Abstract

A Mueller analysis has been done of IR ellipsometry performed with imperfect optical components. Equations linking experimental and calculated Fourier coefficients have been derived and consistently solved. Correction routines for permittivity measurements are demonstrated and discussed with gold and SrTiO3 as examples. It is shown that such effects as interferometer polarization, detector dichroism, transmission, and phase changes in polarizers can be calculated and effectively removed from the spectra. The problems of calibration and multiple reflections between IR polarizers are discussed, and error propagation in permittivity measurements is analyzed.

© 1994 Optical Society of America

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References

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  1. A. Röseler, “Spectroscopic ellipsometry in the infrared,” Infrared Phys. 21, 349–355 (1981).
    [CrossRef]
  2. A. Röseler, “Spectroscopic infrared ellipsometry with the Fourier transform spectrometer,” Ph.D. dissertation (Zentralinstitut für Optik, Akademie der Wissenschaften der Deutsche Demokratische Republik, Berlin, 1985).
  3. R. W. Stobie, B. Rao, M. J. Dignam, “Automatic ellipsometer with high sensitivity and special advantages for infrared spectroscopy of adsorbed species,” Appl. Opt. 14, 999–1003 (1975).
    [PubMed]
  4. A. Röseler, “Improvement in accuracy of spectroscopic IR ellipsometry by the use of IR retarders,” Infrared Phys. 24, 1–5 (1984).
    [CrossRef]
  5. G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
    [CrossRef]
  6. J. Bremer, O. Hunderi, K. Fanping, T. Skauli, E. Wold, “Infrared ellipsometer for the study of surfaces, thin films, and superlattices,” Appl. Opt. 31, 471–478 (1992).
    [CrossRef] [PubMed]
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1979).
  8. D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).
  9. A. Röseler, “Die Berechnung der Polarisationseigenschaften eines Fourierspektrometers mit Mueller-Matrizen,” Optik 60, 237–246 (1982).
  10. A. Röseler, “Die Anwendung der Mueller-Matrix auf die spectroscopische Infrarot-Ellipsometrie mit dem Fourier-spektrometer,” Optik 61, 177–186 (1982).
  11. W. L. Hyde, “Polarization techniques in the infrared,” J. Opt. Soc. Am. 38, 663 (1948).
  12. K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry assessment of intersubband transitions in n-GaAs/Al0.3Ga0.7As quantum wells,” Thin Solid Films 233, 194–198 (1993).
    [CrossRef]
  13. J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–245 (1955).
  14. E. Wold, Ph. D. dissertation(Norwegian Institute of Technology, N-7034 Trondheim, Norway, 1993).
  15. H. Nyquist, “Certain topics in telegraph transmission theory,” AIIE Trans. 47, 617–644 (1928).
  16. W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press. Cambridge, England, 1989).
  17. D. E. Aspnes, “The accurate determination of optical properties by ellipsometry,” in Handbook of Optical Constants of Solids,E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 89–112.
  18. R. W. Stobie, M. J. Dignam, “Transmission properties of grid polarizers,” Appl. Opt. 12, 1390–1391 (1973).
    [CrossRef] [PubMed]

1993

G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
[CrossRef]

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry assessment of intersubband transitions in n-GaAs/Al0.3Ga0.7As quantum wells,” Thin Solid Films 233, 194–198 (1993).
[CrossRef]

1992

1984

A. Röseler, “Improvement in accuracy of spectroscopic IR ellipsometry by the use of IR retarders,” Infrared Phys. 24, 1–5 (1984).
[CrossRef]

1982

A. Röseler, “Die Berechnung der Polarisationseigenschaften eines Fourierspektrometers mit Mueller-Matrizen,” Optik 60, 237–246 (1982).

A. Röseler, “Die Anwendung der Mueller-Matrix auf die spectroscopische Infrarot-Ellipsometrie mit dem Fourier-spektrometer,” Optik 61, 177–186 (1982).

1981

A. Röseler, “Spectroscopic ellipsometry in the infrared,” Infrared Phys. 21, 349–355 (1981).
[CrossRef]

1975

1973

1955

J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–245 (1955).

1948

W. L. Hyde, “Polarization techniques in the infrared,” J. Opt. Soc. Am. 38, 663 (1948).

1928

H. Nyquist, “Certain topics in telegraph transmission theory,” AIIE Trans. 47, 617–644 (1928).

Aspnes, D. E.

D. E. Aspnes, “The accurate determination of optical properties by ellipsometry,” in Handbook of Optical Constants of Solids,E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 89–112.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1979).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1979).

