Abstract

Polarization mode dispersion is the effect of signal broadening in a fiber with birefringent disorder. The disorder, frozen into the fiber, is characterized by the so-called vector of birefringence (VB). In a linear medium a pulse broadens as the two principal states of polarization split. It is well-known that, under the action of short-correlated disorder, naturally present in fibers, the dispersion vector (DV), characterizing the split, performs a Brownian random walk. We discuss a strategy of passive (i.e., pulse-independent) control of the DV broadening. The suggestion is to pin (compensate) periodically or quasi-periodically the integral of VB to zero. As a result of the influence of pinning, the probability distribution function of the DV becomes statistically steady in the linear case. Moreover, pinning improves confinement of the pulse in the weakly nonlinear case. The theoretical findings are confirmed by numerical analysis.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  37. M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
    [CrossRef]

2003 (1)

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

2002 (1)

2001 (3)

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in an imperfect medium,” JETP Lett. 74, 357–361 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Solitons in a disordered anisotropic optical medium,” JETP Lett. 74, 535–538 (2001).
[CrossRef]

2000 (4)

1999 (2)

M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939–941 (1999).
[CrossRef]

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

1997 (3)

A. Carena, V. Curri, R. Gausino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. 15, 751–765 (1997).
[CrossRef]

P. K. Wai, W. L. Kath, C. R. Menyuk, and J. W. Zhang, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 14, 2967–2979 (1997).
[CrossRef]

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

1996 (1)

N. Gisin, B. Gisin, J. P. Von der Weid, and R. Passy, “How accurately can one measure a statistical quantity like polarization-mode dispersion,” IEEE Photon. Technol. Lett. 8, 1671–1673 (1996).
[CrossRef]

1992 (1)

N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316–323 (1992).
[CrossRef]

1991 (3)

1990 (1)

F. C. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1165 (1990).
[CrossRef]

1989 (2)

1988 (2)

C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
[CrossRef] [PubMed]

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190 (1988).
[CrossRef]

1987 (2)

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. LT-5, 1618–1622 (1987).
[CrossRef]

D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurements of the group velocity difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
[CrossRef] [PubMed]

1986 (1)

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibres,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

1981 (3)

S. Machida, I. Sakai, and T. Kimura, “Polarization conservation in single-mode fibres,” Electron. Lett. 17, 494–495 (1981).
[CrossRef]

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
[CrossRef]

W. Eickhoff, Y. Yen, and R. Ulrich, “Wavelength dependence of birefringence in single-mode fiber,” Appl. Opt. 20, 3428–3435 (1981).
[CrossRef] [PubMed]

1979 (1)

1978 (1)

1977 (1)

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Andresciani, D.

Bao, X.

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Benedetto, S.

A. Carena, V. Curri, R. Gausino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. 15, 751–765 (1997).
[CrossRef]

Bergano, N. S.

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190 (1988).
[CrossRef]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. LT-5, 1618–1622 (1987).
[CrossRef]

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Brentel, J.

Cameron, J.

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Carena, A.

A. Carena, V. Curri, R. Gausino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. 15, 751–765 (1997).
[CrossRef]

Chen, H. H.

Chen, L.

L. Chen, J. Cameron, and X. Bao, “Statistics of polarization mode dispersion in presence of the polarization dependent loss in single mode fibers,” Opt. Commun. 169, 69–73 (1999).
[CrossRef]

Chertkov, M.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

M. Chertkov, I. Gabitov, P. Lushnikov, J. Moeser, and Z. Toroczkai, “Pinning method of pulse confinement in optical fiber with random dispersion,” J. Opt. Soc. Am. B 19, 2538–2550 (2002).
[CrossRef]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in an imperfect medium,” JETP Lett. 74, 357–361 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Solitons in a disordered anisotropic optical medium,” JETP Lett. 74, 535–538 (2001).
[CrossRef]

Chung, Y.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

Curri, V.

A. Carena, V. Curri, R. Gausino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. 15, 751–765 (1997).
[CrossRef]

Curti, F.

Curti, F. C.

F. C. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1165 (1990).
[CrossRef]

Daino, B.

