Abstract

Cholesteric liquid crystals, consisting of chiral molecules, form self-assembled periodic structures exhibiting a photonic bandgap. Their selective reflectivity makes them well suited for a variety of applications; their optical response is therefore of considerable interest. The reflectance and transmittance of finite cholesteric cells is usually calculated numerically. Evanescent modes in the bandgap make the calculations challenging; existing matrix propagation methods cannot describe the reflection and transmission coefficients of thick cholesteric cells accurately. Here we present analytic solutions for the electromagnetic fields in cholesteric cells of finite thickness, and use them to calculate the transmission and reflection spectra. The use of analytic solutions allows for the accurate description of arbitrarily thick cholesteric cells, which would not be possible with only direct numerical methods.

© 2013 Chinese Laser Press

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References

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  1. P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).
  2. V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).
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    [CrossRef]
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    [CrossRef]
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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

2006

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[CrossRef]

2000

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

1998

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[CrossRef]

1989

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef]

1988

H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
[CrossRef]

H. Wöhler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4x4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5, 1554–1557 (1988).
[CrossRef]

1983

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

1973

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[CrossRef]

1972

1951

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[CrossRef]

1933

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

1911

C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

Beliakov, V.

V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).

Berreman, D. W.

Cao, W.

W. Cao, “Fluorescence and lasing in liquid crystalline photonic bandgap materials,” Ph.D. thesis (Kent State University, 2005).

Collings, P.

P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

de Vries, H.

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[CrossRef]

Dessaud, N.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[CrossRef]

Dreher, R.

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[CrossRef]

Fritsch, M.

Glebov, L. B.

Haas, G.

Hird, M.

P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

Lu, Z.

Lumeau, J.

Mauguin, C.

C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

Meier, G.

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[CrossRef]

Mi, X.-D.

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

Miraldi, E.

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Mitov, M.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[CrossRef]

Mlynski, D. A.

Mokhov, S.

Oldano, C.

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef]

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Ong, H. L.

H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
[CrossRef]

Oseen, C. W.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

Smirnov, V.

Valabrega, P. T.

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Wöhler, H.

Xu, F.

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[CrossRef]

Xu, M.

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[CrossRef]

Yang, D.-K.

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[CrossRef]

Zeldovich, B. Y.

Acta Crystallogr.

H. de Vries, “Rotatory power and other optical properties of certain liquid crystals,” Acta Crystallogr. 4, 219–226 (1951).
[CrossRef]

Bull. Soc. Fr. Mineral. Cristallogr.

C. Mauguin, “Sur la représentation géométrique de Poincaré relative aux propriétés optiques des piles de lames,” Bull. Soc. Fr. Mineral. Cristallogr. 34, 6–15 (1911).

J. Appl. Phys.

M. Xu, F. Xu, and D.-K. Yang, “Effects of cell structure on the reflection of cholesteric liquid crystal displays,” J. Appl. Phys. 83, 1938–1944 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

D.-K. Yang and X.-D. Mi, “Modelling of the reflection of cholesteric liquid crystals using the Jones matrix,” J. Phys. D 33, 672–676 (2000).
[CrossRef]

Nat. Mater.

M. Mitov and N. Dessaud, “Going beyond the reflectance limit of cholesteric liquid crystals,” Nat. Mater. 5, 361–364 (2006).
[CrossRef]

Opt. Lett.

Phys. Rev. A

R. Dreher and G. Meier, “Optical properties of cholesteric liquid crystals,” Phys. Rev. A 8, 1616–1623 (1973).
[CrossRef]

H. L. Ong, “Wave propagation in cholesteric and chiral smectic-C liquid crystals: exact and generalized geometrical-optics-approximation solutions,” Phys. Rev. A 37, 3520–3529 (1988).
[CrossRef]

C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef]

C. Oldano, E. Miraldi, and P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

Trans. Faraday Soc.

C. W. Oseen, “The theory of liquid crystals,” Trans. Faraday Soc. 29, 883–899 (1933).
[CrossRef]

Other

P. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, the Liquid Crystals Book Series (Taylor & Francis, 1997).

V. Beliakov, Diffraction Optics of Complex-Structured Periodic Media, Partially Ordered Systems (Springer-Verlag, 1992).

W. Cao, “Fluorescence and lasing in liquid crystalline photonic bandgap materials,” Ph.D. thesis (Kent State University, 2005).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic representation of the helical structure of the CLC. The orientation of the ellipsoids represents the average orientation of the molecules. The coordinate axes of the lab frame and the rotating frame are shown.

