Abstract

This research proposed a dual-frequency heterodyne ellipsometer (DHE) in which a dual-frequency collinearly polarized laser beam with equal amplitude and zero phase difference between p- and s-polarizations is setup. It is based on the polarizer-sample-analyzer, PSA configuration of the conventional ellipsometer. DHE enables to characterize a generalized elliptical phase retarder by treating it as the combination of a linear phase retarder and a polarization rotator. The method for measuring elliptical birefringence of an elliptical phase retarder based on the equivalence theorem of an unitary optical system was derived and the experimental verification by use of DHE was demonstrated too. The experimental results show the capability of DHE on characterization of a generalized phase retardation plate accurately.

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References

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  1. J. F. Nye, Physical Properties of Crystals (Oxford University Press, Oxford, 1957), pp. 261–268.
  2. F. Ratajczyk and P. Kurzynowski, “Phase difference superposition law for elliptically birefringent media,” Optik (Stuttg.) 99, 92–94 (1995).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. S. Berezhna, I. Berezhnyy, and M. Takashi, “Integrated photoelasticity through imaging Fourier polarimetry of an elliptic retarder,” Appl. Opt. 40(5), 644–651 (2001).
    [CrossRef]
  7. K. Pietraszkiewicz, W. A. Woźniak, and P. Kurzynowski, “Effect of multiple reflections in retardation plates with elliptical birefringence,” J. Opt. Soc. Am. A 12(2), 420–424 (1995).
    [CrossRef]
  8. S. Berezhna, I. Berezhnyy, and M. Takashi, “Determination of the normalized Jones matrix of elliptical retarder,” Proc. SPIE 4317, 129–134 (2001).
    [CrossRef]
  9. P. Kurzynowski, “Senarmont compensator for elliptically birefringent media,” Opt. Commun. 197(4-6), 235–238 (2001).
    [CrossRef]
  10. P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006).
    [CrossRef]
  11. C. Chou, Y. C. Huang, and M. Chang, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14(6), 1367–1372 (1997).
    [CrossRef]
  12. C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996).
    [CrossRef]
  13. H. Hurwitz and R. C. Jones, “A new calculus for the treatment of optical systems. II. Proof of three general equivalence theorems,” J. Opt. Soc. Am. 31, 493–499 (1941).
  14. S. T. Tang and H. S. Kwok, “3 × 3 Matrix for unitary optical systems,” J. Opt. Soc. Am. A 18(9), 2138–2145 (2001).
    [CrossRef]
  15. V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
    [CrossRef]
  16. V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
    [CrossRef]
  17. R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), pp. 98–99.
  18. P. Yeh, and C. Gu, Optics of liquid crystal displays (Wiley, New York, 1999), pp. 120–122.
  19. C. Chou, C. W. Lyu, and L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40(1), 95–99 (2001).
    [CrossRef]
  20. H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
    [CrossRef]
  21. D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996).
    [CrossRef]
  22. C. J. Yu, C. E. Lin, L. P. Yu, and C. Chou, “Paired circularly polarized heterodyne ellipsometer,” Appl. Opt. 48(4), 758–764 (2009).
    [CrossRef] [PubMed]
  23. W. Mao, S. Zhang, L. Cui, and Y. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006).
    [CrossRef] [PubMed]

2009 (1)

2006 (4)

W. Mao, S. Zhang, L. Cui, and Y. Tan, “Self-mixing interference effects with a folding feedback cavity in Zeeman-birefringence dual frequency laser,” Opt. Express 14(1), 182–189 (2006).
[CrossRef] [PubMed]

P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
[CrossRef]

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

2005 (1)

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
[CrossRef]

2001 (6)

1999 (1)

1997 (1)

1996 (2)

C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996).
[CrossRef]

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996).
[CrossRef]

1995 (2)

F. Ratajczyk and P. Kurzynowski, “Phase difference superposition law for elliptically birefringent media,” Optik (Stuttg.) 99, 92–94 (1995).

K. Pietraszkiewicz, W. A. Woźniak, and P. Kurzynowski, “Effect of multiple reflections in retardation plates with elliptical birefringence,” J. Opt. Soc. Am. A 12(2), 420–424 (1995).
[CrossRef]

1941 (2)

Berezhna, S.

