Abstract

For the measurement of Mueller matrix in an optical system with birefringence and small polarization-dependent loss or gain (PDL/G), we theoretically derive the statistical relationship between the Mueller matrix measurement error and three input states of polarization (SOP). Based on this theoretical relation and simulation results, it can be concluded that the three input SOPs, that are coplanar with an angle of 120° between any two of them in Stokes space, can be considered as a substitute for the best input SOPs which can statistically lead to the minimum measurement error. This conclusion is valid when the PDL/G of the optical system under test is less than 0.35dB.

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008).
[CrossRef]

2007 (2)

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

Y. Takakura and J. E. Ahmad, “Noise distribution of Mueller matrices retrieved with active rotating polarimeters,” Appl. Opt. 46(30), 7354–7364 (2007).
[CrossRef] [PubMed]

2006 (1)

2004 (1)

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

1998 (1)

R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998).
[CrossRef]

1997 (1)

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

1996 (1)

1995 (2)

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995).
[CrossRef]

1992 (4)

D. B. Chenault, R. A. Chipman, K. M. Johnson, and D. Doroski, “Infrared linear diattenuation and birefringence spectra of ferroelectric liquid crystals,” Opt. Lett. 17(6), 447–449 (1992).
[CrossRef] [PubMed]

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. 4(5), 451–454 (1992).
[CrossRef]

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4(9), 1066–1069 (1992).
[CrossRef]

D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31(31), 6676–6683 (1992).
[CrossRef] [PubMed]

1990 (1)

1981 (1)

R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981).
[CrossRef]

1979 (1)

1963 (1)

1947 (1)

Ahmad, J. E.

Allen, S. J.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Ambirajan, A.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995).
[CrossRef]

Barakat, R.

R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981).
[CrossRef]

R. Barakat, “Theory of the coherency matrix for light of arbitrary spectral bandwidth,” J. Opt. Soc. Am. 53(3), 317–323 (1963).
[CrossRef]

Birge, R. R.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Chenault, D. B.

Chipman, R. A.

Craig, R. M.

R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998).
[CrossRef]

Dong, H.

H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).

Doroski, D.

Galan, J.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Gilbert, S. L.

R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998).
[CrossRef]

Gisin, N.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

Goldstein, D. H.

Gong, Y. D.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).

Hale, P. D.

R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998).
[CrossRef]

Heffner, B. L.

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. 4(5), 451–454 (1992).
[CrossRef]

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4(9), 1066–1069 (1992).
[CrossRef]

Hong, M. H.

H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).

Howell, B. J.

Huttner, B.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

Johnson, K. M.

Jones, R. C.

Look, D. C.

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995).
[CrossRef]

Lu, S.-Y.

Nee, S.-M. F.

Ning, G. X.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

Plaxco, K.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Ramin, G.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Savvidis, P.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Scopatz, A.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Shum, P.

H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

Takakura, Y.

Tyo, J. S.

Wu, C. Q.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

Xu, J.

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

Yan, M.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

Zhou, J. Q.

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain,” Opt. Express 14(12), 5067–5072 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-12-5067 .
[CrossRef] [PubMed]

Appl. Opt. (5)

IEEE J. Lightwave Technol. (1)

R. M. Craig, S. L. Gilbert, and P. D. Hale, “High-resolution, nonmechanical approach to polarization-dependent transmission measurements,” IEEE J. Lightwave Technol. 16(7), 1285–1294 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

B. L. Heffner, “Deterministic, analytically complete measurement of polarization-dependent transmission though optical devices,” IEEE Photon. Technol. Lett. 4(5), 451–454 (1992).
[CrossRef]

B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4(9), 1066–1069 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (4)

H. Dong, Y. D. Gong, and M. H. Hong, “Polarization state and Mueller matrix measurements in terahertz time domain spectroscopy,” Opt. Commun. (Accepted).

