Abstract

Calibration and data-reduction procedures have been developed for use in rotating-polarizer ellipsometry. The procedures are demonstrated with a novel rotating-polarizer multichannel ellipsometer designed for real-time spectroscopic investigations of film growth, etching, and surface phenomena. This instrument employs a photodiode array detection system, permitting the collection of spectra from 1.5 to 4.5 eV with a maximum time resolution of 40 ms. Analyses of errors specific to the detection system, including nonlinearity and stray light, are outlined, and simple correction procedures are applied to calibrations and measurements. Source and polarization system imperfections are determined in expanded calibration procedures that are designed to provide polarizer phase and analyzer azimuth versus photon energy, corrected to first order in the imperfections. Two alternative approaches to calibration are demonstrated, depending on the value of the ellipsometric parameter Δ. Exact data-reduction equations are also derived to include source and polarization system imperfections in the calculation of (Ψ, Δ). Although the overall procedures can also be applied to ellipsometers with single-channel detection, the advantages of the multichannel ellipsometer for characterization and correction of systematic errors are apparent.

© 1991 Optical Society of America

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References

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  1. H. Takasaki, “Photoelectric measurement of polarized light by means of an ADP polarization modulator. I. Photoelectric polarimeter,” J. Opt. Soc. Am. 51, 462–463 (1961); “An automatic retardation meter for automatic polarimetry by means of an ADP polarization modulator,” Appl. Opt. 3, 345–350 (1964); “Automatic ellipsometer. Automatic polarimetry by means of an ADP polarization modulator III,” Appl. Opt. 5, 759–764 (1966).
    [Crossref] [PubMed]
  2. B. D. Cahan, R. F. Spanier, “A high speed precision automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
    [Crossref]
  3. S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
    [Crossref]
  4. For a review in materials science, see J. B. Theeten, D. E. Aspnes, “Ellipsometry in thin film analysis,” Annu. Rev. Mater. Sci. 11, 97–122 (1981).
    [Crossref]
  5. For a review in electrochemistry, see S. Gottesfeld, “Ellipsometry: principles and recent applications in electrochemistry,” in Electroanalytical Chemistry: A Series of Advances, A. J. Bard, ed. (Dekker, New York, 1989), Vol. 15, pp. 143–265.
  6. D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), pp. 800–846.
  7. K. Vedam, “Non-destructive depth profiling of multilayer structures by spectroscopic ellipsometry,” Mater. Res. Soc. Bull. 12, 21–23 (1987).
  8. H. J. Mathieu, D. E. McClure, R. H. Muller, “Fast self-compensating ellipsometer,” Rev. Sci. Instrum. 45, 798–802 (1974).
    [Crossref]
  9. R. H. Muller, J. C. Farmer, “Fast, self-compensating spectral-scanning ellipsometer,” Rev. Sci. Instrum. 55, 371–374 (1984).
    [Crossref]
  10. Y.-T. Kim, R. W. Collins, K. Vedam, “Fast scanning spectroelectrochemical ellipsometry: in-situcharacterization of gold oxide,” Surf. Sci. 223, 341–350 (1990).
    [Crossref]
  11. R. W. Collins, “Automatic rotating element ellipsometers: calibration, operation, and real-time applications,” Rev. Sci. Instrum. 61, 2029–2062 (1990).
    [Crossref]
  12. EG&G Princeton Applied Research, Princeton, N.J. 08543; OMA III Detector Interface Model 1461.
  13. Y. Talmi, R. W. Simpson, “Self-scanned photodiode array: a multichannel spectrometric detector,” Appl. Opt. 19, 1401–1414 (1980).
    [Crossref] [PubMed]
  14. D. E. Aspnes, A. A. Studna, “Photomultiplier linearization and system stabilization for photometric ellipsometers and polarimeters,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 62–66 (1977); “Methods for drift stabilization and photomultiplier linearization for photometric ellipsometers and polarimeters,” Rev. Sci. Instrum. 49, 291–297 (1978).
    [Crossref] [PubMed]
  15. D. E. Aspnes, A. A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [PubMed]
  16. 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]
  17. R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
    [Crossref]
  18. D. E. Aspnes, “Optimizing precision of rotating analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
    [Crossref]
  19. R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
    [Crossref]
  20. S. H. Russev, “Correction for nonlinearity and polarization-dependent sensitivity in the detection system of rotating analyzer ellipsometers,” Appl. Opt. 28, 1504–1507 (1989).
    [Crossref] [PubMed]
  21. J. M. M. de Nijs, A. H. M. Holtslag, A. Hoeksta, A. van Silfhout, “Calibration method for rotating-analyzer ellipsometers,” J. Opt. Soc. Am. A 5, 1466–1471 (1988).
    [Crossref]
  22. 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).
    [Crossref] [PubMed]

