Abstract

Systematic error sources for the rotating-compensator ellipsometer are discussed. Starting from a general formalism, we derive explicit first-order expressions for the errors δΨ and δΔ caused by azimuthal errors and residual ellipticity introduced by imperfect polarizers or compensators and windows. Comparison is made with measurement results. It is shown that all azimuthal-angle errors can be eliminated by a two-zone measurement and that the natural optical rotation of the rotating quartz compensator has no effect on the measurements.

© 1994 Optical Society of America

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References

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  1. D. E. Aspnes, “Fourier transform detection system for rotating-analyser ellipsometers,” Opt. Commun. 8, 222–225 (1973).
    [Crossref]
  2. P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
    [Crossref]
  3. D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [PubMed]
  4. P. S. Hauge, F. H. Dill, “A rotating-compensator Fourier ellipsometer,” Opt. Commun. 14, 431–437 (1975).
    [Crossref]
  5. P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
    [Crossref]
  6. S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Istrum. 40, 761–767 (1969).
    [Crossref]
  7. A. Moritani, Y. Okuda, J. Nakai, “Use of an ADP four-crystal electrooptic modulator in ellipsometry,” Appl. Opt. 22, 1329–1336 (1983).
    [Crossref] [PubMed]
  8. R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2061 (1990).
    [Crossref]
  9. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [Crossref]
  10. J. M. M. de Nijs, A. van Silfhout, “Systematic and random errors in rotating-analyzer ellipsometry,” J. Opt. Soc. Am. A 5, 773–781 (1988).
    [Crossref]
  11. D. E. Aspnes, P. S. Hauge, “Rotating-compensator/analyzer fixed-analyzer ellipsometer: analysis and comparison to other automatic ellipsometers,”J. Opt. Soc. Am. 66, 949–954 (1976).
    [Crossref]
  12. A. Moritani, J. Nakai, “High-speed retardation modulation ellipsometry,” Appl. Opt. 21, 3231–3232 (1982).
    [Crossref] [PubMed]
  13. O. Archer, E. Bigan, B. Drevillon, Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
    [Crossref]
  14. G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
    [Crossref] [PubMed]
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  16. Sofie Instruments, 7 route d’Egly, 91290 Arpajon (France).
  17. mathematica 2.1, Wolfram Research Inc., 100 Trade Center Drive, Champaign, Ill. 61820–72370.
  18. D. E. Aspnes, “Effects of component optical activity in data reduction and calibration of rotating-analyzer ellipsometers,”J. Opt. Soc. Am. 64, 812–819 (1974).
    [Crossref]
  19. H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
    [Crossref]
  20. R. M. A. Azzam, N. M. Bashara, “Ellipsometry with imperfect components including incoherent effects,”J. Opt. Soc. Am. 61, 1380–1391 (1971).
    [Crossref]
  21. J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978).
  22. A. Ehringhaus, “Drehkompensatoren mit besonders grossem Messbereich,”Z. Kristallogr. 102, 85–111 (1939).
  23. S. A. Henck, “In situ real-time ellipsometry for film thickness measurement and control,” J. Vac. Sci. Technol. A 10, 934–938 (1992).
    [Crossref]
  24. R. M. A. Azzam, N. M. Bashara, “Analysis of systematic errors in rotating-analyzer ellipsometers,”J. Opt. Soc. Am. 64, 1459–1469 (1974).
    [Crossref]
  25. D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,”J. Opt. Soc. Am. 64, 639–646 (1974).
    [Crossref]
  26. See, e.g., B. N. Taylor, W. H. Parker, D. N. Langenberg, The Fundamental Constants and Quantum Electrodynamics (Academic, New York, 1969), pp. 337–341.

1992 (1)

S. A. Henck, “In situ real-time ellipsometry for film thickness measurement and control,” J. Vac. Sci. Technol. A 10, 934–938 (1992).
[Crossref]

1991 (1)

H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
[Crossref]

1990 (2)

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[Crossref] [PubMed]

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2061 (1990).
[Crossref]

1989 (1)

O. Archer, E. Bigan, B. Drevillon, Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[Crossref]

1988 (1)

1983 (2)

A. Moritani, Y. Okuda, J. Nakai, “Use of an ADP four-crystal electrooptic modulator in ellipsometry,” Appl. Opt. 22, 1329–1336 (1983).
[Crossref] [PubMed]

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

1982 (1)