Beattie, J. R.

J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–245 (1955).

Bremer, J.

Dignam, M. J.

Dittmar, G.

G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
[CrossRef]

Fanping, K.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press. Cambridge, England, 1989).

Grosse, P.

G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
[CrossRef]

Hunderi, O.

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry assessment of intersubband transitions in n-GaAs/Al0.3Ga0.7As quantum wells,” Thin Solid Films 233, 194–198 (1993).
[CrossRef]

J. Bremer, O. Hunderi, K. Fanping, T. Skauli, E. Wold, “Infrared ellipsometer for the study of surfaces, thin films, and superlattices,” Appl. Opt. 31, 471–478 (1992).
[CrossRef] [PubMed]

Hyde, W. L.

W. L. Hyde, “Polarization techniques in the infrared,” J. Opt. Soc. Am. 38, 663 (1948).

Kliger, D. S.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

Lewis, J. W.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” AIIE Trans. 47, 617–644 (1928).

Offermann, V.

G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
[CrossRef]

Ozanyan, K. B.

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry assessment of intersubband transitions in n-GaAs/Al0.3Ga0.7As quantum wells,” Thin Solid Films 233, 194–198 (1993).
[CrossRef]

Pohlen, M.

G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press. Cambridge, England, 1989).

Randall, C. E.

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

Rao, B.

Röseler, A.

A. Röseler, “Improvement in accuracy of spectroscopic IR ellipsometry by the use of IR retarders,” Infrared Phys. 24, 1–5 (1984).
[CrossRef]

A. Röseler, “Die Anwendung der Mueller-Matrix auf die spectroscopische Infrarot-Ellipsometrie mit dem Fourier-spektrometer,” Optik 61, 177–186 (1982).

A. Röseler, “Die Berechnung der Polarisationseigenschaften eines Fourierspektrometers mit Mueller-Matrizen,” Optik 60, 237–246 (1982).

A. Röseler, “Spectroscopic ellipsometry in the infrared,” Infrared Phys. 21, 349–355 (1981).
[CrossRef]

A. Röseler, “Spectroscopic infrared ellipsometry with the Fourier transform spectrometer,” Ph.D. dissertation (Zentralinstitut für Optik, Akademie der Wissenschaften der Deutsche Demokratische Republik, Berlin, 1985).

Skauli, T.

Stobie, R. W.

Teukolski, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press. Cambridge, England, 1989).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press. Cambridge, England, 1989).

Wold, E.

AIIE Trans.

H. Nyquist, “Certain topics in telegraph transmission theory,” AIIE Trans. 47, 617–644 (1928).

Appl. Opt.

Infrared Phys.

A. Röseler, “Improvement in accuracy of spectroscopic IR ellipsometry by the use of IR retarders,” Infrared Phys. 24, 1–5 (1984).
[CrossRef]

A. Röseler, “Spectroscopic ellipsometry in the infrared,” Infrared Phys. 21, 349–355 (1981).
[CrossRef]

J. Opt. Soc. Am.

W. L. Hyde, “Polarization techniques in the infrared,” J. Opt. Soc. Am. 38, 663 (1948).

Optik

A. Röseler, “Die Berechnung der Polarisationseigenschaften eines Fourierspektrometers mit Mueller-Matrizen,” Optik 60, 237–246 (1982).

A. Röseler, “Die Anwendung der Mueller-Matrix auf die spectroscopische Infrarot-Ellipsometrie mit dem Fourier-spektrometer,” Optik 61, 177–186 (1982).

Philos. Mag.

J. R. Beattie, “Optical constants of metals in the infrared—experimental methods,” Philos. Mag. 46, 235–245 (1955).

Thin Solid Films

K. B. Ozanyan, O. Hunderi, “Spectroscopic transmission ellipsometry assessment of intersubband transitions in n-GaAs/Al0.3Ga0.7As quantum wells,” Thin Solid Films 233, 194–198 (1993).
[CrossRef]

G. Dittmar, V. Offermann, M. Pohlen, P. Grosse, “Extension of spectroscopic ellipsometry to the far infared,” Thin Solid Films 234, 346–351 (1993).
[CrossRef]

Other

A. Röseler, “Spectroscopic infrared ellipsometry with the Fourier transform spectrometer,” Ph.D. dissertation (Zentralinstitut für Optik, Akademie der Wissenschaften der Deutsche Demokratische Republik, Berlin, 1985).

W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press. Cambridge, England, 1989).

D. E. Aspnes, “The accurate determination of optical properties by ellipsometry,” in Handbook of Optical Constants of Solids,E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 89–112.