1990 (2)

Y.-T. Kim, R. W. Collins, K. Vedam, “Fast scanning spectroelectrochemical ellipsometry: in-situcharacterization of gold oxide,” Surf. Sci. 223, 341–350 (1990).
[Crossref]

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

1989 (1)

1988 (1)

1987 (1)

K. Vedam, “Non-destructive depth profiling of multilayer structures by spectroscopic ellipsometry,” Mater. Res. Soc. Bull. 12, 21–23 (1987).

1985 (1)

R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
[Crossref]

1984 (1)

R. H. Muller, J. C. Farmer, “Fast, self-compensating spectral-scanning ellipsometer,” Rev. Sci. Instrum. 55, 371–374 (1984).
[Crossref]

1981 (1)

For a review in materials science, see J. B. Theeten, D. E. Aspnes, “Ellipsometry in thin film analysis,” Annu. Rev. Mater. Sci. 11, 97–122 (1981).
[Crossref]

1980 (1)

1975 (2)

1974 (3)

1969 (3)

B. D. Cahan, R. F. Spanier, “A high speed precision automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[Crossref]

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

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

1961 (1)

Aspnes, D. E.

For a review in materials science, see J. B. Theeten, D. E. Aspnes, “Ellipsometry in thin film analysis,” Annu. Rev. Mater. Sci. 11, 97–122 (1981).
[Crossref]

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

D. E. Aspnes, “Optimizing precision of rotating analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974).
[Crossref]

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]

D. E. Aspnes, A. A. Studna, “Photomultiplier linearization and system stabilization for photometric ellipsometers and polarimeters,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 62–66 (1977); “Methods for drift stabilization and photomultiplier linearization for photometric ellipsometers and polarimeters,” Rev. Sci. Instrum. 49, 291–297 (1978).
[Crossref] [PubMed]

D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), pp. 800–846.

Biter, W. J.

R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
[Crossref]

Cahan, B. D.

B. D. Cahan, R. F. Spanier, “A high speed precision automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[Crossref]

Clark, A. H.

R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
[Crossref]

Collins, R. W.

Y.-T. Kim, R. W. Collins, K. Vedam, “Fast scanning spectroelectrochemical ellipsometry: in-situcharacterization of gold oxide,” Surf. Sci. 223, 341–350 (1990).
[Crossref]

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

R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
[Crossref]

de Nijs, J. M. M.

Dignam, M. J.

Farmer, J. C.

R. H. Muller, J. C. Farmer, “Fast, self-compensating spectral-scanning ellipsometer,” Rev. Sci. Instrum. 55, 371–374 (1984).
[Crossref]

Gottesfeld, S.

For a review in electrochemistry, see S. Gottesfeld, “Ellipsometry: principles and recent applications in electrochemistry,” in Electroanalytical Chemistry: A Series of Advances, A. J. Bard, ed. (Dekker, New York, 1989), Vol. 15, pp. 143–265.

Hoeksta, A.

Holtslag, A. H. M.

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. Instrum. 40, 761–767 (1969).
[Crossref]

Kim, Y.-T.

Y.-T. Kim, R. W. Collins, K. Vedam, “Fast scanning spectroelectrochemical ellipsometry: in-situcharacterization of gold oxide,” Surf. Sci. 223, 341–350 (1990).
[Crossref]

Mathieu, H. J.

H. J. Mathieu, D. E. McClure, R. H. Muller, “Fast self-compensating ellipsometer,” Rev. Sci. Instrum. 45, 798–802 (1974).
[Crossref]

McClure, D. E.

H. J. Mathieu, D. E. McClure, R. H. Muller, “Fast self-compensating ellipsometer,” Rev. Sci. Instrum. 45, 798–802 (1974).
[Crossref]

Muller, R. H.

R. H. Muller, J. C. Farmer, “Fast, self-compensating spectral-scanning ellipsometer,” Rev. Sci. Instrum. 55, 371–374 (1984).
[Crossref]

H. J. Mathieu, D. E. McClure, R. H. Muller, “Fast self-compensating ellipsometer,” Rev. Sci. Instrum. 45, 798–802 (1974).
[Crossref]

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

Rao, B.

Russev, S. H.

