Abstract

A third-order aberration theory has been developed for a plane-symmetric double-element optical system that consists of an extended source, two ellipsoidal gratings, and an image plane. The gratings can have any of the groove patterns producible by means of currently available technologies. Analytic formulas of spot diagrams are derived for the system by analytically following a ray-tracing formalism. With these formulas coma, spherical aberration, astigmatism, field curvature, and distortion of the system are discussed in detail together with the focusing conditions. The spot-diagram formulas are critically evaluated in comparison with ray tracing. The result shows their capability in reproducing ray-traced spots with a high degree of accuracy.

© 1999 Optical Society of America

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References

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  1. H. Petersen, “The plane grating and elliptical mirror: a new optical configuration for monochromators,” Opt. Commun. 40, 402–406 (1982).
    [CrossRef]
  2. M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
    [CrossRef]
  3. M. Koike, 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]
  4. K. Ito, T. Namioka, Y. Morioka, T. Sasaki, H. Noda, K. Goto, T. Katayama, M. Koike, “High-resolution VUV spectroscopic facility at the Photon Factory,” Appl. Opt. 25, 837–847 (1986).
    [CrossRef] [PubMed]
  5. T. Namioka, H. Noda, K. Goto, T. Katayama, “Design studies of mirror-grating systems for use with an electron storage ring source at the Photon Factory,” Nucl. Instrum. Methods Phys. Res. 208, 215–222 (1983).
    [CrossRef]
  6. T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3993 (1980).
    [CrossRef] [PubMed]
  7. M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 96–101 (1987).
    [CrossRef]
  8. T. Namioka, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” Appl. Opt. 34, 2180–2186 (1995). Misprints in Eqs. (23)–(25) should be corrected as follows. VC in the last line of Eqs. (23e) should read VD.KC in the third line of Eq. (23f) should read KD. Both a1 and a2 in the first and second of Eqs. (24) should read a. The minus signs in the third and fourth lines of Eq. (25b) should be plus signs. The term in the third line of Eq. (25c) should be multiplied by cos ηC.
    [CrossRef] [PubMed]
  9. D. Bajuk, R. Kestner, “Fabrication and testing of EUVL optics,” in JSPE Proceedings on Soft X-Ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), pp. 325–335.
  10. C. T. Chen, “Concept and design procedure for cylindrical element monochromators for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 256, 595–604 (1987).
    [CrossRef]
  11. For example, see T. Namioka, M. Seya, “Optical properties of a system consisting of a mirror and a grating,” Appl. Opt. 9, 459–464 (1970), and references therein.
    [CrossRef] [PubMed]
  12. G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. I. Ray tracing formulas for meridional plane,” J. Opt. Soc. Am. 51, 1–3 (1961);“Contributions to the optics of mirror systems and gratings with oblique incidence. II. A discussion of aberrations,” J. Opt. Soc. Am. 52, 408–411 (1962);“Contributions to the optics of mirror systems and gratings with oblique incidence. III. Some applications,” J. Opt. Soc. Am. 52, 412–415 (1962).
    [CrossRef]
  13. M. P. Crisp, “Aberration of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [CrossRef]
  14. M. P. Crisp, “X-ray spectrograph design,” Appl. Opt. 22, 1519–1529 (1983).
    [CrossRef]
  15. T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994). The following misprints should be corrected. The numerator in the first line of Eq. (4) should read M-M′+ mλ(∂n/∂w). λ in the last line of Eq. (21) should read λ0. The last terms in Eqs. (23) and (24) should read O(w4/R4). The second line of Eq. (26e) should read +2 tan2 ϕ)-[(3/2)F300F020+F120F200]sec β0 tan ϕ. The last term in the fifth line of Eq. (26e) should read +2 tan2 ϕ. The term 3/r0′ in the last line of Eq. (26g) should read (1/r0′)(3-2 tan β0 tan ϕ). A bracket [ ] is missing at the end of Eq. (35). O(w3/R) in the last term of Eq. (36) should read O(w3/R2).
    [CrossRef] [PubMed]
  16. C. H. F. Velzel, “A general theory of the aberrations of diffraction gratings and gratinglike optical instruments,” J. Opt. Soc. Am. A 66, 346–353 (1976).
    [CrossRef]
  17. H. G. Beutler, “The theory of the concave grating,” J. Opt. Soc. Am. 35, 311–350 (1945).
    [CrossRef]

1996 (1)

M. Koike, 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)

1994 (1)

1990 (1)

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

1987 (1)

C. T. Chen, “Concept and design procedure for cylindrical element monochromators for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 256, 595–604 (1987).
[CrossRef]

1986 (1)

1983 (3)

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

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

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

1982 (1)

H. Petersen, “The plane grating and elliptical mirror: a new optical configuration for monochromators,” Opt. Commun. 40, 402–406 (1982).
[CrossRef]

1980 (1)

1976 (1)

C. H. F. Velzel, “A general theory of the aberrations of diffraction gratings and gratinglike optical instruments,” J. Opt. Soc. Am. A 66, 346–353 (1976).
[CrossRef]

1970 (1)

1961 (1)

1945 (1)

Bajuk, D.

D. Bajuk, R. Kestner, “Fabrication and testing of EUVL optics,” in JSPE Proceedings on Soft X-Ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), pp. 325–335.

