Abstract

Uniform line spacing plane gratings are introduced into a recording system to generate aspherical wavefronts for recording varied line spacing plane holographic gratings. Analytical expressions of groove parameters are derived to the fourth order. A ray-tracing validation algorithm is provided based on Fermat's principle and a local search method. The recording parameters are optimized to record a varied line spacing plane holographic grating with the aid of derived analytical expressions. A design example demonstrates the exactness of the analytical expressions and the superiority of recording optics with auxiliary gratings.

© 2006 Optical Society of America

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  1. T. Namioka, M. Seya, and H. Noda, "Design and performance of holographic concave gratings," Jpn. J. Appl. Phys. 15, 1181-1197 (1976).
    [CrossRef]
  2. M. Koike and Y. Harada, "New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method," Proc. SPIE 815, 96-101 (1987).
  3. M. Duban, "Holographic aspheric gratings printed with aberrant waves," Appl. Opt. 26, 4263-4273 (1987).
  4. T. Namioka and M. Koike, "Aspheric wave-front recording optics for holographic gratings," Appl. Opt. 34, 2180-2186 (1995).
  5. H. Noda, Y. Harada, and M. Koike, "Holographic grating recorded using aspheric wavefronts for a Seya-Namioka monochromator," Appl. Opt. 28, 4375-4380 (1989).
  6. M. Koike and T. Namioka, "Optimization and evaluation of varied line spacing plane grating monochromators for third generation synchrotron radiation sources," J. Electron Spectrosc. Relat. Phenom. 80, 303-308 (1996).
    [CrossRef]
  7. K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
    [CrossRef]
  8. M. Koike, T. Yamazaki, and Y. Harada, "Design of holographic gratings recorded with aspheric wavefront recording optics for soft x-ray flat-field spectrographs," J. Electron Spectrosc. Relat. Phenom. 101, 913-918 (1999).
    [CrossRef]
  9. M. Duban, "Third-generation Rowland holographic mounting," Appl. Opt. 30, 4019-4025 (1991).
  10. M. Duban, "Theory of spherical holographic gratings recorded by use of a multimode deformable mirror," Appl. Opt. 37, 7209-7213 (1998).
  11. M. Duban, "Recording high-dispersion spherical holographic gratings in a modified Rowland mounting by use of a multimode deformable mirror," Appl. Opt. 39, 16-19 (2000).
  12. M. Duban, G. Lemaitre, and R. Malina, "Recording method for obtaining high-resolution holographic gratings through use of multimode deformable plane mirrors," Appl. Opt. 37, 3438-3439 (1998).
  13. M. Duban, K. Dohlen, and G. Lemaitre, "Illustration of the use of multimode deformable plane mirrors to record high-resolution concave gratings: results for the Cosmic Origins Spectrograph gratings of the Hubble Space Telescope," Appl. Opt. 37, 7214-7217 (1998).
  14. G. Lemaitre and M. Duban, "Corrected grating recording by active optics compensator case of HST Cosmic Origins Spectrometer," Proc. SPIE 4854, 447-456 (2003).
    [CrossRef]
  15. M. Duban, "Third-generation Rowland holographic mounting: fourth-order theory," Appl. Opt. 38, 3443-3449 (1999).
  16. M. Duban, "High-dispersion spherical holographic gratings in a modified Rowland mounting," Appl. Opt. 40, 1599-1608 (2001).
  17. E. Sokolova, "Geometric theory of two-steps recorded holographic diffraction gratings," Proc. SPIE 3450, 113-124 (1998).
    [CrossRef]
  18. E. Sokolova, B. Kruizinga, and I. Golubenko, "Recording of concave diffraction gratings in a two-step process using spatially incoherent light," Opt. Eng. 43, 2613-2622 (2004).
    [CrossRef]
  19. B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
    [CrossRef]
  20. Q. Ling, G. Wu, and Q. Wang, "Restricted evolution based multimodal function optimization in holographic grating design," in Proceedings of IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 789-794.

