Abstract

We present experiments that show simultaneous achievement of suppression of modulational instability and substantial reduction of stimulated Raman scattering in birefringent optical fibers under dual-frequency, orthogonal polarization pumping. The suppression mechanism is based on careful control of the group-velocity match between the pumps and on an appropriate choice of the input power distribution among the components of the pump field. The use of orthogonally polarized pump beams permits suppression of the parametric and Raman Stokes beams in any one of the fiber axes, with a frequency spacing between the pumps (2.4 THz) that is reduced by more than 1 order of magnitude compared with the pump-frequency spacing required for a linearly polarized pump field (26 THz).

© 2000 Optical Society of America

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References

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  1. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, Chur, Switzerland, 1989).
  2. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  3. P. Tchofo Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
    [CrossRef]
  4. E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
    [CrossRef] [PubMed]
  5. E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
    [CrossRef]
  6. P. V. Mamyshev and A. P. Vertikov, “Stimulated Raman scattering in fibers: influence of parametric interaction,” in Quantum Electronics Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 43–130.
  7. S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
    [CrossRef]
  8. P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433–1441 (1998).
    [CrossRef]
  9. P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching of stimulated Raman scattering in optical fibers by dual-frequency pumping,” Opt. Lett. 22, 1595–1597 (1997).
    [CrossRef]
  10. T. Sylvestre, H. Maillotte, and E. Lantz, “Stimulated Raman suppression under dual-frequency pumping in single-mode fibers,” Electron. Lett. 34, 1417–1418 (1998).
    [CrossRef]
  11. P. Tchofo Dinda, S. Wabnitz, E. Coquet, T. Sylvestre, H. Maillotte, and E. Lantz, “Demonstration of stimulated-Raman-scattering suppression in optical fibers in a multifrequency pumping configuration,” J. Opt. Soc. Am. B 16, 757–767 (1999).
    [CrossRef]
  12. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
    [CrossRef]
  13. E. A. Golovchenko and A. N. Pilipetskii, “Unified analysis of four-photon mixing, modulation instability, and stimulated Raman scattering under various polarization conditions in fibers,” J. Opt. Soc. Am. B 11, 92–101 (1994).
    [CrossRef]
  14. E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulation-induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
    [CrossRef]
  15. R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  16. B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
    [CrossRef]

2000 (1)

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulation-induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

1999 (1)

1998 (2)

T. Sylvestre, H. Maillotte, and E. Lantz, “Stimulated Raman suppression under dual-frequency pumping in single-mode fibers,” Electron. Lett. 34, 1417–1418 (1998).
[CrossRef]

P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433–1441 (1998).
[CrossRef]

1997 (1)

1996 (2)

P. Tchofo Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
[CrossRef]

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

1990 (3)

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

1977 (1)

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Berger, H.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Bilbault, J. M.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Bonamy, J.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Bonamy, L.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Coquet, E.

Dianov, E. M.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Golovchenko, E.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Golovchenko, E. A.

Haelterman, M.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

P. Tchofo Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
[CrossRef]

Harvey, J. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Hellwarth, R.

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Lantz, E.

Lavorel, B.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Leonhardt, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Maillotte, H.

Mamyshev, P. V.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Millot, G.

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulation-induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching and suppression of stimulated Raman scattering in birefringent optical fibers,” J. Opt. Soc. Am. B 15, 1433–1441 (1998).
[CrossRef]

P. Tchofo Dinda, G. Millot, and S. Wabnitz, “Polarization switching of stimulated Raman scattering in optical fibers by dual-frequency pumping,” Opt. Lett. 22, 1595–1597 (1997).
[CrossRef]

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

P. Tchofo Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
[CrossRef]

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Pilipetskii, A. N.

E. A. Golovchenko and A. N. Pilipetskii, “Unified analysis of four-photon mixing, modulation instability, and stimulated Raman scattering under various polarization conditions in fibers,” J. Opt. Soc. Am. B 11, 92–101 (1994).
[CrossRef]

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

Remoissenet, M.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Robert, D.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Saint-Loup, R.

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

Seve, E.

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulation-induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

P. Tchofo Dinda, G. Millot, E. Seve, and M. Haelterman, “Demonstration of nonlinear gap in the modulational instability spectra of wave propagation in highly birefringent fibers,” Opt. Lett. 21, 1640–1642 (1996).
[CrossRef]

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Sylvestre, T.

Tchofo Dinda, P.

Trillo, S.

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulation-induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
[CrossRef]

Wabnitz, S.