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. Instrum. 40, 761–767 (1969).
[Crossref]

Simpson, R. W.

Spanier, R. F.

B. D. Cahan, R. F. Spanier, “A high speed precision automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[Crossref]

Stobie, R. W.

Studna, A. A.

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

D. E. Aspnes, A. A. Studna, “Photomultiplier linearization and system stabilization for photometric ellipsometers and polarimeters,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 62–66 (1977); “Methods for drift stabilization and photomultiplier linearization for photometric ellipsometers and polarimeters,” Rev. Sci. Instrum. 49, 291–297 (1978).
[Crossref] [PubMed]

Takasaki, H.

Talmi, Y.

Theeten, J. B.

For a review in materials science, see J. B. Theeten, D. E. Aspnes, “Ellipsometry in thin film analysis,” Annu. Rev. Mater. Sci. 11, 97–122 (1981).
[Crossref]

van Silfhout, A.

Vedam, K.

Y.-T. Kim, R. W. Collins, K. Vedam, “Fast scanning spectroelectrochemical ellipsometry: in-situcharacterization of gold oxide,” Surf. Sci. 223, 341–350 (1990).
[Crossref]

K. Vedam, “Non-destructive depth profiling of multilayer structures by spectroscopic ellipsometry,” Mater. Res. Soc. Bull. 12, 21–23 (1987).

Windischmann, H.

R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
[Crossref]

Annu. Rev. Mater. Sci. (1)

For a review in materials science, see J. B. Theeten, D. E. Aspnes, “Ellipsometry in thin film analysis,” Annu. Rev. Mater. Sci. 11, 97–122 (1981).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

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

Mater. Res. Soc. Bull. (1)

K. Vedam, “Non-destructive depth profiling of multilayer structures by spectroscopic ellipsometry,” Mater. Res. Soc. Bull. 12, 21–23 (1987).

Rev. Sci. Instrum. (4)

H. J. Mathieu, D. E. McClure, R. H. Muller, “Fast self-compensating ellipsometer,” Rev. Sci. Instrum. 45, 798–802 (1974).
[Crossref]

R. H. Muller, J. C. Farmer, “Fast, self-compensating spectral-scanning ellipsometer,” Rev. Sci. Instrum. 55, 371–374 (1984).
[Crossref]

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

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

Surf. Sci. (3)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

Y.-T. Kim, R. W. Collins, K. Vedam, “Fast scanning spectroelectrochemical ellipsometry: in-situcharacterization of gold oxide,” Surf. Sci. 223, 341–350 (1990).
[Crossref]

B. D. Cahan, R. F. Spanier, “A high speed precision automatic ellipsometer,” Surf. Sci. 16, 166–176 (1969).
[Crossref]

Thin Solid Films (1)

R. W. Collins, W. J. Biter, A. H. Clark, H. Windischmann, “A study of the microstructure of a–Si:H using spectroscopic ellipsometry measurements,” Thin Solid Films 129, 127–137 (1985).
[Crossref]

Other (4)

EG&G Princeton Applied Research, Princeton, N.J. 08543; OMA III Detector Interface Model 1461.

D. E. Aspnes, A. A. Studna, “Photomultiplier linearization and system stabilization for photometric ellipsometers and polarimeters,” in Optical Polarimetry: Instrumentation and Applications, R. M. A. Azzam, D. L. Coffeen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 62–66 (1977); “Methods for drift stabilization and photomultiplier linearization for photometric ellipsometers and polarimeters,” Rev. Sci. Instrum. 49, 291–297 (1978).
[Crossref] [PubMed]

For a review in electrochemistry, see S. Gottesfeld, “Ellipsometry: principles and recent applications in electrochemistry,” in Electroanalytical Chemistry: A Series of Advances, A. J. Bard, ed. (Dekker, New York, 1989), Vol. 15, pp. 143–265.

D. E. Aspnes, “Spectroscopic ellipsometry of solids,” in Optical Properties of Solids: New Developments, B. O. Seraphin, ed. (North-Holland, Amsterdam, 1976), pp. 800–846.

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Figures (14)

Fig. 1
Fig. 1

Optical configuration of the rotating polarizer ellipsometer with multichannel detection system (PC, personal computer; OMA, optical multichannel analyzer).

Fig. 2
Fig. 2

Ellipsometry spectra for two calibration samples: (a) Au, (b) c–Si with a 590-Å thermally grown SiO2. These data, collected in ~3 s, were not corrected for detector, polarizer, or source errors.