1976 (2)

1975 (2)

D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
[PubMed]

P. S. Hauge, F. H. Dill, “A rotating-compensator Fourier ellipsometer,” Opt. Commun. 14, 431–437 (1975).
[Crossref]

1974 (3)

1973 (2)

D. E. Aspnes, “Fourier transform detection system for rotating-analyser ellipsometers,” Opt. Commun. 8, 222–225 (1973).
[Crossref]

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[Crossref]

1971 (1)

1969 (1)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Istrum. 40, 761–767 (1969).
[Crossref]

1939 (1)

A. Ehringhaus, “Drehkompensatoren mit besonders grossem Messbereich,”Z. Kristallogr. 102, 85–111 (1939).

Archer, O.

O. Archer, E. Bigan, B. Drevillon, Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[Crossref]

Aspnes, D. E.

Azzam, R. M. A.

Bashara, N. M.

Becker, H.

H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
[Crossref]

Bennet, H. E.

J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978).

Bennet, J. M.

J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978).

Bigan, E.

O. Archer, E. Bigan, B. Drevillon, Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[Crossref]

Brach, D.

H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
[Crossref]

Collins, R. W.

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2061 (1990).
[Crossref]

de Nijs, J. M. M.

Dill, F. H.

P. S. Hauge, F. H. Dill, “A rotating-compensator Fourier ellipsometer,” Opt. Commun. 14, 431–437 (1975).
[Crossref]

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[Crossref]

Drevillon, B.

O. Archer, E. Bigan, B. Drevillon, Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[Crossref]

Ehringhaus, A.

A. Ehringhaus, “Drehkompensatoren mit besonders grossem Messbereich,”Z. Kristallogr. 102, 85–111 (1939).

Hauge, P. S.

D. E. Aspnes, P. S. Hauge, “Rotating-compensator/analyzer fixed-analyzer ellipsometer: analysis and comparison to other automatic ellipsometers,”J. Opt. Soc. Am. 66, 949–954 (1976).
[Crossref]

P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[Crossref]

P. S. Hauge, F. H. Dill, “A rotating-compensator Fourier ellipsometer,” Opt. Commun. 14, 431–437 (1975).
[Crossref]

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[Crossref]

Henck, S. A.

S. A. Henck, “In situ real-time ellipsometry for film thickness measurement and control,” J. Vac. Sci. Technol. A 10, 934–938 (1992).
[Crossref]

Jasperson, S. N.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Istrum. 40, 761–767 (1969).
[Crossref]

Jellison, G. E.

Langenberg, D. N.

See, e.g., B. N. Taylor, W. H. Parker, D. N. Langenberg, The Fundamental Constants and Quantum Electrodynamics (Academic, New York, 1969), pp. 337–341.

Modine, F. A.

Moritani, A.

Nakai, J.

Okuda, Y.

Otto, A.

H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
[Crossref]

Parker, W. H.

See, e.g., B. N. Taylor, W. H. Parker, D. N. Langenberg, The Fundamental Constants and Quantum Electrodynamics (Academic, New York, 1969), pp. 337–341.

Schnatterly, S. E.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Istrum. 40, 761–767 (1969).
[Crossref]

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
[PubMed]

Taylor, B. N.

See, e.g., B. N. Taylor, W. H. Parker, D. N. Langenberg, The Fundamental Constants and Quantum Electrodynamics (Academic, New York, 1969), pp. 337–341.

van Silfhout, A.

Weber, H. J.

H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
[Crossref]

Appl. Opt. (4)

IBM J. Res. Dev. (1)

P. S. Hauge, F. H. Dill, “Design and operation of ETA, an automated ellipsometer,” IBM J. Res. Dev. 17, 472–489 (1973).
[Crossref]

J. Opt. Soc. Am. (5)

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

J. Vac. Sci. Technol. A (1)

S. A. Henck, “In situ real-time ellipsometry for film thickness measurement and control,” J. Vac. Sci. Technol. A 10, 934–938 (1992).
[Crossref]

Opt. Commun. (2)

D. E. Aspnes, “Fourier transform detection system for rotating-analyser ellipsometers,” Opt. Commun. 8, 222–225 (1973).
[Crossref]

P. S. Hauge, F. H. Dill, “A rotating-compensator Fourier ellipsometer,” Opt. Commun. 14, 431–437 (1975).
[Crossref]

Phys. Rev. B (1)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

Rev. Sci. Instrum. (3)

