Abstract

We describe the design of a high-speed multichannel ellipsometer in the optical configuration PC1r(ω1)SC2r(ω2)A having frequency-coupled rotating compensators (C1r and C2r) and a fixed polarizer and analyzer (P and A) symmetrically placed about the sample (S) on the polarization generation and detection arms of the instrument. For this instrument the frequency-coupled compensators rotate continuously at ω1 =5ω and ω2=3ω, where π/ω is the fundamental optical period. Although the dual rotating-compensator configuration has been proposed and demonstrated earlier, we focus on its extension to real-time Mueller matrix spectroscopy of surface modification and thin-film growth utilizing high-speed multichannel detection with a wide spectral range. The proposed instrument design provides the capability of extracting all 16 elements of the unnormalized Mueller matrix of an evolving sample at 1024 points from 1.5 to 6.5 eV with potential acquisition and repetition times of 0.2 s. Techniques of data acquisition, data reduction, and instrument calibration are described for the general case of arbitrary compensator retardances and polarizer and analyzer angles. We expect that the proposed instrument will have important applications in studies of surfaces and thin films that exhibit anisotropy and inhomogeneity.

© 1999 Optical Society of America

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References

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  1. P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
    [CrossRef]
  2. R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
    [CrossRef]
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  4. G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313-314, 33–39 (1998).
    [CrossRef]
  5. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978).
    [CrossRef] [PubMed]
  6. P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
    [CrossRef]
  7. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992).
    [CrossRef] [PubMed]
  8. R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
    [CrossRef]
  9. J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
    [CrossRef]
  10. J. A. Zapien, R. W. Collins, R. Messier, “Extensions of multichannel spectroscopic ellipsometry into the ultraviolet for real time characterization of the growth of wide bandgap materials from 1.5 to 6.5 eV,” Mater. Res. Soc. Symp. Proc. (to be published).
  11. J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
    [CrossRef]
  12. N. V. Nguyen, B. S. Pudliner, I. An, R. W. Collins, “Error correction for calibration and data reduction in rotating polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
    [CrossRef]
  13. Manual for PDA-1024 Spectrometric Detector, Princeton Instruments, Inc., Princeton, N.J., 1998.
  14. Y. Talmi, R. W. Simpson, “Self-scanned photodiode array: a multichannel spectrometric detector,” Appl. Opt. 19, 1401–1414 (1980).
    [CrossRef] [PubMed]
  15. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: experiment and calibration,” Appl. Opt. 36, 8184–8189 (1997).
    [CrossRef]
  16. G. E. Jellison, F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
    [CrossRef]
  17. E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
    [CrossRef]
  18. S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
    [CrossRef]

1998

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313-314, 33–39 (1998).
[CrossRef]

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

1997

1993

S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
[CrossRef]

1992

1991

1990

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

1980

1978

An, I.

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

N. V. Nguyen, B. S. Pudliner, I. An, R. W. Collins, “Error correction for calibration and data reduction in rotating polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
[CrossRef]

Aspnes, D. E.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

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

Bouree, J. E.

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

Butler, S. W.

S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
[CrossRef]

Chen, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Collins, R. W.

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

N. V. Nguyen, B. S. Pudliner, I. An, R. W. Collins, “Error correction for calibration and data reduction in rotating polarizer ellipsometry: applications to a novel multichannel ellipsometer,” J. Opt. Soc. Am. A 8, 919–931 (1991).
[CrossRef]

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

J. A. Zapien, R. W. Collins, R. Messier, “Extensions of multichannel spectroscopic ellipsometry into the ultraviolet for real time characterization of the growth of wide bandgap materials from 1.5 to 6.5 eV,” Mater. Res. Soc. Symp. Proc. (to be published).

Compain, E.

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

Drevillon, B.

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

Duncan, W. M.

S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
[CrossRef]

Fanton, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Fujiwara, H.

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

Goldstein, D. H.

Hauge, P. S.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

P. S. Hauge, “Mueller matrix ellipsometry with imperfect compensators,” J. Opt. Soc. Am. 68, 1519–1528 (1978).
[CrossRef]

Henck, S. A.

S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
[CrossRef]

Huc, J.

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

Jellison, G. E.

Koh, J.

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

Lee, J.

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Leng, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Lowenstein, L. M.

S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
[CrossRef]

Lu, Y.

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

Messier, R.

J. A. Zapien, R. W. Collins, R. Messier, “Extensions of multichannel spectroscopic ellipsometry into the ultraviolet for real time characterization of the growth of wide bandgap materials from 1.5 to 6.5 eV,” Mater. Res. Soc. Symp. Proc. (to be published).

Modine, F. A.

Nguyen, N. V.

Opsal, J.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Parey, J. Y.

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

Pudliner, B. S.

Rovira, P. I.

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

Senko, M.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Simpson, R. W.

Talmi, Y.

Uhrich, C.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Wei, L.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Zaiser, C.

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

Zapien, J. A.

