Abstract

We develop from first principles the coupled wave equations that describe polarization-sensitive parametric amplification based on four-wave mixing (FWM) in standard (randomly birefringent) optical fibers. We show that in the small-signal case these equations can be solved analytically, and permit us to predict the gain experienced by the signal beam as well as its state of polarization (SOP) at the fiber output. We find that, independently of its initial value, the output SOP of a signal within the parametric gain bandwidth is solely determined by the pump SOP. We call this effect of pulling the polarization of the signal towards a reference SOP the polarization attraction, and we call the parametric amplifier the FWM polarizer (which can equivalently be called the fiber-optic parametric amplifier polarizer). Our theory is valid beyond the zero polarization mode dispersion (PMD) limit, and it takes into account moderate deviations of the PMD from zero. In particular, our theory is capable of analytically predicting the rate of degradation of the efficiency of the parametric amplifier, which is caused by the detrimental PMD effect.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. W. Magnus, “On the exponential solution of differential equations for a linear operator,” Commun. Pure Appl. Math. 7, 649–673 (1954).
    [CrossRef]
  35. V. V. Kozlov, J. Nun¯o, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers: erratum,” J. Opt. Soc. Am. B 29, 153–154 (2012).
    [CrossRef]
  36. V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
    [CrossRef]

2012 (4)

2011 (10)

V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

Nelson J. Muga, Mario F. S. Ferreira, and Armando N. Pinto, “Broadband polarization pulling using Raman amplification,” Opt. Express 19, 18707–18712 (2011).
[CrossRef]

V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
[CrossRef]

P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40  Gbit/s regeneration applications,” Opt. Express 19, 17158–17166 (2011).
[CrossRef]

V. V. Kozlov, J. Nun¯o, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[CrossRef]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[CrossRef]

V. V. Kozlov, Javier Nuño, Diego Juan Ania-Castañón, and S. Wabnitz, “Theoretical study of optical fiber Raman polarizers with counterpropagating beams,” J. Lightwave Technol. 29, 341–347 (2011).
[CrossRef]

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[CrossRef]

F. Chiarello, L. Ursini, L. Palmieri, and M. Santagiustina, “Polarization attraction in counterpropagating fiber Raman amplifiers,” IEEE Photon. Technol. Lett. 23, 1457–1459 (2011).
[CrossRef]

V. V. Kozlov, and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

2010 (5)

2009 (2)

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef]

M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express 17, 947–955 (2009).
[CrossRef]

2008 (2)

2007 (1)

2006 (1)

2005 (1)

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88–94 (2005).
[CrossRef]

2004 (3)

2003 (1)

2001 (1)

2000 (1)

1996 (1)

P. K. A. Wai, and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

1954 (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Commun. Pure Appl. Math. 7, 649–673 (1954).
[CrossRef]

Agrawal, G. P.

Ania-Castañón, Diego Juan

Ania-Castañón, J. D.

Ania-Castañón, J.-D.

V. V. Kozlov, J. Nun¯o, J.-D. Ania-Castañón, and S. Wabnitz, Trapping Polarization of Light in Nonlinear Optical Fibers: An Ideal Raman Polarizer (Springer-Verlag, 2012).

Assémat, E.

Bennink, R. S.

Boyd, R. W.

Chiarello, F.

F. Chiarello, L. Ursini, L. Palmieri, and M. Santagiustina, “Polarization attraction in counterpropagating fiber Raman amplifiers,” IEEE Photon. Technol. Lett. 23, 1457–1459 (2011).
[CrossRef]

Cirigliano, M.

Claveau, R.

Costa e Silva, M. B.

Dargent, D.

de Matos, C. J. S.

Eyal, A.

Fatome, J.

Ferrario, M.

Ferreira, Mario F. S.

Finot, C.

Fisher, R. A.

Freitas, J. F. L.

Gomes, A. S. L.

Guasoni, M.

M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high birefringence optical fibers,” J. Opt. Soc. Am. B29, in press (2012).

Haelterman, M.

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88–94 (2005).
[CrossRef]

Heebner, J. E.

