Abstract

A third-order geometric aberration theory of the ellipsoidal grating has been developed by analytically following an exact ray-tracing formalism with the aid of power series expansions. The theory takes into account all the possible aberrations up to third order and provides analytic formulas for the spot diagram of a spectral image formed by a modified or a nonmodified ellipsoidal grating with any of the groove patterns producible by means of mechanical ruling or conventional holographic recording. The present analytic formulas and other analytic ray-deviation formulas used in designing grating instruments have been evaluated in comparison with exact ray tracing. The results show the validity of the present theory and the limitation of the ray-deviation formulas based on the light path function and wave-front-aberration theory.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Namioka, “Theory of the ellipsoidal concave grating. I,” J. Opt. Soc. Am. 51, 4–12 (1961).
    [Crossref]
  2. T. Namioka, “Theory of the ellipsoidal concave grating. II. Application of the theory to the specific grating mountings,” J. Opt. Soc. Am. 51, 13–16 (1961).
    [Crossref]
  3. T. Namioka, “Choice of grating mountings suitable for a space telescope,” in Space Astrophysics, W. Liller, ed. (McGraw-Hill, New York, 1961), Chap. 15, pp. 228–268.
  4. H. W. Moos, ed., LYMAN, the Far Ultraviolet Spectroscopic Explorer, Phase A Study Final Report, (NASA Goddard Space Flight Center, Greenbelt, Md., 1989).
  5. D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).
  6. W. C. Cash, “Aspheric concave grating spectrographs,” Appl. Opt. 23, 4518–4522 (1984).
    [Crossref] [PubMed]
  7. D. Content, C. Trout, P. Davila, M. Wilson, “Aberration corrected aspheric gratings for far ultraviolet spectrographs: conventional approach,” Appl. Opt. 30, 801–806 (1991).
    [Crossref] [PubMed]
  8. C. Trout, D. Content, P. Davila, “Aberration-corrected aspheric grating designs for the Lyman/Far-Ultraviolet Spectroscopic Explorer high-resolution spectrograph: a comparison,” Appl. Opt. 31, 943–948 (1992).
    [Crossref] [PubMed]
  9. M. Duban, “Comparison of grating designs for the Lyman Far-Ultraviolet Spectroscopic Explorer spectrograph,” Appl. Opt. 32, 4253–4264 (1993).
    [Crossref] [PubMed]
  10. D. Content, T. Namioka, “Deformed ellipsoidal gratings for far-ultraviolet spectrographs: analytic optimization,” Appl. Opt. 32, 4881–4889 (1993).
    [Crossref] [PubMed]
  11. 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]
  12. P. Davila, D. Content, C. Trout, “Aberration-corrected aspheric gratings for far-ultraviolet spectrographs: holographic approach,” Appl. Opt. 31, 949–954 (1992).
    [Crossref] [PubMed]
  13. M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
    [Crossref] [PubMed]
  14. M. Duban, “Some reflections about a high-resolution spectrograph,” Appl. Opt. 31, 443–445 (1992).
    [Crossref] [PubMed]
  15. R. Grange, “Aberration-reduced holographic spherical gratings for Rowland circle spectrographs,” Appl. Opt. 31, 3744–3749 (1992).
    [Crossref] [PubMed]
  16. T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 1992; in UV and X-Ray Spectroscopy of Laboratory and Astrophysical Plasmas, E. Silver, S. Kahn, eds. (Cambridge U. Press, Cambridge, England, 1993), pp. 357–360.
  17. R. Grange, “Holographic spherical gratings: a new family of quasistigmatic designs for the Rowland-circle mounting,” Appl. Opt. 32, 4875–4880 (1993).
    [Crossref] [PubMed]
  18. H. G. Beutler, “The theory of the concave grating,” J. Opt. Soc. Am. 35, 311–350 (1945).
    [Crossref]
  19. T. Namioka, “Theory of the concave grating. I,” J. Opt. Soc. Am. 49, 446–460 (1959).
    [Crossref]
  20. T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
    [Crossref]
  21. C. H. F. Velzel, “On the imaging properties of holographic gratings,” J. Opt. Soc. Am. 67, 1021–1027 (1977).
    [Crossref]
  22. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [Crossref] [PubMed]
  23. W. R. McKinney, C. Palmer, “Numerical design method for aberration-reduced concave grating spectrometers,” Appl. Opt. 26, 3108–3118(1987).
    [Crossref] [PubMed]
  24. T. Namioka, S. Morozumi, “Design of constant-deviation monochromators,” Nucl. Instrum. Methods 177, 141–146 (1980).
    [Crossref]
  25. T. Namioka, “A merit function for the design of toroidal holographic gratings (in Japanese),” Bull. Res. Inst. Sci. Meas. Tohoku Univ. 29, 65–75 (1980).
  26. 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]
  27. M. Koike, T. Namoika, “Merit function for the design of grating instruments,” Appl. Opt. 33, 2048–2056 (1994).
    [Crossref] [PubMed]
  28. K. Goto, T. Kurosaki, “Canonical formulation for the geometrical optics of concave gratings,” J. Opt. Soc. Am. A 10, 452–465 (1993).
    [Crossref]
  29. T. Namioka, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” submitted to Appl. Opt.
  30. T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3993 (1980).
    [Crossref] [PubMed]

