Abstract

We present an eight-dimensional matrix formulation of vector four-wave mixing. We study the case of a nonoptically active photorefractive for the transmission grating. The coupled equations are represented in a group formalism and irreducible forms are found. We show that when cross-polarization coupling is neglected, the system of equations is governed by two coupled SU(2) groups. We develop a theory that allows for pump depletion and electric-field-induced bulk birefringence. Analytical solutions are found when cubic 4¯3m materials are considered in two standard experimental geometries, i.e., electric fields perpendicular to the (110) crystal faces and electric fields perpendicular to the (001) crystal faces. Results are presented that show the effects of linear and circular polarized beams, and the use of these materials for vector phase conjugation is discussed. Solution routes to other geometries and noncubic photorefractive materials are also presented.

© 1997 Optical Society of America

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References

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  1. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
    [CrossRef]
  2. D. A. Fish, A. K. Powell, and T. J. Hall, “A steady state solution to four-wave mixing in the transmission geometryutilising the SU(2) group symmetry,” Opt. Commun. 88, 281–290 (1992).
    [CrossRef]
  3. P. Stojkov and M. R. Belić, “Symmetries of photorefractive four-wave mixing,” Phys. Rev. A 45, 5061–5064 (1992).
    [CrossRef] [PubMed]
  4. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  5. M. Petrović and M. R. Belić, “Vectorial two-beam mixing in photorefractive crystals,” Opt. Commun. 109, 338–347 (1994).
    [CrossRef]
  6. P. Stojkov, D. Timotijević, and M. R. Belić, “Symmetries of two-wave mixing in photorefractive crystals,” Opt. Lett. 17, 1406–1408 (1992).
    [CrossRef] [PubMed]
  7. P. D. Foote, “Optically induced anisotropic light diffractionin photorefractive crystals,” Ph.D. dissertation (King's College, Universityof London, London, UK, 1987).
  8. C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
    [CrossRef]
  9. S. Trillo and S. Wabnitz, “Nonlinear phase distortion in phase conjugation by degenerate four-wavemixing in Kerr media,” J. Opt. Soc. Am. B 5, 195–201 (1988).
    [CrossRef]
  10. R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
    [CrossRef] [PubMed]
  11. M. Zgonik and P. Gunter, “Cascaded nonlinearities in optical four-wave mixing,” J. Opt. Soc. Am. B 13, 570–576 (1996).
    [CrossRef]
  12. S. W. James and R. W. Eason, “Extraordinary-polarized light does not always yield the highest selectivityin self-pumped BaTiO3,” Opt. Lett. 16, 633–635 (1991).
    [CrossRef] [PubMed]
  13. W. Wu, S. Campbell, S. Zhou, and P. Yeh, “Polarization-encoded optical logic operations in photorefractive media,” Opt. Lett. 18, 1742–1744 (1993).
    [CrossRef] [PubMed]
  14. D. A. Fish and A. K. Powell, “Irreducible subgroups of SU(2, 2) in four-wave mixing,” J. Opt. Soc. Am. B 14, 1–13 (1997).
    [CrossRef]
  15. T. J. Hall, A. K. Powell, and C. Stace, “Vector four-wave mixing in cubic, optically active photorefractivemedia,” Opt. Commun. 75, 159–164 (1990).
    [CrossRef]
  16. M. W. McCall, “A study of vector coupled wave theoryin photorefractive media,” Ph.D. dissertation (King's College, Universityof London, London, UK, 1987).
  17. G. R. Barrett, A. K. Powell, and T. J. Hall, “The SU(2) dynamics of DFWM,” Opt. Commun. 89, 477–483 (1992).
    [CrossRef]
  18. M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependentsaturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
    [CrossRef] [PubMed]

1997 (1)

1996 (3)

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependentsaturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

M. Zgonik and P. Gunter, “Cascaded nonlinearities in optical four-wave mixing,” J. Opt. Soc. Am. B 13, 570–576 (1996).
[CrossRef]

1994 (1)

M. Petrović and M. R. Belić, “Vectorial two-beam mixing in photorefractive crystals,” Opt. Commun. 109, 338–347 (1994).
[CrossRef]

1993 (1)

1992 (4)

G. R. Barrett, A. K. Powell, and T. J. Hall, “The SU(2) dynamics of DFWM,” Opt. Commun. 89, 477–483 (1992).
[CrossRef]

P. Stojkov, D. Timotijević, and M. R. Belić, “Symmetries of two-wave mixing in photorefractive crystals,” Opt. Lett. 17, 1406–1408 (1992).
[CrossRef] [PubMed]