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation and real-time applications,” Rev. Sci. Instrum. 61, 2029–2061 (1990).
[Crossref]

O. Archer, E. Bigan, B. Drevillon, Improvements of phase-modulated ellipsometry,” Rev. Sci. Instrum. 60, 65–77 (1989).
[Crossref]

H. Becker, D. Brach, A. Otto, H. J. Weber, “Sensitive and selective polarimeter for application in crystal optics,” Rev. Sci. Instrum. 62, 1196–1205 (1991).
[Crossref]

Rev. Sci. Istrum. (1)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Istrum. 40, 761–767 (1969).
[Crossref]

Surf. Sci. (1)

P. S. Hauge, “Generalized rotating compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976).
[Crossref]

Z. Kristallogr. (1)

A. Ehringhaus, “Drehkompensatoren mit besonders grossem Messbereich,”Z. Kristallogr. 102, 85–111 (1939).

Other (5)

See, e.g., B. N. Taylor, W. H. Parker, D. N. Langenberg, The Fundamental Constants and Quantum Electrodynamics (Academic, New York, 1969), pp. 337–341.

J. M. Bennet, H. E. Bennet, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Sofie Instruments, 7 route d’Egly, 91290 Arpajon (France).

mathematica 2.1, Wolfram Research Inc., 100 Trade Center Drive, Champaign, Ill. 61820–72370.

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Figures (8)

Fig. 1
Fig. 1

Diagram of the RCE.

Fig. 2
Fig. 2

Error on Ψ caused by the azimuthal-angle error δA of the analyzer. The calculated curve uses formula (28) and experimental values from Table 1.

Fig. 3
Fig. 3

Error on Ψ caused by the azimuthal-angle error δP of the polarizer. The calculated curve uses formula (31) and experimental values from Table 1.

Fig. 4
Fig. 4

Error on Δ caused by the azimuthal-angle error δP of the polarizer. The calculated curve uses formula (31) and experimental values from Table 1.

Fig. 5
Fig. 5

Error on Ψ caused by the azimuthal-angle error δC of the compensator. The calculated curve uses formula (34) and experimental values from Table 1.

Fig. 6
Fig. 6

Error on Δ caused by the azimuthal-angle error δC of the compensator. The calculated curve uses formula (34) and experimental values from Table 1.

Fig. 7
Fig. 7

Error on Δ caused by a small polarizer ellipticity γP = 0.11o (sample: Si) as a function of the polarizer azimuthal-angle P.

Fig. 8
Fig. 8

Error on Δ caused by a small retardation introduced by the exit window as a function of its orientation θWb. The calculated curve corresponds to an amplitude of 4°.

Tables (3)

Tables Icon

Table 1 Ellipsometric Parameters of the Samples Used to Test the Validity of the Systematic-Error Theoretical Formulas

Tables Icon

Table 2 Systematic Errors in the PCSA Ellipsometer for A = 45° (sgn A = 1) and A = −45° (sgn A = −1)

Tables Icon

Table 3 Systematic Errors in the PSCA Ellipsometer for P = 45° (sgn P = 1) and P = −45° (sgn P = −1)

Equations (59)

