Abstract

A novel approach, based on a scalar-corrected formalism, is proposed for investigating cross-phase modulation phenomena in waveguides with arbitrary cross sections. The proposed approach is applied to investigate the evolution of the polarization state in asymmetric angle-facet AlGaAs rib waveguides. It is seen that the orientation of the optical axes of the waveguides should be a determining parameter in optimizing the power transfer between the polarization modes and that asymmetric angle-facet AlGaAs rib waveguides could be promising devices for application in all-optical switching.

© 1997 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
  2. A. D. Boardman and G. S. Cooper, “Power-dependent polarization of optical pulses,” J. Opt. Soc. Am. B 5, 403–417 (1988).
    [CrossRef]
  3. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
    [CrossRef]
  4. H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 521–523 (1985).
    [CrossRef]
  5. A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
    [CrossRef]
  6. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  7. Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
    [CrossRef]
  8. J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
    [CrossRef]
  9. D. C. Hutchings and B. S. Wherrett, “Theory of the anisotropy of ultrafast nonlinear refraction in zinc-blende semiconductors,” Phys. Rev. B 52, 8150–8159 (1995).
    [CrossRef]
  10. D. C. Hutchings, “Nonlinear-optical activity owing to an-isotropy of ultrafast nonlinear refraction in cubic materi-als,” Opt. Lett. 20, 1607–1609 (1995).
    [CrossRef] [PubMed]
  11. D. C. Hutchings, J. S. Aitchison, B. S. Wherrett, G. T. Kennedy, and W. Sibbett, “Polarization dependence of ultrafast nonlinear refraction in an AlGaAs waveguide at the half-band gap,” Opt. Lett. 20, 991–993 (1995).
    [CrossRef] [PubMed]
  12. V. P. Tzolov and M. Fontaine, “Nonlinear modal parameters of optical fibers: a full-vectorial approach,” J. Opt. Soc. Am. B 12, 1933–1941 (1995).
    [CrossRef]
  13. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Newcastle upon Tyne, UK, 1993).
  14. V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
    [CrossRef]
  15. V. P. Tzolov and M. Fontaine, “Theoretical analysis of birefringence and form-induced polarization mode dispersion in birefringent optical fibers: a full-vectorial approach,” J. Appl. Phys. 77, 1–6 (1995).
    [CrossRef]
  16. M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
    [CrossRef]

1996 (2)

V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
[CrossRef]

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

1995 (7)

V. P. Tzolov and M. Fontaine, “Theoretical analysis of birefringence and form-induced polarization mode dispersion in birefringent optical fibers: a full-vectorial approach,” J. Appl. Phys. 77, 1–6 (1995).
[CrossRef]

A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
[CrossRef]

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

D. C. Hutchings and B. S. Wherrett, “Theory of the anisotropy of ultrafast nonlinear refraction in zinc-blende semiconductors,” Phys. Rev. B 52, 8150–8159 (1995).
[CrossRef]

D. C. Hutchings, “Nonlinear-optical activity owing to an-isotropy of ultrafast nonlinear refraction in cubic materi-als,” Opt. Lett. 20, 1607–1609 (1995).
[CrossRef] [PubMed]

D. C. Hutchings, J. S. Aitchison, B. S. Wherrett, G. T. Kennedy, and W. Sibbett, “Polarization dependence of ultrafast nonlinear refraction in an AlGaAs waveguide at the half-band gap,” Opt. Lett. 20, 991–993 (1995).
[CrossRef] [PubMed]

V. P. Tzolov and M. Fontaine, “Nonlinear modal parameters of optical fibers: a full-vectorial approach,” J. Opt. Soc. Am. B 12, 1933–1941 (1995).
[CrossRef]

1991 (1)

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

1988 (1)

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[CrossRef]

1985 (1)

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 521–523 (1985).
[CrossRef]

Aitchison, J. S.