1994 (1)

1993 (4)

1992 (5)

1991 (3)

1987 (1)

1984 (1)

1983 (1)

1980 (3)

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 (in Japanese),” Bull. Res. Inst. Sci. Meas. Tohoku Univ. 29, 65–75 (1980).

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

1977 (1)

1976 (1)

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

1961 (2)

1959 (1)

1945 (1)

Beutler, H. G.

Cash, W. C.

Chrisp, M. P.

Content, D.

Content, D. A.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Davila, P.

Davila, P. M.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Duban, M.

Goto, K.

Grange, R.

Harada, T.

Kita, T.

Koike, M.

M. Koike, T. Namoika, “Merit function for the design of grating instruments,” Appl. Opt. 33, 2048–2056 (1994).
[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]

T. Namioka, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” submitted to Appl. Opt.

T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 1992; in UV and X-Ray Spectroscopy of Laboratory and Astrophysical Plasmas, E. Silver, S. Kahn, eds. (Cambridge U. Press, Cambridge, England, 1993), pp. 357–360.

Kurosaki, T.

Laget, M.

McKinney, W. R.

Morozumi, S.

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

Namioka, T.

D. Content, T. Namioka, “Deformed ellipsoidal gratings for far-ultraviolet spectrographs: analytic optimization,” Appl. Opt. 32, 4881–4889 (1993).
[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]

T. Namioka, “A merit function for the design of toroidal holographic gratings (in Japanese),” 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]

T. Namioka, “Theory of the ellipsoidal concave grating. II. Application of the theory to the specific grating mountings,” J. Opt. Soc. Am. 51, 13–16 (1961).
[Crossref]

T. Namioka, “Theory of the ellipsoidal concave grating. I,” J. Opt. Soc. Am. 51, 4–12 (1961).
[Crossref]

T. Namioka, “Theory of the concave grating. I,” J. Opt. Soc. Am. 49, 446–460 (1959).
[Crossref]

T. Namioka, “Choice of grating mountings suitable for a space telescope,” in Space Astrophysics, W. Liller, ed. (McGraw-Hill, New York, 1961), Chap. 15, pp. 228–268.

T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 1992; in UV and X-Ray Spectroscopy of Laboratory and Astrophysical Plasmas, E. Silver, S. Kahn, eds. (Cambridge U. Press, Cambridge, England, 1993), pp. 357–360.

T. Namioka, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” submitted to Appl. Opt.

Namoika, T.

Noda, H.

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

Osantowski, J. F.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Palmer, C.

Saha, T. T.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Seya, M.

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

Trout, C.

Velzel, C. H. F.

Wilson, M.

Wilson, M. E.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Appl. Opt. (15)

W. C. Cash, “Aspheric concave grating spectrographs,” Appl. Opt. 23, 4518–4522 (1984).
[Crossref] [PubMed]

D. Content, C. Trout, P. Davila, M. Wilson, “Aberration corrected aspheric gratings for far ultraviolet spectrographs: conventional approach,” Appl. Opt. 30, 801–806 (1991).
[Crossref] [PubMed]

C. Trout, D. Content, P. Davila, “Aberration-corrected aspheric grating designs for the Lyman/Far-Ultraviolet Spectroscopic Explorer high-resolution spectrograph: a comparison,” Appl. Opt. 31, 943–948 (1992).
[Crossref] [PubMed]

M. Duban, “Comparison of grating designs for the Lyman Far-Ultraviolet Spectroscopic Explorer spectrograph,” Appl. Opt. 32, 4253–4264 (1993).
[Crossref] [PubMed]

D. Content, T. Namioka, “Deformed ellipsoidal gratings for far-ultraviolet spectrographs: analytic optimization,” Appl. Opt. 32, 4881–4889 (1993).
[Crossref] [PubMed]

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]

P. Davila, D. Content, C. Trout, “Aberration-corrected aspheric gratings for far-ultraviolet spectrographs: holographic approach,” Appl. Opt. 31, 949–954 (1992).
[Crossref] [PubMed]

M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
[Crossref] [PubMed]