F. C. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1165 (1990).
[CrossRef]

D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurements of the group velocity difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
[CrossRef] [PubMed]

de Marchis, G.

F. C. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1165 (1990).
[CrossRef]

Dyachenko, A.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

Eickhoff, W.

Foschini, G. J.

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fiber,” J. Lightwave Technol. 9, 1439–1456 (1991).
[CrossRef]

Gabitov, I.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

M. Chertkov, I. Gabitov, P. Lushnikov, J. Moeser, and Z. Toroczkai, “Pinning method of pulse confinement in optical fiber with random dispersion,” J. Opt. Soc. Am. B 19, 2538–2550 (2002).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in an imperfect medium,” JETP Lett. 74, 357–361 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Solitons in a disordered anisotropic optical medium,” JETP Lett. 74, 535–538 (2001).
[CrossRef]

Gausino, R.

A. Carena, V. Curri, R. Gausino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. 15, 751–765 (1997).
[CrossRef]

Gisin, B.

N. Gisin, B. Gisin, J. P. Von der Weid, and R. Passy, “How accurately can one measure a statistical quantity like polarization-mode dispersion,” IEEE Photon. Technol. Lett. 8, 1671–1673 (1996).
[CrossRef]

Gisin, N.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

N. Gisin, B. Gisin, J. P. Von der Weid, and R. Passy, “How accurately can one measure a statistical quantity like polarization-mode dispersion,” IEEE Photon. Technol. Lett. 8, 1671–1673 (1996).
[CrossRef]

N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316–323 (1992).
[CrossRef]

Gordon, J. P.

Huttner, B.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119–125 (1997).
[CrossRef]

Jopson, R. M.

Kaminow, I. P.

I. P. Kaminow, “Polarization in optical fibers,” IEEE J. Quantum Electron. QE-17, 15–22 (1981).
[CrossRef]

Karlsson, M.

Kath, W. L.

Kimura, T.

S. Machida, I. Sakai, and T. Kimura, “Polarization conservation in single-mode fibres,” Electron. Lett. 17, 494–495 (1981).
[CrossRef]

Kogelnik, H.

Kolokolov, I.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in an imperfect medium,” JETP Lett. 74, 357–361 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Solitons in a disordered anisotropic optical medium,” JETP Lett. 74, 535–538 (2001).
[CrossRef]

Lebedev, V.

M. Chertkov, Y. Chung, A. Dyachenko, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in weakly disordered optical fibers,” Phys. Rev. E 67, 036615 (2003).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Shedding and interaction of solitons in an imperfect medium,” JETP Lett. 74, 357–361 (2001).
[CrossRef]

M. Chertkov, I. Gabitov, I. Kolokolov, and V. Lebedev, “Solitons in a disordered anisotropic optical medium,” JETP Lett. 74, 535–538 (2001).
[CrossRef]

Li, Y.

Lushnikov, P.

Machida, S.

S. Machida, I. Sakai, and T. Kimura, “Polarization conservation in single-mode fibres,” Electron. Lett. 17, 494–495 (1981).
[CrossRef]

Matera, F.

F. C. Curti, B. Daino, G. de Marchis, and F. Matera, “Statistical treatment of the evolution of the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1165 (1990).
[CrossRef]

D. Andresciani, F. Curti, F. Matera, and B. Daino, “Measurements of the group velocity difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
[CrossRef] [PubMed]

Menyuk, C. R.

Midrio, M.

Moeser, J.

M. Chertkov, I. Gabitov, P. Lushnikov, J. Moeser, and Z. Toroczkai, “Pinning method of pulse confinement in optical fiber with random dispersion,” J. Opt. Soc. Am. B 19, 2538–2550 (2002).
[CrossRef]

M. Chertkov, I. Gabitov, and J. Moeser, “Pulse confinement in optical fibers with random dispersion,” Proc. Natl. Acad. Sci. U.S.A. 98, 14208–14211 (2001).
[CrossRef] [PubMed]

Mollenauer, L. F.

Nagel, J. A.

Nelson, L. E.

Passy, R.