A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
[CrossRef]

D. C. Hutchings, J. S. Aitchison, B. S. Wherrett, G. T. Kennedy, and W. Sibbett, “Polarization dependence of ultrafast nonlinear refraction in an AlGaAs waveguide at the half-band gap,” Opt. Lett. 20, 991–993 (1995).
[CrossRef] [PubMed]

Alferness, R.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Boardman, A. D.

Bock, W. J.

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

Cooper, G. S.

Fontaine, M.

V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
[CrossRef]

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

V. P. Tzolov and M. Fontaine, “Theoretical analysis of birefringence and form-induced polarization mode dispersion in birefringent optical fibers: a full-vectorial approach,” J. Appl. Phys. 77, 1–6 (1995).
[CrossRef]

V. P. Tzolov and M. Fontaine, “Nonlinear modal parameters of optical fibers: a full-vectorial approach,” J. Opt. Soc. Am. B 12, 1933–1941 (1995).
[CrossRef]

Groen, F. H.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Hakimzadeh, F.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Hutchings, D. C.

Kang, J. U.

A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
[CrossRef]

Kennedy, G. T.

Koch, T.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Koren, U.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Menyuk, C. R.

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[CrossRef]

Metaal, E. G.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Miller, B. I.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Moerman, I.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Oei, Y. S.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Oron, M.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Pedersen, J. W.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Shani, Y.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Sibbett, W.

Stegeman, G. I.

A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
[CrossRef]

Tzolov, V. P.

V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
[CrossRef]

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

V. P. Tzolov and M. Fontaine, “Theoretical analysis of birefringence and form-induced polarization mode dispersion in birefringent optical fibers: a full-vectorial approach,” J. Appl. Phys. 77, 1–6 (1995).
[CrossRef]

V. P. Tzolov and M. Fontaine, “Nonlinear modal parameters of optical fibers: a full-vectorial approach,” J. Opt. Soc. Am. B 12, 1933–1941 (1995).
[CrossRef]

Urbanczyk, W.

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

van der Tol, J. J. G. M.

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

Villeneuve, A.

A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
[CrossRef]

Wherrett, B. S.

Winful, H. G.

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 521–523 (1985).
[CrossRef]

Wu, B.

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

Young, M. G.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

Appl. Phys. Lett. (3)

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 521–523 (1985).
[CrossRef]

A. Villeneuve, J. U. Kang, J. S. Aitchison, and G. I. Stegeman, “Unity ratio of cross-to self-phase modulation in bulk AlGaAs and AlGaAsGaAs multiple quantum well waveguides at half the band gap,” Appl. Phys. Lett. 67, 760–762 (1995).
[CrossRef]

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, “Polarization rotation in asymmetric periodic loaded rib waveguides,” Appl. Phys. Lett. 59, 1278–1280 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. J. G. M. van der Tol, J. W. Pedersen, E. G. Metaal, F. Hakimzadeh, Y. S. Oei, F. H. Groen, and I. Moerman, “Realization of a short integrated optic passive polarization converter,” IEEE Photon. Technol. Lett. 7, 893–895 (1995).
[CrossRef]

J. Appl. Phys. (1)

V. P. Tzolov and M. Fontaine, “Theoretical analysis of birefringence and form-induced polarization mode dispersion in birefringent optical fibers: a full-vectorial approach,” J. Appl. Phys. 77, 1–6 (1995).
[CrossRef]

J. Lightwave Technol. (1)

M. Fontaine, B. Wu, V. P. Tzolov, W. J. Bock, and W. Urbanczyk, “Theoretical and experimental analysis of thermal stress effects on modal polarization properties of highly birefringent optical fibers,” J. Lightwave Technol. 14, 585–591 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

V. P. Tzolov and M. Fontaine, “A passive polarization converter free of longitudinally-periodic structure,” Opt. Commun. 127, 7–13 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (1)

D. C. Hutchings and B. S. Wherrett, “Theory of the anisotropy of ultrafast nonlinear refraction in zinc-blende semiconductors,” Phys. Rev. B 52, 8150–8159 (1995).
[CrossRef]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, Newcastle upon Tyne, UK, 1993).