Fig. 3
Fig. 3

(a) Residual function and (b) phase function at 2.51 eV for the Au calibration sample. The vertical line in (a) identifies the analyzer angle corresponding to the residual function minimum, and the horizontal line in (b) identifies the value of the phase function evaluated this analyzer angle.

Fig. 4
Fig. 4

Calibration results for thermally oxidized c–Si: (a) analyzer angle corresponding to the residual function minima near the p direction (A1, for A′ ≅ 0°) and the s direction (A2, for A′ ≅ 90°), (b) difference in the phase functions evaluated at the analyzer angles corresponding to the p and s residual function minima (P1P2). Also included in (b) is the quantity δP1 that represents the deviation of P1 from the linear function obtained in a best fit versus pixel number. In the absence of system errors, A1 = A2, P1 = P2, and δP1 = 0.

Fig. 5
Fig. 5

Average effective dc–ac gain ratio ηave obtained in the p and s directions for the Au calibration sample. In the absence of errors this should be unity independent of photon energy.

Fig. 6
Fig. 6

Optical activity coefficient for the rotating polarizer deduced from the residual and phase function calibration data set for the Au sample (circles). The relation of Eq. (6) is shown (solid line).

Fig. 7
Fig. 7

Error parameters, (a) Ss and (b) γSc, that characterize the source polarization [from Eqs. (9)] deduced from the residual and phase function calibration data set for 590-Å SiO2 on c–Si.

Fig. 8
Fig. 8

Measured sin 4ω Fourier coefficient (open circles) for the Au sample (corrected for detector errors) along with that calculated from a −4PS rotation transformation of the coefficients in Eqs. (11a) and (11b) with the use of γSc and γSs determined in calibration (filled circles).

Fig. 9
Fig. 9

Analyzer offset As determined in calibration for (a) Au and (b) SiO2/c–Si and corrected to first order in source and polarizer imperfections. The observed residual function minimum A1 is shown for comparison. In the absence of remaining errors AS should be independent of photon energy.

Fig. 10
Fig. 10

Polarizer phase angle (filled circles) PS determined in calibration for (a) Au and (b) SiO2/c–Si and corrected to first order in source and polarizer imperfections. The pixel number range covers 1.5 eV < < 3.5 eV. In the absence of remaining errors the phase angle should be a linear function of pixel group number. δPS is the deviation from the best-fitting linear function in each case (open circles). δP1 is the deviation from the best-fitting line for the phase function evaluated at the observed residual function minimum.

Fig. 11
Fig. 11

(a) Residual function and (b) phase function at 2.51 eV for a (native oxide)/c–Si calibration sample, The vertical line in (a) identifies the analyzer angle corresponding to the residual function minimum, and the horizontal line in (b) identifies the value of the phase function evaluated at this analyzer angle.

Fig. 12
Fig. 12

(a) Zone-difference phase function Φ(A) ≡ Θ(A) − Θ(A + π/2) and (b) phase function Θ(A) at 2.51 eV for the (native oxide)/c–Si calibration sample. The vertical line in (a) identifies A0, the analyzer angle at which Φ(A) vanishes, and the horizontal line in (b) identifies P0, the value of the phase function evaluated at this analyzer angle.

Fig. 13
Fig. 13

(a) Analyzer azimuth A0 and (b) polarizer phase P0 obtained from the zone-difference phase function calibration method for the (native oxide)/c–Si sample. δP0 is the deviation from the best-fitting linear function of P0 versus pixel number. The solid circles for A0, P0, and δP0 include corrections of detector errors for comparison with corresponding uncorrected values (open circles). The pixel number range here is limited to 1.5 eV < < 2.9 eV. The linear fittings to P0 were performed over a wider pixel number range (thus explaining a fixed sign for the uncorrected δP0 over the limited range).

Fig. 14
Fig. 14

Effects of source and polarizer imperfections on (a) Ψ and (b) Δ for the thermally oxidized c–Si sample. The results are obtained by subtracting corrected data from uncorrected data. For the open circles, the corrected data are adjusted for source and polarizer errors in calibration alone; for the filled circles, the corrected data are adjusted for these errors in calibration and data reduction.