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S f = [ M A R ( A ) ] M S [ R - 1 ( C ) M C R ( C ) ] [ R - 1 ( P ) M P ] S i ,
M A = M P = [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] .
M S = r p r p * + r s r s * 2 [ 1 - cos 2 Ψ 0 0 - cos 2 Ψ 1 0 0 0 0 sin 2 Ψ cos Δ sin 2 Ψ sin Δ 0 0 - sin 2 Ψ sin Δ sin 2 Ψ cos Δ ] .
M C = [ 1 0 0 0 0 1 0 0 0 0 cos δ c sin δ c 0 0 - sin δ c cos δ c ] .
R ( θ ) = [ 1 0 0 0 0 cos 2 θ sin 2 θ 0 0 - sin 2 θ cos 2 θ 0 0 0 0 1 ] .
I = I 0 ( a 0 + a 2 c cos 2 C + a 2 s sin 2 C + a 4 c cos 4 C + a 4 s sin 4 C ) ,
a 0 = ( 1 / 2 ) ( 1 + cos δ c ) ( cos 2 A cos 2 P - cos 2 P cos 2 Ψ + sin 2 A sin 2 P sin 2 Ψ cos Δ ) - cos 2 A cos 2 Ψ + 1 ,
a 2 c = - sin 2 A sin 2 P sin δ c sin 2 Ψ sin Δ ,
a 2 s = sin 2 A cos 2 P sin δ c sin 2 Ψ sin Δ ,
a 4 c = ( 1 / 2 ) ( 1 - cos δ c ) ( cos 2 A cos 2 P - cos 2 P cos 2 Ψ - sin 2 A sin 2 P sin 2 Ψ cos Δ ) ,
a 4 s = ( 1 / 2 ) ( 1 - cos δ c ) ( cos 2 A sin 2 P - sin 2 P cos 2 Ψ + sin 2 A cos 2 P sin 2 Ψ cos Δ ) .
a = [ a 0 a 2 c a 2 s a 4 c a 4 s ] ,             a T = ( a 0             a 2 c             a 2 s             a 4 c             a 4 s ) .
a = [ ½ ( 1 + cos δ c ) ( - cos 2 P cos 2 Ψ + sgn A sin 2 P sin 2 Ψ cos Δ ) + 1 - sgn A sin 2 P sin δ c sin 2 Ψ sin Δ + sgn A cos 2 P sin δ c sin 2 Ψ sin Δ ½ ( 1 - cos δ c ) ( - cos 2 P cos 2 Ψ - sgn A sin 2 P sin 2 Ψ cos Δ ) ½ ( 1 - cos δ c ) ( - sin 2 P cos 2 Ψ + sgn A cos 2 P sin 2 Ψ cos Δ ) ] .
i ph = α 0 + α 2 c cos 2 C + α 2 s sin 2 C + α 4 c cos 4 C + α 4 s sin 4 C ,
α k = G a k             k = 0 , 2 c , 2 s , 4 c , 4 s .
tan 2 Ψ = - [ ( α 2 s 2 + α 2 c 2 ) ( 1 - cos δ c ) 2 / sin 2 δ c + 4 ( α 4 s cos 2 P - α 4 c sin 2 P ) 2 ] 1 / 2 2 ( α 4 c cos 2 P + α 4 s sin 2 P ) ,
tan Δ = ( 1 - cos δ c 2 sin δ c ) α 2 c sin 2 P - α 2 s cos 2 P α 4 c sin 2 P - α 4 s cos 2 P .
δ Δ α 2 c = ( 1 - cos δ c ) 2 sin δ c cos 2 Δ sin 2 P ( α 4 c sin 2 P - α 4 s cos 2 P ) δ α 2 c .
δ Δ α 2 c = - sin 2 P sgn A cos Δ sin δ c sin 2 Ψ δ a 2 c .
δ Ψ = δ a T × [ 0 - sin 2 P sgn A cos 2 Ψ sin Δ 2 sin δ c + cos 2 P sgn A cos 2 Ψ sin Δ 2 sin δ c cos 2 P sin 2 Ψ - sgn A sin 2 P cos 2 Ψ cos Δ ( 1 - cos δ c ) sin 2 P sin 2 Ψ + sgn A cos 2 P cos 2 Ψ cos Δ ( 1 - cos δ c ) ] ,
δ Δ = δ a T [ 0 - sin 2 P sgn A cos Δ sin δ c sin 2 Ψ cos 2 P sgn A cos Δ sin δ c sin 2 Ψ 2 sgn A sin 2 P sin Δ ( 1 - cos δ c ) sin 2 Ψ - 2 sgn A cos 2 P sin Δ ( 1 - cos δ c ) sin 2 Ψ ] .