D. A. Fish, A. K. Powell, and T. J. Hall, “A steady state solution to four-wave mixing in the transmission geometryutilising the SU(2) group symmetry,” Opt. Commun. 88, 281–290 (1992).
[CrossRef]

P. Stojkov and M. R. Belić, “Symmetries of photorefractive four-wave mixing,” Phys. Rev. A 45, 5061–5064 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

T. J. Hall, A. K. Powell, and C. Stace, “Vector four-wave mixing in cubic, optically active photorefractivemedia,” Opt. Commun. 75, 159–164 (1990).
[CrossRef]

1989 (2)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

1988 (1)

1984 (1)

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Barrett, G. R.

G. R. Barrett, A. K. Powell, and T. J. Hall, “The SU(2) dynamics of DFWM,” Opt. Commun. 89, 477–483 (1992).
[CrossRef]

Belic, M. R.

M. Petrović and M. R. Belić, “Vectorial two-beam mixing in photorefractive crystals,” Opt. Commun. 109, 338–347 (1994).
[CrossRef]

P. Stojkov and M. R. Belić, “Symmetries of photorefractive four-wave mixing,” Phys. Rev. A 45, 5061–5064 (1992).
[CrossRef] [PubMed]

P. Stojkov, D. Timotijević, and M. R. Belić, “Symmetries of two-wave mixing in photorefractive crystals,” Opt. Lett. 17, 1406–1408 (1992).
[CrossRef] [PubMed]

Camacho-Lopez, S.

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependentsaturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Campbell, S.

Crofts, G. J.

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

Cronin-Golomb, M.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Damzen, M. J.

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependentsaturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Eason, R. W.

Fischer, B.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Fish, D. A.

D. A. Fish and A. K. Powell, “Irreducible subgroups of SU(2, 2) in four-wave mixing,” J. Opt. Soc. Am. B 14, 1–13 (1997).
[CrossRef]

D. A. Fish, A. K. Powell, and T. J. Hall, “A steady state solution to four-wave mixing in the transmission geometryutilising the SU(2) group symmetry,” Opt. Commun. 88, 281–290 (1992).
[CrossRef]

Green, R. P. M.

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependentsaturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Gunter, P.

Hall, T. J.

D. A. Fish, A. K. Powell, and T. J. Hall, “A steady state solution to four-wave mixing in the transmission geometryutilising the SU(2) group symmetry,” Opt. Commun. 88, 281–290 (1992).
[CrossRef]

G. R. Barrett, A. K. Powell, and T. J. Hall, “The SU(2) dynamics of DFWM,” Opt. Commun. 89, 477–483 (1992).
[CrossRef]

T. J. Hall, A. K. Powell, and C. Stace, “Vector four-wave mixing in cubic, optically active photorefractivemedia,” Opt. Commun. 75, 159–164 (1990).
[CrossRef]

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

James, S. W.

Kim, D. H.

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

Petrovic, M.

M. Petrović and M. R. Belić, “Vectorial two-beam mixing in photorefractive crystals,” Opt. Commun. 109, 338–347 (1994).
[CrossRef]

Powell, A. K.

D. A. Fish and A. K. Powell, “Irreducible subgroups of SU(2, 2) in four-wave mixing,” J. Opt. Soc. Am. B 14, 1–13 (1997).
[CrossRef]

G. R. Barrett, A. K. Powell, and T. J. Hall, “The SU(2) dynamics of DFWM,” Opt. Commun. 89, 477–483 (1992).
[CrossRef]

D. A. Fish, A. K. Powell, and T. J. Hall, “A steady state solution to four-wave mixing in the transmission geometryutilising the SU(2) group symmetry,” Opt. Commun. 88, 281–290 (1992).
[CrossRef]

T. J. Hall, A. K. Powell, and C. Stace, “Vector four-wave mixing in cubic, optically active photorefractivemedia,” Opt. Commun. 75, 159–164 (1990).
[CrossRef]

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

Stace, C.

T. J. Hall, A. K. Powell, and C. Stace, “Vector four-wave mixing in cubic, optically active photorefractivemedia,” Opt. Commun. 75, 159–164 (1990).
[CrossRef]

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

Stojkov, P.