Equations (55)

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I ( t ) = I 0 ( 1 + α cos 2 ω t + β sin 2 ω t ) ,
S j = I 0 ( j - 1 ) π / 4 ω j π / 4 ω ( 1 + α cos 2 ω t + β sin 2 ω t ) d t ,
α = ( π / 2 ) ( S 1 - S 2 - S 3 + S 4 ) / ( S 1 + S 2 + S 3 + S 4 ) ,
β = ( π / 2 ) ( S 1 + S 2 - S 3 - S 4 ) / ( S 1 + S 2 + S 3 + S 4 ) ,
S 1 - S 2 + S 3 - S 4 = 0.
S j ( k ) = I 0 [ ( j - 1 ) π / 4 ] + P S ( k ) ( j π / 4 ) + P S ( k ) [ 1 + α ( k ) cos 2 P + β ( k ) sin 2 P ] d P ,
S j , str ( k ) = I 0 [ ( j - 1 ) π / 4 ] + P S ( k ) ( j π / 4 ) + P S ( k ) [ 1 + α ( k ) cos 2 P + β ( k ) sin 2 P ] d P ,
[ E u E v ] = [ 1 - i γ A 0 0 ] [ cos A sin A - sin A cos A ] [ ρ 0 0 1 ] × [ cos P - sin P sin P cos P ] [ 1 - i γ P i γ P γ P 2 ] × [ cos ( P - S ) sin ( P - S ) - sin ( P - S ) cos ( P - S ) ] [ 1 i ( 1 - ξ ) ] E 0 ,
γ i = ± 0.0010 h ν ( eV - 1 ) ,
I ( t ) = I 0 [ 1 + α cos 2 ( ω t - P S ) + β sin 2 ( ω t - P S ) + α 4 cos 4 ( ω t - P S ) + β 4 sin 4 ( ω t - P S ) ] ,
I ( t ) = I 0 ( 1 + α cos 2 ω t + β sin 2 ω t + α 4 cos 4 ω t + β 4 sin 4 ω t ) ,
β 4 = ( π / 2 ) ( S 1 - S 2 + S 3 - S 4 ) / ( S 1 + S 2 + S 3 + S 4 ) .
γ S s = ξ sin 2 S ,
γ S c = ξ cos 2 S ,
α = α 0 - 2 γ P α 0 β 0 tan Δ + γ S c [ 1 - ( α 0 2 / 2 ) ] - ( γ S s α 0 β 0 / 2 ) ,
β = β 0 - 2 γ P β 0 2 tan Δ - 2 γ A β 0 tan Δ csc 2 A - ( γ S c α 0 β 0 / 2 ) + γ S s [ 1 - ( β 0 2 / 2 ) ] ,
α 0 = ( tan 2 Ψ cos 2 A - sin 2 A ) / ( tan 2 Ψ cos 2 A + sin 2 A ) ,
β 0 = tan Ψ cos Δ sin 2 A / ( tan 2 Ψ cos 2 A + sin 2 A ) .
α 4 = ( α 0 γ S c - β 0 γ S s ) / 2 ,
β 4 = ( α 0 γ S s + β 0 γ S c ) / 2.
R ( A ) = 1 - ( α 2 + β 2 ) ,
R ( A ) = 1 - ( α 2 + β 2 ) ,
R ( A ) 4 ( A - A S ) 2 cot 2 Ψ sin 2 Δ + 2 γ S c ( A - A S ) 2 × ( 1 - 2 sin 2 Δ ) cot 2 Ψ + 2 ( A - A S ) ( 4 γ P cot Ψ sin Δ + 2 γ A cot 2 Ψ sin 2 Δ - γ S s cot Ψ cos Δ ) - γ S c .
R ( A ) 4 [ A - ( A S + π / 2 ) ] 2 tan 2 Ψ sin 2 Δ - 2 γ S c [ A - ( A S + π / 2 ) ] 2 ( 1 - 2 sin 2 Δ ) tan 2 Ψ - 2 [ A - ( A S + π / 2 ) ] ( 4 γ P tan Ψ sin Δ + 2 γ A × tan 2 Ψ sin 2 Δ - γ S c tan Ψ cos Δ ) + γ S c .
A S = A 1 + [ ( γ P tan Ψ + γ A cos Δ ) / sin Δ ] - [ γ S s tan Ψ cos Δ / ( 4 sin 2 Δ ) ] ,
A S + π / 2 = A 2 + π / 2 - [ ( γ P cot Ψ + γ A cos Δ ) / sin Δ ] + [ γ S s cot Ψ cos Δ / ( 4 sin 2 Δ ) ] .
Θ ( A ) = [ tan - 1 ( β / α ) ] / 2 ,