E. Wold, Ph. D. dissertation(Norwegian Institute of Technology, N-7034 Trondheim, Norway, 1993).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1979).

D. S. Kliger, J. W. Lewis, C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, San Diego, Calif., 1990).

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Figures (9)

Fig. 1
Fig. 1

Normalized second-order Fourier coefficients as measured for a straight-through beam in an IR ellipsometer with an empty sample holder. Curves 1 and 2 are the A2/A0 coefficients from RP and RA measurements, respectively. The two almost identical curves labeled 3 are the corresponding B2/A0 coefficients. These coefficients correspond formally to the ellipsometric parameters α and β and should in reality equal zero and unity. Correcting the data for interferometer polarization or detector dichroism in a perfect polarizer approximation yields the overlapping curves 4 for α. There are only small changes in the formal β parameter on correction.

Fig. 2
Fig. 2

Effect of the three correction schemes discussed in the text on the experimental α (lower triplet) and β (upper triplet) curves. Curves 1 are the results in a perfect polarizer approximation. Including the polarizer loss factor C2p, but not the leak factor S2p, yields the curves that are labeled 2. Taking both the finite extinction and the phase change into account, we obtained curves 3. Note the change in scale on the ordinate axis. The data set is recorded in the RA mode.

Fig. 3
Fig. 3

Flow chart for IR ellipsometric measurements. Fourier coefficients A0,A2,B2, …, are recorded with either an RP or an RA mode measurement. Stored data for Stokes components and Mueller matrix elements are denoted by f, g, C2d, τ, …, and have been found from separate RPA recordings as explained in the text [see Eqs. (12) and Appendix A]. We can produce satisfactory initial values α′ and β′ by solving the perfect polarizer equations [Eqs. (10) or (11)].

Fig. 4
Fig. 4

Phase change and extinction parameter of our far-IR polarizers as calculated by the RPA method discussed in the text.

Fig. 5
Fig. 5

Calculated permittivity from a gold sample measurement. The curves labeled 1 show the results when perfect polarizers are assumed. Curves 2 show the results after polarizer imperfections are corrected for.

Fig. 6
Fig. 6

Effect of correction on the α parameter for SrTiO3. The curve numbering is the same as that in Fig. 2.

Fig. 7
Fig. 7

Consistency of solutions. The orientations of the detector (1) and the interferometer (2) are treated as free variables. Curves 1 and 2 show that these angles vary over 1–2 deg.

Fig. 8
Fig. 8

Curve 1 shows the B6/A0 coefficient recorded with both polarizers aligned normal to the IR beam. Curve 2 shows the coefficient after the analyzer is tilted ∼2°.

Fig. 9
Fig. 9

Surface plots show the relative errors in the permittivity-related quantity e = e1 + ie2 (see text). It is assumed that the absolute experimental errors are Δα = Δβ = 0.01. The surfaces are cut when the relative errors exceed 25%. The two plots are symmetrical around the β axis and cover the whole physically meaningful α, β plane. For the sake of clarity the two surface nets are slightly shifted along the a directions. In (a) |Δe1/e1| is large near the ellipse 2α2 + β2 = 1. According to (b) the critical regions for |Δe1/e1| are found near α = 0 and α2 + β2 = 1. In both cases the errors diverge as one moves along the ridges toward more negative values for β.

Equations (18)