R. Barakat, “Bilinear constraints between elements of the 4x4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38(3), 159–161 (1981).
[CrossRef]

H. Dong, P. Shum, M. Yan, J. Q. Zhou, G. X. Ning, Y. D. Gong, and C. Q. Wu, “Measurement of Mueller matrix for an optical fiber system with birefringence and polarization-dependent loss or gain,” Opt. Commun. 274(1), 116–123 (2007).
[CrossRef]

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142(1-3), 119–125 (1997).
[CrossRef]

Opt. Eng. (3)

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part I,” Opt. Eng. 34(6), 1651–1655 (1995).
[CrossRef]

A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter: part II,” Opt. Eng. 34(6), 1656–1658 (1995).
[CrossRef]

H. Dong and P. Shum, “Effect of input polarization states on the error of polarization measurement,” Opt. Eng. 47(6), 065007 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

J. Xu, J. Galan, G. Ramin, P. Savvidis, A. Scopatz, R. R. Birge, S. J. Allen, and K. Plaxco, “Terahertz circular dichroism spectroscopy of biomolecules,” Proc. SPIE 5268, 19–26 (2004).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(α,β,γ) can only take values on the surface and the inner space of the red tetrahedron.

Fig. 2
Fig. 2

Curved surfaces formed by (α,β,γ) with the same values of KΔ|M| .

Fig. 3
Fig. 3

KΔ|M| of Eq. (17) as a function of angles between three inputs (a) three angles are identical and (b) three inputs are coplanar and two of them are identical. The insets show the “zoom in” views of the same data.

Fig. 4
Fig. 4

Curved surfaces formed by (α,β,γ) with the same values of KΔM .

Fig. 5
Fig. 5

KΔM of Eq. (29) as a function of angles between three inputs (a) three angles are identical and (b) three inputs are coplanar and two of them are identical. The insets show the “zoom in” views of the same data.

Fig. 6
Fig. 6

Simulation results of ΔM˜ as a function of angles between three inputs corresponding to different values of D (a) three angles are identical and (b) three inputs are coplanar and two of them are identical.

Fig. 7
Fig. 7

Simulation results of ΔM˜ as a function of angles between three inputs corresponding to different values of D (a) three angles are identical and (b) three inputs are coplanar and two of them are identical.

Equations (32)