Electron. Lett. (1)

T. Sylvestre, H. Maillotte, and E. Lantz, “Stimulated Raman suppression under dual-frequency pumping in single-mode fibers,” Electron. Lett. 34, 1417–1418 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 26, 1815–1820 (1990).
[CrossRef]

J. Chem. Phys. (1)

B. Lavorel, G. Millot, R. Saint-Loup, H. Berger, L. Bonamy, J. Bonamy, and D. Robert, “Study of collisional effects on band shapes of the ν1/2ν2 Fermi dyad in CO2 gas with stimulated Raman spectroscopy: rotational and vibrational relaxation in the 2ν2 band,” J. Chem. Phys. 93, 2185–2191 (1990).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519–3534 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (1)

E. Seve, G. Millot, and S. Trillo, “Strong four-photon conversion regime of cross-phase-modulation-induced modulational instability,” Phys. Rev. E 61, 3139–3150 (2000).
[CrossRef]

Prog. Quantum Electron. (1)

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Other (3)

P. V. Mamyshev and A. P. Vertikov, “Stimulated Raman scattering in fibers: influence of parametric interaction,” in Quantum Electronics Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 43–130.

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and V. N. Serkin, Nonlinear Effects in Optical Fibers (Harwood Academic, Chur, Switzerland, 1989).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

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Figures (10)

Fig. 1
Fig. 1

MI effects in the birefringent fiber under orthogonal polarization pumping, with ωP1=522 THz, ωP2=521.5 THz (Δω=0.5 THz), and various power distributions [R=P1(0)/P2(0)] at the fiber input: (a) optimum MI gain G(Ωopt) as a function of total input power P, (b) optimum MI frequency Ωopt versus power P. Normalized intensities for Stokes and anti-Stokes sidebands on (c) the fast axis and (d) the slow axis. Here β1β259 ps2/km, γ1γ20.04 W-1 m-1, and δ=1.08 ps/m.

Fig. 2
Fig. 2

Suppression of MI for three power distributions at the fiber input: R=0.1 (dashed curve) and R=10 (solid curve). Here ωP1=522 THz and ωP2=ωP1-Δω. (a) Each point (Δω, P) above the curve represents modulationally stable operating conditions. The curves represent the critical power Pc versus Δω. (b) GVM versus Δω. (c) Optimum gain versus Δω. (d) Optimum MI frequency versus Δω.

Fig. 3
Fig. 3

Frequency dependence of the orthogonal and parallel components of the Raman susceptibility: J(χ˜1212) (dotted–dashed curve), R(χ˜1212) (dotted curve), J(χ˜1111) (solid curve), and R(χ˜1111) (dashed curve).

Fig. 4
Fig. 4

Reduction of SRS for ωP1=522 THz and Δω=2.4 THz obtained from Eqs. (12). Crosses, the total power of the Raman Stokes waves S/P=[S1(L)+S2(L)]/P normalized to P=175 W as a function of the input power distribution between the two components of the pump field. P1(0)=P-P2(0). Here, the fiber length is L=2 m.

Fig. 5
Fig. 5

Variation of normalized pump and Stokes powers Qj/[P1(0)+P2(0)], Q=S, P, j=1, 2 versus propagation coordinate z for ωP1=522 THz and Δω=2.4 THz. (a), (b), (c) the wave-coupling behavior corresponding to a, b, and c, respectively, in Fig. 4.

Fig. 6
Fig. 6

Schematic of the experimental setup used for recording Raman spectra: P’s, Glan polarizers; BS, beam splitter; MO’s, 20× microscope objectives; F’s, filters; SL, lens; PM, photomultiplier.

Fig. 7
Fig. 7

Schematic of the experimental setup used for measurement of the total power of Raman Stokes beams: P’s, Glan polarizers; BS, beam splitter; MO’s, 20× microscope objectives; F, filters; CL, cylindrical lens; D, diaphragm; PD, photodiode.

Fig. 8
Fig. 8

Effect of the orthogonal component of the Raman gain in the process of reduction of the total power of SRS for ωP1=522 THz and Δω=2.4 THz. Crosses, total power of Raman Stokes waves S=S1(L)+S2(L) as a function of input power on slow axis P2(0). Solid curve, linear interpolation of the experimental measurements.

Fig. 9
Fig. 9

Reduction of SRS for ωP1=522 THz and ωP2=519.6 THz (Δω=2.4 THz). Crosses, total power of the Raman Stokes waves S/P normalized to P=175 W as a function of the input power distribution between the two components of the pump field.