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Figures (7)

Fig. 1
Fig. 1

Cross section of an angle-facet rib waveguide. The geometrical parameters are d1=1 µm, d2=0.8 µm, and s=0.2 µm. The refractive indices are n1 =3.342591 (Al0.18Ga0.82As), n2=3.209206 (Al0.44Ga0.56As), and n3=1.0 (air). The facet angle α and the rib width w are variable parameters.

Fig. 2
Fig. 2

Discretized finite cross section of an angle-facet rib waveguide. The total number of triangles is 660.

Fig. 3
Fig. 3

Isointensity contours of the electrical field, |ψ(x,y)|2 (a) for structure 1 in Table 1 and (b) for 2. The parameter T is the effective refractive index β˜/k.

Fig. 4
Fig. 4

Rotation angle as a function of the propagation distance for propagation within structure 1 with a linearly x-polarized input. Solid curve, Pin=0 W; dashed curve, Pin=500 W.

Fig. 5
Fig. 5

Power distribution along the xˆ and yˆ axes, 1/[1 +|u(z)|2] and |u(z)|2/[1+|u(z)|2], respectively, as a function of the propagation distance for propagation within structure 1 with a linearly x-polarized input. Solid curve, Pin=0 W; dashed curve, Pin=500 W.

Fig. 6
Fig. 6

Rotation angle as a function of the propagation distance for propagation within structure 2 with a linearly x-polarized input. Solid curve, Pin=0 W; dotted curve, Pin=250 W; dashed curve, Pin=500 W.

Fig. 7
Fig. 7

Power distribution along the xˆ axis, 1/[1+|u(z)|2], as a function of the propagation distance for propagation within structure 2 with a linearly x-polarized input. Solid curve, Pin =0 W; dashed curve, Pin=500 W.

Tables (1)

Tables Icon

Table 1 Comparison of the Orientation of the Optical Axes and the Linear Beat Length for Various Geometries of Angle-Facet Rib Waveguides

Equations (58)