M. Duban, “Some reflections about a high-resolution spectrograph,” Appl. Opt. 31, 443–445 (1992).
[Crossref] [PubMed]

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

R. Grange, “Holographic spherical gratings: a new family of quasistigmatic designs for the Rowland-circle mounting,” Appl. Opt. 32, 4875–4880 (1993).
[Crossref] [PubMed]

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

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

M. Koike, T. Namoika, “Merit function for the design of grating instruments,” Appl. Opt. 33, 2048–2056 (1994).
[Crossref] [PubMed]

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

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

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

J. Opt. Soc. Am. (5)

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

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 (1)

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 (5)

T. Namioka, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” submitted to Appl. Opt.

T. Namioka, “Choice of grating mountings suitable for a space telescope,” in Space Astrophysics, W. Liller, ed. (McGraw-Hill, New York, 1961), Chap. 15, pp. 228–268.

H. W. Moos, ed., LYMAN, the Far Ultraviolet Spectroscopic Explorer, Phase A Study Final Report, (NASA Goddard Space Flight Center, Greenbelt, Md., 1989).

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy VII, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 1992; in UV and X-Ray Spectroscopy of Laboratory and Astrophysical Plasmas, E. Silver, S. Kahn, eds. (Cambridge U. Press, Cambridge, England, 1993), pp. 357–360.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the optical system. A and A0 are point sources. B and B0 are the image points of A and A0 formed in an image plane ∑ by rays APB and A 0 OB 0, respectively. Point O is the vertex of the grating, and P is a point on the nth groove. For reference the holographic recording point sources C and D are also shown.

Fig. 2
Fig. 2

Schematic diagram showing two image planes ∑0 and ∑ in relation to the diffracted principal ray OB 0. Both image planes are perpendicular to the xy plane and pass through a point B0 on the diffracted principal ray of wavelength λ in mth order. Plane ∑0 is perpendicular to ray OB 0, whereas plane ∑ makes angles ϕ and χ with plane ∑0 and the xz plane, respectively.

Fig. 3
Fig. 3

Spot diagrams constructed for model optical system I with 2000 rays of 91, 95, 99, and 103 nm with (a) exact ray tracing and (b) the LPF formulas. The standard deviations s Y and s Z of the spots in the Y and Z directions are indicated in the respective diagrams. The ΔZ LPF-versus-ΔY LPF plots also are given in (c) to show the deviations of individual spots generated by the LPF formulas from the corresponding spots constructed by exact ray tracing. The rms errors (σΔ Y and σΔ Z of the LPF formulas are indicated in (c).

Fig. 4
Fig. 4

Spot diagrams constructed for model optical system II with 2000 rays of 15, 60, 105, and 150 nm with (a) exact ray tracing, (b) SD formulas, (c) LPF formulas, and (d) WFA formulas. The standard deviations, s Y and s Z , of the spots in the Y and Z directions are also indicated in the respective diagrams.

Fig. 5
Fig. 5

ΔZ-versus-ΔY plots corresponding to Fig. 4. In (a), (b), and (c) are the deviations of individual spots generated by the SD, LPF, and WFA formulas from the corresponding spots constructed by exact ray tracing, respectively. The rms errors, σΔ Y and σΔ Z , of the SD, LPF, and WFA formulas are indicated in the respective diagrams.

Fig. 6
Fig. 6

Spot diagrams constructed for model optical system III with 2000 rays of 15, 60, 105, and 150 nm with (a) exact ray tracing, (b) SD formulas, (c) LPF formulas, and (d) WFA formulas. The standard deviations, s Y and s Z , of the spots in the Y and Z directions are also indicated in the respective diagrams.

Fig. 7
Fig. 7

ΔZ-versus-ΔY plots corresponding to Fig. 6. In (a), (b), and (c) are the deviations of individual spots generated by the SD, LPF, and WFA formulas from the corresponding spots constructed by exact ray tracing, respectively. The rms errors, σΔ Y and σΔ Z , of the SD, LPF, and WFA formulas are indicated in the respective diagrams.

Fig. 8
Fig. 8

Scheme for ruling a grating with varied spacing and curved grooves (cross section at l = constant). The grooves are ruled by constraining the movement of the diamond tool in reference plane II and by advancing the reference plane by a predetermined amount σ n in they direction.