2004 (1)

E. Sokolova, B. Kruizinga, and I. Golubenko, "Recording of concave diffraction gratings in a two-step process using spatially incoherent light," Opt. Eng. 43, 2613-2622 (2004).
[CrossRef]

2003 (1)

G. Lemaitre and M. Duban, "Corrected grating recording by active optics compensator case of HST Cosmic Origins Spectrometer," Proc. SPIE 4854, 447-456 (2003).
[CrossRef]

2001 (1)

2000 (1)

1999 (2)

M. Koike, T. Yamazaki, and Y. Harada, "Design of holographic gratings recorded with aspheric wavefront recording optics for soft x-ray flat-field spectrographs," J. Electron Spectrosc. Relat. Phenom. 101, 913-918 (1999).
[CrossRef]

M. Duban, "Third-generation Rowland holographic mounting: fourth-order theory," Appl. Opt. 38, 3443-3449 (1999).

1998 (5)

1997 (1)

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

1996 (1)

M. Koike and T. Namioka, "Optimization and evaluation of varied line spacing plane grating monochromators for third generation synchrotron radiation sources," J. Electron Spectrosc. Relat. Phenom. 80, 303-308 (1996).
[CrossRef]

1995 (1)

1991 (1)

1989 (1)

1987 (2)

M. Duban, "Holographic aspheric gratings printed with aberrant waves," Appl. Opt. 26, 4263-4273 (1987).

M. Koike and Y. Harada, "New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method," Proc. SPIE 815, 96-101 (1987).

1976 (1)

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

Amemiya, K.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Bonnemason, F.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Deville, B.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Dohlen, K.

Duban, M.

Flamand, J.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Gattino, V.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Golubenko, I.

E. Sokolova, B. Kruizinga, and I. Golubenko, "Recording of concave diffraction gratings in a two-step process using spatially incoherent light," Opt. Eng. 43, 2613-2622 (2004).
[CrossRef]

Harada, Y.

M. Koike, T. Yamazaki, and Y. Harada, "Design of holographic gratings recorded with aspheric wavefront recording optics for soft x-ray flat-field spectrographs," J. Electron Spectrosc. Relat. Phenom. 101, 913-918 (1999).
[CrossRef]

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

H. Noda, Y. Harada, and M. Koike, "Holographic grating recorded using aspheric wavefronts for a Seya-Namioka monochromator," Appl. Opt. 28, 4375-4380 (1989).

M. Koike and Y. Harada, "New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method," Proc. SPIE 815, 96-101 (1987).

Ito, K.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Kitajima, Y.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Koeda, M.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Koike, M.

M. Koike, T. Yamazaki, and Y. Harada, "Design of holographic gratings recorded with aspheric wavefront recording optics for soft x-ray flat-field spectrographs," J. Electron Spectrosc. Relat. Phenom. 101, 913-918 (1999).
[CrossRef]

M. Koike and T. Namioka, "Optimization and evaluation of varied line spacing plane grating monochromators for third generation synchrotron radiation sources," J. Electron Spectrosc. Relat. Phenom. 80, 303-308 (1996).
[CrossRef]

T. Namioka and M. Koike, "Aspheric wave-front recording optics for holographic gratings," Appl. Opt. 34, 2180-2186 (1995).

H. Noda, Y. Harada, and M. Koike, "Holographic grating recorded using aspheric wavefronts for a Seya-Namioka monochromator," Appl. Opt. 28, 4375-4380 (1989).

M. Koike and Y. Harada, "New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method," Proc. SPIE 815, 96-101 (1987).

Kruizinga, B.

E. Sokolova, B. Kruizinga, and I. Golubenko, "Recording of concave diffraction gratings in a two-step process using spatially incoherent light," Opt. Eng. 43, 2613-2622 (2004).
[CrossRef]

Lemaitre, G.

Ling, Q.

Q. Ling, G. Wu, and Q. Wang, "Restricted evolution based multimodal function optimization in holographic grating design," in Proceedings of IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 789-794.

Malina, R.

Millet, V.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Nagano, T.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Namioka, T.

M. Koike and T. Namioka, "Optimization and evaluation of varied line spacing plane grating monochromators for third generation synchrotron radiation sources," J. Electron Spectrosc. Relat. Phenom. 80, 303-308 (1996).
[CrossRef]

T. Namioka and M. Koike, "Aspheric wave-front recording optics for holographic gratings," Appl. Opt. 34, 2180-2186 (1995).