N. Gisin, B. Gisin, J. P. Von der Weid, and R. Passy, “How accurately can one measure a statistical quantity like polarization-mode dispersion,” IEEE Photon. Technol. Lett. 8, 1671–1673 (1996).
[CrossRef]

Pellaux, J. P.

N. Gisin and J. P. Pellaux, “Polarization mode dispersion: time versus frequency domains,” Opt. Commun. 89, 316–323 (1992).
[CrossRef]

Poggiolini, P.

A. Carena, V. Curri, R. Gausino, P. Poggiolini, and S. Benedetto, “A time-domain optical transmission system simulation package accounting for nonlinear and polarization-related effects in fiber,” IEEE J. Sel. Areas Commun. 15, 751–765 (1997).
[CrossRef]

Poole, C. D.

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fiber,” J. Lightwave Technol. 9, 1439–1456 (1991).
[CrossRef]

C. D. Poole, J. H. Winters, and J. A. Nagel, “Dynamical equation for polarization dispersion,” Opt. Lett. 16, 372–374 (1991).
[CrossRef] [PubMed]

C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,” Opt. Lett. 14, 523–525 (1989).
[CrossRef] [PubMed]

C. D. Poole, “Statistical treatment of polarization dispersion in single-mode fiber,” Opt. Lett. 13, 687–689 (1988).
[CrossRef] [PubMed]

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190 (1988).
[CrossRef]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. LT-5, 1618–1622 (1987).
[CrossRef]

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibres,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Rashleigh, S. C.

Sakai, I.

S. Machida, I. Sakai, and T. Kimura, “Polarization conservation in single-mode fibres,” Electron. Lett. 17, 494–495 (1981).
[CrossRef]

Schulte, H. J.

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190 (1988).
[CrossRef]

Simon, A.

Smith, K.

Toroczkai, Z.

Ulrich, R.

Von der Weid, J. P.

N. Gisin, B. Gisin, J. P. Von der Weid, and R. Passy, “How accurately can one measure a statistical quantity like polarization-mode dispersion,” IEEE Photon. Technol. Lett. 8, 1671–1673 (1996).
[CrossRef]

Wagner, R. E.

C. D. Poole, N. S. Bergano, R. E. Wagner, and H. J. Schulte, “Polarization dispersion and principal states in a 147-km undersea lightwave cable,” J. Lightwave Technol. 6, 1185–1190 (1988).
[CrossRef]

N. S. Bergano, C. D. Poole, and R. E. Wagner, “Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers,” J. Lightwave Technol. LT-5, 1618–1622 (1987).
[CrossRef]

C. D. Poole and R. E. Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibres,” Electron. Lett. 22, 1029–1030 (1986).
[CrossRef]

Wai, P. K.

Wai, P. K. A.

Winters, J. H.

Yariv, A.

Yen, Y.

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M. Chertkov, I. Gabitov, I. Kolokolov, V. Lebedev, and K. Turitsyn, “Solitons, birefringence and disorder,” in preparation.

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Figures (3)

Fig. 1
Fig. 1

Pinning in the case of δ-correlated noise. The amplitude of the noise is D=0.3, corresponding to a PMD parameter of δ=0.03 ps/km, 23 if the propagation distance is normalized with a length L=100 km. The growth of the pulse width is periodically suppressed. If the system is linear and the noise is δ correlated, this suppression is complete. The pinning period is here (and in the following numerical simulations) 0.5 (in dimensionless units), corresponding to 50 km.

Fig. 2
Fig. 2

Numerical results for the linear case for short-correlated noise. The amplitude of the noise is D=0.3. The solid line represents the analytical result (3Dz) of the δ-correlated case, the dashes represent the numerical simulations in the unpinned case for short-correlated noise, and the dots represent the pinned case for short-correlated noise. TPMD2 is averaged over 500 realizations. The correlation length is 0.002. Again, the pinning period is 0.5, and the total length is normalized to unity.

Fig. 3
Fig. 3

Numerical results for the nonlinear case with the initial condition being a perfect soliton. The effect of PMD is considerably reduced by the application of pinning. The solid curve is for the unpinned case, and the dashed curve shows the pinned-case curve. The other parameters of the simulation are the same as those in the linear case.