Jauslin, H. R.

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
[CrossRef]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[CrossRef]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[CrossRef]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88–94 (2005).
[CrossRef]

Jopson, R.

Kanaev, A.

Kogelnik, H.

Kozlov, V. V.

V. V. Kozlov, and S. Wabnitz, “Silicon Raman polarizer,” Opt. Lett. 37, 737–739 (2012).
[CrossRef]

V. V. Kozlov, J. Nun¯o, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers: erratum,” J. Opt. Soc. Am. B 29, 153–154 (2012).
[CrossRef]

V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

V. V. Kozlov, and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
[CrossRef]

V. V. Kozlov, J. Nun¯o, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[CrossRef]

V. V. Kozlov, Javier Nuño, Diego Juan Ania-Castañón, and S. Wabnitz, “Theoretical study of optical fiber Raman polarizers with counterpropagating beams,” J. Lightwave Technol. 29, 341–347 (2011).
[CrossRef]

V. V. Kozlov, J. Nun¯o, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35, 3970–3972 (2010).
[CrossRef]

V. V. Kozlov, and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
[CrossRef]

V. V. Kozlov, J. Nun¯o, J.-D. Ania-Castañón, and S. Wabnitz, Trapping Polarization of Light in Nonlinear Optical Fibers: An Ideal Raman Polarizer (Springer-Verlag, 2012).

Lagrange, S.

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[CrossRef]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef]

Lin, Q.

Magnus, W.

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Commun. Pure Appl. Math. 7, 649–673 (1954).
[CrossRef]

Marazzi, L.

Martelli, P.

Martinelli, M.

McKinstrie, C.

Menyuk, C. R.

P. K. A. Wai, and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

Millot, G.

Morin, P.

Muga, Nelson J.

Nun¯o, J.

Nuño, Javier

Palmieri, L.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[CrossRef]

F. Chiarello, L. Ursini, L. Palmieri, and M. Santagiustina, “Polarization attraction in counterpropagating fiber Raman amplifiers,” IEEE Photon. Technol. Lett. 23, 1457–1459 (2011).
[CrossRef]

Picozzi, A.

E. Assémat, A. Picozzi, H. R. Jauslin, and D. Sugny, “Hamiltonian tools for the analysis of optical polarization control,” J. Opt. Soc. Am. B 29, 559–571 (2012).
[CrossRef]

E. Assémat, D. Dargent, A. Picozzi, H. R. Jauslin, and D. Sugny, “Polarization control in spun and telecommunication optical fibers,” Opt. Lett. 36, 4038–4040 (2011).
[CrossRef]

E. Assémat, S. Lagrange, A. Picozzi, H. R. Jauslin, and D. Sugny, “Complete nonlinear polarization control in an optical fiber system,” Opt. Lett. 35, 2025–2027 (2010).
[CrossRef]

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef]

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88–94 (2005).
[CrossRef]

Pinto, Armando N.

Pitois, S.

Popov, S.

S. Sergeyev, and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48, 56–60 (2012).
[CrossRef]

Radic, S.

Santagiustina, M.

F. Chiarello, L. Ursini, L. Palmieri, and M. Santagiustina, “Polarization attraction in counterpropagating fiber Raman amplifiers,” IEEE Photon. Technol. Lett. 23, 1457–1459 (2011).
[CrossRef]

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[CrossRef]

Sergeyev, S.

S. Sergeyev, and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48, 56–60 (2012).
[CrossRef]

Sugny, D.

Thevenaz, L.

Tur, M.

Turitsyn, K.

Ursini, L.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[CrossRef]

F. Chiarello, L. Ursini, L. Palmieri, and M. Santagiustina, “Polarization attraction in counterpropagating fiber Raman amplifiers,” IEEE Photon. Technol. Lett. 23, 1457–1459 (2011).
[CrossRef]

Wabnitz, S.

