Abstract

The wave propagation in symmetric, nonlinear, integrated-optical Y junctions is investigated with consideration of the nonlinear polarization coupling. The propagation of the field in the waveguide structure is described by a coupled pair of nonlinear two-dimensional wave equations, which are solved numerically by the beam-propagation method. The nonlinear Y junction is analyzed with respect to its application for all-optical modulation of a strong pump wave by a weak control beam. It is shown that sensitive continuous amplitude-to-amplitude modulation as well as phase-to-amplitude modulation is possible. This operation is stable with respect to the total powers in each output waveguide, despite the nonlinear-polarization coupling that, for example, is caused by a polarization error of the input field. Further, the results indicate that the conditions for a polarization-maintaining operation can be obtained from the known closed-form solutions for nonlinear polarization coupling in birefringent optical fibers.

© 1997 Optical Society of America

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References

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  1. G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
    [CrossRef]
  2. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
    [CrossRef]
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    [CrossRef] [PubMed]
  5. H. Fouckhardt and Y. Silberberg, “All-optical switching in waveguide X junctions,” J. Opt. Soc. Am. B 7, 803–809 (1990).
    [CrossRef]
  6. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
    [CrossRef]
  7. B. Diano, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
    [CrossRef]
  8. Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
    [CrossRef]
  9. A. T. Pham and L. N. Binh, “All-optical modulation and switching using a nonlinear-optical directional coupler,” J. Opt. Soc. Am. B 8, 1914–1931 (1991).
    [CrossRef]
  10. A. W. Snyder, D. J. Mitchel, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
    [CrossRef]
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  13. U. Hempelmann, “Polarization coupling and transverse interaction of spatial optical solitons in a slab waveguide,” J. Opt. Soc. Am. B 12, 77–86 (1995).
    [CrossRef]
  14. H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
    [CrossRef]
  15. B. Diano, G. Gregori, and S. Wabnitz, “New all-optical devices based on third order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
    [CrossRef]
  16. Y. Chen, “Nonlinear power coupling in the birefringent fiber form birefringence effect,” J. Appl. Phys. 66, 43–46 (1989).
    [CrossRef]
  17. S. Trillo and S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarization fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
    [CrossRef]
  18. Y. Chen and A. W. Snyder, “Chaos in a conventional nonlinear coupler,” Opt. Lett. 14, 1237–1239 (1989).
    [CrossRef] [PubMed]
  19. Y. Chen and A. W. Snyder, “Stochastic instability in nonlinear anisotropic fiber couplers,” J. Lightwave Technol. 8, 802–810 (1990).
    [CrossRef]
  20. S. Trillo and S. Wabnitz, “Nonlinear dynamics of parametric wave-mixing interactions in optics:Instabilities and chaos,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluver Academic, Dordrecht, 1992).
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    [CrossRef]
  23. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation and beam breakup in self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  24. K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
    [CrossRef] [PubMed]
  25. W. K. Burns and A. F. Milton, “An analytic solution for mode coupling in optical waveguide branches,” IEEE J. Quantum Electron. QE-16, 446–454 (1980).
    [CrossRef]
  26. F. Dios, X. Nogues, and F. Canal, “Critical power in a symmetric nonlinear directional coupler,” Opt. Quantum Electron. 24, 1191–1201 (1992).
    [CrossRef]
  27. D. Artigas and F. Dios, “Phase space description of nonlinear directional couplers,” IEEE J. Quantum Electron. QE-30, 1587–1595 (1994).
    [CrossRef]
  28. S. Wabnitz, “Spatial chaos in the polarization for a birefringent optical fiber with periodic coupling,” Phys. Rev. Lett. 58, 1415–1418 (1987).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

D. Artigas and F. Dios, “Phase space description of nonlinear directional couplers,” IEEE J. Quantum Electron. QE-30, 1587–1595 (1994).
[CrossRef]

1992 (1)

F. Dios, X. Nogues, and F. Canal, “Critical power in a symmetric nonlinear directional coupler,” Opt. Quantum Electron. 24, 1191–1201 (1992).
[CrossRef]

1991 (3)

1990 (4)

K. Hayata, A. Misawa, and M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems,” J. Opt. Soc. Am. B 7, 1268–1280 (1990).
[CrossRef]

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

H. Fouckhardt and Y. Silberberg, “All-optical switching in waveguide X junctions,” J. Opt. Soc. Am. B 7, 803–809 (1990).
[CrossRef]

Y. Chen and A. W. Snyder, “Stochastic instability in nonlinear anisotropic fiber couplers,” J. Lightwave Technol. 8, 802–810 (1990).
[CrossRef]

1989 (2)