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

M˜=(m11im12im13im14im21m22m23m24im31m32m33m34im41m42m43m44)
M˜TM˜=|M˜|I
M˜=Tu(10T0mR)(     1iDTiDTmD)
mR=cosϕI+(1cosϕ)rrsinϕr×
mD=1D2I+(11D2)DD
|M˜|=Tu4(1D2)2and  M˜=2Tu
sout0=Tusin0Ds
cosαout=11D2DsDt(1cosαin)
|M˜|=SoutTout+SoutUout+ToutUoutSinTin+SinUin+TinUin
FoutM˜=Fin
|M˜|+Δ|M˜|=[(Sout+ΔSout)(Tout+ΔTout)+(Sout+ΔSout)(Uout+ΔUout)+(Tout+ΔTout)(Uout+ΔUout)]SinTin+SinUin+TinUin
Δ|M˜|=(Tout+Uout)ΔSout+(Sout+Uout)ΔTout+(Sout+Tout)ΔUoutSinTin+SinUin+TinUin
(Fout+ΔFout)(M˜+ΔM˜)=Fin
ΔM˜=Fout1ΔFoutM˜=M˜Fin1ΔFoutM˜
{αβγα+β          when  αβ  and  α+β180βαγα+β          when  α<β  and  α+β180αβγ360αβ  when  αβ  and  α+β>180βαγ360αβ  when  α<β  and  α+β>180
Var(Δ|M˜|)=(Δ|M˜|)2=2Tu2σ2[(Ds+Dt)2+(Ds+Du)2+(Dt+Du)2(3cosαcosβcosγ)](3cosαcosβcosγ)2
Var(Δ|M˜|)D<<12Tu2σ2KΔ|M|
ΔM˜Tu2Fin1ΔFout
ΔFout=1|M˜|(ΔFout1ΔFout2)
ΔM˜Fin1ΔFout1Fin1ΔFout2=Tr(ΔFout1HFΔFout1+ΔFout2HFΔFout2ΔFout1HFΔFout2ΔFout2HFΔFout1)
ΔM˜ΔM˜2=Tr(ΔFout1HFΔFout1)+Tr(ΔFout2HFΔFout2)Tr(ΔFout1HFΔFout2)Tr(ΔFout2HFΔFout1)
Tr(ΔFout1HFΔFout1)=2σ2{2j=13fjj+f44[4(3cosαcosβcosγ)(1cosα)2(1cosβ)2(1cosγ)2]}
Tr(ΔFout2HFΔFout2)=Var(Δ|M˜|)D<<1|M˜|j=14k=14fjkΩjk
Tr(ΔFout1HFΔFout2)=Tr(ΔFout2HFΔFout1)=σ23cosαcosβcosγj=13k=13fjkΨjk
{f11={[4(1+cosγ)2](Bin12+Bin22)8(1cosγ)Bin12}/(Bin12Bin22)2f22={[4(1+cosβ)2](Bin12+Bin22)8(1cosβ)Bin12}/(Bin12Bin22)2f33={[4(1+cosα)2](Bin12+Bin22)8(1cosα)Bin12}/(Bin12Bin22)2f44=2[4+2(1+cosα)(1+cosβ)(1+cosγ)(1+cosα)2(1+cosβ)2(1+cosγ)2]/(Bin12Bin22)2f12=f21={[(1+cosβ)(1+cosγ)2(1+cosα)](Bin12+Bin22)+4[(1+cosα)(1+cosβ)(1+cosγ)+2]}/(Bin12Bin22)2f13=f31={[(1+cosα)(1+cosγ)2(1+cosβ)](Bin12+Bin22)+4[(1+cosβ)(1+cosα)(1+cosγ)+2]}/(Bin12Bin22)2f23=f32={[(1+cosα)(1+cosβ)2(1+cosγ)](Bin12+Bin22)+4[(1+cosγ)(1+cosα)(1+cosβ)+2]}/(Bin12Bin22)2f14=f41=2iBin1[2(1+cosα)+2(1+cosβ)+(1+cosγ)2(1+cosα)(1+cosγ)(1+cosβ)(1+cosγ)4]/(Bin12Bin22)2f24=f42=2iBin1[2(1+cosα)+2(1+cosγ)+(1+cosβ)2(1+cosα)(1+cosβ)(1+cosβ)(1+cosγ)4]/(Bin12Bin22)2f34=f43=2iBin1[2(1+cosβ)+2(1+cosγ)+(1+cosα)2(1+cosα)(1+cosβ)(1+cosα)(1+cosγ)4]/(Bin12Bin22)2
{Bin1=|sin(tin×uin)|=1cos2αcos2βcos2γ+2cosαcosβcosγBin2=|(tinsin)×(uinsin)|=4(1cosβ)(1cosγ)(1+cosαcosβcosγ)2
{Ω11=Ω22=Ω33=2,Ω44=4(Bin12+Bin22)Ω12=Ω21=1+cosα,Ω13=Ω31=1+cosβ,Ω23=Ω32=1+cosγΩ14=Ω24=Ω34=Ω41=Ω42=Ω43=4iBin1
{Ψ11=2cosαcosβ,Ψ22=2cosαcosγ,Ψ33=2cosβcosγΨ12=Ψ13=1cosγ,Ψ21=Ψ23=1cosβ,Ψ31=Ψ32=1cosα
ΔM˜ΔM˜2=KΔMσ
KΔM=2{2j=13fjj+f44[4(3cosαcosβcosγ)(1cosα)2(1cosβ)2(1cosγ)2]+KΔ|M|j=14k=14fjkΩjkj=13k=13fjkΨjk3cosαcosβcosγ}12
Uin=(i,cosα,cosαcos2αsinα,sin4α(cosαcos2α)2sinα)T
Uin=(i,cosα,sinα,0)T

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