Equations (114)

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izΨα+ΔαβΨβ+imαβtΨβ+d(z)t2Ψα=-δWδΨα*.
6W=[E2 exp(2iκ0z)+2|E|2+E*2 exp(-2iκ0z)]2=2E2E*2+4|E|4=2(Ψ12+Ψ22)(Ψ1*2+Ψ2*2)+4(|Ψ1|2+|Ψ2|2)2,
δWδΨα*=23Ψα*(Ψ12+Ψ22)+43Ψα(|Ψ1|2+|Ψ2|2).
Ψ1Ψ2cos θsin θ-sin θcos θΨ1Ψ2.
mˆ=jhj(z)σˆj,
Dh(z)exp-12Dh2dz,
hi(z)hj(z)=Dδijδ(z-z).
Dh(z)exp-12Dh2dznδL(n)L(n+1)dzh,
hi(z1)hj(z2)=Dδijδ(z1-z2)-1L(n+1)-L(n)
izΨ+[imˆ(z)t+d(z)t2]Ψ=0.
Ψ(t; z)=-+dω-+dt2πexp[-iω(t-t)-iω2R(z)]Wˆ(z|0; ω)Ψ(t; 0),
Wˆ(z|0; ω)T expiω0zdzmˆ(z), R(z)=0zd(z)dz,
Wˆ=u1u2-u2*u1*,|u1(z; ω)|2+|u2(z; ω)|2=1.
Ψ(z; ω)=Wˆ(z|0; ω)Ψ0,
Ψz-iωmˆΨ=0.
ωΨ(z; ω)=ωWˆ(z|0; ω)Wˆ-1(z|0; ω)Ψ(z; ω).
JˆωWˆ(z|0; ω)Wˆ-1(z|0; ω)
Tr(Jˆ)=ω(|u1|2+|u2|2)=0.
z(Wˆ-1)=-iωWˆ-1mˆ
zJˆ=imˆ+iω[mˆ, Jˆ].
Ψ(l; ω)=Ψ1(l; ω)Ψ2(l; ω)=ψp(l, ω)ep(l, ω),
ddωep(l, ω)=0,
dΨ(l, ω)dω=dψp(l, ω)dωep(l, ω)=Jˆ(l, ω)Ψ(l, ω)=Jˆ(l, ω)ψp(l, ω)ep(l, ω);
Jˆ(l, ω)ep(l, ω)=1ψp(l, ω)dψp(l, ω)dωep(l, ω).
λ=±i[|u1(l; ω)|2+|u2(l; ω)|2]1/2,
ψp(l, ω)=ap(l, ω)exp[iϕp(l, ω)].
λ=1ψp(l, ω)dψp(l, ω)dω=1ap(l, ω)dap(l, ω)dω+idϕp(l, ω)dω.
Δt=dϕp+(l, ω)dω-dϕp-(l, ω)dω=2[|u1(l; ω)|2+|u2(l; ω)|2]1/2.
det(Jˆ)=|u1(l; ω)|2+|u2(l; ω)|2=Δt24.
σˆ1=0110,σˆ2=0-ii0,σˆ3=100-1.
Πi=(Ψ1*Ψ2*)σˆiΨ1Ψ2.
Π(z; ω)=R(z; ω)Π0,
2R(z; ω)=u1*2-u22+u12-u2*2u1*2-u22-u12+u2*2-2(u1u2+u1*u2*)i(u12+u2*2-u1*2-u22)u12+u2*2+u1*2+u222i(u1*u2*-u1u2)2(u1u2*+u1*u2)2i(u1*u2-u1u2*)2(|u1|2-|u2|2).
Jˆ=iiΩiσi=iΩ·σ,
2iΩ1=u1u2-u1u2+u1*u2*-u1*u2*,
2Ω2=-u1u2+u1u2+u1*u2*-u1*u2*,
iΩ3=u1u1*+u2u2*=-u1u1*-u2u2*,