Y. Chen, “Nonlinear power coupling in the birefringent fiber form birefringence effect,” J. Appl. Phys. 66, 43–46 (1989).
[CrossRef]

Y. Chen and A. W. Snyder, “Chaos in a conventional nonlinear coupler,” Opt. Lett. 14, 1237–1239 (1989).
[CrossRef] [PubMed]

1988 (4)

1987 (3)

K. J. Blow, N. J. Doran, and D. Wood, “Polarization instabilities for solitons in birefringent fibers,” Opt. Lett. 12, 202–204 (1987).
[CrossRef] [PubMed]

S. Wabnitz, “Spatial chaos in the polarization for a birefringent optical fiber with periodic coupling,” Phys. Rev. Lett. 58, 1415–1418 (1987).
[CrossRef] [PubMed]

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

1986 (1)

1985 (2)

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

B. Diano, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

1980 (1)

W. K. Burns and A. F. Milton, “An analytic solution for mode coupling in optical waveguide branches,” IEEE J. Quantum Electron. QE-16, 446–454 (1980).
[CrossRef]

1974 (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1974).
[CrossRef]

Artigas, D.

D. Artigas and F. Dios, “Phase space description of nonlinear directional couplers,” IEEE J. Quantum Electron. QE-30, 1587–1595 (1994).
[CrossRef]

Binh, L. N.

Blow, K. J.

Burns, W. K.

W. K. Burns and A. F. Milton, “An analytic solution for mode coupling in optical waveguide branches,” IEEE J. Quantum Electron. QE-16, 446–454 (1980).
[CrossRef]

Canal, F.

F. Dios, X. Nogues, and F. Canal, “Critical power in a symmetric nonlinear directional coupler,” Opt. Quantum Electron. 24, 1191–1201 (1992).
[CrossRef]

Chen, Y.

A. W. Snyder, D. J. Mitchel, L. Poladian, D. R. Rowland, and Y. Chen, “Physics of nonlinear fiber couplers,” J. Opt. Soc. Am. B 8, 2102–2118 (1991).
[CrossRef]

Y. Chen and A. W. Snyder, “Stochastic instability in nonlinear anisotropic fiber couplers,” J. Lightwave Technol. 8, 802–810 (1990).
[CrossRef]

Y. Chen and A. W. Snyder, “Chaos in a conventional nonlinear coupler,” Opt. Lett. 14, 1237–1239 (1989).
[CrossRef] [PubMed]

Y. Chen, “Nonlinear power coupling in the birefringent fiber form birefringence effect,” J. Appl. Phys. 66, 43–46 (1989).
[CrossRef]

de Sterke, C. M.

Diano, B.

B. Diano, G. Gregori, and S. Wabnitz, “New all-optical devices based on third order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[CrossRef]

B. Diano, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Dios, F.

D. Artigas and F. Dios, “Phase space description of nonlinear directional couplers,” IEEE J. Quantum Electron. QE-30, 1587–1595 (1994).
[CrossRef]

F. Dios, X. Nogues, and F. Canal, “Critical power in a symmetric nonlinear directional coupler,” Opt. Quantum Electron. 24, 1191–1201 (1992).
[CrossRef]

Doran, N. J.

Feit, M. D.

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Fleck , Jr., J. A.

Fouckhardt, H.

Gregori, G.

B. Diano, G. Gregori, and S. Wabnitz, “New all-optical devices based on third order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[CrossRef]

B. Diano, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Hayata, K.

Hempelmann, U.

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Koshiba, M.

Marcuse, D.

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1974).
[CrossRef]

Milton, A. F.

W. K. Burns and A. F. Milton, “An analytic solution for mode coupling in optical waveguide branches,” IEEE J. Quantum Electron. QE-16, 446–454 (1980).
[CrossRef]

Misawa, A.

Mitchel, D. J.

Nogues, X.

F. Dios, X. Nogues, and F. Canal, “Critical power in a symmetric nonlinear directional coupler,” Opt. Quantum Electron. 24, 1191–1201 (1992).
[CrossRef]

Pham, A. T.

Poladian, L.

Rowland, D. R.

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Sfez, B. G.

Silberberg, Y.

Sipe, J. E.

Snyder, A. W.

Stegeman, G. I.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

Trillo, S.

Wabnitz, S.

S. Trillo and S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarization fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1988).
[CrossRef]

S. Wabnitz, “Spatial chaos in the polarization for a birefringent optical fiber with periodic coupling,” Phys. Rev. Lett. 58, 1415–1418 (1987).
[CrossRef] [PubMed]

B. Diano, G. Gregori, and S. Wabnitz, “New all-optical devices based on third order nonlinearity of birefringent fibers,” Opt. Lett. 11, 42–44 (1986).
[CrossRef]

B. Diano, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Winful, H. G.

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

Wood, D.