J. A. Zapien, R. W. Collins, R. Messier, “Extensions of multichannel spectroscopic ellipsometry into the ultraviolet for real time characterization of the growth of wide bandgap materials from 1.5 to 6.5 eV,” Mater. Res. Soc. Symp. Proc. (to be published).

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. A

S. A. Henck, W. M. Duncan, L. M. Lowenstein, S. W. Butler, “In situ spectral ellipsometry for real time thickness measurement: etching multilayer stacks,” J. Vac. Sci. Technol. A 11, 1179–1185 (1993).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
[CrossRef]

J. Lee, P. I. Rovira, I. An, R. W. Collins, “Rotating compensator multichannel ellipsometry: applications for real time Stokes vector spectroscopy of thin film growth,” Rev. Sci. Instrum. 69, 1800–1810 (1998).
[CrossRef]

Surf. Sci.

P. S. Hauge, “Recent developments in instrumentation in ellipsometry,” Surf. Sci. 96, 108–140 (1980).
[CrossRef]

Thin Solid Films

E. Compain, B. Drevillon, J. Huc, J. Y. Parey, J. E. Bouree, “Complete Mueller matrix measurement with a single high-frequency modulation,” Thin Solid Films 313-314, 47–52 (1998).
[CrossRef]

J. Opsal, J. Fanton, J. Chen, J. Leng, L. Wei, C. Uhrich, M. Senko, C. Zaiser, D. E. Aspnes, “Broadband spectral operation of a rotating-compensator ellipsometer,” Thin Solid Films 313-314, 58–61 (1998).
[CrossRef]

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313-314, 33–39 (1998).
[CrossRef]

R. W. Collins, I. An, H. Fujiwara, J. Lee, Y. Lu, J. Koh, P. I. Rovira, “Advances in multichannel spectroscopic ellipsometry,” Thin Solid Films 313-314, 18–32 (1998).
[CrossRef]

Other

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

Manual for PDA-1024 Spectrometric Detector, Princeton Instruments, Inc., Princeton, N.J., 1998.

J. A. Zapien, R. W. Collins, R. Messier, “Extensions of multichannel spectroscopic ellipsometry into the ultraviolet for real time characterization of the growth of wide bandgap materials from 1.5 to 6.5 eV,” Mater. Res. Soc. Symp. Proc. (to be published).

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Figures (1)

Fig. 1
Fig. 1

Schematic optical configuration for the dual rotating-compensator multichannel ellipsometer. The Mueller matrices for (left to right) the polarizer, the first compensator, the sample, the second compensator, and the analyzer are also provided. For the polarizer, compensators, and analyzer, these matrices are normalized, and the transmitted fraction of the incident irradiance is taken into account through the parameter I00 that describes the overall ellipsometer spectral response function (see Subsection 2.D).

Tables (4)

Tables Icon

Table 1 System Design Parameters for a Dual Rotating-Compensator Multichannel Ellipsometera

Tables Icon

Table 2 Listing of the Nonzero, dc-Normalized Fourier Coefficients and Calibration Phase Angles of Eq. (3), Where cj=cos2(δj/2) and sj=sin2(δj/2) (j=1, 2)

Tables Icon

Table 3 Product of the Square of the dc Term and the Amplitude Function (a0R2n)2 of Eq. (7a), Where cj=cos2(δj/2) and sj=sin2(δj/2) (j=1, 2)

Tables Icon

Table 4 Phase Functions Θ2n in the Measured Fourier Coefficients of Eq. (7b)

Equations (45)