Equations (72)

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

F / w = ( L - L ) ( ξ / w ) + ( M - M ) + m λ ( n / w ) = 0 , F / l = ( L - L ) ( ξ / l ) + ( N - N ) + m λ ( n / l ) = 0 ,
F = A P + P B + n m λ .
L ( ξ - x ) / A P , M ( w - y ) / A P , N ( l - z ) / A P , L ( x - ξ ) / P B , M ( y - w ) / P B , N ( z - l ) / P B ,
L - L = M - M + ( n / w ) ξ / w = N - N + m λ ( n / l ) ξ / l T
L = L + T ,             M = M + m λ n w - T ξ w , N = N + m λ n l - T ξ l .
e T 2 - 2 p T + q = 0 ,
e = 1 + ( ξ / w ) 2 + ( ξ / l ) 2 , p = - L + [ M + m λ ( n / w ) ] ( ξ / w ) + [ N + m λ ( n / l ) ] ( ξ / l ) , q = 2 m λ [ M ( n / w ) + N ( n / l ) ] + ( m λ ) 2 [ ( n / w ) 2 + ( n / l ) 2 ] .
T = 1 e [ p + ( p 2 - e q ) 1 / 2 ]
x - ξ L = y - w M = z - l N d ,
x cos ( β 0 + ϕ ) + y sin ( β 0 + ϕ ) = r 0 cos ϕ .
σ ( sin α + sin β 0 ) = m λ ,
σ 1 / ( n / w ) w = l = 0 .
d = r 0 cos ϕ - ξ cos ( β 0 + ϕ ) - w sin ( β 0 + ϕ ) L cos ( β 0 + ϕ ) + M sin ( β 0 + ϕ ) .
x = ξ + L d ,             y = w + M d ,             z = l + N d .
Y = ( y - r 0 sin β 0 ) sec ( β 0 + ϕ ) ,             Z = z .
ξ = A - A [ 1 - ( w 2 B 2 + l 2 C 2 ) ] 1 / 2 + 30 w 3 + 12 w l 2 + 40 w 4 + 22 w 2 l 2 + 04 l 4 ,
ξ = w 2 2 R + l 2 2 ρ + 30 w 3 + 12 w l 2 + w 4 8 A R 2 ( 1 + 8 40 A R 2 ) + w 2 l 2 4 A R ρ ( 1 + 4 22 A R ρ ) + l 4 8 A ρ 2 ( 1 + 8 04 A ρ 2 ) + O ( w 6 R 5 ) ,
R = B 2 / A ,             ρ = C 2 / A .
n σ = w + Γ ( ½ n 20 w 2 + ½ n 02 l 2 + ½ n 30 w 3 + ½ n 12 w l 2 + n 40 w 4 + ¼ n 22 w 2 l 2 + n 04 l 4 + ) .
n 20 = - ( tan θ / R ) - J , n 02 = - tan θ / ρ , n 30 = J ( tan θ / R ) + J 2 - K - 2 30 tan θ , n 12 = J ( tan θ / ρ ) - 2 12 tan θ , n 40 = - J ( tan θ / R ) 2 + [ 6 K - 6 J 2 - ( 1 / R A ) - 8 ( 40 - 30 J ) R ] ( tan θ / R ) + 5 J ( 2 K - J 2 ) - ( 8 c / σ 4 ) , n 22 = - J ( tan 2 θ / R ρ ) + ( tan θ / ρ ) [ 3 K - 3 J 2 - 4 ( 22 - J 12 ) ρ - 1 / ( R A ) ] , n 04 = - J ( tan θ / ρ ) 2 - ( tan θ / ρ 2 A ) - 8 04 tan θ , Γ = 1 , J = 2 ( a - 3 b + c ) σ 2 , K = 4 ( b - c ) / σ 3 .
n 20 = T C - T D , n 02 = S ¯ C - S ¯ D , n 30 = ( T C sin γ / r C ) - ( T D sin δ / r D ) - 2 30 ( cos γ - cos δ ) , n 12 = ( S ¯ C sin γ / r C ) - ( S ¯ D sin δ / r D ) - 2 12 ( cos γ - cos δ ) , n 40 = ( 4 T C sin 2 γ / r C 2 ) - ( 4 T D sin 2 δ / r D 2 ) - ( T C 2 / r C ) + ( T D 2 / r D ) + ( S C - S D ) / R 2 - 8 30 [ ( sin γ cos γ / r C ) - ( sin δ cos δ / r D ) ] - 8 40 ( cos γ - cos δ ) , n 22 = ( 2 S ¯ C sin 2 γ / r C 2 ) - ( 2 S ¯ D sin 2 δ / r D 2 ) - ( T C S ¯ C / r C ) + ( T D S ¯ D / r D ) + ( S C - S D ) / R ρ - 4 12 [ ( sin γ cos γ / r C ) - ( sin δ cos δ / r D ) ] - 4 22 ( cos γ - cos δ ) , n 04 = - ( S ¯ C 2 / r C ) + ( S ¯ D 2 / r D ) + ( S C - S D ) / ρ 2 - 8 04 ( cos γ - cos δ ) , Γ = σ / λ ,