Beutler, H. G.

Chen, C. T.

C. T. Chen, “Concept and design procedure for cylindrical element monochromators for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 256, 595–604 (1987).
[CrossRef]

Content, D.

Crisp, M. P.

Domke, M.

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

Goto, K.

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

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

Harada, T.

Harada, Y.

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 96–101 (1987).
[CrossRef]

Ito, K.

Kaindl, G.

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

Katayama, T.

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

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

Kestner, R.

D. Bajuk, R. Kestner, “Fabrication and testing of EUVL optics,” in JSPE Proceedings on Soft X-Ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), pp. 325–335.

Kita, T.

Koike, M.

M. Koike, 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, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” Appl. Opt. 34, 2180–2186 (1995). Misprints in Eqs. (23)–(25) should be corrected as follows. VC in the last line of Eqs. (23e) should read VD.KC in the third line of Eq. (23f) should read KD. Both a1 and a2 in the first and second of Eqs. (24) should read a. The minus signs in the third and fourth lines of Eq. (25b) should be plus signs. The term in the third line of Eq. (25c) should be multiplied by cos ηC.
[CrossRef] [PubMed]

T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994). The following misprints should be corrected. The numerator in the first line of Eq. (4) should read M-M′+ mλ(∂n/∂w). λ in the last line of Eq. (21) should read λ0. The last terms in Eqs. (23) and (24) should read O(w4/R4). The second line of Eq. (26e) should read +2 tan2 ϕ)-[(3/2)F300F020+F120F200]sec β0 tan ϕ. The last term in the fifth line of Eq. (26e) should read +2 tan2 ϕ. The term 3/r0′ in the last line of Eq. (26g) should read (1/r0′)(3-2 tan β0 tan ϕ). A bracket [ ] is missing at the end of Eq. (35). O(w3/R) in the last term of Eq. (36) should read O(w3/R2).
[CrossRef] [PubMed]

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

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 96–101 (1987).
[CrossRef]

Morioka, Y.

Namioka, T.

M. Koike, 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, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” Appl. Opt. 34, 2180–2186 (1995). Misprints in Eqs. (23)–(25) should be corrected as follows. VC in the last line of Eqs. (23e) should read VD.KC in the third line of Eq. (23f) should read KD. Both a1 and a2 in the first and second of Eqs. (24) should read a. The minus signs in the third and fourth lines of Eq. (25b) should be plus signs. The term in the third line of Eq. (25c) should be multiplied by cos ηC.
[CrossRef] [PubMed]

T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994). The following misprints should be corrected. The numerator in the first line of Eq. (4) should read M-M′+ mλ(∂n/∂w). λ in the last line of Eq. (21) should read λ0. The last terms in Eqs. (23) and (24) should read O(w4/R4). The second line of Eq. (26e) should read +2 tan2 ϕ)-[(3/2)F300F020+F120F200]sec β0 tan ϕ. The last term in the fifth line of Eq. (26e) should read +2 tan2 ϕ. The term 3/r0′ in the last line of Eq. (26g) should read (1/r0′)(3-2 tan β0 tan ϕ). A bracket [ ] is missing at the end of Eq. (35). O(w3/R) in the last term of Eq. (36) should read O(w3/R2).
[CrossRef] [PubMed]

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

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

For example, see T. Namioka, M. Seya, “Optical properties of a system consisting of a mirror and a grating,” Appl. Opt. 9, 459–464 (1970), and references therein.
[CrossRef] [PubMed]

Noda, H.

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

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

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 96–101 (1987).
[CrossRef]

Petersen, H.

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

H. Petersen, “The plane grating and elliptical mirror: a new optical configuration for monochromators,” Opt. Commun. 40, 402–406 (1982).
[CrossRef]

Puschmann, A.

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

Rosendahl, G. R.

Sasaki, T.

Seya, M.

Shirley, D. A.

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

Velzel, C. H. F.

C. H. F. Velzel, “A general theory of the aberrations of diffraction gratings and gratinglike optical instruments,” J. Opt. Soc. Am. A 66, 346–353 (1976).
[CrossRef]

Xue, C.

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

Appl. Opt. (7)

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

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

T. Namioka, M. Koike, “Aspheric wave-front recording optics for holographic gratings,” Appl. Opt. 34, 2180–2186 (1995). Misprints in Eqs. (23)–(25) should be corrected as follows. VC in the last line of Eqs. (23e) should read VD.KC in the third line of Eq. (23f) should read KD. Both a1 and a2 in the first and second of Eqs. (24) should read a. The minus signs in the third and fourth lines of Eq. (25b) should be plus signs. The term in the third line of Eq. (25c) should be multiplied by cos ηC.
[CrossRef] [PubMed]

For example, see T. Namioka, M. Seya, “Optical properties of a system consisting of a mirror and a grating,” Appl. Opt. 9, 459–464 (1970), and references therein.
[CrossRef] [PubMed]