S f = S f 0 + k S f x k δ x k = S f 0 + k ( δ S f ) x k .
δ Ψ = - ( 1 - cos 2 P cos 2 Ψ ) 2 2 sin 2 2 P sin 2 Ψ δ α ¯ 2 c ,
δ Δ = - cos Δ ( cos 2 P - cos 2 Ψ ) ( 1 - cos 2 P cos 2 Ψ ) sin 2 2 P sin Δ sin 2 2 Ψ δ α ¯ 2 c - ( 1 - cos 2 P cos 2 Ψ ) sin 2 P sin Δ sin 2 Ψ δ α ¯ 2 s .
δ R ( θ ) = 2 δ θ [ 0 0 0 0 0 - sin 2 θ cos 2 θ 0 0 - cos 2 θ - sin 2 θ 0 0 0 0 0 ] .
( δ S f ) A = [ M A δ R ( A ) ] M S [ R - 1 ( C ) M C R ( C ) ] × [ R - 1 ( P ) M P ] S i .
δ a A = δ A [ ( - cos 2 P sin 2 A + sin 2 P cos 2 A cos Δ sin 2 Ψ ) ( 1 + cos δ c ) + 2 sin 2 A cos 2 Ψ - 2 cos 2 A sin 2 P sin δ c sin Δ sin 2 Ψ 2 cos 2 A cos 2 P sin δ c sin Δ sin 2 Ψ - ( 1 - cos δ c ) ( sin 2 A cos 2 P + cos 2 A sin 2 P cos Δ sin 2 Ψ ) - ( 1 - cos δ c ) ( sin 2 A sin 2 P - cos 2 A cos 2 P cos Δ sin 2 Ψ ) ] .
[ δ Ψ δ Δ ] = δ A [ - sgn A sin 2 Ψ 0 ] .
( δ S f ) P = [ M A R ( A ) ] M S [ R - 1 ( C ) M C R ( C ) ] [ δ R - 1 ( P ) M P ] S i ,
δ a P = δ P [ - ( cos 2 A sin 2 P - cos 2 Ψ sin 2 P - sin 2 A cos 2 P cos Δ sin 2 Ψ ) ( 1 + cos δ c ) - 2 sin 2 A cos 2 P sin δ c sin Δ sin 2 Ψ - 2 sin 2 A sin 2 P sin δ c sin Δ sin 2 Ψ ( 1 - cos δ c ) ( sin 2 P cos 2 Ψ - sin 2 A cos 2 P cos Δ sin 2 Ψ - cos 2 A sin 2 P ) ( 1 - cos δ c ) ( - cos 2 P cos 2 Ψ - sin 2 A sin 2 P cos Δ sin 2 Ψ + cos 2 A cos 2 P ) ] .
[ δ Ψ δ Δ ] = δ P [ - sgn A cos Δ 2 sgn A sin Δ cot 2 Ψ ] .
( δ S f ) C = [ M A R ( A ) ] M S [ δ R - 1 ( C ) M C R ( C ) + R - 1 ( C ) M C δ R ( C ) ] [ R - 1 ( P ) M P ] S i ,
δ a C = 2 δ C [ 0 sin 2 A cos 2 P sin δ c sin Δ sin 2 Ψ sin 2 A sin 2 P sin δ c sin Δ sin 2 Ψ ( 1 - cos δ c ) ( - sin 2 P cos 2 Ψ + sin 2 A cos 2 P cos Δ sin 2 Ψ + cos 2 A sin 2 P ) - ( 1 - cos δ c ) ( cos 2 A cos 2 P - cos 2 P cos 2 Ψ - sin 2 A sin 2 P cos Δ sin 2 Ψ ) ]
[ δ Ψ δ Δ ] = 2 δ C [ sgn A cos Δ - 2 sgn A sin Δ cot 2 Ψ ] .
M C = [ 1 0 0 0 0 Z 2 cos δ c + Y 2 Z sin δ c Y Z ( 1 - cos δ c ) 0 - Z sin δ c cos δ c Y sin δ c 0 Y Z ( 1 - cos δ c ) - Y sin δ c Y 2 cos δ c + Z 2 ] ,
Y = 1 - γ C 2 1 + γ C 2 ,
Z = 2 γ C 1 + γ C 2 .
M C = [ 1 0 0 0 0 1 2 γ C sin δ c 2 γ C ( 1 - cos δ c ) 0 - 2 γ C sin δ c cos δ c sin δ c 0 2 γ C ( 1 - cos δ c ) - sin δ c cos δ c ] .
M C 0 = [ 1 0 0 0 0 1 0 0 0 0 cos δ c sin δ c 0 0 - sin δ c cos δ c ]