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

Noda, H.

H. Noda, Y. Harada, and M. Koike, "Holographic grating recorded using aspheric wavefronts for a Seya-Namioka monochromator," Appl. Opt. 28, 4375-4380 (1989).

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

Ohata, T.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Sano, K.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Sasai, H.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Seya, M.

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

Sokolova, E.

E. Sokolova, B. Kruizinga, and I. Golubenko, "Recording of concave diffraction gratings in a two-step process using spatially incoherent light," Opt. Eng. 43, 2613-2622 (2004).
[CrossRef]

E. Sokolova, "Geometric theory of two-steps recorded holographic diffraction gratings," Proc. SPIE 3450, 113-124 (1998).
[CrossRef]

Thevenon, A.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Touzet, B.

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

Wang, Q.

Q. Ling, G. Wu, and Q. Wang, "Restricted evolution based multimodal function optimization in holographic grating design," in Proceedings of IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 789-794.

Wu, G.

Q. Ling, G. Wu, and Q. Wang, "Restricted evolution based multimodal function optimization in holographic grating design," in Proceedings of IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 789-794.

Yamazaki, T.

M. Koike, T. Yamazaki, and Y. Harada, "Design of holographic gratings recorded with aspheric wavefront recording optics for soft x-ray flat-field spectrographs," J. Electron Spectrosc. Relat. Phenom. 101, 913-918 (1999).
[CrossRef]

Yonamoto, Y.

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Appl. Opt. (10)

J. Electron Spectrosc. Relat. Phenom. (2)

M. Koike and T. Namioka, "Optimization and evaluation of varied line spacing plane grating monochromators for third generation synchrotron radiation sources," J. Electron Spectrosc. Relat. Phenom. 80, 303-308 (1996).
[CrossRef]

M. Koike, T. Yamazaki, and Y. Harada, "Design of holographic gratings recorded with aspheric wavefront recording optics for soft x-ray flat-field spectrographs," J. Electron Spectrosc. Relat. Phenom. 101, 913-918 (1999).
[CrossRef]

Jpn. J. Appl. Phys. (1)

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

Opt. Eng. (1)

E. Sokolova, B. Kruizinga, and I. Golubenko, "Recording of concave diffraction gratings in a two-step process using spatially incoherent light," Opt. Eng. 43, 2613-2622 (2004).
[CrossRef]

Proc. SPIE (5)

B. Deville, F. Bonnemason, J. Flamand, V. Millet, A. Thevenon, V. Gattino, and B. Touzet, "Holographically recorded, ion etched variable line space gratings," Proc. SPIE 3450, 24-35 (1998).
[CrossRef]

G. Lemaitre and M. Duban, "Corrected grating recording by active optics compensator case of HST Cosmic Origins Spectrometer," Proc. SPIE 4854, 447-456 (2003).
[CrossRef]

E. Sokolova, "Geometric theory of two-steps recorded holographic diffraction gratings," Proc. SPIE 3450, 113-124 (1998).
[CrossRef]

M. Koike and Y. Harada, "New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method," Proc. SPIE 815, 96-101 (1987).

K. Amemiya, Y. Kitajima, Y. Yonamoto, T. Ohata, K. Ito, K. Sano, T. Nagano, M. Koeda, H. Sasai, and Y. Harada, "Fabrication of a varied-line-spacing plane grating with aspheric wavefront holographic recording for a new grazing incidence monochromator at Photon Factory," Proc. SPIE 3150, 171-182 (1997).
[CrossRef]

Other (1)

Q. Ling, G. Wu, and Q. Wang, "Restricted evolution based multimodal function optimization in holographic grating design," in Proceedings of IEEE Congress on Evolutionary Computation (IEEE, 2005), pp. 789-794.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the recording system consists of two coherent point sources, C and D, two uniform line spacing plane gratings, G 1 and G 2, and a plane grating blank G. C, D, G 1, G 2, and G are arranged so that C, D, and the normal of G 1, G 2, and G at their vertices O 1, O 2, and O, lie in a common plane. The incident principal rays CO 1 and DO 2 pass through O after being diffracted at O 1 and O 2, respectively. The distances pC , qC , pD , and qD , the angles of incidence η C , γ, η D , and δ of the principle rays CO 1, O 1 O, DO 2, and O 2 O, and the angles of diffraction ζ C and ζ D of the principle rays O 1 O and O 2 O are named as recording parameters.