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

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

T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994). The following misprints should be corrected. The numerator in the first line of Eq. (4) should read M-M′+ mλ(∂n/∂w). λ in the last line of Eq. (21) should read λ0. The last terms in Eqs. (23) and (24) should read O(w4/R4). The second line of Eq. (26e) should read +2 tan2 ϕ)-[(3/2)F300F020+F120F200]sec β0 tan ϕ. The last term in the fifth line of Eq. (26e) should read +2 tan2 ϕ. The term 3/r0′ in the last line of Eq. (26g) should read (1/r0′)(3-2 tan β0 tan ϕ). A bracket [ ] is missing at the end of Eq. (35). O(w3/R) in the last term of Eq. (36) should read O(w3/R2).
[CrossRef] [PubMed]

J. Electron Spectrosc. Relat. Phenom. (1)

M. Koike, 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]

J. Opt. Soc. Am. (2)

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

C. H. F. Velzel, “A general theory of the aberrations of diffraction gratings and gratinglike optical instruments,” J. Opt. Soc. Am. A 66, 346–353 (1976).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. (1)

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

Nucl. Instrum. Methods Phys. Res. A (1)

C. T. Chen, “Concept and design procedure for cylindrical element monochromators for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 256, 595–604 (1987).
[CrossRef]

Opt. Commun. (1)

H. Petersen, “The plane grating and elliptical mirror: a new optical configuration for monochromators,” Opt. Commun. 40, 402–406 (1982).
[CrossRef]

Synchrotron Radiat. News (1)

M. Domke, A. Puschmann, C. Xue, D. A. Shirley, G. Kaindl, H. Petersen, “Spectral resolution in the soft-x-ray region up to 11,000,” Synchrotron Radiat. News 3 (No. 5), 21–22 (1990).
[CrossRef]

Other (2)

D. Bajuk, R. Kestner, “Fabrication and testing of EUVL optics,” in JSPE Proceedings on Soft X-Ray Optics: Technical Challenges, T. Namioka, H. Kinoshita, K. Ito, eds. (Japan Society for Precision Engineering, Tokyo, 1997), pp. 325–335.

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. SPIE815, 96–101 (1987).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Optical scheme and the coordinate systems of a double-grating system.

Fig. 2
Fig. 2

Coma curves of the M-G monochromator in the tangential focal plane, illustrating changes in the coma pattern with r˜. s=-1 mm and z=5 mm are assumed.

Fig. 3
Fig. 3

Spherical aberration curves of the M-G monochromator in the tangential focal plane. r˜=20 mm is assumed.

Fig. 4
Fig. 4

Spherical aberration curves of the 6VOPE predisperser in the tangential focal plane. r˜=30 mm is assumed. Each inset is an enlargement of a central portion of the curve.

Fig. 5
Fig. 5

Astigmatic curvatures of the M-G monochromator in the tangential focal plane.

Fig. 6
Fig. 6

Distortion patterns of the M-G monochromator for a meshlike source of dimension 100 μm (s)×40 mm (z) with line separations of Δs=25 μm and Δz=10 mm.

Fig. 7
Fig. 7

Spot diagrams and ΔZ-versus-ΔY plots constructed for the M-G monochromator tuned to λ=2.5 nm. Spot diagrams in (a) are generated from Eqs. (19), and those in (b) are constructed by ray tracing. Deviations, ΔY and ΔZ, of individual spots in (a) from the corresponding spots in (b) are plotted in (c). The point-source positions are given at the top of the respective diagrams. σY and σZ are the standard deviations of spots in the Y and Z directions, and RMSΔY and RMSΔZ are the root mean squares of the differences ΔY and ΔZ.

Fig. 8
Fig. 8

Plots of RMSΔY and RMSΔZ against the angle of incidence α2 constructed for the M-G monochromator with a point source at s=z=0 mm. The ruled area of G2 is assumed to be (a) 120 mm (W)×40 mm (H) and (b) large enough to accept all the rays reflected from G1.

Fig. 9
Fig. 9

Plots of RMSΔY and RMSΔZ against the angle of incidence α2 constructed for the 6VOPE predisperser with a point source at (a) s=z=0 mm and (b) s=6 mm and z=30 mm.

Equations (174)