Fig. 2.
Fig. 2.

Schematic representation of a CLC film sandwiched between two substrates, the helical configuration in the lab coordinates systems and the electric fields inside and outside the CLC.

Fig. 3.
Fig. 3.

Reflection and transmission spectra of a CLC slab: (a) numerical inversion of matrix P¯¯ used and (b) P¯¯1 and Q¯¯P¯¯1 evaluated from analytic expressions.

Fig. 4.
Fig. 4.

Reflection and transmission spectra of the CLC slab. Cell thickness to the helical pitch ratio is 104.

Fig. 5.
Fig. 5.

Reflection spectrum of the CLC slab near the edge of the reflection band. Cell thickness to the helical pitch ratio is 104.

Fig. 6.
Fig. 6.

Reflection spectrum of the CLC slab at the center of the reflection band. Cell thickness to the helical pitch ratio is 104.

Equations (68)

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η^=cos(qz)x^+sin(qz)y^ξ^=sin(qz)x^+cos(qz)y^.
E(z)+2ikEk2E(z)=ω2c2ε¯¯rE(z),
(α2ε+n2)E=2inαE(α2ε+n2)E=2inαE,
n2=ε+ε2+α2±[(εε)24+2(ε+ε)α2]12.
n1±=±[ε¯+α2+(δ2+4ε¯α2)12]12,
n2±=±[ε¯+α2(δ2+4ε¯α2)12]12,
E1±=E±[1±iA]ei(±k1zωt),
E2±=E±[iB1]ei(±k2zωt).
A=2α2εε2+[(εε)24+2(ε+ε)α2]122α{ε+ε2+α2+[(εε)24+2(ε+ε)α2]12}12,
B=2α2+εε2[(εε)24+2(ε+ε)α2]122α{ε+ε2+α2[(εε)24+2(ε+ε)α2]12}12,
A=n12+α2ε2n1α,B=n22+α2ε2n2α.
E1+=E1+eiωt((1+A)[1i]ei(k1z+qz)+(1A)[1i]ei(k1zqz)),
E1=E1eiωt((1+A)[1i]ei(k1z+qz)+(1A)[1i]ei(k1zqz)),
E2+=E2+eiωt((1+B)[i1]ei(k2z+qz)+(1B)[i1]ei(k2zqz)),
E2=E2eiωt((1+B)[i1]ei(k2z+qz)+(1B)[i1]ei(k2zqz)),
Ei=[EixEiy]ei(kszωt)Er=[ErxEry]ei(kszωt)Et=[EtxEty]ei(ks(zdp)ωt).
[EixEiyErxEry]=[P¯¯0¯¯Q¯¯0¯¯][EtxEty00].
[EtxEty]=P¯¯1[EixEiy],
[ErxEry]=Q¯¯P¯¯1[EixEiy].
[ExiEyiExrEyr]=[P¯¯0¯¯Q¯¯0¯¯][ExtEyt00],
P¯¯=[eiδ1p11(1)+eiδ1p11(2)+eiδ2p11(3)+eiδ2p11(4)i(eiδ1p12(1)+eiδ1p12(2)+eiδ2p12(3)+eiδ2p12(4))i(eiδ1p21(1)+eiδ1p21(2)+eiδ2p21(3)+eiδ2p21(4))eiδ1p22(1)+eiδ1p22(2)+eiδ2p22(3)+eiδ2p22(4)],
Q¯¯=[eiδ1q11(1)+eiδ1q11(2)+eiδ2q11(3)+eiδ2q11(4)i(eiδ1q12(1)+eiδ1q12(2)+eiδ2q12(3)+eiδ2q12(4))i(eiδ1q21(1)+eiδ1q21(2)+eiδ2q21(3)+eiδ2q21(4))eiδ1q22(1)+eiδ1q22(2)+eiδ2q22(3)+eiδ2q22(4)],
p11(1)=18(1γiA+N1i)[(eiqdeiqd)(BN2tXγtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p11(2)=18(1+γiAN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p11(3)=18(Bγi+BN2i)[(eiqdeiqd)(N1tX+γtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p11(4)=18(B+γiBN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p12(1)=18(1γiA+N1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