S. Berezhna, I. Berezhnyy, and M. Takashi, “Determination of the normalized Jones matrix of elliptical retarder,” Proc. SPIE 4317, 129–134 (2001).
[CrossRef]

S. Berezhna, I. Berezhnyy, and M. Takashi, “Integrated photoelasticity through imaging Fourier polarimetry of an elliptic retarder,” Appl. Opt. 40(5), 644–651 (2001).
[CrossRef]

Berezhnyy, I.

S. Berezhna, I. Berezhnyy, and M. Takashi, “Integrated photoelasticity through imaging Fourier polarimetry of an elliptic retarder,” Appl. Opt. 40(5), 644–651 (2001).
[CrossRef]

S. Berezhna, I. Berezhnyy, and M. Takashi, “Determination of the normalized Jones matrix of elliptical retarder,” Proc. SPIE 4317, 129–134 (2001).
[CrossRef]

Chang, H. F.

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

Chang, M.

C. Chou, Y. C. Huang, and M. Chang, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14(6), 1367–1372 (1997).
[CrossRef]

C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996).
[CrossRef]

Chartier, T.

Chen, C. D.

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996).
[CrossRef]

Chiu, M. H.

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996).
[CrossRef]

Chou, C.

C. J. Yu, C. E. Lin, L. P. Yu, and C. Chou, “Paired circularly polarized heterodyne ellipsometer,” Appl. Opt. 48(4), 758–764 (2009).
[CrossRef] [PubMed]

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

C. Chou, C. W. Lyu, and L. C. Peng, “Polarized differential-phase laser scanning microscope,” Appl. Opt. 40(1), 95–99 (2001).
[CrossRef]

C. Chou, Y. C. Huang, and M. Chang, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14(6), 1367–1372 (1997).
[CrossRef]

C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996).
[CrossRef]

Cui, L.

Drobczynski, S.

P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006).
[CrossRef]

Durán, V.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
[CrossRef]

Hideur, A.

Huang, Y. C.

C. Chou, Y. C. Huang, and M. Chang, “Effect of elliptical birefringence on the measurement of the phase retardation of a quartz wave plate by an optical heterodyne polarimeter,” J. Opt. Soc. Am. A 14(6), 1367–1372 (1997).
[CrossRef]

C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996).
[CrossRef]

Hurwitz, H.

Jaroszewicz, Z.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
[CrossRef]

Jones, R. C.

Kurzynowski, P.

P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006).
[CrossRef]

P. Kurzynowski, “Senarmont compensator for elliptically birefringent media,” Opt. Commun. 197(4-6), 235–238 (2001).
[CrossRef]

F. Ratajczyk and P. Kurzynowski, “Phase difference superposition law for elliptically birefringent media,” Optik (Stuttg.) 99, 92–94 (1995).

K. Pietraszkiewicz, W. A. Woźniak, and P. Kurzynowski, “Effect of multiple reflections in retardation plates with elliptical birefringence,” J. Opt. Soc. Am. A 12(2), 420–424 (1995).
[CrossRef]

Kwok, H. S.

Lancis, J.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
[CrossRef]

Lin, C. E.

Lyu, C. W.

Mao, W.

Özkul, C.

Peng, L. C.

Pietraszkiewicz, K.

Ratajczyk, F.

F. Ratajczyk and P. Kurzynowski, “Phase difference superposition law for elliptically birefringent media,” Optik (Stuttg.) 99, 92–94 (1995).

Roy, R.

Sanchez, F. O.

Stéphan, G. M.

Su, D. C.

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996).
[CrossRef]

Tajahuerce, E.

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
[CrossRef]

Takashi, M.

S. Berezhna, I. Berezhnyy, and M. Takashi, “Determination of the normalized Jones matrix of elliptical retarder,” Proc. SPIE 4317, 129–134 (2001).
[CrossRef]

S. Berezhna, I. Berezhnyy, and M. Takashi, “Integrated photoelasticity through imaging Fourier polarimetry of an elliptic retarder,” Appl. Opt. 40(5), 644–651 (2001).
[CrossRef]

Tan, Y.

Tang, S. T.

Teng, H. K.

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

VanWiggeren, G. D.