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

niσi=wi+(ni)20wi2+(ni)02li2+(ni)30wi3+(ni)12wili2+(ni)40wi4+(ni)22wi2li2+(ni)04li4+,
ξi=ai-ai[1-(wi2/bi2+li2/ci2)]1/2,
σi(sin αi+sin βi)=miλ,
F1=AP+PQ+n1m1λ,
L1=L1+t1,
M1=M1+m1λn1w1-t1ξ1w1,
N1=N1+m1λn1l1-t1ξ1l1,
t1=1e1 (p1+p12-e1q1),
e1=1+ξ1w12+ξ1l12,
p1=-L1+M1+m1λ n1w1 ξ1w1+N1+m1λ n1l1 ξ1l1,
q1=2m1λM1 n1w1+N1 n1l1+(m1λ)2×n1w12+n1l12,
L1=ξ1-xAAP,M1=w1-yAAP,N1=l1-zAAP,
ξ2-ξ¯1L2=w2-w¯1M2=l2-l¯1N2,
x cos β2+y sin β2=r2.
xB=ξ2+L2K,yB=w2+M2K,
zB=l2+N2K,
K=r2-ξ2 cos β2-w2 sin β2L2 cos β2+M2 sin β2.
Y=(r2 sin β2-yB)sec β2,Z=zB.
ξi=wi22Ri+li22ρi+wi48aiRi2+wi2li24aiRiρi+li48aiρi2+O(wi6/Ri5),
Ri=bi2/ai,ρi=ci2/ai.
w2=h+i+j+k=13Ahijkw1hl1izjsk,
l2=h+i+j+k=13Bhijkw1hl1izjsk.
w2=A1000w1+A0001s+A2000w12+A1001w1s+A0200l12+A0110l1z+a0020z2+A0002s2+A3000w13+A2001w12s+A1200w1l12+A1110w1l1z+A1020w1z2+A1002w1s2+A0201l12s+A0111l1zs+A0021z2s+A0003s3,
l2=B0100l1+B0010z+B1100w1l1+B1010w1z+B0101l1s+B0011zs+B2100w12l1+B2010w12z+B1101w1l1s+B1011w1zs+B0300l13+B0210l12z+B0120l1z2+B0102l1s2+B0030z3+B0012zs2.
Y=C1000w1+C0001s+C2000w12+C1001w1s+C0200l12+C0110l1z+C0020z2+C0002s2+C3000w13+C2001w12s+C1200w1l12+C1110w1l1z+C1020w1z2+C1002w1s2+C0201l12s+C0111l1zs+C0021z2s+C0003s3,
Z=D0100l1+D0010z+D1100w1l1+D1010w1z+D0101l1s+D0011zs+D2100w12l1+D2010w12z+D1101w1l1s+D1011w1zs+D0300l13+D0210l12z+D0120l1z2+D0102l1s2+D0030z3+D0012zs2.
Ahijk=Ahijk|r1d,dr2,α20,R2=ρ2,u1u2,
Bhijk=Bhijk|r1d,dr2,α20,R2=ρ2,u1u2,
4(F1)20(F2*)20=cos2 β1 cos2 α2d2,
2(F1)20=cos2 α1r1+cos2 β1d-cos α1+cos β1R1+2(n1)20(sin α1+sin β1),
2(F2)20=cos2 α2d+cos2 β2r2-cos α2+cos β2R2+2(n2)20(sin α2+sin β2).
2(F1)20=cos2 β1d-cos2 β1(r1)T,
2(F2)20=cos2 α2d-cos2 α2(r2)T,
2(F1)02=1d-1(r1)S,2(F2*)02=1d-1(r2)S,
Ycoma=C2000w12+C2001w12s+C0201l12s+C1110w1l1z,
Zcoma=D1100w1l1+D1101w1l1s+D2010w12z+D0210l12z.
w1=r˜ cos θ,l1=r˜ sin θ
2Ycomar˜2-[C2000+s(C2001+C0201)]=[C2000+s(C2001-C0201)]cos 2θ+zC1110 sin 2θ,
2Zcomar˜2-z(D2010+D0210)=z(D2010-D0210)cos 2θ+(D1100+sD1101)sin 2θ.
α2Ycomar˜2-[C2000+s(C2001+C0201)]2
+b2Zcomar˜2-z(D2010+D0210)2
-2h2Ycomar˜2-[C2000+s(C2001+C0201)]
×2Zcomar˜2-z(D2010+D0210)=c2,
a=(D1100+sD1101)2+z2(D2010-D0210)2,
b=[C2000+s(C2001-C0201)]2+z2C11102,
c=(D1100+sD1101)[C2000+s(C2001-C0201)]-z2C1110(D2010-D0210),
h=z{C1110(D1100+sD1101)+(D2010-D0210)×[C2000+s(C2001-C0201)]}.
Ycoma=12r˜2s(D1101)G(2+cos 2θ),
Zcoma=12r˜2s(D1101)G sin 2θ,
[Ycoma-r˜2s(D1101)G]2+Zcoma2=[12r˜2s(D1101)G]2,
Ysph=C3000w13+C1200w1l12,
Zsph=D0300l13+D2100w12l1.
Ysph=r˜3(cos θ)(C3000 cos2 θ+C1200 sin2 θ),
Zsph=r˜3(sin θ)(D0300 sin2 θ+D2100 cos2 θ),
Yast=C0001s+C0200l12+C0110l1z+C0020z2,
Zast=D0100l1+D0010z,
Last=|D0100|L+|D0010|H=2r2d2L(F1)02(F2)02-1d2+Hr1 |(F2)02|,
Yast=C0200ZastD0100+C01102C0200-D0010D0100z2+C0020-C011024C0200z2+C0001s.
Yast=C0020D00102 Zast2+C0001s.