Timotijevic, D.

Trillo, S.

Udaiyan, D.

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

Wabnitz, S.

Walsh, K.

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

White, J. O.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Wu, W.

Yariv, A.

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

Yeh, P.

Zgonik, M.

Zhou, S.

IEEE J. Quantum Electron. (2)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
[CrossRef]

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

Opt. Commun. (5)

T. J. Hall, A. K. Powell, and C. Stace, “Vector four-wave mixing in cubic, optically active photorefractivemedia,” Opt. Commun. 75, 159–164 (1990).
[CrossRef]

G. R. Barrett, A. K. Powell, and T. J. Hall, “The SU(2) dynamics of DFWM,” Opt. Commun. 89, 477–483 (1992).
[CrossRef]

C. Stace, A. K. Powell, K. Walsh, and T. J. Hall, “Coupling modulation in photorefractive materials by applying ac electricfields,” Opt. Commun. 70, 509–514 (1989).
[CrossRef]

D. A. Fish, A. K. Powell, and T. J. Hall, “A steady state solution to four-wave mixing in the transmission geometryutilising the SU(2) group symmetry,” Opt. Commun. 88, 281–290 (1992).
[CrossRef]

M. Petrović and M. R. Belić, “Vectorial two-beam mixing in photorefractive crystals,” Opt. Commun. 109, 338–347 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

P. Stojkov and M. R. Belić, “Symmetries of photorefractive four-wave mixing,” Phys. Rev. A 45, 5061–5064 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

R. P. M. Green, D. Udaiyan, G. J. Crofts, D. H. Kim, and M. J. Damzen, “Holographic laser oscillator which adaptively corrects for polarizationand phase distortions,” Phys. Rev. Lett. 77, 3533–3536 (1996).
[CrossRef] [PubMed]

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependentsaturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Other (2)

M. W. McCall, “A study of vector coupled wave theoryin photorefractive media,” Ph.D. dissertation (King's College, Universityof London, London, UK, 1987).

P. D. Foote, “Optically induced anisotropic light diffractionin photorefractive crystals,” Ph.D. dissertation (King's College, Universityof London, London, UK, 1987).

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Figures (7)

Fig. 1
Fig. 1

Definition of beams in vector four-wave mixing.

Fig. 2
Fig. 2

Ring phase-conjugate resonator.

Fig. 3
Fig. 3

Three-dimensional plot of the phase-conjugate reflectivity versus the complex coupling constant. The boundary conditions are I1=0.1, I2=1.0, and I3=1.0, with each beam having the following linear polarizations: θ1=0.0°, θ2=0.0°, and θ3=0.0°. Electric fields are perpendicular to the (110) crystal faces.

Fig. 4
Fig. 4

Phase-conjugate reflectivity versus the complex coupling constant. Electric fields are perpendicular to the (110) crystal faces. (a) Linear polarizations: θ1=0.0°, θ2=0.0°, and θ3 =50.0°. (b) Two-dimensional section of Fig. 4(a) showing reflectivity plotted against the intensity modulus of the coupling constant for a grating phase shift of -50°.

Fig. 5
Fig. 5

Variation of phase-conjugate reflectivity with linear polarization angle. Electric fields are perpendicular to the (110) crystal faces. (a) Grating phase shift of -90°, θ1=0.0, and θ2 =0.0. (b) Grating phase shift of +90°, θ1=0.0, and θ3 =0.0.

Fig. 6
Fig. 6

Phase-conjugate reflectivity versus the coupling constant. Electric fields are perpendicular to the (110) crystal faces, and the polarization is circular.

Fig. 7
Fig. 7

Phase-conjugate reflectivity versus the coupling constant. Electric fields are perpendicular to the (001) crystal faces. (a) Linear polarization with variation of θ3. θ1=0.0° and θ2=45.0°. (b) Linear polarization with variation of θ3. θ1=45.0° and θ2=45.0°. (c) Circular polarization.