V. V. Kozlov, and S. Wabnitz, “Silicon Raman polarizer,” Opt. Lett. 37, 737–739 (2012).
[CrossRef]

V. V. Kozlov, J. Nun¯o, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers: erratum,” J. Opt. Soc. Am. B 29, 153–154 (2012).
[CrossRef]

V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

V. V. Kozlov, and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

V. V. Kozlov, Javier Nuño, Diego Juan Ania-Castañón, and S. Wabnitz, “Theoretical study of optical fiber Raman polarizers with counterpropagating beams,” J. Lightwave Technol. 29, 341–347 (2011).
[CrossRef]

V. V. Kozlov, K. Turitsyn, and S. Wabnitz, “Nonlinear repolarization in optical fibers: polarization attraction with copropagating beams,” Opt. Lett. 36, 4050–4052 (2011).
[CrossRef]

V. V. Kozlov, J. Nun¯o, and S. Wabnitz, “Theory of lossless polarization attraction in telecommunication fibers,” J. Opt. Soc. Am. B 28, 100–108 (2011).
[CrossRef]

V. V. Kozlov, and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. 35, 3949–3951 (2010).
[CrossRef]

V. V. Kozlov, J. Nun¯o, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35, 3970–3972 (2010).
[CrossRef]

S. Wabnitz, “Broadband parametric amplification in photonic crystal fibers with two zero-dispersion wavelengths,” J. Lightwave Technol. 24, 1732–1738 (2006).
[CrossRef]

S. Pitois, G. Millot, and S. Wabnitz, “Nonlinear polarization dynamics of counterpropagating waves in an isotropic optical fiber: theory and experiments,” J. Opt. Soc. Am. B 18, 432–443 (2001).
[CrossRef]

V. V. Kozlov, J. Nun¯o, J.-D. Ania-Castañón, and S. Wabnitz, Trapping Polarization of Light in Nonlinear Optical Fibers: An Ideal Raman Polarizer (Springer-Verlag, 2012).

M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high birefringence optical fibers,” J. Opt. Soc. Am. B29, in press (2012).

Wai, P. K. A.

P. K. A. Wai, and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
[CrossRef]

Zadok, A.

Zilka, E.

Commun. Pure Appl. Math. (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Commun. Pure Appl. Math. 7, 649–673 (1954).
[CrossRef]

Europhys. Lett. (1)

S. Pitois, A. Picozzi, G. Millot, H. R. Jauslin, and M. Haelterman, “Polarization and modal attractors in conservative counterpropagating four-wave interaction,” Europhys. Lett. 70, 88–94 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Sergeyev, and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48, 56–60 (2012).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23, 254–256 (2011).
[CrossRef]

F. Chiarello, L. Ursini, L. Palmieri, and M. Santagiustina, “Polarization attraction in counterpropagating fiber Raman amplifiers,” IEEE Photon. Technol. Lett. 23, 1457–1459 (2011).
[CrossRef]

V. V. Kozlov, and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23, 1088–1090 (2011).
[CrossRef]

J. Lightwave Technol. (3)

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

Opt. Express (7)

Opt. Lett. (8)

Phys. Rev. E (1)

S. Lagrange, D. Sugny, A. Picozzi, and H. R. Jauslin, “Singular tori as attractors of four-wave-interaction systems,” Phys. Rev. E 81, 016202 (2010).
[CrossRef]

Phys. Rev. Lett. (1)

D. Sugny, A. Picozzi, S. Lagrange, and H. R. Jauslin, “Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems,” Phys. Rev. Lett. 103, 034102 (2009).
[CrossRef]

Other (2)

M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high birefringence optical fibers,” J. Opt. Soc. Am. B29, in press (2012).

V. V. Kozlov, J. Nun¯o, J.-D. Ania-Castañón, and S. Wabnitz, Trapping Polarization of Light in Nonlinear Optical Fibers: An Ideal Raman Polarizer (Springer-Verlag, 2012).

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Figures (5)

Fig. 1.
Fig. 1.

Dependence of signal gain gs on its frequency detuning from the pump, with L=300m. Curves and circles were obtained with z-varying or average M coefficients, respectively. Moreover Dp=0 (solid curve), Dp=0.50pskm1/2 (dashed curve), Dp=0.75pskm1/2 (dotted curve), and Dp=5pskm1/2 (dash-dotted curve).