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S = ( I tot I 0 I 90 I 45 I 45 I r I l ) ,
= 1 2 ( T 1 + T 2 ) [ 1 C 2 γ 0 0 C 2 γ 1 0 0 0 0 S 2 γ cos Γ S 2 γ sin Γ 0 0 S 2 γ sin Γ S 2 γ cos Γ ] ,
C 2 γ = T 1 T 2 T 1 + T 2 ,
S 2 γ = 2 ( T 1 T 2 ) 1 / 2 T 1 + T 2 .
( θ ) = [ 1 0 0 0 0 C 2 θ S 2 θ 0 0 S 2 θ C 2 θ 0 0 0 0 1 ] .
S 0 = m = 2 2 n = 2 2 C m n exp [ 2 i ( m θ + n ϕ ) ] ,
S 0 = m = 0 2 [ A 2 m ( ϕ ) cos 2 m θ + B 2 m ( ϕ ) sin 2 m θ ] .
A 2 m ( ϕ ) = n = 0 2 [ A 2 m a 2 n cos 2 n ϕ + A 2 m b 2 n sin 2 n ϕ ] .
S 0 = A 0 [ 1 α ( cos 2 θ + cos 2 ϕ ) + cos 2 θ cos 2 ϕ + β sin 2 θ sin 2 ϕ ] .
A 0 = 1 + ½ ( α f + β g ) , A 2 = ( α + f ) , B 2 = ( β + g ) , A 4 = ½ ( α f + β g ) , B 4 = ½ ( α g β f ) .
A 0 = 1 + ½ ( α D C + β D S ) , A 2 = ( α + D C ) , B 2 = ( β + D S ) , A 4 = ½ ( α D C + β D S ) , B 4 = ½ ( α D S β D C ) .
A 0 = 1 + D S C 2 p α D C C δ S 2 p + ½ ( 1 + C δ S 2 p ) × [ α f ( 1 + D S C 2 p ) + β g ( C 2 p + D S ) + D C S 2 p ( f C δ + β tan Δ g S δ ) ] , A 2 = C 2 p [ ( α + f ) ( 1 + D S C 2 p ) + D C C δ S 2 p ( 1 α f ) ] + g β S δ S 2 p [ D C S δ S 2 p tan Δ ( C 2 p + D S ) ] , B 2 = C 2 p [ β ( C 2 p + D S ) + g ( 1 + D S C 2 p α D C C δ S 2 p ) ] β f D C ( S δ S 2 p ) 2 + β tan Δ S δ S 2 p × ( D C C 2 p + f C 2 p + f D S ) , A 4 = ½ ( 1 C δ S 2 p ) [ α f ( 1 + D C C 2 p ) + β g ( C 2 p + D S ) D C S 2 p ( f C δ β tan Δ g S δ ) ] , B 4 = ½ ( 1 C δ S 2 p ) [ α g ( 1 + D C C 2 p ) β f ( C 2 p + D S ) D C S 2 p [ g C δ + β tan Δ f S δ ) ] .
C 2 p 2 A 2 a 2 A 0 a 0 + A 0 a 4 + A 4 a 0 + A 4 a 4 .
sin 2 θ sin 2 θ tan 2 θ = 2 α 2 + β 2 1 ( 1 + β ) 2 + i 2 α ( 1 α 2 β 2 ) 1 / 2 ( 1 + β ) 2 .
S = S + SA R SP R S + SA R SP R SA R SP R S + ,
S = S I A R SP R S ,
S 0 = m , n = 0 2 ( A 2 m a 2 n cos 2 m θ cos 2 n ϕ + A 2 m b 2 n cos 2 m θ sin 2 n ϕ + B 2 m a 2 n sin 2 m θ cos 2 n ϕ + B 2 m b 2 n sin 2 m θ sin 2 n ϕ ) .
A 0 a 0 = 1 ½ C 2 ψ ( 1 + C δ S 2 p ) ( C 2 d C 2 τ + f ) + ¼ ( C 2 d C 2 τ f + S 2 ψ cos Δ g C 2 d S 2 τ ) ( 1 + C δ S 2 p ) 2 , A 0 a 2 = C 2 p ( C 2 ψ + C 2 d C 2 τ ) + ½ ( 1 + C δ S 2 p ) [ C 2 p f × ( 1 C 2 ψ C 2 d C 2 τ ) S 2 ψ C 2 d g sin Δ S δ S 2 τ S 2 p ] , A 0 b 2 = C 2 d C 2 p S 2 τ + ½ ( 1 + C δ S 2 p ) ( S 2 ψ cos Δ C 2 p g C 2 ψ C 2 d C 2 p f S 2 τ + S 2 ψ C 2 d C 2 τ g sin Δ S δ S 2 p ) , A 0 a 4 = ½ ( 1 C δ S 2 p ) C 2 ψ C 2 d C 2 τ + ¼ ( f C 2 d C 2 τ S 2 ψ cos Δ g C 2 d