T C = ( cos 2 γ / r C ) - ( cos γ / R ) , T D = ( cos 2 δ / r D ) - ( cos δ / R ) , S C = ( 1 / r C ) - ( cos γ / A ) , S D = ( 1 / r D ) - ( cos δ / A ) , S ¯ C = ( 1 / r C ) - ( cos γ / ρ ) , S ¯ D = ( 1 / r D ) - ( cos δ / ρ ) .
Y = r 0 sec ( β 0 + ϕ ) ( 1 - tan β 0 tan ϕ ) × [ 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 = r 0 [ z g 001 + l g 010 + w l g 110 + w z g 101 + w 2 l g 210 + w 2 z g 201 + l 3 g 030 + l 2 z g 021 + l z 2 g 012 + z 3 g 003 + O ( w 4 / R 3 ) ] .
f 100 = F 200 ,
f 200 = 3 2 F 300 + 1 2 ( F 200 ) 2 sec β 0 ( tan β 0 - 2 tan ϕ ) + F 200 [ tan β 0 R - cos β 0 r 0 ( 2 tan β 0 - tan ϕ ) ] ,
f 020 = 1 2 F 120 + 1 2 ( F 020 ) 2 sin β 0 - F 020 ( sin β 0 / r 0 ) ,
f 011 = F 111 - F 020 ( sin β 0 / r ) ,
f 002 = 1 2 F 102 ,
f 300 = 1 2 F 400 + 3 2 F 300 F 200 sec β 0 ( tan β 0 - 2 tan ϕ ) + 3 2 F 300 [ tan β 0 R - cos β 0 r 0 ( 2 tan β 0 - tan ϕ ) ] + 1 2 ( F 200 ) 3 sec 2 β 0 [ sec 2 β 0 + 2 tan ϕ ( tan ϕ - tan β 0 ) ] + 1 2 ( F 200 ) 2 { sec β 0 R ( 1 + 3 tan 2 β 0 - 4 tan β 0 tan ϕ ) - 1 r 0 [ 3 + 4 ( tan β 0 - tan ϕ ) 2 + tan β 0 tan ϕ ] } + 1 2 F 200 { 2 R 2 tan 2 β 0 - sec β 0 R r 0 × ( 5 - 2 cos 2 β 0 - sin β 0 cos β 0 tan ϕ ) + 1 r 0 2 × [ 2 + cos 2 β 0 + 2 cos 2 β 0 tan ϕ ( tan ϕ - tan β 0 ) ] + 6 30 tan β 0 } ,
f 120 = 1 2 F 220 + 1 2 F 120 F 200 sec β 0 ( tan β 0 - 2 tan ϕ ) + F 120 F 020 sin β 0 + 1 2 F 200 ( F 020 ) 2 × ( sec 2 β 0 - 2 tan β 0 tan ϕ ) + 1 2 F 120 × [ tan β 0 R - cos β 0 r 0 ( 4 tan β 0 - tan ϕ ) ] + 1 2 ( F 020 ) 2 { sec β 0 R - 1 r 0 [ 1 + sin β 0 cos β 0 × ( tan β 0 - tan ϕ ) ] } + F 200 F 020 [ 1 ρ sin β 0 × tan β 0 - 1 r 0 ( sec 2 β 0 - 2 tan β 0 tan ϕ ) ] + 1 2 F 200 [ 1 r 0 2 - 1 ρ r 0 ( sec β 0 + sin β 0 tan ϕ ) + 2 12 tan β 0 ] + F 020 × [ 1 r 0 2 ( 1 - sin β 0 cos β 0 tan ϕ ) - sec β 0 R r 0 ] ,
f 111 = F 211 + F 111 F 200 sec β 0 ( tan β 0 - 2 tan ϕ ) + F 111 F 020 sin β 0 - F 120 sin β 0 r + F 111 [ tan β 0 R - cos β 0 r 0 ( 3 tan β 0 - tan ϕ ) ] - F 200 F 020 sec 2 β 0 r ( 1 - 2 sin β 0 cos β 0 tan ϕ ) + F 200 ( 1 r r 0 - sin β 0 tan β 0 ρ r ) + F 020 [ 1 r r 0 ( 1 - sin β 0 cos β 0 tan ϕ ) - sec β 0 R r ] ,