p12(2)=18(1+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tXγtXBY)],
p12(3)=18(Bγi+BN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
p12(4)=18(B+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tX+γtAX+Y)],
p21(1)=18(Aγi+AN1i)[(eiqdeiqd)(BN2tX+γtXBY)+(eiqd+eiqd)(X+γtBYN2tY)],
p21(2)=18(A+γiAN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
p21(3)=18(1γiB+N2i)[(eiqdeiqd)(N1tXγtAX+Y)+(eiqd+eiqd)(AX+AN1tYγtY)],
p21(4)=18(1+γiBN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
p22(1)=18(Aγi+AN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
p22(2)=18(A+γiAN1i)[(eiqdeiqd)(XγtBY+N2tY)+(eiqd+eiqd)(BN2tX+γtX+BY)],
p22(3)=18(1γiB+N2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
p22(4)=18(1+γiBN2i)[(eiqdeiqd)(AXAN1tY+γtY)+(eiqd+eiqd)(N1tXγtAXY)],
q11(1)=18(1+γiAN1i)[(eiqdeiqd)(BN2tXγtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q11(2)=18(1γiA+N1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q11(3)=18(B+γiBN2i)[(eiqdeiqd)(N1tX+γtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q11(4)=18(Bγi+BN2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q12(1)=i8(1+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
q12(2)=i8(1γiA+N1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tXγtXBY)],
q12(3)=i8(B+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
q12(4)=i8(Bγi+BN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tX+γtAX+Y)],
q21(1)=i8(A+γiAN1i)[(eiqdeiqd)(BN2tX+γtXBY)+(eiqd+eiqd)(X+γtBYN2tY)],
q21(2)=i8(Aγi+AN1i)[(eiqdeiqd)(BN2tX+γtX+BY)+(eiqd+eiqd)(XγtBY+N2tY)],
q21(3)=i8(1+γiBN2i)[(eiqdeiqd)(N1tXγtAX+Y)+(eiqd+eiqd)(AX+AN1tYγtY)],
q21(4)=i8(1γiB+N2i)[(eiqdeiqd)(N1tXγtAXY)+(eiqd+eiqd)(AXAN1tY+γtY)],
q22(1)=18(A+γiAN1i)[(eiqdeiqd)(X+γtBYN2tY)+(eiqd+eiqd)(BN2tX+γtXBY)],
q22(2)=18(Aγi+AN1i)[(eiqdeiqd)(XγtBY+N2tY)+(eiqd+eiqd)(BN2tX+γtX+BY)],
q22(3)=18(1+γiBN2i)[(eiqdeiqd)(AX+AN1tYγtY)+(eiqd+eiqd)(N1tXγtAX+Y)],
q22(4)=18(1γiB+N2i)[(eiqdeiqd)(AXAN1tY+γtY)+(eiqd+eiqd)(N1tXγtAXY)].
detP¯¯=P11P22P12P21=(p11(1)p22(2)+p11(2)p22(1)+p11(3)p22(4)+p11(4)p22(3)+p12(1)p21(2)+p12(2)p21(1)+p12(3)p21(4)+p12(4)p21(3))+ei(δ1+δ2)(p11(1)p22(3)+p11(3)p22(1)+p12(1)p21(3)+p12(3)p21(1))+ei(δ1δ2)(p11(1)p22(4)+p11(4)p22(1)+p12(1)p21(4)+p12(4)p21(1))+ei(δ1δ2)(p11(2)p22(3)+p11(3)p22(2)+p12(2)p21(3)+p12(3)p21(2))+ei(δ1+δ2)(p11(2)p22(4)+p11(4)p22(2)+p12(2)p21(4)+p12(4)p21(2))+ei2δ1(p11(1)p22(1)+p12(1)p21(1))+ei2δ1(p11(2)p22(2)+p12(2)p21(2))+ei2δ2(p11(3)p22(3)+p12(3)p21(3))+ei2δ2(p11(4)p22(4)+p12(4)p21(4)).