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

Sout=MAR(A)R(-C2)MC2(δ2)R(C2)×MSR(-C1)MC1(δ1)R(C1)R(-P)MPR(P)Sin,
I=I0{K1+[c2 cos 2A+s2 cos(4C2-2A)]K2+[c2 sin 2A+s2 sin(4C2-2A)]K3-[sin δ2 sin(2C2-2A)]K4},
Kj=Mj1+[c1 cos 2P+s1 cos(4C1-2P)]Mj2+[c1 sin 2P+s1 sin(4C1-2P)]Mj3+[sin δ1 sin(2C1-2P)]Mj4.
I(t)=I01+n=116[α2n cos(2nC-ϕ2n)+β2n sin(2nC-ϕ2n)],
M12=a0[-α8 cos(2P-4A)-β8 sin(2P-4A)+t2α20 cos 2P+t2β20 sin 2P-α32 cos(2P+4A)-β32 sin(2P+4A)]/s1t2,
M13=a0[α8 sin(2P-4A)-β8 cos(2P-4A)-t2α20 sin 2P+t2β20 cos 2P+α32 sin(2P+4A)-β32 cos(2P+4A)]/s1t2,
M14=a0[2α2 sin(2P-4A)+2β2 cos(2P-4A)-t2α10 sin 2P+t2β10 cos 2P]/[(sin δ1)t2],
M21=a0[-α8 cos(4P-2A)-β8 sin(4P-2A)+t1α12 cos 2A+t1β12 sin 2A-α32 cos(4P+2A)-β32 sin(4P+2A)]/t1s2,
M22=a0[α8 cos(2P-2A)+β8 sin(2P-2A)+α32 cos(2P+2A)+β32 sin(2P+2A)]/s1s2,
M23=a0[-α8 sin(2P-2A)+β8 cos(2P-2A)-α32 sin(2P+2A)+β32 cos(2P+2A)]/s1s2,
M24=2a0[-α2 sin(2P-2A)-β2 cos(2P-2A)]/[(sin δ1)s2]
=2a0[-α22 sin(2P+2A)+β22 cos(2P+2A)]/[(sin δ1)s2],
M31=a0[-α8 sin(4P-2A)+β8 cos(4P-2A)-t1α12 sin 2A+t1β12 cos 2A+α32 sin(4P+2A)-β32 cos(4P+2A)]/t1s2,
M32=a0[α8 sin(2P-2A)-β8 cos(2P-2A)-α32 sin(2P+2A)+β32 cos(2P+2A)]/s1s2,
M33=a0[α8 cos(2P-2A)+β8 sin(2P-2A)-α32 cos(2P+2A)-β32 sin(2P+2A)]/s1s2,
M34=2a0[α2 cos(2P-2A)-β2 sin(2P-2A)]/[(sin δ1)s2]
=2a0[-α22 cos(2P+2A)-β22 sin(2P+2A)]/[(sin δ1)s2],
M41=a0[2α14 sin(4P-2A)-2β14 cos(4P-2A)+t1α6 sin 2A-t1β6 cos 2A]/t1 sin δ2,
M42=2a0[-α14 sin(2P-2A)+β14 cos(2P-2A)]/s1 sin δ2
=2a0[α26 sin(2P+2A)-β26 cos(2P+2A)]/s1 sin δ2,
M43=2a0[-α14 cos(2P-2A)-β14 sin(2P-2A)]/s1 sin δ2
=2a0[α26 cos(2P+2A)+β26 sin(2P+2A)]/s1 sin δ2,
M44=2a0[-α4 cos(2P-2A)-β4 sin(2P-2A)]/sin δ1 sin δ2
=2a0[α16 cos(2P+2A)+β16 sin(2P+2A)]/sin δ1 sin δ2.(4o)
a0=t1t2[t1t2+α8 cos(4P-4A)+β8 sin(4P-4A)-t1α12 cos 4A-t1β12 sin 4A-t2α20 cos 4P-t2β20 sin 4P+α32 cos(4P+4A)+β32 sin(4P+4A)]-1,
I(t)=I01+n=116(α2n cos 2nωt+β2n sin 2nωt),
Sj=(j-1)π/36ωjπ/36ωI01+n=116(α2n cos 2nωt+β2n sin 2nωt)dt(j=1, , 36)
=πI036ω+n=116 I0nω sin nπ36×α2n cos (2j-1)nπ36+β2n sin (2j-1)nπ36.
α2n=α2n cos ϕ2n+β2n sin ϕ2n,
β2n=-α2n sin ϕ2n+β2n cos ϕ2n,
(R2n)2=(α2n)2+(β2n)2=(α2n)2+(β2n)2,
Θ2n=tan-1(β2n/α2n)=tan-1(β2n/α2n)+ϕ2n.
P-PS+5CS1=(1/2)Θ10±(π/4)=(1/2)(Θ4+Θ6)±(π/4)=(1/4)(Θ4+Θ16)=(1/2)(Θ16-Θ6)±(π/4)=(1/4)(Θ22-Θ2),
A-AS+3CS2=(1/2)Θ6±(π/4)=(1/2)(Θ10-Θ4)±(π/4)=(1/4)(Θ16-Θ4)=(1/2)(Θ16-Θ10)±(π/4)=(1/4)(Θ26-Θ14).
(α6)2+(β6)2(α4)2+(β4)2=a6,0+n=12(a6,2n cos 2nP+b6,2n sin 2nP),
(α10)2+(β10)2(α4)2+(β4)2=a10,0+n=12(a10,2n cos 2nA+b10,2n sin 2nA).
tan 4PS=2 2Zb6,4-a6,2b6,2(b6,2)2-(a6,2)2+4Za6,4,
Z=a6,0-{(a6,0)2-(1/2)[(a6,2)2+(b6,2)2]}1/2.
(α2)2+(β2)2(α22)2+(β22)2=1,
(α4)2+(β4)2(α16)2+(β16)2=1,
(α14)2+(β14)2(α26)2+(β26)2=1,
CS1=(1/10)(Θ2+Θ8)=(1/10)(Θ14-Θ4)=(1/10)(Θ26-Θ16)=(1/10)(Θ32-Θ22),
CS2=(1/6)(Θ2+Θ4)=(1/6)(Θ14-Θ8)=(1/6)(Θ22-Θ16)=(1/6)(Θ32-Θ26).
M11(t)=I0(t)a0(0)I0(0)a0(t) M11(0),
M11(0)=(1/2)(|rp|2+|rs|2),

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