f 102 = 1 2 F 202 - F 111 sin β 0 r + 1 2 F 102 [ tan β 0 R - cos β 0 r 0 ( 2 tan β 0 - tan ϕ ) ] + 1 2 F 102 F 200 sec β 0 ( tan β 0 - 2 tan ϕ ) + 1 2 r 2 F 200 ,
g 001 = - 1 / r ,
g 010 = F 020 ,
g 110 = F 120 - F 200 F 020 sec β 0 tan ϕ + F 200 × sec β 0 ( sin β 0 ρ + tan ϕ r 0 ) - F 020 sin β 0 r 0 ,
g 101 = F 111 + 1 r F 200 sec β 0 tan ϕ ,
g 210 = 1 2 F 220 + 1 2 ( F 200 ) 2 F 020 sec 2 β 0 ( 1 - tan β 0 tan ϕ ) + tan 2 ϕ ) - ( 3 2 F 300 + F 120 ) F 200 sec β 0 tan ϕ + 3 2 F 300 sec β 0 ( sin β 0 ρ + tan ϕ r 0 ) - F 120 sin β 0 r 0 + 1 2 ( F 200 ) 2 sec 2 β 0 [ 1 ρ ( sec β 0 - 2 sin β 0 tan ϕ ) - 1 r 0 ( 1 - tan β 0 tan ϕ - 2 tan 2 ϕ ) ] - F 200 F 020 [ 1 r 0 ( 1 - 2 tan β 0 tan ϕ + tan 2 ϕ ) + 1 R sec β 0 tan β 0 tan ϕ ] + F 200 × [ 1 r 0 2 ( 1 - tan β 0 tan ϕ + tan 2 ϕ ) - sec β 0 ρ r 0 × ( 1 + sin 2 β 0 ) + 1 R r 0 sec β 0 tan β 0 tan ϕ + tan 2 β 0 R ρ + 12 tan β 0 ] + 1 2 r 0 F 020 × ( cos 2 β 0 r 0 - cos β 0 R ) + 12 r 0 sin β 0 cos β 0 ,
g 201 = 1 2 F 211 - F 111 F 200 sec β 0 tan ϕ - F 111 sin β 0 r 0 - 1 2 r ( F 200 ) 2 sec 2 β 0 ( 1 - tan β 0 tan ϕ + 2 tan 2 ϕ ) + 3 2 r F 300 sec β 0 tan ϕ + F 200 × [ 1 r r 0 ( 1 - tan β 0 tan ϕ + tan 2 ϕ ) + 1 R r sec β 0 tan β 0 tan ϕ ] ,
g 030 = 1 2 F 040 + 1 2 F 120 sec β 0 [ sin β 0 ρ + ( 1 r 0 - F 020 ) tan ϕ ] + 1 2 ( F 020 ) 3 ( 1 - tan β 0 tan ϕ ) + 1 2 ( F 020 ) 2 × [ sec β 0 ρ - 3 r 0 ( 1 - tan β 0 tan ϕ ) ] + 1 2 r 0 F 020 × [ 3 r 0 - 1 ρ ( 2 sec β 0 + cos β 0 ) ] ,
g 021 = 3 2 F 031 + F 111 sec β 0 [ sin β 0 ρ + ( 1 r 0 - F 020 ) tan ϕ ] + 1 2 r F 120 sec β 0 tan ϕ - 3 2 r ( F 020 ) 2 × ( 1 - tan β 0 tan ϕ ) + 1 r F 020 × [ 1 r 0 ( 3 - 2 tan β 0 tan ϕ ) - sec β 0 ρ ] ,
g 012 = 1 2 F 022 + 1 2 F 102 sec β 0 × [ sin β 0 ρ + ( 1 r 0 - F 020 ) tan ϕ ] + 1 r F 111 sec β 0 × tan ϕ + 1 2 r 2 F 020 ( 3 - 2 tan β 0 tan ϕ ) ,
g 003 = 1 2 r F 102 sec β 0 tan ϕ .
F 200 = T A + T B + n 20 Λ ,
F 020 = S ¯ A + S ¯ B + n 02 Λ ,
F 300 = ( T A sin α / r ) + ( T B sin β 0 / r 0 ) - 2 30 ( cos α + cos β 0 ) + n 30 Λ ,
F 120 = ( S ¯ A sin α / r ) + ( S ¯ B sin β 0 / r 0 ) - 2 12 ( cos α + cos β 0 ) + n 12 Λ ,
F 111 = - ( sin α / r 2 ) + ( sin β 0 / r r 0 ) ,
F 102 = ( sin α + sin β 0 ) / r 2 ,
F 400 = 4 T A ( sin α / r ) 2 + 4 T B ( sin β 0 / r 0 ) 2 - ( T A 2 / r ) - ( T B 2 / r 0 ) + [ ( S A + S B ) / R 2 ] - 8 30 [ ( sin α cos α / r ) + ( sin β 0 cos β 0 / r 0 ) ] - 8 40 ( cos α + cos β 0 ) + n 40 Λ ,
