2 + 1 R 2 ( 1 L S G cos i R ) + 4 sin 2 θ cos θ L G I 2 ( cos θ L G I 1 R ) cos 2 θ L G I ( cos θ L G I 1 R ) 2 + 1 R 2 ( 1 L G I cos θ R ) ] .
Δ λ D = 2 d b m L G I cos θ cos σ ,
Δ λ S = s d m L S G cos i cos α ,
Δ λ A max = d c 8 m R 2 cos i cos θ | 2 W sin ( i θ ) c cos i ( sin i + sin θ ) | ,

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