Θ ( A ) = P S + { [ tan - 1 ( β / α ) ] / 2 } ,
Θ ( A ) P S + ( A - A S ) cot Ψ cos Δ - γ S c ( A - A S ) cot Ψ cos Δ - γ A cot Ψ sin Δ + ( γ S s / 2 ) ,
Θ ( A ) P S + [ A - ( A S + π / 2 ) ] tan Ψ cos Δ + γ S c [ A - ( A S + π / 2 ) ] tan Ψ cos Δ + γ A tan Ψ sin Δ - ( γ S s / 2 ) .
P S = P 1 + [ ( γ A cot Ψ + γ P cos Δ ) / sin Δ ] - { γ S s [ 1 + ( cot 2 Δ ) / 2 ] / 2 } ,
P S = P 2 - [ ( γ A tan Ψ + γ P cos Δ ) / sin Δ ] + { γ S s [ 1 + ( cot 2 Δ ) / 2 ] / 2 } .
γ P = { [ ( P 1 - P 2 ) cot Δ ] / 4 } - { [ ( A 1 - A 2 ) ( csc Δ + sin Δ ) × sin 2 Ψ ] / 4 } ,
γ S s = ( P 1 - P 2 ) - ( A 1 - A 2 ) cos Δ sin 2 Ψ ,
γ S c = ( R min , 90 - R min , 0 ) / 2.
Φ ( A ) Θ ( A ) - Θ ( A + π / 2 ) ,
Φ ( A ) = [ tan - 1 ( β / α ) A - tan - 1 ( β / α ) A + π / 2 ] / 2 ,
Φ ( A ) 2 ( A - A S ) cot 2 Ψ cos Δ - 2 γ S c ( A - A S ) csc 2 Ψ cos Δ - 2 γ A csc 2 Ψ sin Δ + γ S s ,
A S = A 0 - ( γ A tan Δ / cos 2 Ψ ) + [ γ S s tan 2 Ψ / ( 2 cos Δ ) ] ,
P S = P 0 - γ A sin Δ tan 2 Ψ + [ γ S s / ( 2 cos 2 Ψ ) ] .
α 0 = α cos 2 P S + β sin 2 P S ,
β 0 = - α sin 2 P S + β cos 2 P S .
tan Ψ = [ ( 1 + α 0 ) / ( 1 - α 0 ) ] 1 / 2 tan ( A - A S ) ,
cos Δ = β 0 / ( 1 - α 0 2 ) 1 / 2 .
[ 1 - i γ A 0 0 ] [ cos A sin A - sin A cos A ] [ ρ 0 0 1 ] = [ 1 - i a 0 0 ] [ cos Q sin Q - sin Q cos Q ] ,
[ E u E v ] = [ 1 - i a 0 0 ] [ cos ( Q - P ) sin ( Q - P ) - sin ( Q - P ) cos ( Q - P ) ] [ 1 - i γ P i γ P γ P 2 ] × [ cos ( P - S ) sin ( P - S ) - sin ( P - S ) cos ( P - S ) ] [ 1 i ( 1 - ξ ) ] E 0 .
Γ S c = ( { [ 1 - ( 1 - ξ ) 2 ] ( 1 - γ P 2 ) } / { [ 1 + ( 1 - ξ ) 2 ] ( 1 + γ P 2 ) + 4 γ P ( 1 - ξ ) } ) cos 2 S ,
Γ S s = ( { [ 1 - ( 1 - ξ ) 2 ] ( 1 - γ P 2 ) } / { [ 1 + ( 1 - ξ ) 2 ] ( 1 + γ P 2 ) + 4 γ P ( 1 - ξ ) } ) sin 2 S ,
Q = { [ tan - 1 ( y / x ) ] + [ π u ( - x ) sgn ( y ) ] } / 2 ,
y = ( 2 - Γ S c 2 ) β - ( 2 - Γ S c α ) Γ S s ,
x = ( 2 - Γ S s 2 ) α - ( 2 - Γ S s β ) Γ S c .
ζ = 2 ( α - Γ S c ) [ ( 2 - Γ S c α ) cos 2 Q - Γ S s α sin 2 Q ] - 1 ,
ζ = 2 ( β - Γ S s ) [ ( 2 - Γ S s β ) sin 2 Q - Γ S c β cos 2 Q ] - 1 .
a = [ 2 γ P ζ ± ( 1 - γ P 2 ) ( 1 - ζ 2 ) 1 / 2 ] [ ( 1 + ζ ) - γ P 2 ( 1 - ζ ) ] - 1 .
ρ = { ( cot Q - i a ) [ tan ( A - A S ) - i γ A ] } { ( 1 + i a cot Q ) × [ 1 + i γ A tan ( A - A S ) ] } - 1 .

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