ωΨ(z; ω)=iΩ3iΩ1+Ω2iΩ1-Ω2-iΩ3Ψ(z; ω)
ωΠ(z; ω)=2[Π(z; ω)×Ω(z; ω)].
zΠ(z; ω)=2ω[Π(z; ω)×h(z)].
[mˆ, Jˆ]=ik,jhkΩj[σˆk, σˆj]=-2[(h1Ω2-h2Ω1)σˆ3-(h1Ω3-h3Ω1)σˆ2+(h2Ω3-h3Ω2)σˆ1],
zΩ(z; ω)=h(z)-2ω[h(z)×Ω(z; ω)].
|Ω(l, ω)|=det(Jˆ)=Δt2,
Ψ(0; ω)=f(ω)0
ϕ1(ω)ϕ2(ω)=Wˆ(z, ω)Ψ(0; ω),
T˜2=12π[|ϕ1(ω)|2+ϕ2(ω)|2]dω=t2[|ϕ1(t)|2+|ϕ2(t)|2]dt,
|ϕ1|2+|ϕ2|2=(|u1|2+|u2|2)|f|2+|f|2,
T˜PMD2=12π(|ϕ1|2+|ϕ2|2-|f|2)dω
ψ1(ω, z)ψ2(ω, z)=exp[-iω2R(z)]u1(ω, z)f(ω)-u2*(ω, z)f(ω),
|ψ1|2+|ψ2|2=(|u1|2+|u2|2)|f|2+|f|2+4ω2R2|f|2+2iωR(u1*u1-u1u1*+u2*u2-u2u2*)|f|2,
TPMD2=12π(|ψ1|2+|ψ2|2)dω-(4ω2R2|f|2+|f|2)dω.
TPMD2=|Ω|2|f|2dω.
P(z|{Ωn})={Qn(z)=Ωn}n=1NDQnDpn×exp0zdz-in=1NpnddzQn-D2×n=1N[pn-2ωn(Qn×pn)]2×P(0|{Qn(0)})={Qn(z)=Ωn}n=1NDQn(z)Dpn(z)×exp-0zdzSP(0|{Qn(0)}),
S=in=1NpnddzQn+D2k,n=1N{pkpn+2(ωnQn-ωkQk)(pk×pn)+4ωnωk[(pkpn)(QkQn)-(Qkpn)(Qnpk)]}.
z-D2k,n=1N[Ω;kiΩ;ni+2(Ω;k×Ω;n)(ωnΩn-ωkΩk)+4ωnωk(Ω;kiΩkjΩ;niΩnj-Ω;kiΩkjΩ;njΩni)]P=0.
zΩ2n=n(2n+1)DΩ2n-2
P(|Ω|)=4|Ω|2Δ3πexp-Ω2Δ2,
Δ223Ω02+2Dz.
z(Ω1Ω2)n=D2(Ω;2+Ω;1)2(Ω1Ω2)n+2D(ω1Ω1-ω2Ω2)(Ω;2×Ω;1)(Ω1Ω2)n+2D2ω1ω2[(Ω1Ω2)n(Ω;1Ω;2)-Ω1iΩ;1jΩ2jΩ;2i]+k=12ωk2(ΩkjΩ;kiΩkjΩ;ki-ΩkjΩ;kiΩkiΩ;kj)(Ω1Ω2)n=n(n-1)2D(Ω1Ω2)n-2(Ω22+Ω12)+Dn(n+2)(Ω1Ω2)n-1+2Dn(n-1)×(ω1-ω2)2(Ω1Ω2)n-2Ω22Ω12-2Dn(n+1)(ω1-ω2)2(Ω1Ω2)n,
zΩ22nΩ12m=D2(Ω;2+Ω;1)2Ω22nΩ12m=D[n(2n+1)Ω22n-2Ω12m+m(2m+1)Ω22nΩ12m-2]+4Dnm(Ω1Ω2)Ω22n-2Ω12m-2.
(Ω1Ω2)(z)=exp[-4D(ω1-ω2)2z](Ω1Ω2)(0)+31-exp[-4D(ω1-ω2)2z]4(ω1-ω2)2,
z(Ω1Ω2)2=DΩ12+Ω22+8D(Ω1Ω2)+4D(ω1-ω2)2Ω22Ω12-12D(ω1-ω2)2(Ω1Ω2)2.
Ω22Ω12=0zdz[3DΩ12+Ω22+4D(Ω1Ω2)]=9D2z2+3D(Ω12+Ω22)0z+(Ω1Ω2)01-exp[-4D(ω1-ω2)2z](ω1-ω2)2+34D(ω1-ω2)2z-{1-exp[-4D(ω1-ω2)2z]}4(ω1-ω2)4,