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Zanoni, R.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Appl. Phys. Lett. (2)

Y. Silberberg and G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, “Coupled-mode theory for anisotropic optical waveguides,” Bell Syst. Tech. J. 54, 985–995 (1974).
[CrossRef]

IEEE J. Quantum Electron. (3)

W. K. Burns and A. F. Milton, “An analytic solution for mode coupling in optical waveguide branches,” IEEE J. Quantum Electron. QE-16, 446–454 (1980).
[CrossRef]

D. Artigas and F. Dios, “Phase space description of nonlinear directional couplers,” IEEE J. Quantum Electron. QE-30, 1587–1595 (1994).
[CrossRef]

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

J. Appl. Phys. (2)

B. Diano, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[CrossRef]

Y. Chen, “Nonlinear power coupling in the birefringent fiber form birefringence effect,” J. Appl. Phys. 66, 43–46 (1989).
[CrossRef]

J. Lightwave Technol. (2)

Y. Chen and A. W. Snyder, “Stochastic instability in nonlinear anisotropic fiber couplers,” J. Lightwave Technol. 8, 802–810 (1990).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Lett. (5)

Opt. Quantum Electron. (2)

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

F. Dios, X. Nogues, and F. Canal, “Critical power in a symmetric nonlinear directional coupler,” Opt. Quantum Electron. 24, 1191–1201 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

S. Wabnitz, “Spatial chaos in the polarization for a birefringent optical fiber with periodic coupling,” Phys. Rev. Lett. 58, 1415–1418 (1987).
[CrossRef] [PubMed]

Other (3)

G. Assanto, “Third order nonlinear integrated optics,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer Academic, Dordrecht, 1992).

S. Trillo and S. Wabnitz, “Nonlinear dynamics of parametric wave-mixing interactions in optics:Instabilities and chaos,” in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluver Academic, Dordrecht, 1992).

D. Marcuse, Theory of Dielectric Waveguides (Academic, New York, 1974).

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Figures (13)

Fig. 1
Fig. 1

Sketch of the waveguide structure, including the even and odd local modes Φμ,e and Φμ,o, the complete field Aμ, the powers of the pump Pp, and the control wave Pc and the output powers Pμ,1, Pμ,2 in the slow or the fast polarization (μ =s, f).

Fig. 2
Fig. 2

(A) Sectional view of the three-dimensional refractive-index distribution. (B) Corresponding linear effective index profile.

Fig. 3
Fig. 3

Difference of the phase constants Δβμ,eo and coupling coefficients Γ.. versus waveguide separation s for the effective waveguide parameters in Table 1. The identity Δβs,eo =Δβf,eo=Δβμ,eo is a result of the approximation δnsδnf. All coupling coefficients approach the asymptotic value Γ =lims(Γe, Γo, Γeo)3.62×10-6 (W μm)-1 for large s.

Fig. 4
Fig. 4

Phase–space portrait of the power and phase differences in the NLCC for the initial values (Uμ,eo0=0.5, Θμ,eo0=π/18) and the waveguide separation s=8 µm. The parameter is the relation P/Pμ,eoc of the total power and the critical power.

Fig. 5
Fig. 5

Nonlinear coupling length lnl in the NLCC related to the linear beat length π/Δβμ,eo=5695.5 µm versus power of the control field and initial phase difference. The pump power is Pp=152 W, and the waveguide separation is s=8 µm. In the interval [0, 0.26π) there is P>Pμ,eoc and in (0.26π, 0.5π] there is P<Pμ,eoc.

Fig. 6
Fig. 6

Phase–space portrait for the Y junction for the initial values (a) (Uμ,eo0=0.99, Θμ,eo0=0), (b) (Uμ,eo0=0.9, Θμ,eo0 =0) and (c) (Uμ,eo0=0.99, Θμ,eo0=3π/4) at the pump power Pp=1000 W.

Fig. 7
Fig. 7

Output powers of the Y junction versus initial phase difference at different control powers Pc=0.5, 1, 2, 5, 10, 20, 50 W at Pp=1000 W.

Fig. 8
Fig. 8

Output power Pμ,2 and complete output power P =Pμ,1+Pμ,2 versus amplitude Cc of the control field for Pp =1000 W and Θμ,eo0=π/12 at positive Cc, corresponding to Θμ,eo0=13π/12 at negative Cc.