Fig. 10
Fig. 10

Experimental output spectra, showing the simultaneous suppression of MI and SRS for ωP1=522 THz and ωP2=519.6 THz (Δω=2.4 THz). (a), (b), (c) Spectra corresponding to the labels a, b, and c, respectively, in Fig. 9.

Equations (30)

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

Ejz+(-1) j2δ Ejt+12iβj 2Ejt2
=iγj|Ej|2+23|E3-j|2Ej,j=1, 2,
δ1/Vgy(ωP2)-1/Vgx(ωP1)
δ=δ0+β(ωP2-ωP1),
Ej=Pj exp[iγj(Pj+2P3-j/3)z],j=1, 2,
E1=(P1+u)exp[iγ1(P1+2P2/3)z],
E2=(P2+v)exp[iγ2(P2+2P1/3)z],
u=us(z)exp[i(Ωt)]+ua(z)exp[i(-Ωt)],
v=vs(z)exp[i(Ωt)]+va(z)exp[i(-Ωt)],
d[Y]/dz=i[M][Y],[Y]T[ua, us*, va, vs*].
[M]-Ωδ2+β1 Ω22+γ1P1γ1P123γ1P1P223γ1P1P2-γ1P1-Ωδ2-β1 Ω22-γ1P1-23γ1P1P2-23γ1P1P223γ2P1P223γ2P1P2Ωδ2+β2 Ω22+γ2P2γ2P2-23γ2P1P2-23γ2P1P2-γ2P2Ωδ2-β2 Ω22-γ2P2.
G(Ω)=2|I(K)|.
E=12 j=12Q=P,S EQj+c.c.=12 j=12Q=P,S 1(αNQj)1/2 AQj(z)ψQj(x, y)×exp[i(kQjz-ωQjt)]+c.c.,
Pjz=-2LPj{I(HP3-jP3-j*PjPj)PjP3-j+I(HSjSj*PjPj)SjPj+I(HS3-jS3-j*PjPj)S3-jPj+[(-1) jR(HP3-jS3-j*SjPj)sin θ+I(HP3-jS3-j*SjPj)cos θ](SjS3-jPjP3-j)1/2},
ϕPjz=LPj{R(HPjPj*PjPj)Pj+R(HP3-jP3-j*PjPj)P3-j+R(HSjSj*PjPj)Sj+R(HS3-jS3-j*PjPj)S3-j+[(-1) jR(HP3-jS3-j*SjPj)cos θ-I(HP3-jS3-j*SjPj)sin θ](SjS3-jP3-j/Pj)1/2},
Sjz=-2LSj{I(HS3-jS3-j*SjSj)SjS3-j+I(HPjPj*SjSj)PjSj+I(HP3-jP3-j*SjSj)P3-jSj+[(-1) j+1R(HS3-jP3-j*PjSj)sin θ+I(HS3-jP3-j*PjSj)cos θ](SjS3-jPjP3-j)1/2},
ϕSjz=LSj{R(HSjSj*SjSj)Sj+R(HS3-jS3-j*SjSj)S3-j+R(HPjPj*SjSj)Pj+R(HP3-jP3-j*SjSj)P3-j+[(-1) j+1R(HS3-jP3-j*PjSj)cos θ-I(HS3-jP3-j*PjSj)sin θ](S3-jPjP3-j/Sj)1/2},
HQjUlVmWnηQjUlVmξQjUlVmWn,(U, V, W)=S, P,
(l, m, n)=1, 2
ξQjUlVmWn(NQjNUlNVmNWn)1/2ψQjψUlψVmψWndxdy,
(U, V, W)=S, P(l, m, n)=1, 2.
ηPjPj*Pj=ηSjSj*Sj=χ˜1111(0)+3σ/4,
ηSjSj*Pj=χ˜1111(-Ω˜)+χ˜1111(0)+3σ/4,
ηP3-jP3-j*Pj=ηS3-jS3-j*Sj=χ˜1212[(-1) jΔω]+χ˜1122(0)+σ/2,
ηS3-jS3-j*Pj=χ˜1212[(-1) jΔω-Ω˜]+χ˜1122(0)+σ/2,
ηPjPj*Sj=χ˜1111(Ω˜)+χ˜1111(0)+3σ/4,
ηP3-jP3-j*Sj=χ˜1212[(-1) jΔω+Ω˜]+χ˜1122(0)+σ/2,
ηP3-jS3-j*Sj=χ˜1122(-Ω˜)+χ˜1212[(-1) jΔω]+σ/2,
ηS3-jP3-j*Pj=χ˜1122(Ω˜)+χ˜1212[(-1) jΔω]+σ/2,
χ˜1122(ω)χ˜1111(ω)-2χ˜1212(ω).

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