F 220 = 2 S ¯ A ( sin α / r ) 2 + 2 S ¯ B ( sin β 0 / r 0 ) 2 - ( T A S ¯ A / r ) - ( T B S ¯ B / r 0 ) + [ ( S A + S B ) / R ρ ] - 4 12 [ ( sin α cos α / r ) + ( sin β 0 cos β 0 / r 0 ) ] - 4 22 ( cos α + cos β 0 ) + n 22 Λ ,
F 211 = ( T A / r 2 ) - ( T B / r r 0 ) - ( 2 sin 2 α / r 3 ) + ( 2 sin 2 β 0 / r r 0 2 ) ,
F 202 = [ - T A - T B + 2 ( sin 2 α / r ) + 2 ( sin 2 β 0 / r 0 ) ] / r 2 ,
F 040 = - ( S ¯ A 2 / r ) - ( S ¯ B 2 / r 0 ) + [ ( S A + S B ) / ρ 2 ] - 8 04 ( cos α + cos β 0 ) + n 04 Λ ,
F 031 = [ ( S ¯ A / r ) - ( S ¯ B / r 0 ) ] / r ,
F 022 = - [ S ¯ A + S ¯ B + ( 2 / r ) + ( 2 / r 0 ) ] / r 2 ,
T A = ( cos 2 α / r ) - ( cos α / R ) , T B = ( cos 2 β 0 / r 0 ) - ( cos β 0 / R ) , S A = ( 1 / r ) - ( cos α / A ) , S B = ( 1 / r 0 ) - ( cos β 0 / A ) , S ¯ A = ( 1 / r ) - ( cos α / ρ ) , S ¯ B = ( 1 / r 0 ) - ( cos β 0 / ρ ) , Λ = Γ ( sin α + sin β 0 ) .
Y = r 0 sec β 0 ( F / w ) ,             Z = - ( z r 0 / r ) + r 0 ( F / l ) ,
F = r + r + w [ ( m λ / σ ) - ( sin α + sin β ) ] + l [ ( z / r ) + ( z / r ) ] + ½ w 2 F ^ 200 + ½ l 2 F ^ 020 + ½ z 2 F ^ 002 + ½ w 3 F ^ 300 + ½ w l 2 F ^ 120 + w l z F ^ 111 + ½ w z 2 F ^ 102 + w 4 F ^ 400 + ¼ w 2 l 2 F ^ 220 + l 4 F ^ 040 + ½ w 2 l z F ^ 211 + ¼ w 2 z 2 F ^ 202 + ½ l 3 z F ^ 031 + ¼ l 2 z 2 F ^ 022 + ½ l z 3 F ^ 013 + z 4 F ^ 004 + O ( w 5 / R 4 ) .
F ^ i j k ( F ^ i j k ) r = r 0 , β = β 0 , z = - z r 0 / r = F i j k
Y = r 0 sec β 0 ( F / w ) r = r 0 β = β 0 , z = - z r 0 / r = r 0 sec β 0 [ w F 200 + 3 2 w 2 F 300 + 1 2 l 2 F 120 + l z F 111 + 1 2 z 2 F 102 + 1 2 w 3 F 400 + 1 2 w l 2 F 220 + w l z F 211 + 1 2 w z 2 F 202 + O ( w 4 / R 4 ) ] ,
Z = - ( z r 0 / r ) + r 0 ( F / l ) r = r 0 , β = β 0 , z = - z r 0 / r = r 0 [ - z r + l F 020 + w l F 120 + w z F 111 + 1 2 w 2 l F 220 + 1 2 l 3 F 040 + 1 2 w 2 z F 211 + 3 2 l 2 z F 031 + 1 2 l z 2 F 022 + O ( w 4 / R 4 ) ] .
Y = r 0 sec β 0 [ 1 - ( ξ / r 0 ) sec β 0 ] - 1 { [ 1 - ( w / r 0 ) sin β 0 - ( ξ / r 0 ) cos β 0 ] 2 ( F / w ) r = r 0 , β = β 0 , z = - z r 0 / r - ( l / r 0 ) [ 1 - ( w / r 0 ) sin β 0 - ( ξ / r 0 ) cos β 0 ] × sin β 0 ( F / l ) r = r 0 , β = β 0 , z = - z r 0 / r } = r 0 sec β 0 { w F 200 + w 2 ( 3 2 F 300 - 2 F 200 sin β 0 r 0 ) + l 2 ( 1 2 F 120 - F 020 sin β 0 r 0 ) + l z F 111 + 1 2 z 2 F 102 + w 3 [ 1 2 F 400 - 3 F 300 sin β 0 r 0 + F 200 ( sin 2 β 0 r 0 2 + sec β 0 - 2 cos β 0 2 R r 0 ) ] + w l 2 [ 1 2 F 220 - 2 F 120 sin β 0 r 0 + 1 2 F 200 × ( sec β 0 - 2 cos β 0 ) ρ r 0 + F 020 sin 2 β 0 r 0 2 ] + w l z ( F 211 - 3 F 111 sin β 0 r 0 ) + w z 2 ( 1 2 F 202 - F 102 sin β 0 r 0 ) + O ( w 4 R 4 ) } ,