Y=C0001s+C1000w1+C1001w1s+C0200l12+C0110l1z+C0020z2+C1020w1z2+C1002w1s2+C0111l1zs,
Z=D0010z+D0100l1+D1010w1z+D0101l1s+D1011w1zs+D0120l1z2+D0102l1s2.
Yτ=C0020z2+w1(C1000+C1020z2),
Zτ=D0010z+D1010w1z,
Yσ=C0110l1z+C0200l12+C0020z2,
Zσ=D0010z+l1(D0100+D0120z2).
C1000+C1020z2=0(orD0100+D0120z2=0)
C1000=-(Δr2)τf+O((Δr2)τ2),
f=1(r2)T 1-d(r1)Tcos β1 sec α2 cos β2.
(Δr2)τ=C1020*f(D0010*)2 Zτ2=C1020*fC0020* Yτ.
ρT=2C0020*(D0010*)2 1+C1020*fC0020*21/2at(Δr2)τ=0.
ρS=2C0020*(D0010*)2 1+D0120*gC0020*21/2at(Δr2)σ=0,
g=1(r2)S 1-d(r1)S.
Ydist=C0001s+C0020z2+C0002s2+C0021z2s+C0003s3,
Zdist=D0010z+D0011zs+D0030z3+D0012zs2.
A1000=-2d(F1)20 sec β1 sec α2,
A0001=-dr1 cos α1 sec β1 sec α2,
A2000=[-3d(F1)30-2dΓ1(F1)202 sec β1+2Λ1(F1)20]sec β1 sec α2,
A1001=A00012Γ1(F1)20 sec β1+2 sin α1r1-Λ1d-tan α1R1,
A0200=[-d(F1)12-2dΦ1(F1)022+2(F1)02 sin β1]sec β1 sec α2,
A0110=dr1 [2Φ1(F1)02-r1(F1)111]sec β1 sec α2,
A0020=d2 dr12ρ2 cos β1 sin α2-(F1)102sec β1 sec α2,
A0002=12r1 A0001(tan α1+Γ1 cos α1 sec β1),
A3000=d(sec β1 sec α2)-4(F1)40-6Γ1(F1)30(F1)20 sec β1+3Λ1(F1)30d-(F1)203(sec2 β1)4Ψ1+4(tan β1)(tan β1+tan α2)-4dR2 (sec α2 tan α2)(tan β1+2 tan α2)+4d2R22 sec2 α2 tan2 α2+2(F1)2021d (1+2Ψ1+4 tan2 β1+5 tan β1 tan α2)-sec β1R1 (1+3 tan2 β1+4 tan β1 tan α2)+sec3 α2R2-2Λ1R2 sec β1 sec α2 tan α2-(F1)201d2 [3-(cos2 β1)(tan2 β1-tan2 α2)]-cos β1R1d (3+5 tan2 β1)+Λ1d2 cos β1 tan α2+2R12 tan2 β1,
A2001=A00013Γ1(F1)30 sec β1+6(F1)202(sec2 β1)Ψ1+(Γ1-tan α2)tan β1+dR2 (sec α2 tan2 α2)-2+dR2 sec α2-2(F1)201d (1+2Ψ1+4Γ1 tan β1-3 tan β1 tan α2)+sec α2R2 (1-tan2 α2)-sec β1R1 [1+tan2 β1-Γ1(tan β1)(tan α1-2 tan β1)]-2r1 Γ1 sin α1 sec β1+3 cos2 α12r12× (2 tan2 α1-1)-2Λ1 sin α1r1d+cos2 β12d2 (3+2 tan2 β1+2 tan β1 tan α2+2 tan2 α2)-cos β12R1d (3+tan β1 tan α2+5 tan2 β1)+tan2 β1R12+Λ1 tan α1R1d+3 cos α12r1R1 (1-tan2 α1),
A1200=-2d(F1)22 sec β1 sec α2+A02002Γ1(F1)20 sec β1+tan β1R1-cos β1d (4 tan β1+tan α2)-A10002(F1)022×-1+d cos α2ρ2 (1+tan2 α2-tan β1 tan α2)+2(F1)021d-sin β1 tan β1ρ1-Ω1 sin α2ρ2-12d2+cos β12ρ1d (sec2 β1-tan β1 tan α2)+2d(F1)02(sec β1 sec α2)-2Φ1(F1)12+(F1)02×1d (1-3 sin2 β1)-cos β1R1+cos2 β1 cos α2ρ2 (2 tan β1 tan α2-1)-d sin β1 sin α2R1ρ2+2(F1)02(sec α2)1R1-sec β1d (1-4 sin2 β1),
A1110=d(sec β1 sec α2)-2(F1)211+2r1 Φ1(F1)12+2Γ1(F1)022(F1)20 tan β1r1-(F1)111 sec β1+4r1 Θ1(F1)20(F1)02+(F1)111Λ1d-2(F1)02 sin β1+sin β1d-2r1 (F1)201d-sin β1 tan β1ρ1-Ω1 sin α2ρ2-2Γ1(F1)111[(F1)20-(F1)02]sec β1-2r1 (F1)021d (1+sin β1 cos β1 tan α2)-sec β1R1-cos2 β1 cos α2ρ2×(1+2 tan β1 tan α2+tan2 α2)+dr1ρ2 sin α1 cos β1 sin α2,
A1020=d(sec β1 sec α2)-2(F1)202+(F1)111 sin β1r1+Λ1(F1)1022d-Γ1(F1)102(F1)20 sec β1-Θ1r12 (F1)20-cos β12r12 cos β1 cos α2ρ2 (sec2 α2+2 tan β1 tan α2)-2dr1ρ2 sin α1 sin α2,