Fig. 2
Fig. 2

Groove number error between H A (computed from analytical expressions) and H N (computed from exact ray-tracing) l = 0 mm is less than 1.5 lines.

Fig. 3
Fig. 3

Two-dimensional schematic recording optics is designed for the varied line spacing holographic grating. Light originating from D is diffracted at auxiliary grating G 2, and the +1 order diffractive light reaches G. Light originating from C reaches G directly because the auxiliary plane mirror can be omitted. It is clear that the other diffractive light will not disturb the recording process when the recording parameters are properly selected.

Tables (3)

Tables Icon

Table 1 Optimized Recording Parameters of Holographic Grating

Tables Icon

Table 2 Corresponding Groove Density Parameters of Optimization Results

Tables Icon

Table 3 Design Results Considering Tolerances for Recording Parameters Group 2

Equations (53)

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x C = p C cos η C ,     x D = p D cos η D ,
y C = p C sin η C ,     y D = p D sin η D .
sin η C + sin ζ C = m 1 λ 0 / σ 1 ,
sin η D + sin ζ D = m 2 λ 0 / σ 2 ,
H = 1 λ 0 [ ( C Q 1 + Q 1 P ) + w 1 m 1 λ 0 / σ 1 ( C O 1 + O 1 O ) ] 1 λ 0 [ ( D Q 2 + Q 2 P ) + w 2 m 2 λ 0 / σ 2 ( D O 2 + O 2 O ) ] .
H = 1 λ 0 ( H C H D ) ,                                                                              
H C = x C       2 = ( w 1 y C ) 2 + l 1     2 + ξ ¯ 1     2 + ( w ¯ 1 w ) 2 + ( l ¯ 1 l ) 2 ( p C + q C ) + w 1 ( sin η C + sin ζ C ) ,
H D = x D       2 + ( w 2 y D ) 2 + l 2     2 + ξ ¯ 2     2 + ( w ¯ 2 w ) 2 + ( l ¯ 2 l ) 2 ( p D + q D ) + w 2 ( sin η D + sin ζ D ) .
ξ ¯ 1 = w 1 sin ( ζ C + γ ) + q C cos γ ,
w ¯ 1 = w 1 cos ( ζ C + γ ) + q C sin γ ,
l ¯ 1 = l 1 .
H = 1 λ 0 i = 0 4 j = 0 4 i c i j H i j w i l j = 1 λ 0 i = 0 4 j = 0 4 i c i j ( H i j ) C w i l j  −  1 λ 0 i = 0 4 j = 0 4 i c i j ( H i j ) D w i l j ,
H C = i = 0 4 j = 0 4 i c i j ( H i j ) C w i l j ,     H D = i = 0 4 j = 0 4 i c i j ( H i j ) D w i l j .
w 1 = i = 0 4 j = 0 4 i ( A i j ) C w i l j ,     l 1 = i = 0 4 j = 0 4 i ( B i j ) C w i l j .
Min   F C = C Q 1 + Q 1 P + σ 1 w 1 m 1 λ 0 = C Q 1 + Q 1 P + w 1 ( sin η C + sin ζ C ) ,
L 1 = L 1     2 2 ( sin η C + sin ζ C ) M 1 ( sin η C + sin ζ C ) 2 ,
M 1 = M 1 + sin η C + sin ζ C ,
N 1 = N 1 ,
L 1 = x C / C Q 1 ,     M 1 = ( w 1 y C ) / C Q 1 ,     N 1 = l 1 / C Q 1 .
L C = L 1 cos ( ζ C + γ ) + M 1 sin ( η C + γ ) ,
M C = L 1 sin ( ζ C + γ ) + M 1 cos ( η C + γ ) ,
N C = N 1 .
L C = ξ ¯ 1 / Q 1 P ,