Z = - ( z r 0 / r ) + r 0 [ 1 - ( w / r 0 ) sin β 0 - ( ξ / r 0 ) cos β 0 ] × ( F / l ) r = r 0 , β = β 0 , z = - z r 0 / r = r 0 [ - z r + l F 020 + w l ( F 120 - F 020 sin β 0 r 0 ) + w z F 111 + w 2 l ( 1 2 F 220 - F 120 sin β 0 r 0 - 1 2 F 020 cos β 0 R r 0 ) + w 2 z ( 1 2 F 211 - F 111 sin β 0 r 0 ) + 1 2 l 3 ( F 040 - F 020 cos β 0 ρ r 0 ) + 3 2 l 2 z F 031 + 1 2 l z 2 F 022 + O ( w 4 R 4 ) .
[ Eq . ( 23 ) - Eq . ( 34 ) ] r 0 sec β 0 = F 200 [ w 2 tan β 0 ( 1 R + 1 2 F 200 sec β 0 ) + O ( w 3 R 2 ) ] + tan β 0 R × O ( w 3 R 2 ) + F 020 { sin β 0 [ 1 2 l 2 F 020 - l z r + O ( w 3 R 2 ) ] + ( 1 r 0 - sec β 0 R ) × O ( w 3 R ) } ,
[ Eq . ( 24 ) - Eq . ( 35 ) ] / r 0 = ( tan β 0 / ρ ) [ w l F 200 + O ( w 3 / R 2 ) ] + F 200 × O ( w 3 / R 2 ) + F 020 × O ( w 3 / R 2 ) .
Δ Y SD = Y SD - Y RT , Δ Y LPF = Y LPF - Y RT , Δ Y WFA = Y WFA - Y RT , Δ Z SD = Z SD - Z RT , Δ Z LPF = Z LPF - Z RT , Δ Z WFA = Z WFA - Z RT ,
σ n = σ 0 + 2 a n + 6 b n 2 + 4 c n 3 ,
w n = w - ξ tan θ = n = 0 n - 1 σ n = n ( σ 0 - a + b ) + n 2 ( a - 3 b + c ) + 2 n 3 ( b - c ) + c n 4 .
σ = σ 0 - a + b .
n σ = w + ½ ( n 20 w 2 + n 02 l 2 + n 30 w 3 + n 12 w l 2 ) + ( n 40 w 4 + 2 n 22 w 2 l 2 + n 04 l 4 ) + ( n 50 w 5 + 2 n 32 w 3 l 2 + n 14 w l 4 ) + / 16 1 ( n 60 w 6 + n 42 w 4 l 2 + n 24 w 2 l 4 + n 06 l 6 ) + ,
n 50 = - [ J ( n 40 + 2 n 30 n 20 ) + 3 K ( 2 n 30 + n 20 2 ) + ( 16 c / σ 4 ) n 20 ] , n 32 = - [ J ( n 22 + n 30 n 02 + n 12 n 20 ) + 3 K ( n 12 + n 20 n 02 ) + ( 8 c / σ 4 ) n 02 ] , n 14 = - [ J ( n 04 + 2 n 12 n 02 ) + 3 n 02 2 K ] , n 60 = - [ J ( 2 n 50 + n 40 n 20 + 2 n 30 2 ) + K ( 3 n 40 + 12 n 30 n 20 + n 20 3 ) + ( 8 c / σ 4 ) ( 4 n 30 + 3 n 20 2 ) + ( tan θ / R 3 A 2 ) ] , n 42 = - [ J ( 4 n 32 + n 40 n 02 + 2 n 22 n 20 + 4 n 30 n 12 ) + 3 K ( 2 n 22 + 4 n 30 n 02 + 4 n 12 n 20 + n 20 2 n 02 ) + ( 16 c / σ 4 ) ( 2 n 12 + 3 n 20 n 02 ) + ( 3 tan θ / R 2 ρ A 2 ) ] , n 24 = - [ J ( 2 n 14 + 2 n 22 n 02 + n 04 n 20 + 2 n 12 2 ) + 3 K ( n 04 + 4 n 12 n 02 + n 20 n 02 2 ) + ( 24 c / σ 4 ) n 02 2 + ( 3 tan θ / R ρ 2 A 2 ) ] , n 06 = - [ n 04 n 02 J + n 02 3 K + ( tan θ / ρ 3 A 2 ) ] .

Metrics