Wozniak, W. A.

P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006).
[CrossRef]

K. Pietraszkiewicz, W. A. Woźniak, and P. Kurzynowski, “Effect of multiple reflections in retardation plates with elliptical birefringence,” J. Opt. Soc. Am. A 12(2), 420–424 (1995).
[CrossRef]

Wu, H. T.

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

Yau, H. F.

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

Yu, C. J.

Yu, L. P.

Zhang, S.

Appl. Opt. (5)

J. Appl. Phys. (2)

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Cell parameter determination of a twisted-nematic liquid crystal display by single-wavelength polarimetry,” J. Appl. Phys. 97(4), 043101 (2005).
[CrossRef]

V. Durán, J. Lancis, E. Tajahuerce, and Z. Jaroszewicz, “Equivalent retarder-rotator approach to on-state twisted nematic liquid crystal displays,” J. Appl. Phys. 99(11), 113101 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

C. Chou, Y. C. Huang, and M. Chang, “Polarized common path optical heterodyne interferometer for measuring the elliptical birefringence of a quartz wave plate,” Jpn. J. Appl. Phys. 35(Part 1, No. 10), 5526–5529 (1996).
[CrossRef]

Opt. Commun. (3)

H. F. Chang, C. Chou, H. K. Teng, H. T. Wu, and H. F. Yau, “The use of polarization and amplitude-sensitive optical heterodyne interferometry for linear birefringence parameters measurement,” Opt. Commun. 260(2), 420–426 (2006).
[CrossRef]

P. Kurzynowski, “Senarmont compensator for elliptically birefringent media,” Opt. Commun. 197(4-6), 235–238 (2001).
[CrossRef]

P. Kurzynowski, W. A. Woźniak, and S. Drobczyński, “A new phase difference compensation method for elliptically birefringent media,” Opt. Commun. 267(1), 44–49 (2006).
[CrossRef]

Opt. Express (1)

Optik (Stuttg.) (1)

F. Ratajczyk and P. Kurzynowski, “Phase difference superposition law for elliptically birefringent media,” Optik (Stuttg.) 99, 92–94 (1995).

Precis. Eng. (1)

D. C. Su, M. H. Chiu, and C. D. Chen, “Simple two-frequency laser,” Precis. Eng. 18(2-3), 161–163 (1996).
[CrossRef]

Proc. SPIE (1)

S. Berezhna, I. Berezhnyy, and M. Takashi, “Determination of the normalized Jones matrix of elliptical retarder,” Proc. SPIE 4317, 129–134 (2001).
[CrossRef]

Other (3)

R. M. A. Azzam, and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), pp. 98–99.

P. Yeh, and C. Gu, Optics of liquid crystal displays (Wiley, New York, 1999), pp. 120–122.

J. F. Nye, Physical Properties of Crystals (Oxford University Press, Oxford, 1957), pp. 261–268.

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Figures (2)

Fig. 1
Fig. 1

The elliptically eigen-polarizations of (a) the fast light wave and (b) the slow light wave of an elliptically birefringent medium.

Fig. 2
Fig. 2

Optical setup of DHE. H: half wave plate, PL and P: polarizers, S: specimen, A: analyzer, D: photo detector, DVM: digital voltmeter.

Tables (4)

Tables Icon

Table 1 Jones matrix elements of circularly, linearly and elliptically birefringent materials.

Tables Icon

Table 2 Measured intensities at different combinations of the azimuth angles of polarizer and analyzer.

Tables Icon

Table 3 Linear birefringence and circular birefringence measurement of an unitary optical system composed by a quarter wave plate combining with a Faraday rotator.

Tables Icon

Table 4 Measured and predicted results of the equivalently linear birefringence, circular birefringence, and elliptical birefringence of a TNLCD.