δ M C = [ 0 0 0 0 0 0 2 γ C sin δ c 2 γ C ( 1 - cos δ c ) 0 - 2 γ C sin δ c 0 0 0 2 γ C ( 1 - cos δ c ) 0 0 ] .
( δ S f ) γ C = [ M A R ( A ) ] M S [ R - 1 ( C ) δ M C R ( C ) ] [ R - 1 ( P ) M P ] S i ,
δ a γ C = 2 γ C [ sin δ c ( cos 2 A sin 2 P - sin 2 P cos 2 Ψ - sin 2 A cos 2 P cos Δ sin 2 Ψ ) ( 1 - cos δ c ) sin 2 A cos 2 P sin Δ sin 2 Ψ ( 1 - cos δ c ) sin 2 A sin 2 P sin Δ sin 2 Ψ 0 0 ] .
[ δ Ψ δ Δ ] = γ C [ 0 0 ] .
M A = [ 1 1 0 2 γ A 1 1 0 2 γ A 0 0 0 0 2 γ A 2 γ A 0 0 ] .
δ M A = [ 0 0 0 2 γ A 0 0 0 2 γ A 0 0 0 0 2 γ A 2 γ A 0 0 ]
( δ S f ) γ A = [ δ M A R ( A ) ] M S [ R - 1 ( C ) M C R ( C ) ] × [ R - 1 ( P ) M P ] S i ,
δ a γ A = γ A [ - ( 1 + cos δ c ) sin 2 P sin Δ sin 2 Ψ - 2 sin 2 P sin δ c cos Δ sin 2 Ψ 2 cos 2 P sin δ c cos Δ sin 2 Ψ ( 1 - cos δ c ) sin 2 P sin Δ sin 2 Ψ - ( 1 - cos δ c ) cos 2 P sin Δ sin 2 Ψ ] ,
[ δ Ψ δ Δ ] = 2 γ A [ 0 sgn A ] .
δ a γ P = 2 γ P [ cos δ c sin 2 A sin Δ sin 2 Ψ sin 2 A sin δ c cos Δ sin 2 Ψ sin δ c ( - cos 2 A + cos 2 Ψ ) 0 0 ] ,
[ δ Ψ δ Δ ] = γ P [ ( cos 2 P sgn A sin Δ cos 2 2 Ψ - sin 2 P cos Δ sin Δ cos 2 Ψ sin 2 Ψ ) 2 ( - sin 2 P cos 2 Δ + cos 2 P sgn A cos Δ cot 2 Ψ ) ] .
S f = [ M A R ( A ) M W b M S M W a [ R - 1 ( C ) M C R ( C ) ] × [ R - 1 ( P ) M P ] S i ,
M W a = R - 1 ( θ W a ) M C a R ( θ W a ) ,
M C a = [ 1 0 0 0 0 1 0 0 0 0 cos δ W a sin δ W a 0 0 - sin δ W a cos δ W a ] = [ 1 0 0 0 0 1 0 0 0 0 1 δ W a 0 0 - δ W a 1 ] .
δ a W a = δ W a 2 [ - 1 ( 1 + cos δ c ) sin 2 A sin Δ sin 2 Ψ sin 2 ( P - θ W a ) 2 sin δ c sin 2 P [ - sin 2 A cos Δ sin 2 Ψ cos 2 θ W a + ( cos 2 A - cos 2 Ψ ) sin 2 θ W a ] 2 sin δ c cos 2 P [ sin 2 A cos Δ sin 2 Ψ cos 2 θ W a - ( cos 2 A - cos 2 Ψ ) sin 2 θ W a ] ( 1 - cos δ c ) sin 2 A sin Δ sin 2 Ψ sin 2 ( P + θ W a ) - ( 1 - cos δ c ) sin 2 A sin Δ sin 2 Ψ cos 2 ( P + θ W a ) ] ,
[ δ Ψ δ Δ ] = δ W a [ ( sgn A sin 2 θ W a sin Δ ) / 2 cos 2 θ W a + sgn A sin 2 θ W a cos Δ cot 2 Ψ ] .
δ a W b = sin 2 Ψ sin 2 ( A - θ W b ) δ W b 2 × [ - ( 1 + cos δ c ) sin 2 P sin Δ - 2 sin δ c sin 2 P cos Δ 2 sin δ c cos 2 P cos Δ ( 1 - cos δ c ) sin 2 P sin Δ - ( 1 - cos δ c ) cos 2 P sin Δ ] ,
[ δ Ψ δ Δ ] = δ W b [ 0 cos 2 θ W b ] .
[ δ Ψ δ Δ ] = δ ( δ c ) [ - sin 2 Δ sin 4 Ψ 4 sin δ c - sin 2 Δ 2 sin δ c ] .
σ 2 c = σ 2 s = σ 4 c = σ 4 s = 2 1 / 2 σ 0 .

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