Equations (154)

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ddzA1+iHA1=iκSA2,
ddzA2+iHA2=iκ*SA1,
-ddzA3+iHA3=iκSA4,
-ddzA4+iHA4=iκ*SA3,
κ=γI0[(A1, A2)+(A3, A4)]
exp(iLz)=J0+n=1 (iLz)nn!,
ddzA1=iκDA2
ddzA2=iκ*DA1,
-ddzA3=iκDA4,
-ddzA4=iκ*DA3,
κ=γI0[(A1, A2)+(A3, A4)]
D=exp(iβSAz)S exp(-iβSAz),
D=exp(-iβSAz)S exp(iβSAz).
A1A2A3A4
=exp(iβSz)OOOOexp(iβSz)OOOOexp(-iβSz)OOOOexp(-iβSz)×A1A2A3A4,
dUdz=i(J3KS)U
K=0κ*κ0,
J0=1001,J1=0110,J2=0i-i0,
J3=100-1.
dUdz=i[H(-H)]U,
U=X(J3J0)X(J3J0),
dXdz=iHX,
P1=1000,P2=0010,P3=0100,
P4=0001
e1=10,e2=01.
S=s1P1+s2P4
X=w1P1+w2P4,
dw1dz=is1Kw1,
dw2dz=is2Kw2,
κ=γI0{[(H11-J3H41J3)w1e2, w1e1]+[(H14-J3H44J3)w2e2, w2e1]},
dwdz=iKw.
dwdz=iKw.
A=B-1A=B-1UA(z0).
g=A1(0)A2(0)A3(L)A4(L)=TA(z0),
T=(P1J0J0)B-1(0)U(0)+(P4J0J0)B-1(L)U(L).
H=A(z0)A(z0)=T-1gg(T-1)=T-1G(T-1).
H14+H11=w(0)(G11+G14)w(0),
H44+H41=J3w(L)J3(G41+G44)J3w(L)J3.
H11=w(0)G11w(0),
H41=J3w(L)J3G41J3w(L)J3,
H14=G14,
H44=G44.
A1(0)=a1(0)cos(θ1)sin(θ1),A2(0)=a2(0)cos(θ2)sin(θ2),
A3(L)=a3(L)cos(θ3)sin(θ3),A4(L)=a4(L)cos(θ4)sin(θ4).
G=A1(0)A1(0)A1(0)A2(0)A1(0)A3(L)A1(0)A4(L)A2(0)A1(0)A2(0)A2(0)A2(0)A3(L)A2(0)A4(L)A3(L)A1(0)A3(L)A2(0)A3(L)A3(L)A3(L)A4(L)A4(L)A1(0)A4(L)A2(0)A4(L)A3(L)A4(L)A4(L);
G11=I1(0)cos2(θ1)a1(0)a2*(0)cos(θ1)cos(θ2)a2(0)a1*(0)cos(θ2)cos(θ1)I2(0)cos2(θ2),
G14=I1(0)sin2(θ1)a1(0)a2*(0)sin(θ1)sin(θ2)a2(0)a1*(0)sin(θ2)sin(θ1)I2(0)sin2(θ2),
G41=I3(L)cos2(θ3)a3(L)a4*(L)cos(θ3)cos(θ4)a4(L)a3*(L)cos(θ4)cos(θ3)I4(L)cos2(θ4),
G44=I3(L)sin2(θ3)a3(L)a4*(L)sin(θ3)sin(θ4)a4(L)a3*(L)sin(θ4)sin(θ3)I4(L)sin2(θ4),
A1(0)=a1(0)21±i,A2(0)=a2(0)21±i,
A3(L)=a3(L)21±i,A4(L)=a4(L)21±i,
G11=G14=12I1(0)a1(0)a2*(0)a2(0)a1*(0)I2(0),
G41=G44=12I3(L)a3(L)a4*(L)a4(L)a3*(L)I4(L).
R=(A4(0), A4(0))(A1(0), A1(0)),
A4(0)=a4(1)(0)a4(2)(0)=(A(0), e221)(A(0), e222)=(B-1(0)U(0)A(z0), e221)(B-1(0)U(0)A(z0), e222),
A4(0)=-exp(-iβs1L)(w1(0)w1(L)e1,e2)00exp(-iβs2L)(w2(0)w2(L)e1,e2)A3(L).
A4(0)=exp(-iβL)00-exp(iβL)A3(L)sin(ξ)×exp[i(μ-ν)].