Equations (45)

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JU=(a+ibc+idc+idaib),
Jeq=JCB(Φeq)JLB(Γeq,ψeq)=(aeq+ibeqceq+ideqceq+ideqaeqibeq),
aeq=cosΓeq2cosΦeq,
beq=sinΓeq2cos(2ψeqΦeq),
ceq=cosΓeq2sinΦeq,
deq=sinΓeq2sin(2ψeqΦeq).
I˜3
JCB(Φeq)=(cosΦeqsinΦeqsinΦeqsinΦeq).
JEB=(cosγ2+isinγ2cos2εcos2θsinγ2sin2ε+isinγ2cos2εsin2θsinγ2sin2ε+isinγ2cos2εsin2θcosγ2isinγ2cos2εcos2θ).
JEB(γ,θ,ε)=Jeq(Γeq,ψeq,Φeq).
cosγ2=cosΓeq2cosΦeq,
sinγ2cos2εcos2θ=sinΓeq2cos(2ψeqΦeq),
sinγ2sin2ε=cosΓeq2sinΦeq,
sinγ2cos2εsin2θ=sinΓeq2sin(2ψeqΦeq).
γ=2cos1(cosΓeq2cosΦeq),
θ=ψeq(Φeq/2),
sinγ2cos2ε=sinΓeq2,
ε=12tan1(cotΓeq2sinΦeq).
JLC0=m=1NR(mρ)JLB0(iβ/2N)R(mρ),
R(mρ)=(cosmρsinmρsinmρcosmρ),
JLB0(iβ/2N)=(exp(iβ/2N)00exp(iβ/2N)),
JLC0=(cosΩsinΩsinΩcosΩ)(cosχiβ2χsinχΩχsinχΩχsinχcosχ+iβ2χsinχ).
JLC=R(D)JLC0R(D)=(aLC+ibLCcLC+idLCcLC+idLCaLCibLC),
aLC=cosχcosΩ+(Ω/χ)sinχsinΩ,
bLC=(β/2χ)sinχcos(2D+Ω),
cLC=cosχsinΩ+(Ω/χ)sinχcosΩ,
dLC=(β/2χ)sinχsin(2D+Ω),
Γeq=cos1[2(aLC2+cLC2)1],
ψeq=14tan1{2[aLCcLC(bLC2dLC2)+bLCdLC(aLC2cLC2)](aLC2cLC2)(bLC2dLC2)4aLCbLCcLCdLC},
Φeq=12tan1(2aLCcLCaLC2cLC2).
EZL=(Epexp[i(ωpt+φp)]Esexp[i(ωst+φs)]),
EZL=(Epexp[i(ωt+Δφ)/2]Esexp[i(ωt+Δφ)/2])exp{i[ω0t+(φ0/2)]}.
EDCLL=JPL(α)JH(Θ)EZL=(cos2αsinαcosαsinαcosαsin2α)(cos2Θsin2Θsin2Θcos2Θ)×(Epexp[i(ωt+Δφ)/2]Esexp[i(ωt+Δφ)/2])exp{i[ω0t+(φ0/2)]}=(EpEs)exp{i[ω0t+(φ0/2)]},
Ep=Epcos(2Θα)cosαexp[i(ωt+Δφ)/2]+Essin(2Θα)cosαexp[i(ωt+Δφ)/2],
Es=Epcos(2Θα)sinαexp[i(ωt+Δφ)/2]+Essin(2Θα)sinαexp[i(ωt+Δφ)/2],
Epcos(2Θα)=Essin(2Θα)=E0,
EDCLL(α)=(cosαsinα)(2E0)cos(ωt+Δφ2)exp{i[ω0t+(φ02)]},
EDCLL(α)=(cosαsinα)E0{exp[i(ω0+ω2)t+i(φ0+Δφ2)]+exp[i(ω0ω2)t+i(φ0Δφ2)]}=(cosαsinα)E0exp[i(ωpt+φp)]+(cosαsinα)E0exp[i(ωst+φs)].
EDCLL(45°)=(11)2E0cos(ωt+Δφ2)exp{i[ω0t+(φ02)]}.
En=JA(A)JeqJP(P)EDCLL(45°),
In=En*.En=FnI0[1+cos(ωt+Δφ)].
A=I˜1I˜3I˜1+I˜3,B=I˜6I˜8I˜6+I˜8,C=I˜5I˜7I˜5+I˜7,  and  D=I˜2I˜4I˜2+I˜4.
Γeq=cos1{[(A+B)2+(CD)2]1/21},
ψeq=14tan1[2(AC+BD)(A2C2)(B2D2)],
Φeq=12tan1(CDA+B),

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