M C = ( w w ¯ 1 ) / Q 1 P ,
N C = ( l l ¯ 1 ) / Q 1 P .
w = w ¯ 1 ξ ¯ 1 M C / L C ,     l = l ¯ 1 ξ ¯ 1 N C / L C .
w = f w [ w , l , ( A i j ) C , ( B i j ) C , η C , ζ C , γ , p C , q C ] ,
l = f l [ w , l , ( A i j ) C , ( B i j ) C , η C , ζ C , γ , p C , q C ] ,
w = i = 0 4 j = 0 4 i ( f w ) i j w i l j ,     l = i = 0 4 j = 0 4 i ( f l ) i j w i l j .
( f w ) i j = { 1 i = 1 , j = 0 0 else ,     ( f l ) i j = { 1 i = 1 , j = 0 0 else ,
H = 1 λ 0 [ H 10 w + ( H 20 w 2 + H 02 l 2 + H 30 w 3 + H 12 w l 2 ) + ( H 40 w 4 + 2 H 22 w 2 l 2 + H 04 l 4 ) ] .
H C = f H C ( w , l , w 1 , l 1 , η C , ζ C , γ , p C , q C ) .
H C = g H C ( w 1 , l 1 ) .
H C = Min   g H C ( w 1 , l 1 ) .
n 0 = H 10 / λ 0 , b 2 = H 20 / λ 0 n 0 ,
b 3 = 3 H 30 / 2 λ 0 n 0 , b 4 = H 40 / 2 λ 0 n 0 .
( A 01 ) C = ( A 11 ) C = 0 ,
( A 10 ) C = p C cos γ cos ζ C / r C ,
( A 20 ) C = ( A 10 ) 2 cos 2 η C [ cos 2 η C ( 2 q C cos ζ C sin γ q C sin ζ C cos γ ) + cos 2 ζ C ( 2 p C cos ζ C × sin γ + 2 p C sin ζ C cos γ 3 q C cos γ × sin η C ) ] / r C p C cos 2 ζ C cos γ ,
( A 02 ) C = ( B 01 ) 2 q C ( sin η C + sin ζ C ) / r C p C ,
( B 10 ) C = ( B 20 ) C = ( B 02 ) C = 0 ,
( B 01 ) C = 1 / p C ( p C + q C ) ,
( B 11 ) C = ( A 10 ) C [ r C sin γ + cos ζ C ( q C cos γ sin η C p C sin ζ C cos γ ) ] / cos γ cos ζ C ,
( H 10 ) C = sin γ ,
( H 20 ) C = [ ( A 10 ) C       2 r C + p cos γ ( cos γ + 2 ( A 10 ) C cos ζ C ) ] / p C q C ,
( H 30 ) C = [ 2 ( A 10 ) C ( A 20 ) C r C p C q C + ( A 10 ) C       3 ( q C       2 sin η C cos 2 η C + p C       2 sin ζ C cos 2 ζ C ) + ( A 10 ) C       2 p C       2 cos ζ C ( 2 cos γ sin ζ C + sin γ cos ζ C ) + ( A 10 ) C p C       2 cos γ ( cos γ sin ζ C + 2 sin γ cos ζ C ) + p C       2 cos γ ( 2 q C ( A 20 ) C cos ζ C + sin γ cos γ ) ] / p C       2 q C       2 ,
( H 40 ) C = [ 4 ( A 20 ) C       2 r C p C       2 q C       2 + 4 ( A 10 ) C       2 p C       3 ( cos 2 γ + cos 2 ζ C ) + 12 ( A 10 ) C       2 ( A 20 ) C p C q C ( q C       2 cos 2 η C sin η C + p C       2 cos 2 ζ C sin ζ C ) 5 ( A 10 ) C       4 ( q C       3 cos 4 η C + p C       3 cos 4 ζ C ) + p C       3 cos 2 γ ( 4 5 cos 2 γ ) + ( A 10 ) C       3 p C       3 cos ζ C ( 8 cos γ 12 cos 2 ζ C cos γ + 8 sin γ sin ζ C cos ζ C ) + ( A 10 ) C       2 p C       3 cos ζ C cos γ ( 14 cos ζ C cos γ + 16 sin ζ C sin γ ) + ( A 10 ) C p C       3 cos γ ( 8 cos ζ C 12 cos 2 ζ C cos γ + 8 sin γ sin ζ C cos γ ) + 8 ( A 10 ) C ( A 20 ) C p C       3 q C cos ζ C ( sin γ + 2 cos γ sin ζ C ) + 4 ( A 10 ) C       4 ( q C       3 cos 2 η C + p C       3 cos 2 ζ C ) + 4 ( A 20 ) C       q C p C       3 cos γ ( cos γ sin ζ C + 2 sin γ cos ζ C ) ] / p C       2 q C       2 ,