R=I3 sin2(ξ)/I1linearpolarizationI3 sin2(ξ)/I1circularpolarization.
A4(0)=exp(-iβL)000A3(L)sin(ξ)exp[i(μ-ν)].
R=I3 cos2(θ3)sin2(ξ)/I1,linearpolarizationI3 sin2(ξ)/2I1,circularpolarization.
ddzve1=iγI0(Mve1, ve2)ve2,
ddzve2=iγ*I0(Mve2, ve1)ve1.
M=v(0)Fv(0)+v(L)Bv(L),
F=-J2(G11-J3G14J3)J2,
B=-J2(G44-J3G41J3)J2.
F=[I1(0)cos(2θ1)-I2(0)cos(2θ2)]/2a2(0)a1*(0)cos(θ1-θ2)a1(0)a2*(0)cos(θ1-θ2)-[I1(0)cos(2θ1)-I2(0)cos(2θ2)]/2,
B=-[I3(L)cos(2θ3)-I4(L)cos(2θ4)]/2a4(L)a3*(L)cos(θ3-θ4)a3(L)a4*(L)cos(θ3-θ4)[I3(L)cos(2θ3)-I4(L)cos(2θ4)]/2.
I1(0)=0.1,I2(0)=1.0,I3(L)=1.0.
F=0a2(0)a1*(0)a1(0)a2*(0)0,
B=0a4(L)a3*(L)a3(L)a4*(L)0.
v(L)=cos(ξ)exp[i(μ+ν)]sin(ξ)exp[i(μ-ν)]-sin(ξ)exp[-i(μ-ν)]cos(ξ)exp[-i(μ+ν)],
v(0)=J0,
v=yx[Tr(yy)/2]1/2,
y=exp(iγML/I0)P1+exp(iγ*ML/I0)P4,
x=exp(iψJ3),
(v(L)e2, e1)=sin(ξ)exp[i(μ-ν)]=exp(-iψ)[cosh(2γifL/I0)]1/2(y(L)e2, e1),
(y(L)e2, e1)=sinh(iγ*fL/I0) (Fe2, e1)f.
R=I3I1cos2(γrfL/I0)sinh2(γifL/I0)+sin2(γrfL/I0)cosh2(γifL/I0)cosh(2γifL/I0).
ddzve1=iγI0[(M1ve1, ve2)ve2+(M2e2, e1)ve2],
ddzve2=iγ*I0[(M1ve2, ve1)ve1+(M2e1, e2)ve1].
M1=v(0)Fv(0)+v(L)Bv(L),
M2=G14+G44.
F=-J2G11J2,
B=+J1G41J1.
F=[I1(0)cos2(θ1)-I2(0)cos2(θ2)]/2a2(0)a1*(0)cos(θ1)cos(θ2)a1(0)a2*(0)cos(θ1)cos(θ2)-[I1(0)cos2(θ1)-I2(0)cos2(θ2)]/2,
B=-[I3(L)cos2(θ3)-I4(L)cos2(θ4)]/2a4(L)a3*(L)cos(θ3)cos(θ4)a3(L)a4*(L)cos(θ3)cos(θ4)[I3(L)cos2(θ3)-I4(L)cos2(θ4)]/2.
F=12[I1(0)-I2(0)]/2a2(0)a1*(0)a1(0)a2*(0)-[I1(0)-I2(0)]/2,
B=12-[I3(L)-I4(L)]/2a4(L)a3*(L)a3(L)a4*(L)[I3(L)-I4(L)]/2
v=cos(ξ)sin(ξ)-sin(ξ)cos(ξ).
dvdz=ivKT,
κ=(Ke2, e1)=γI0[(M1ve1, ve2)+(M2e2, e1)].
dξdz=-iκ.
dξdz=γiI0[(M1ve1, ve2)+(M2e2, e1)].
M1=m11m12m12-m11,M2=m21m22m22-m21,
dξdz=γiI0[m11 sin(2ξ)+m12 cos(2ξ)+m22].
m11=A1+B1 cos[2ξ(L)]+C1 sin[2ξ(L)],
m12=A2+C1 cos[2ξ(L)]-B1 sin[2ξ(L)],
m22=A3+B3,
A1=12[I1(0)cos2(θ1)-I2(0)cos2(θ2)],
B1=12[I3(L)cos2(θ3)-I4(L)cos2(θ4)],
C1=[I3(L)I4(L)]1/2 cos(θ3)cos(θ4),
A2=[I1(0)I2(0)]1/2 sin(θ1)sin(θ2),
A3=[I1(0)I2(0)]1/2 cos(θ1)cos(θ2),
B3=[I3(L)I4(L)]1/2 sin(θ3)sin(θ4),