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

×E=iω0μ0H,
×H=-iω0(0E+PNL),
·(0E+PNL)=0,
·H=0.
2Ex,yx2+2Ex,yy2+2Ex,yz2+k02Ex,y-x, y×Exx+Eyy+Ezz+ω02μ0Px,yNL=0.
2Exz2-iω0μ0Hzy+k02Ex+ω02μ0PxNL=0,
2Eyz2+iω0μ0Hzx+k02Ey+ω02μ0PyNL=0.
AEx2Eyz2-Ey2Exz2dA+iω0μ0 A(Et·tHz)dA
+ω02μ0 A(ExPyNL-EyPxNL)dA=0.
Ex=ψ(x, y)fx(z)exp(iβ˜z),
Ey=ψ(x, y)fy(z)exp(iβ˜z).
t2ψ(x, y)+(k02-β˜2)ψ(x, y)=0,
AEx2Eyz2-Ey2Exz2dA=exp(2iβ˜z)2iβ˜fx(z)fy(z)z-fy(z)fx(z)z×Aψ2(x, y)dA.
t·(UAt)=U(t·At)+At·tU,
iω0μ0 A(Et·tHz)dA=iω0μ0 At·(HzEt)dA-iω0μ0 AHz(t·Et)dA.
iω0μ0 A(Et·tHz)dA=iω0μ0 l(HzEt)·nˆdl-iω0μ0 AHz(t·Et)dA,
iω0μ0 A(Et·tHz)dA=-iω0μ0 AHz(t·Et)dA.
·(0E+PNL)·(0E)=t·(0Et)=0,
t·Et=Exx+Eyy=-Ex ln()x+Ey ln()y.
iω0μ0 A(Et·tHz)dA=exp(2iβ˜z)[fx(z)fy(z)Dxx+fy(z)2Dxy-fx(z)2Dyx-fx(z)fy(z)Dyy]Aψ2(x, y)dA,
Dp,q=Aψψp ln()qdAAψ2dA.
PxNL=34χxxxx(3)0|Ex|2+1-δ-σ2|Ey|2Ex+δ-σ2Ey2Ex*,
PyNL=34χxxxx(3)0|Ey|2+1-δ-σ2|Ex|2Ey+δ-σ2Ex2Ey*,
δ=χxxxx(3)+χxxyy(3)-2χxyxy(3)2χxxxx(3),
σ=χxxxx(3)-χxxyy(3)-2χxyxy(3)χxxxx(3).
fx(z)dfy(z)dz-fy(z)dfx(z)dz-ifx(z)fy(z)(Dxx-Dyy)2β˜-fx(z)2Dyx2β˜-fy(z)2Dxy2β˜+i2γPin×δ+σ2(|fx|2-|fy|2)fxfy+δ-σ2×(fy3fx*-fx3fy*)=0.
Pin=β˜k00μ01/2 Aψ2(x, y)dA,
γ=k034χxxxx(3)(β˜/k0)2(0/μ0)1/2A ψ4(x, y)dAA ψ2(x, y)dA2=k034χxxxx(3)n02c0 1Aeff=k0n2L[001]1Aeff,
n2L[001]=34χxxxx(3)n02c0
u(z)=fy(z)fx(z),
dudz-iu2Dxy2β˜+(Dxx-Dyy)2β˜u-Dyx2β˜+i2γPin×δ+σ21-|u|21+|u|2u+δ-σ2×u31+|u|2-|u|21+|u|21u=0.
PxNL=34χxxxx(3)0|Ex|2+1-δ-σ2|Ey|2Ex+δ-σ2Ey2Ex*,
PyNL=34χxxxx(3)01-σ2|Ey|21-δ-σ2|Ex|2×Ey+δ-σ2Ex2Ey*.
dudz-iu2Dxy2β˜+(Dxx-Dyy)2β˜u-Dyx2β˜+i2γPinσ2|u|21+|u|2u+δ+σ2×1-|u|21+|u|2u+δ-σ2
×u31+|u|2-|u|21+|u|21u=0.
dudz=i2β˜[u2Dxy+(Dxx-Dyy)u-Dyx].
a+=(Dxx-Dyy)2Dxy+[(Dyy-Dxx)2+4DxyDyx]1/22Dxy,
a-=(Dxx-Dyy)2Dxy-[(Dyy-Dxx)2+4DxyDyx]1/22Dxy,
dudz=i2β˜Dxy(u+a+)(u+a-).
u(z)=-a+a-[1-exp(iθz)]a--a+ exp(iθz)=[1-exp(iθz)]a--a+ exp(iθz),
θ=12β˜(a--a+)Dxy.
|φ|=|tan-1(a+)|,|η|=|tan-1(a-)|.
L=4πβ˜|(a--a+)Dxy|=2π|(a--a+)Dxy|/2β˜=2π|δβ|,
L=4πβ˜|Dxx-Dyy|=2π|Dxx-Dyy|/2β˜=2π|δβ|.
β˜2-β2=A(t·e˜t)et·t ln()dAAet·etdA.
et=(axˆ+byˆ)ψ,
e˜t=ψxˆore˜t=ψyˆ,
aDxx-a(β˜2-β2)+bDxy=0,
aDyx-b(β˜2-β2)+bDyy=0,
(β˜2-β2)±=(Dxx+Dyy)2±[(Dyy-Dxx)2+4DxyDyx]1/22,
(β˜2-β2)+-(β˜2-β2)-=[(Dyy-Dxx)2+4DxyDyx]1/2=(a+-a-)Dxy,
(β˜2-β2)±=-(β˜+β)±(β-β˜)±-2β˜δβ±,
δβ±=(β-β˜)±=(β˜+δβ±)-β˜=β±-β˜
β+=β˜+δβ+,
β-=β˜+δβ-
|δβ|=|β+-β-|=|δβ+-δβ-|=(β˜2-β2)+-(β˜2-β2)-2β˜=(a--a+)Dxy2β˜.
L=2π|δβ|=2π|β+-β-|=4πβ˜|(a--a+)Dxy|,
L=2π|δβ|=2π|Dxx-Dyy|/2β˜=4πβ˜|Dxx-Dyy|.

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