A1002=-1r1 A0001-(F1)20(sec2 β1)Γ1tan α1 cos β1-3dR2 cos α1 sec α2 tan α2+3(cos α1)(Ψ1+tan2 β1+tan β1 tan α2)+cos α12r1 (3-2 tan2 α1-4Γ1 sin α1 sec β1)+cos α12d 3Ψ1+Γ1(4 tan β1-tan α2)-3 tan β1 tan α2+dR2 (sec α2)(1-tan2 β1-tan2 α2)-cos α12R1 (sec β1)×[1+tan2 β1-2Γ1(tan α1-tan β1)+sec α1 cos β1]+Λ1 tan α12d+cos α12R2 sec2 β1 sec α2,
A0201=1r1 A0200Γ1 cos α1 sec β1+B0101(F1)02dρ2 tan α2-tan β1 sec α2+A00012(F1)0221-tan β1 tan α2+d cos α2ρ2 (tan β1 tan α2-1)+2(F1)0212d (tan β1 tan α2-2)+cos β12ρ1 d sin α2ρ2-tan β1×(tan α1-tan β1)+12ρ2 sin α2 tan α2-12r12+12d2+cos α12r1ρ1 (1-tan2 α1)+cos β12ρ1d (2 tan α1 tan β1+tan β1 tan α2-sec2 β1),
A0111=1r1 A00012(F1)02Φ1d sec α2R2-2sec β1 tan α2+dρ2 sec α2-sec2 β1+1r1 [1-(sin α1 sec β1)×(tan β1+2 tan α2)]+1d (2 tan β1 tan α2+sec2 β1)+1ρ1 Φ1(tan α1-tan β1)-rdR2 (F1)111 sec β1 sec α2 tan α2-1ρ2 sin α2 tan α2,
A0021=12 A0001Γ1(F1)102 sec β1+1r12 (Θ1-1),
A0003=12r12 A0001-1+(cos2 α1 sec2 β1)(Ψ1+tan2 β1+tan β1 tan α2)+Γ1(sec β1)sin α1-dR2 cos2 α1 sec β1 sec α2 tan α2,
B0100=2d(F1)02,
B0010=-dr1,
B1100=2d(F1)12+2(F1)20(F1)02 sec β1 tan α2-Ω1(F1)20 sec β1d-(F1)02 sin β1d,
B1010=dr1 [-2(F1)20 sec β1 tan α2+r1(F1)111],
B0101=-d cos α1r1 Ω1d-2(F1)02 tan α2sec β1+tan α1ρ1,
B0011=A0001 sin α2r1,
B2100=d2(F1)22+4Ψ1(F1)202(F1)02 sec2 β1+[6(F1)30(F1)02+4(F1)12(F1)20]sec β1 tan α2-3Ω1(F1)30 sec β1d-2(F1)12 sin β1d+2(F1)202(sec2 β1)cos β1ρ1 (sec2 β1+2 tan β1 tan α2)-Ψ1d-4(F1)20(F1)021d+Λ1d sec β1 tan α2+2(F1)201d2 (sec2 α2+tan β1 tan α2)-sec β1ρ1d (1+sin2 β1)-Ω1R1d sec β1 tan β1+1d (F1)02T1,
B2010=d(F1)211-2r1 Ψ1(F1)202 sec2 β1-3r1 (F1)30 sec β1 tan α2+2(F1)20(F1)111 sec β1 tan α2-(F1)111 sin β1d+2r1 (F1)201d (sec2 α2+tan β1 tan α2)-1R1 sec β1 tan β1 tan α2,
B1101=-r1d B1100B0011+B01012(F1)20 sec β1 tan α2+2 sin α1r1-sin β1d+4dr1 (F1)20(F1)02(cos α1 sec2 β1)×1+tan β1 tan α2-dR2 sec3 α2-2dr1 (F1)20(cos α1 sec2 β1)1d (1+tan β1 tan α2)-sec β1ρ1-sec3 α2R2-2dr1 (F1)02(cos α1)sec2 α2d+sec β1 tan α2R1 (tan α1-tan β1)+dr12ρ1+d cos α1r1 Ω1 sec β1R1d× (tan α1-tan β1)-1r12+sec2 α2d2-sec β1ρ1d,
B1011=B00112(F1)20(sec β1)(tan β1+2 tan α2)+3 sin α1r1-2 sin β1d-1R1 (tan α1-tan β1)-dr12 (cos α1)2(F1)20(sec2 β1)1-dR2 sec3 α2-1r1-sec2 α2d,
B0300=4d(F1)04+d(F1)12(sec β1)2(F1)02 tan α2-Ω1d+4dϑ1(F1)023+2d(F1)022sec β1ρ1-3ϑ1d-2 sec α2ρ2+(F1)023S1¯+2Ω1d tan β1,
B0210=d32 (F1)031-1r1 (F1)12 sec β1 tan α2+(F1)111(sec β1)2(F1)02 tan α2-Ω1d-6r1 ϑ1(F1)022+2r1 (F1)021d (1+2ϑ1)-sec β1ρ1+1ρ2 sec α2,
B0120=d12 (F1)022+(F1)102(F1)02 sec β1 tan α2-12 Ω1(F1)102 sec β1d-1r1 (F1)111 sec β1 tan α2+1r12 (F1)02(3ϑ1-tan β1 tan α2)+sec α22r12ρ2-S1¯r12,
B0102=12r1 B0101(tan α1+2 cos α1 sec β1 tan α2)+d2r12 2(F1)02-1dcos2 α1 sec2 β1×1+tan β1 tan α2-dR2 sec3 α2-1r1+1ρ1 (sec α1+cos2 α1 sec3 β1),
B0030=d2r1 -(F1)102 sec β1 tan α2+dr12ρ2 sec α2,
B0012=d2r13 (1-sin α1 sec β1 tan α2-Ψ1 cos2 α1 sec2 β1),
Γ1=tan β1+2 tan α2-dR2 sec α2 tan α2,
Θ1=1-d cos α2ρ2 (1+3 tan2 α2)+d2 tan2 α2R2ρ2,
ϑ1=1+tan β1 tan α2-d sec α2ρ2,