Fig. 2.
Fig. 2.

Same as Fig. 1, but for the signal DOP.

Fig. 3.
Fig. 3.

Signal DOP versus fiber length L with Dp=0.75pskm1/2, and different values of the sideband detuning frequency: Δν=0.255THz (solid curve), Δν=0.350THz (dashed curve), Δν=0.365THz (dotted curve), and Δν=0.380THz (dash-dotted curve).

Fig. 4.
Fig. 4.

Tips of input (a) and output (b) signal Stokes vectors on the Poincaré sphere for a fiber length L=500m, Dp=0.75pskm1/2, and Δν=0.255THz. For the sake of clarity, only 225 vectors are represented instead of the 10 000 used in the simulations. Input vectors are distributed uniformly over the sphere. The black triangle represents the input pump Stokes vector.

Fig. 5.
Fig. 5.

Distribution of the output signal Stokes vectors with L=500m and Δν=0.350THz. Panels (a) and (b) display opposite views of the Poincaré sphere.

Equations (26)

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iUpz+ΔB(ωp,z)Up+23γ[(Up·Up*)Up+12(Up·Up)Up*]+23γ[(Us·Us*)Up+(Us·Up)Us*+(Up·Us*)Us+(Ui·Ui*)Up+(Ui·Up)Ui*+(Up·Ui*)Ui]+23γexp(iΔkz)[(Ui·Us)Up*+(Ui·Up*)Us+(Us·Up*)Ui]=0,
iUsz+ΔB(ωs,z)Us+23γ[(Us·Us*)Us+12(Us·Us)Us*]+23γ[(Up·Up*)Us+(Up·Us)Up*+(Us·Up*)Up+(Ui·Ui*)Us+(Ui·Us)Ui*+(Us·Ui*)Ui]+23γexp(iΔkz)[12(Up·Up)Ui*+(Up·Ui*)Up]=0,
iϕ1pz+23(2+23Ca(z))ϕ1iϕ1pϕ1i*+89ϕ1p2ϕ1p*+23(2Ca(z)+23Cb(z))ϕ1iϕ1sϕ1p*eiΔkz+23(2+23Ca(z))ϕ1pϕ1sϕ1s*+23(223Ca(z))ϕ1pϕ2iϕ2i*+89Ca(z)ϕ1iϕ2pϕ2i*+23(Ca(z)+13Cb(z))ϕ1sϕ2iϕ2p*eiΔkz+89ϕ1pϕ2pϕ2p*+23(Ca(z)+13Cb(z))ϕ1iϕ2sϕ2p*eiΔkz+89Ca(z)ϕ1sϕ2pϕ2s*+23(223Ca(z))ϕ1pϕ2sϕ2s*=0,
iϕ1sz+23(Ca(z)+13Cb(z))ϕ1p2ϕ1i*eiΔkz+23(2+23Ca(z))ϕ1iϕ1sϕ1i*+23(2+23Ca(z))ϕ1pϕ1sϕ1p*+89ϕ1s2ϕ1s*+23(223Ca(z))ϕ1sϕ2iϕ2i*+23(Ca(z)+13Cb(z))ϕ1pϕ2pϕ2i*eiΔkz+89Ca(z)ϕ1iϕ2sϕ2i*+23(223Ca(z))ϕ1sϕ2pϕ2p*+89Ca(z)ϕ1pϕ2sϕ2p*+89ϕ1sϕ2sϕ2s*=0,