ΓLI0=1(m2-m222)1/2×loge{tan[ξ(L)+ϕ/2]-α}[tan(ϕ/2)-β][tan(ϕ/2)-α]{tan[ξ(L)+ϕ/2]-β},
α=1m22[-m+(m2-m222)1/2],
β=1m22[-m-(m2-m222)1/2],
dXdz=iHX,
κ=i dθdz,
dXdθ=i(J2S)X.
X=expiJ2Sκdz.
S=s1P1+s2P4.
dw1dz=is1Kw1,
dw2dz=is2Kw2,
s2K=s1vKv-1-i dvdzv-1;
dvdz=i(s2-s1)Kv.
vJ0+i(s2-s1)0zK(z)dz;
w2=w1+i(s2-s1)Mw1,
M=0zK(z)dz.
w2=w1+i(s2+s1)MJ3w1J3.
κ=γI0[(A1, A2)+(A3, A4)].
AA=A1A1A1A2A1A3A1A4A2A1A2A2A2A3A2A4A3A1A3A2A3A3A3A4A4A1A4A2A4A3A4A4,
AiAj=ai1aj1*ai1aj2*ai2aj1*ai2aj2*.
(A1, A2)=(AAe121, e111)+(AAe122, e112),
(A3, A4)=(AAe221, e211)+(AAe222, e212),
AA=U(z)HU(z),
(A1, A2)=(HU(z)e121, U(z)e111)+(HU(z)e122, U(z)e112),
(A3, A4)=(HU(z)e221, U(z)e211)+(HU(z)e222, U(z)e212).
I=(HU(z)eijk, U(z)elmn).
U=XX
H=n=14PnHn;
I=n=14((PnHn)(P1X)eijk, (P1X)elmn)+n=14((PnHn)(P1X)eijk, (P4X)elmn)+n=14((PnHn)(P4X)eijk, (P1X)elmn)+n=14((PnHn)(P4X)eijk, (P4X)elmn).
I=(P1ei, el)(H1Xejk, Xemn)+(P2ei, el)×(H2Xejk, Xemn)+(P3ei, el)×(H3Xejk, Xemn)+(P4ei, el)×(H4Xejk, Xemn).
κ=γI0[(H1Xe21, Xe11)+(H1Xe22, Xe12)+(H4Xe21, Xe11)+(H4Xe22, Xe12)].
X=(w1P1)+(w2P4).
H1=m=14H1mPm,H4=m=14H4mPm.
κ=γI0[((H11-J3H41J3)w1e2, w1e1)+((H14-J3H44J3)w2e2, w2e1)].
κ=γI0[((H11+H44)we2, we1)+((H14+H44)J3wJ3e2, J3wJ3e1)].
κ=γI0[(H11we2, we1)+(H41J3wJ3e2, J3wJ3e1)+((H14+H44)e2, e1)].
H=T-1G(T-1).
T-1=(P1J0J0)U(0)B(0)+(P4J0J0)U(L)B(L).
G=n=14PnGn.
H=[P1X(0)C(0)G1X(0)C(0)]+[P2X(L)C(L)G2X(0)C(0)]+[P3X(0)C(0)G3X(L)C(L)]+[P4X(L)C(L)G4X(L)C(L)].
H1=X(0)C(0)G1X(0)C(0),
H4=X(L)C(L)G4X(L)C(L).
X=(w1P1)+(w2P4).
G1=m=14G1mPm,G4=m=14G4mPm,
H1=[w1(0)G11w1(0)P1]+[w1(0)G12w2(0)P2]+[w2(0)G13w1(0)P3]+[w2(0)G14w2(0)P4],
H4=m=14[J3w1(L)J3G4mJ3w1(L)J3](P1MPmP1M)+m=14[J3w1(L)J3G4mJ3w2(L)J3](P1MPmP1M)+m=14[J3w2(L)J3G4mJ3w1(L)J3](P1MPmP4M)+m=14[J3w2(L)J3G4mJ3w2(L)J3](P4MPmP4M),
H4=J3w1(L)J3G41J3w1(L)J3P1+J3w1(L)J3G42J3w2(L)J3P2 ×exp[iβ(s1-s2)z]+J3w2(L)J3G43J3w1(L)J3P3 exp[iβ(s1-s2)z]+J3w2(L)J3G44J3w2(L)J3P4.
H11=w1(0)G11w1(0),
H14=w2(0)G14w2(0),
H41=J3w1(L)J3G41J3w1(L)J3,
H44=J3w2(L)J3G44J3w2(L)J3.

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