Λ1=(cos β1)(2 tan β1+tan α2)-d tan β1R1,
Π1=sin α1+sin β1,
Φ1=sin β1-dρ2 cos β1 sin α2,
Ψ1=1+tan β1 tan α2+2 tan2 α2-dR2 sec3 α2,
Ω1=tan α2-dρ1 sin β1,
(F1)102=1r12 Π1,
(F1)111=-1r1 sin α1r1-sin β1d,
(F1)211=12r1 T1r1-T1d-2 sin2 α1r12+2 sin2 β1d2,
(F1)031=1r1 S1¯r1-S1¯d,
(F1)022=-1r12 S1¯-S1¯+2r1+2d,
(F1)202=-14r12 T1+T1-2 sin2 α1r1-2 sin2 β1d,
(F1)20=T12+T12+(n1)20Π1,
(F1)02=S1¯2+S1¯2+(n1)02Π1,
(F1)30=T1 sin α12r1+T1 sin β12d+(n1)02Π1,
(F1)12=S1¯ sin α12r1+S1¯ sin β12d+(n1)12Π1,
(F1)22=S1¯ sin2 α12r12-S1¯T14r1+S14R1ρ1+S1¯ sin2 β12d2-S1¯T14d+S14R1ρ1+(n1)22Π1,
(F1)40=T1 sin2 α12r12-T128r1+S18R12+T1 sin2 β12d2-T128d+S18R12+(n1)40Π1,
(F1)04=-S1¯28r1+S18ρ12-S1¯28d+S18ρ12+(n1)04Π1,
T1=cos2 α1r1-cos α1R1,T1=cos2 β1d-cos β1R1,
S1¯=1r1-cos α1ρ1,S1¯=1d-cos β1ρ1,
S1=1r1-cos α1a1,S1=1d-cos β1a1,
(F2)hij=(F1)hij|r1d,dr2,α20,R2=ρ2,u1u2,
(F2)hi=(F1)hi|r1d,dr2,α20,R2=ρ2,u1u2,
T2=cos2 α2d-cos α2R2,T2=cos2 β2r2-cos β2R2,
S2¯=1d-cos α2ρ2,S2¯=1r2-cos β2ρ2,
S2=1d-cos α2a2,S2=1r2-cos β2a2,
Γ2,Θ2,ϑ2,Λ2,Π2,Φ2,Ψ2,Ω2
=Γ1,Θ1,ϑ1,Λ1,Π1,
Φ1,Ψ1,Ω1|dr2,R1R2,ρ1ρ2,R2=ρ2.
C1000=A1000A1000+A0001 cos β1,
C0001=A0001A1000,
C2000=A2000A1000+A10002A2000+A1000A1001 cos β1+A0002 cos2 β1-A0001(sin β1)×12R1-cos β1d-A1000 cos α2d,
C1001=A1001A1000+2A0001A1000A2000+A0001A1001 cos β1+A0001A0001 sin β1 cos α2d,
C0200=A0020+A0200A1000+A0110B0100+A0200B01002-A0001 sin β12ρ1,
C0110=A0110A1000+A0110B0010+2A0200B0010B0100,
C0020=A0020A1000+A0200B00102,
C0002=A0002A1000+A00012A2000,
C3000=A3000A1000+2A1000A2000A2000+A10003A3000+A2000A1001 cos β1+A10002A2001 cos β1+A1000A1002 cos2 β1+A0003 cos3 β1-12R1-cos β1dA0001 sin2 β1d+r2A1000 cos β1 cos2 α2 sec β2d2+A1000A1001 sin β1+2A0002 sin β1 cos β1+A0001 cos2 β12dR1-A10002A0001 sin β1 sin α22dR2+sin β1 cos α2d (A2000A0001+A10002A1001)+A1000sin β1 sin α2 cos α2d2× A1000A0001-2r2 cos β1 sec β2d-r2A1000 sin β1 cos2 α2 sec β2d3(sin β1+cos β1 cos α2 sec β2 tan β2),
C2001=A2001A1000+2A1000A1001A2000+2A0001A2000A2000+3A0001A10002A3000+A1001A1001 cos β1+2A0001A1000A2001 cos β1+A0001A1002 cos2 β1-A000112R1-cos β1d×r2 cos β1 cos2 α2 sec β2d2+A1001 sin β1-2A0001A1000A0001 sin β1 sin α2d×12R2-cos α2d+sin β1 cos α2d (A1001A0001+2A0001A1000A1001)-r2A0001 cos α2 sec β2d3× [cos α2+(sin β1 cos β1)×(2 sin α2+cos2 α2 sec β2 tan β2)],
C1200=A1200A1000+A1000A1020+2A0200A1000A2000+A1000A1110B0100+A1000A1200B01002+A0110B1100+2A0200B0100B1100+A0200A1001 cos β1+A0111B0100 cos β1+A0201B01002 cos β1-A00012dρ1× (1-A1000 cos β1 cos α2-2 cos2 β1)+sin β1d (2A0020-2A0020B0100+A0110B0100-A0110B01002+A0200A0001 cos α2)-sin β12ρ1 (A1000A1001+2A0002 cos β1)+A0020 cos β1 cos α2 sec β2 tan β2d-A0001B01002 sin β1 sin α22dρ2,
C1110=A1110A1000+2A0110A1000A2000+A1000A1110B0010+2A1000A1200B0010B0100+A0110B1010+2A0200B0100B1010+2A0200B0010B1100+A0110A1001 cos β1+A0111B0010 cos β1+2A0201B0010B0100 cos β1+sin β1d A0110A0001 cos α2-B00102A0020-A0110+2A0110B0100+A0001B0100 sin α2ρ2,