Fig. 9
Fig. 9

Nonlinear coupling length lnl and relative coupled fraction of power ΔPsf/P0 for the polarization coupling of two local modes versus birefringence δnsf,0. (A) Initial polarization close to the fast polarization: (a) both modes even, Pf,e(0) =1000 W, Ps,e(0)=10 W; (b) both modes odd, Pf,o(0) =1000 W, Ps,o(0)=10 W; (c) different symmetries, Pf,o(0) =1000 W, Ps,e(0)=10 W; (d) different symmetries, Pf,e(0) =1000 W, Ps,o(0)=10 W; (e) fundamental modes of a single waveguide, Pf(0)=500 W, Ps(0)=5 W; (B) Initial polarization close to the slow polarization: (a),(b) both modes either even or odd, ⇒ no polarization coupling; (c) different symmetries, Pf,o(0)=10 W, Ps,e(0)=1000 W; (d) different symmetries, Pf,e(0)=10 W, Ps,o(0)=1000 W; (e) fundamental modes of a single waveguide, Pf(0)=5 W, Ps(0)=500 W; ⇒ no polarization coupling, ΔPsf/P0<0.01.

Fig. 10
Fig. 10

Output powers of the Y junction versus birefringence δnsf,0 for a polarization error Ppe=Ppemax=100 W at the maximum of the modulation curve (Cc/P0=+0.316 in Fig. 8). Initial polarization close to the fast (A) and slow (B) polarization. The initial polarization error is indicated by the broken lines.

Fig. 11
Fig. 11

Field distribution in the Y junction at Ppe=Ppemax =100 W, Cc/P0=+0.316 and δnsf,0=10-4. Initial polarization close to the fast (A) and slow (B) polarization.

Fig. 12
Fig. 12

Output powers in waveguide 2 versus amplitude of the control field for δnsf,0=10-4 and Ppe=Ppemax=100 W. Initial polarization close to the fast (A) and slow (B) polarization.

Fig. 13
Fig. 13

Total output power P2 in waveguide 2 and respective fraction of power Pμ,2 in the unintended polarization versus polarization error Ppe at the maximum of the modulation curve (Cc/P0=+0.316). Initial polarization close to the fast (a) and slow (b) polarization.

Tables (1)

Tables Icon

Table 1 Effective Parameters of the Symmetric Nonlinear Y Junction and the Stem

Equations (20)

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D=0n2E+n2I0μ01/223|E|2E+13E2E*,
Et(r)=[xAs(x, z)Fs(y)|(x,z)+yAf(x, z)Ff(y)|(x,z)]exp(-jβ0n¯0z),
nμ(x, z)=nμ0+δnμ(x, z),μ{s, f},
jAsz=12β0n¯02Asx2+β0δns+δnsf,02As+β0n2Ideff|As|2As+23|Af|2As+13Af2As*,
jAfz=12β0n¯02Afx2+β0δnf-δnsf,02Af+β0n2Ideff|Af|2Af+23|As|2Af+13As2Af*,
δnsf,0=ns0-nf0,
Aμ(x, z)=m=e,oPμ,m(z)exp[-jϕμ,m(z)]Φμ,m(x)×exp(-jβμ,mz),
Hμ,eo=PΓeo(1-Uμ,eo2)cos(2Өμ,eo)+Γe+Γo-4Γeo2Uμ,eo2+(Γe-Γo)Uμ,eo+2Δβμ,eoUμ,eo,
Pμ,eoc=2Δβμ,eo(Uμ,eo0+1){Γeo[2+cos(2Өμ,eo0)]-(Γe+Γ0)/2}-(Γe-Γo).
Pμ,eob=Pμ,eoc(Uμ,eo0=1,Өμ,eo0=0)=Δβμ,eo3Γeo-Γe,
Pμ,21=P2[1±1-Uμ,eo2 cos(Өμ,eo)]
Hsf,m=4Usf,mPsf,mbP+Usf,m2+(1-Usf,m2)cos(2Өsf,m),
Psf,mb=3Δβsf,m2Γm.
ΔPsfP1-Psf,mbP,ifP>Psf,mb,
Hμν,eo=PΓeo3(1-Uμν,eo2)cos(2Өμν,eo)+Γe+Γo-4/3Γeo2Uμν,eo2+(Γe-Γo)Uμν,eo+2Δβμν,eoUμν,eo,
Pμν,eoc=2Δβμν,eo(Uμν,eo0+1){(Γeo/3)[2+cos(2Өμν,eo0)]-(Γe+Γo)/2}-(Γe-Γo).
Pf,e(0)=Pp-Ppe,Pf,o(0)=Pc,
Ps,e(0)=PpeandPs,0(0)=0,
Pf,e(0)=Ppe,Pf,o(0)=0,
Ps,e(0)=Pp-PpeandPs,o(0)=Pc.

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