iϕ2sz+23(Ca(z)+13Cb(z))ϕ2p2ϕ2i*eiΔkz+23(2+23Ca(z))ϕ2iϕ2sϕ2i*+23(2+23Ca(z))ϕ2pϕ2sϕ2p*+89ϕ2s2ϕ2s*+23(223Ca(z))ϕ2sϕ1iϕ1i*+23(Ca(z)+13Cb(z))ϕ2pϕ1pϕ1i*eiΔkz+89Ca(z)ϕ2iϕ1sϕ1i*+23(223Ca(z))ϕ2sϕ1pϕ1p*+89Ca(z)ϕ2pϕ1sϕ1p*+89ϕ2sϕ1sϕ1s*=0.
Dp=22πωpLcLB.
ϕ1p=Pexp(iθ1p0+iγ(8/9)Ptotz),ϕ2p=Qexp(iθ2p0+iγ(8/9)Ptotz),
ϕ˜z=i89M(z)ϕ˜,
M(z)=[FA(z)P+FB(z)Q+v/2FC(z)PFD(z)PQFC(z)PQFB(z)PFA(z)PFB(z)Qv/2FC(z)PQFD(z)PQFD(z)PQFC(z)PQFB(z)P+FA(z)Q+v/2FC(z)QFC(z)PQFD(z)PQFC(z)QFB(z)PFA(z)Qv/2]
ϕ˜(z=L)=exp(Ω(L))ϕ˜(z=0),
Ω(L)Ω1(L)=z=0LM(z)dzM¯
F¯(A,B,C,D)=1Lz=0LF(A,B,C,D)(z)dz,
ge2=490γ2Ptot2(4+5C¯a2+C¯b28C¯a+6C¯aC¯b)14β22Δω449β2(1+C¯a)γPtotΔω2.
Δωc2=4c(6γLc1Ptot27|β2|c2Lc1+41Dp2γPtotc2Lc1L),Δωp=Δωc/2,ge,peak2=83(γ2Ptot2(3|β2|c2Lc1Dp2γPtotLc2Lc1)27β2c2Lc1+41Dp2γPtotc2Lc1L),
ϕ1iϕ2i=PQeiθ1p0iθ2p0=ϕ1p(z=0)ϕ2p(z=0).
ΔB¯(ωf)=(b(ωf)i2gθ±i2gθb(ωf)).
Tp(z)=(a1(z)a2(z)a2*(z)a1*(z))
iTpz+ΔB¯p·Tp=0.
iΦfz+γ(Nspm+Nxpm+Nex)f=0.
f=limz1z0zdzf(z),
(2Lc12Lc10002Δ()2Lc12Lc1002Δ(+)000002Δ(+)0000002Δ()0Δ(+)Δ(+)0Lc10Δ()00Δ()0Lc1).
(2Lc12Lc10000Δ()Δ()2Lc12Lc100Δ(+)Δ(+)000000Δ(+)Δ(+)00000000Δ()Δ()0Δ(+)Δ(+)0Lc10000Δ(+)Δ(+)00Lc100Δ()00Δ()00Lc10Δ()00Δ()000Lc1).
(u11u19,u12u20,u13u21,u14u22,u13u20,u12u21,u14u19,u11u22)T;(u11u19*,u12u20*,u13u21*,u14u22*,u13u20*,u12u21*,u14u19*,u11u22*)T;(u7u15,u8u16,u9u17,u10u18,u9u16,u8u17,u10u15,u7u18)T;(u7u15*,u8u16*,u9u17*,u10u18*,u9u16*,u8u17*,u10u15*,u7u18*)T.
(00Δ(+)Δ(+)00Δ(+)Lc10Δ(+)00Δ(+)0Lc1Δ(+)000Δ(+)Δ(+)2Lc12Lc1000002Lc12Lc10000Δ()Lc1)
(u6u29,u5u29,u6u28,u5u28,u4u27,u4u30)T;(u6u29*,u5u29*,u6u28*,u5u28*,u4u27*,u4u30*)T;(u3u25,u3u24,u2u25,u2u24,u1u23,u1u26)T;(u3u25*,u3u24*,u2u25*,u2u24*,u1u23*,u1u26*)T.
det(MAλI)=32[Δ()]2[Δ(+)]2Lc1λ+(16[Δ()]2[Δ(+)]2+12[Δ()]2Lc2+12[Δ(+)]2Lc2)λ2+(16[Δ()]2Lc1+16[Δ(+)]2Lc1+4Lc3)λ3+(4[Δ()]2+4[Δ(+)]2+9Lc2)λ4+6Lc1λ5+λ6.

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