p11(1)p22(1)+p12(1)p21(1)=0,
p11(2)p22(2)+p12(2)p21(2)=0,
p11(3)p22(3)+p12(3)p21(3)=0,
p11(4)p22(4)+p12(4)p21(4)=0.
detP¯¯=P11P22P12P21=(p11(1)p22(2)+p11(2)p22(1)+p11(3)p22(4)+p11(4)p22(3)+p12(1)p21(2)+p12(2)p21(1)+p12(3)p21(4)+p12(4)p21(3))+ei(δ1+δ2)(p11(1)p22(3)+p11(3)p22(1)+p12(1)p21(3)+p12(3)p21(1))+ei(δ1δ2)(p11(1)p22(4)+p11(4)p22(1)+p12(1)p21(4)+p12(4)p21(1))+ei(δ1δ2)d(p11(2)p22(3)+p11(3)p22(2)+p12(2)p21(3)+p12(3)p21(2))+ei(δ1+δ2)(p11(2)p22(4)+p11(4)p22(2)+p12(2)p21(4)+p12(4)p21(2)).
P¯¯1=P˜detP¯¯,
P˜=[P22P12P21P11].
detP¯¯=116(ABN1N2)(ABN2N1)(8[A2γN1A(γ2+N121)+γN1][B2γN2B(γ2+N221)+γN2]+ei(δ1+δ2){(AB1)[γ2(N1+1)(N2+1)]+γ(AB)(N1N2)}2ei(δ1δ2){(AB+1)[γ2+(N1+1)(N21)]γ(A+B)(N1+N2)}2ei(δ1δ2){(AB+1)[γ2+(N11)(N2+1)]γ(A+B)(N1+N2)}2+ei(δ1+δ2){(AB1)[γ2+(N11)(N21)]γ(AB)(N1N2)}2),
Q¯¯P¯¯1=Q¯¯P˜detP¯¯.
[Q¯¯P˜]11=(q11(1)p22(2)+q11(2)p22(1)+q11(3)p22(4)+q11(4)p22(3)+q12(1)p21(2)+q12(2)p21(1)+q12(3)p21(4)+q12(4)p21(3))+ei(δ1+δ2)(q11(2)p22(4)+q11(4)p22(2)+q12(2)p21(4)+q12(4)p21(2))+ei(δ1+δ2)(q11(1)p22(3)+q11(3)p22(1)+q12(1)p21(3)+q12(3)p21(1))+ei(δ1δ2)(q11(2)p22(3)+q11(3)p22(2)+q12(2)p21(3)+q12(3)p21(2))+ei(δ1δ2)(q11(1)p22(4)+q11(4)p22(1)+q12(1)p21(4)+q12(4)p21(1)),
[Q¯¯P˜]12=i[(q11(1)p12(2)q11(2)p12(1)q11(3)p12(4)q11(4)p12(3)+q12(1)p11(2)+q12(2)p11(1)+q12(3)p11(4)+q12(4)p11(3))+ei(δ1+δ2)(q11(2)p12(4)q11(4)p12(2)+q12(2)p11(4)+q12(4)p11(2))+ei(δ1+δ2)(q11(1)p12(3)q11(3)p12(1)+q12(1)p11(3)+q12(3)p11(1))+ei(δ1δ2)(q11(2)p12(3)q11(3)p12(2)+q12(2)p11(3)+q12(3)p11(2))+ei(δ1δ2)(q11(1)p12(4)q11(4)p12(1)+q12(1)p11(4)+q12(4)p11(1))],
[Q¯¯P˜]21=i[(q21(1)p22(2)+q21(2)p22(1)+q21(3)p22(4)+q21(4)p22(3)q22(1)p21(2)q22(2)p21(1)q22(3)p21(4)q22(4)p21(3))+ei(δ1+δ2)(q21(2)p22(4)+q21(4)p22(2)q22(2)p21(4)q22(4)p21(2))+ei(δ1+δ2)(q21(1)p22(3)+q21(3)p22(1)q22(1)p21(3)q22(3)p21(1))+ei(δ1δ2)(q21(2)p22(3)+q21(3)p22(2)q22(2)p21(3)q22(3)p21(2))+ei(δ1δ2)(q21(1)p22(4)+q21(4)p22(1)q22(1)p21(4)q22(4)p21(1))],
[Q¯¯P˜]22=(q21(1)p12(2)+q21(2)p12(1)+q21(3)p12(4)+q21(4)p12(3)+q22(1)p11(2)+q22(2)p11(1)+q22(3)p11(4)+q22(4)p11(3))+ei(δ1+δ2)(q21(2)p12(4)+q21(4)p12(2)+q22(2)p11(4)+q22(4)p11(2))+ei(δ1+δ2)(q21(1)p12(3)+q21(3)p12(1)+q22(1)p11(3)+q22(3)p11(1))+ei(δ1δ2)(q21(2)p12(3)+q21(3)p12(2)+q22(2)p11(3)+q22(3)p11(2))+ei(δ1δ2)(q21(1)p12(4)+q21(4)p12(1)+q22(1)p11(4)+q22(4)p11(1)).

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