S 2 τ ) [ 1 ( C δ S 2 p ) 2 ] , A 0 b 4 = ½ ( 1 C δ S 2 p ) C 2 ψ C 2 d S 2 τ + ¼ ( f C 2 d S 2 τ + S 2 ψ cos Δ g C 2 d C 2 τ ) [ 1 ( C δ S 2 p ) 2 ] , A 2 a 0 = C 2 p ( C 2 ψ + f ) + ½ ( 1 + C δ S 2 p ) [ C 2 d C 2 τ C 2 p × ( 1 C 2 ψ f ) S 2 ψ C 2 d g sin Δ S δ S 2 τ S 2 p ] , A 2 a 2 = C 2 p 2 ( 1 C 2 ψ C 2 d C 2 τ C 2 ψ f + C 2 d C 2 τ f ) S 2 ψ cos Δ g C 2 d S 2 τ ( S δ S 2 p ) 2 , A 2 b 2 = C 2 p 2 C 2 d S 2 τ ( C 2 ψ + f ) S 2 ψ g S δ S 2 p ( C 2 p sin Δ cos Δ C 2 d C 2 τ S δ S 2 p ) , A 2 a 4 = ½ ( 1 C δ S 2 p ) [ C 2 d C 2 τ C 2 p ( 1 C 2 ψ f ) + S 2 ψ C 2 d g sin Δ S δ S 2 τ S 2 p ] , A 2 b 4 = ½ ( 1 C δ S 2 p ) [ C 2 p C 2 d S 2 τ ( 1 C 2 ψ f ) S 2 ψ C 2 d C 2 τ g sin Δ S δ S 2 p ] , B 2 a 0 = C 2 p g + ½ ( 1 + C δ S 2 p ) ( C 2 ψ C 2 d C 2 τ C 2 p g + S 2 ψ cos Δ C 2 d C 2 p S 2 τ + S 2 ψ sin Δ f C 2 d S 2 τ S δ S 2 p ) , B 2 a 2 = C 2 p 2 g ( C 2 ψ + C 2 d C 2 τ ) S 2 ψ C 2 d S 2 τ S δ S 2 p ( C 2 p sin Δ cos Δ f S δ S 2 p ) , B 2 b 2 = C 2 p 2 ( S 2 ψ cos Δ + C 2 d g S 2 τ ) + S 2 ψ S δ S 2 p × ( C 2 p sin Δ ( C 2 d C 2 τ + f ) cos Δ C 2 d C 2 τ f S δ S 2 p ) , B 2 a 4 = ½ ( 1 C δ S 2 p ) ( C 2 ψ C 2 d C 2 τ C 2 p g S 2 ψ cos Δ C 2 d C 2 p S 2 τ S 2 ψ C 2 d f sin Δ S δ S 2 τ S 2 p ) , B 2 b 4 = ½ ( 1 C δ S 2 p ) [ C 2 ψ C 2 d S 2 τ C 2 p g + S 2 ψ C 2 d C 2 τ ( cos Δ C 2 p + f sin Δ S δ S 2 p ) ] , A 4 a 0 = ½ ( 1 C δ S 2 p ) C 2 ψ f + ¼ ( C 2 d C 2 τ f S 2 ψ cos Δ C 2 d g S 2 τ ) [ 1 ( C δ S 2 p ) 2 ] , A 4 a 2 = ½ ( 1 C δ S 2 p ) [ C 2 p f ( 1 C 2 ψ C 2 d C 2 τ ) + S 2 ψ C 2 d g sin Δ S δ S 2 τ S 2 p ] , A 4 b 2 = ½ ( 1 C δ S 2 p ) ( S 2 ψ cos Δ C 2 p g + C 2 ψ C 2 d C 2 p f S 2 τ + S 2 ψ C 2 d C 2 τ g sin Δ S δ S 2 p ) , A 4 a 4 = ¼ ( f C 2 d C 2 τ + S 2 ψ cos Δ C 2 d g S 2 τ ) ( 1 C δ S 2 p ) 2 , A 4 b 4 = ¼ ( f C 2 d S 2 τ S 2 ψ cos Δ C 2 d C 2 τ g ) ( 1 C δ S 2 p ) 2 , B 4 a 0 = ½ ( 1 C δ S 2 p ) C 2 ψ g + ¼ ( C 2 d C 2 τ g + S 2 ψ cos Δ C 2 d f S 2 τ ) [ 1 ( C δ S 2 p ) 2 ] , B 4 a 2 = ½ ( 1 C δ S 2 p ) [ C 2 p g ( 1 C 2 ψ C 2 d C 2 τ ) S 2 ψ C 2 d f sin Δ S δ S 2 τ S 2 p ] , B 4 b 2 = ½ ( 1 C δ S 2 p ) ( C 2 ψ C 2 d C 2 p g S 2 τ + S 2 ψ cos Δ C 2 p f + S 2 ψ C 2 d C 2 τ f sin Δ S δ S 2 p ) , B 4 a 4 = ¼ ( C 2 d C 2 τ g S 2 ψ cos Δ f C 2 d S 2 τ ) ( 1 C δ S 2 p ) 2 , B 4 b 4 = ¼ ( C 2 d C 2 τ g + S 2 ψ cos Δ f C 2 d C 2 τ ) ( 1 C δ S 2 p ) 2 .

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