( H 02 ) C = [ ( B 01 ) C       2 q C + ( B 01 ) C       2 p C + p C 2 ( B 01 ) C p C ] / p C q C ,
( H 12 ) C = { ( A 10 ) C ( B 01 ) C       2 ( p C       2 sin ζ C + q C       2 sin η C ) + 2 ( B 10 ) C ( B 11 ) C p C q C ( p C + q C ) + ( A 10 ) C p C       2 sin ζ C ( 1 2 ( B 01 ) C ) + ( B 01 ) C p C       2 sin γ [ ( B 01 ) C 2 ] + 2 ( A 10 ) C ( A 02 ) C r C p C q C 2 ( B 11 ) C p C       2 q C + p C       2 sin γ + 2 ( A 02 ) C p C       2 q C cos γ cos ζ C } / p C       2 q C       2 ,
( H 22 ) C = p C       3 ( 2 4 ( B 01 ) C + 2 ( B 01 ) C       2 + 2 ( A 10 ) C       2 3 cos 2 γ ) + 2 ( B 11 ) C       2 p C       2 q C       2 ( p C + q C ) 3 ( A 10 ) C       2 ( B 01 ) C       2 ( q C       3 cos 2 η C + p C       3 cos 2 ζ C ) + 4 ( A 10 ) C ( B 01 ) C p C       3 ( cos ζ C cos γ 2 sin ζ C sin γ ) 2 ( A 10 ) C ( B 01 ) C       2 p C       3 ( cos ζ C cos γ 2 sin ζ C sin γ ) + 4 ( B 11 ) C p C       3 q C sin γ [ ( B 01 ) C 1 ] 2 ( A 10 ) C p C       3 ( cos ζ C cos γ 2 sin ζ C sin γ ) + 2 ( A 10 ) C       2 ( B 01 ) C       2 ( p C       3 + q C       3 ) + 4 ( A 10 ) C ( B 01 ) C ( B 11 ) C p C q C ( p C       2 sin ζ C + q C       2 sin η C ) 3 ( A 10 ) C       2 p C       3 cos 2 ζ C
+ 2 ( A 10 ) C       2 ( B 01 ) C p C       3 ( 3 cos 2 ζ C 2 ) 4 ( A 10 ) C ( B 11 ) C p C       3 q C sin ζ C + 6 ( A 10 ) C       2 ( A 02 ) C p C q C ( p C       2 cos 2 ζ C sin ζ C + q C       2 cos 2 η C sin η C ) + 4 ( A 10 ) C ( A 02 ) C p C       3 q C cos ζ C ( 2 cos γ sin ζ C + sin γ cos ζ C ) + 4 ( A 20 ) C ( A 02 ) C r C ( p C + q C ) + 2 ( A 20 ) C ( B 01 ) C       2 p C q C ( p C       2 sin ζ C + q C       2 sin η C ) + 2 ( A 02 ) C p C       3 q C cos γ ( cos γ sin ζ C + 2 sin γ cos ζ C ) + 3 ( B 01 ) C p C       3 cos 2 γ [ 2 ( B 01 ) C ] + 2 ( A 20 ) C p C       3 q C sin ζ C ( 1 2 ( B 01 ) C ) } / p C       3 q C       3 ,
( H 04 ) C = [ p C       3 ( 1 + 4 ( B 01 ) C 6 ( B 01 ) C       2 + 4 ( B 01 ) C       3 ) ( B 01 ) C       4 ( p C       3 + q C       3 ) + 4 ( A 02 ) C ( B 01 ) C       2 p C q C ( p C       2 sin ζ C + q C       2 sin η C ) + 4 ( A 02 ) C       2 r C p C q C + 4 ( A 02 ) C p C       3 q C sin ζ C ( 1 2 ( B 01 ) C ) ] / p C       3 q C       3 ,
r C = q C cos 2 η C + p C cos 2 ζ C .

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