(Ω1Ω2)2=exp[-12D(ω1-ω2)2z](Ω1Ω2)02+0zdz exp[-12D(ω1-ω2)2(z-z)][DΩ12+Ω22+8D(Ω1Ω2)+4D(ω1-ω2)2Ω22Ω12]z=3D2z2+16ω124+Dzω122+Dz(Ω12+Ω22)0+(Ω1Ω2)03ω122+exp(-4Dzω122)(Ω1Ω2)02ω122-38ω124+exp(-12Dzω122)(Ω1Ω2)02+5(Ω1Ω2)06ω122+524ω124.
P(z|{Θn})=dqn=1NdΦnexp-q2LD2×n=1Nδ(Ωn-Φn)×exp[-Hˆq(L-z)]×n=1Nδ(Ωn-Θn)n=1Nδ(Ωn-Θn)×exp(-Hˆqz)|P(0|{Ωn})=dq exp-q2LD2Pq(z|{Θn}),
Pq(z|{Ωn})exp(-Hˆqz)|P(0|{Ωn}), HˆqHˆ0+iDn=1NΩ;n[q-2ωn(q×Ωn)],
Hˆ0-D2k,n=1N[Ω;kiΩ;ni+2(Ω;k×Ω;n)(ωnΩn-ωkΩk)-4ωnωk(Ω;kiΩkjΩ;niΩnj-Ω;kiΩkjΩ;njΩni)].
qiqj=δijDL.
zΩq=iDq,zΩ2q=2iDqΩq+3D,
zΩ1Ω2q=iDqΩ1+Ω2q-2iDω12qΩ1×Ω2q+3D-4Dω122Ω1Ω2q,
Ωiq=Ωi(0)+iDqz,
Ωi2q=Ωi2(0)+3Dz-D2q2z2+iDzqΩi(0).
Ω1αΩ2βq=δαβA3+δαβ3-qαqβq2B,
(Ω1Ω2)q;isot=D0zdz exp[-4Dω122(z-z)]×(3-2q2Dz),
(Ω1Ω2)q=(Ω1Ω2)q;isot+(Ω1(0)Ω2(0))×exp(-4Dω122z)+iDq{Ω1(0)+Ω2(0)-2ω12[Ω1(0)×Ω2(0)]}×1-exp(-4Dω122z)4Dω12w,
(Ω1Ω2)=3+q22ω1221-exp(-4Dω122z)4ω122-zDq22ω122+(Ω1(0)Ω2(0))exp(-4Dω122z).
Δpin2=23[Ω02+3Dz(1-z/L)].
TPMD2=|Ω|2=32Δ2
Ψ(t; z)-+dω exp(-iωt)Wˆ(z; ω)Φω(z).
(iz-ω12)Φ1α+2dω2,3,4δ(ω1-ω2+ω3-ω4)×Ξαβνη(ω1,2,3,4|z)Φ2βΦ3*νΦ4η=0,
Ξαβνη(ω, ω1,2,3,4|z)43[Wˆ-1(z; ω1)Wˆ(z; ω2)]αβ×W*μν(z; ω3)Wμη(z; ω4)+23[Wˆ-1(z; ω1)Wˆ*(z; ω3)]αν×Wμβ(z; ω2)Wμη(z; ω4).
1z0zΞαβνη(ω1,2,3,4|z)dz
Wˆ(z+dz; ω)=exp[i(ωmˆ+ω2Iˆ2)dz],
ψin(t)=π-1/4 exp(-t2/2)10
zΨ1=ic(3|Ψ1|2Ψ1+2|Ψ2|2Ψ1+Ψ1*Ψ22),
zΨ2=ic(3|Ψ2|2Ψ2+2|Ψ1|2Ψ2+Ψ2*Ψ12),
|Ψ1|2+|Ψ2|2=const.=E,
Ψ1Ψ2*-Ψ2Ψ1*=const.,
(Ψ12+Ψ22)(z)=exp(6icEz)ζ(0),
ζ(0)=(Ψ12+Ψ22)(0).
zΨα=ic[2EΨα+exp(6icEz)ζ(0)Ψα*].