C1020=A1020A1000+2A0020A1000A2000+A1000A1200B00102+2A0200B0010B1010+A0020A1001 cos β1+A0201B00102 cos β1-sin β1d A0110B00102-A0020A0001 cos α2+A0001B00102 sin α22ρ2,
C1002=A1002A1000+2A0002A1000A2000+2A0001A1001A2000+3A00012A1000A3000+A0002A1001 cos β1+A00012A2001 cos β1+sin β1 cos α2d (A0002A0001+A00012A1001)-A00012A0001 sin β1 sin α2d 12R2-cos α2d,
C0201=A0201A1000+A0001A1020+2A0001A0200A2000+A0001A1110B0100+A0001A1200B01002+A0110B0101+2A0200B0100B0101+A00012ρ1 A0001 cos β1 cos α2d-A1001 sin β1,
C0111=A0111A1000+2A0001A0110A2000+A0001A1110B0010+A0110B0011+2A0001A1200B0010B0100+2A0200B0011B0100+2A0200B0010B0101,
C0021=A0021A1000+2A0001A0020A2000+A0001A1200B00102+2A0200B0010B0011,
C0003=A0003A1000+2A0001A0002A2000+A00013A3000,
D0100=B0100B0100-r2d,
D0010=B0010B0100,
D1100=B1100B0100+A1000B1010+A1000B0100B1100+B0100B0101 cos β1-r2 sin β1d2 (1-B0100),
D1010=B1010B0100+A1000B0010B1100+B0010B0101 cos β1+r2B0010 sin β1d2,
D0101=B0101B0100+A0001B1010+A0001B0100B1100,
D0011=B0011B0100+A0001B0010B1100,
D2100=B2100B0100+A2000B1010+A2000B0100B1100+A1000B1100B1100+A10002B2010+A10002B0100B2100+B1100B0101 cos β1+A1000B1011 cos β1+A1000B0100B1101 cos β1+B0012 cos2 β1+B0100B0102 cos2 β1+r2B1100 sin β1d2-B0100B0101(sin β1)12R1-cos β1d-r2d2 (1-B0100)cos β12R1+sin2 β1d+A1000 sin β1 sin α2d+A1000 sin β1d×(B1010-B0100B1010+B0100B0101 cos α2),
D2010=B2010B0100+A2000B0010B1100+A1000B1010B1100+A10002B0010B2100+B1010B0101 cos β1+A1000B0010B1101 cos β1+β0010B0102 cos2 β1+r2B0010 cos β12d2R1+r2B1010 sin β1d2-B0010B0101 sin β12R1+r2B0010 sin β1d3 (sin β1+A1000 sin α2)-B0010 sin β1d (A1000B1010-B0101 cos β1-A1000B0101 cos α2),
D1101=B1101B0100+A1001B1010+A1001B0100B1100+A1000B0101B1100+A0001B1100B1100+2A0001A1000B2010+2A0001A1000B0100B2100+B0101B0101 cos β1+A0001B1011 cos β1+A0001B0100B1101 cos β1+sin β1d A0001(B1010-B0100B1010+B0100B0101 cos α2)+r2B0101d-r2A0001 sin α2d2 (1-B0100),
D1011=B1011B0100+A1001B0010B1100+A1000B0011B1100+A0001B1010B1100+2A0001A1000B0010B2100+B0011B0101 cos β1+A0001B0010B1101 cos β1-A0001B0010 sin β1d (B1010-B0101 cos α2)+r2 sin β1d2 B0011+A0001B0010 sin α2d,
D0300=B0300B0100+B0100B0120+B01002B0210+B01003B0300+A0200B1010+A0200B0100B1100-12ρ1 B0100B0101 sin β1+r2 cos β1d2 (1-B0100),
D0210=B0210B0100+B0010B0120+2B0010B0100B0210+3B0010B01002B0300+A0110B1010+A0200B0010B1100+A0110B0100B1100+B00102ρ1 r2 cos β1d2-B0101 sin β1,
D0120=B0120B0100+B00102B0210+3B00102B0100B0300+A0020B1010+A0110B0010B1100+A0020B0100B1100,
D0102=B0102B0100+A0002B1010+A0002B0100B1100+A0001B0101B1100+A00012B2010+A00012B0100B2100,
D0030=B0030B0100+B00103B0300+A0020B0010B1100,
D0012=B0012B0100+A0002B0010B1100+A0001B0011B1100+A00012B0010B2100.

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