zηα=ic[-Eηα+ζ(0)ηα*].
T˜PMD2=12π(|ϕ1|2+|ϕ2|2-|f|2)dω.
Vˆ(ω1|ω2)Wˆ-1(z; ω1)Wˆ(ω2).
Σαβ;μν(ω1, ω2; ω3, ω4)
=Vαβ(ω1|ω2)Vμν(ω3|ω4),
Wˆ(z)=[1+iωhi(z)σˆi-12ω22(hiσˆi)2]Wˆ(z-),
hi(z)hj(z)=Dδij.
Wˆ(z|q)={1+iω[hi(z)-Dqi]σˆi-12ω22(hiσˆi)2}×Wˆ(z-|q),
ddzVˆ=-3D2ω122Vˆ.
Vˆ(ω1|ω2)=1ˆ exp(-3Dω122z/2),
ddzVˆqqi[Wˆ1-1(z)σˆiWˆ2(z)]q=D-3ω1222iω12iq2ω12-ω1222Vˆqqi[Wˆ1-1(z)σˆiWˆ2(z)]q.
Vˆ(z)=1ˆ(DL)3/2 exp(-Dzω122)2π0q2dq×exp-q2DL2coshDzω122 ω122-4q2-ω12ω122-4q2 sinhDzω122 ω122-4q2.
Φαβ;μη(z)=[Wˆ1-1(z)σˆiWˆ2(z)]αβ[Wˆ3-1(z)σˆiWˆ4(z)]μη.
σˆiσˆjσˆi=-σˆj,σˆiσˆj=δij+iεijkσˆk,
ddzΣαβ;μηΦαβ;μη=-3D2ω122+ω34223ω12ω342ω12ω34ω12+ω22+ω32+ω42+23ω1ω2+23ω3ω4-43(ω1+ω2)(ω3+ω4)Σαβ;μηΦαβ;μη.
ddzΣqαβ;μηΦqαβ;μη=-3D2ω122+ω34223ω12ω342ω12ω34ω12+ω22+ω32+ω42+23ω1ω2+23ω3ω4-43(ω1+ω2)(ω3+ω4)Σqαβ;μηΦqαβ;μη+Diω12X12;34αβ;μη+iω34X34;12μη;αβiω34X12;34αβ;μη+iω12X34;12μη;αβ+(ω12-ω34)Υ12;34αβ;μη,
X12;34αβ;μη(z|q)qi[Wˆ1-1(z)σˆiWˆ2(z)]αβ[Wˆ3-1(z)Wˆ4(z)]μη,
Υ12;34αβ;μη(z|q)εijkqi[Wˆ1-1(z)σˆjWˆ2(z)]αβ[Wˆ3-1(z)σˆkWˆ4(z)]μη.
ddzX12;34αβ;μη=iq2Dω12Σqαβ;μη+iDω34q2Gαβ;μη-ω12ω34D(X34;12μη;αβ+iΥ12;34αβ;μη)-3D2ω12+ω22+23ω1ω2+ω342X12;34αβ;μη,
Gαβ;μηqiqjq2[Wˆ1-1(z)σˆiWˆ2(z)]αβ[Wˆ3-1(z)σˆjWˆ4(z)]μη,
ddzGαβ;μη=-3D2ω12+ω22+ω32+ω42+23ω1ω2+23ω3ω4-43(ω1+ω2)(ω3+ω4)Gαβ;μη-ω12ω34D(Σqαβ;μη+Gαβ;μη-Φqαβ;μη)+iD(ω34X12;34αβ;μη+ω12X34;12μη;αβ).
Y12;34αβ;μηX12;34αβ;μη+X34;12μη;αβ.
ddzΣΦY=D-3ω122-ω1222iω12-32ω122-ω12+2ω1ω2-5ω22+8ω2ω3-4ω322iω12iq2ω12iq2ω12-4(ω12+ω22-ω1ω2)ΣΦY.

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