Abstract

In this work, polarization attraction is meant to be the conservative nonlinear effect that transforms any arbitrary input state of polarization (SOP) of an intense optical signal beam fed to a nonlinear medium into approximately one and the same SOP at the output, provided that the medium is driven by a relatively stronger counterpropagating pump beam. Essentially, the combination of the nonlinear medium and the pump beam serves as a lossless polarizer for the signal beam. The degree of polarization of the outcoming signal beam can be close to 100% (90% in our present simulations). With an eye toward the development of such lossless polarizers for fiber optics applications, we theoretically study the polarization attraction effect in the optical fibers that are used in telecommunication links; i.e., randomly birefringent fibers. A generic model for the fiber-based lossless polarizers is derived, and a statistical scheme for the quantification of their performance is proposed.

© 2011 Optical Society of America

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Corrections

Victor V. Kozlov, Javier Nuño, and Stefan Wabnitz, "Theory of lossless polarization attraction in telecommunication fibers: erratum," J. Opt. Soc. Am. B 29, 153-154 (2012)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-29-1-153

References

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  1. V. E. Zakharov and A. V. Mikhailov, “Polarization domains in nonlinear optics,” Pis’ma Zh. Eksp. Teor. Fiz., 45, 279–282 (1987) [JETP Lett. 45, 349–352 (1987)].
  2. S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
    [CrossRef]
  3. S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009).
    [CrossRef] [PubMed]
  4. S. Wabnitz, “Cross polarization modulation domain wall solitons for WDM signals in birefringent optical fibers,” IEEE Photonics Technol. Lett. 21, 875–877 (2009).
    [CrossRef]
  5. A. V. Mikhailov and S. Wabnitz, “Polarization dynamics of counterpropagating beams in optical fibers,” Opt. Lett. 15, 1055–1057 (1990).
    [CrossRef] [PubMed]
  6. S. Wabnitz and B. Daino, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289–293(1993).
    [CrossRef]
  7. A. Degasperis, S. V. Manakov, and P. M. Santini, “On the initial-boundary value problems for soliton equations,” JETP Lett. 74, 481–485 (2001).
    [CrossRef]
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    [CrossRef]
  9. 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]
  10. 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] [PubMed]
  11. 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 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  20. 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(1996).
    [CrossRef]
  21. V. V. Kozlov and S. Wabnitz, “Theoretical study of polarization attraction in high-birefringence and spun fibers,” Opt. Lett. , 35, 3949–3951 (2010).
    [CrossRef] [PubMed]
  22. C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. 24, 2806–2826 (2006).
    [CrossRef]
  23. C. Martijn de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
    [CrossRef]

2010 (5)

2009 (5)

2008 (2)

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]

2001 (2)

A. Degasperis, S. V. Manakov, and P. M. Santini, “On the initial-boundary value problems for soliton equations,” JETP Lett. 74, 481–485 (2001).
[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]

2000 (1)

1998 (1)

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
[CrossRef]

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(1996).
[CrossRef]

1993 (1)

S. Wabnitz and B. Daino, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289–293(1993).
[CrossRef]

1991 (1)

1990 (1)

1987 (1)

V. E. Zakharov and A. V. Mikhailov, “Polarization domains in nonlinear optics,” Pis’ma Zh. Eksp. Teor. Fiz., 45, 279–282 (1987) [JETP Lett. 45, 349–352 (1987)].

Assémat, E.

Bennink, R. S.

Boyd, R. W.

Cirigliano, M.

Daino, B.

S. Wabnitz and B. Daino, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289–293(1993).
[CrossRef]

Degasperis, A.

A. Degasperis, S. V. Manakov, and P. M. Santini, “On the initial-boundary value problems for soliton equations,” JETP Lett. 74, 481–485 (2001).
[CrossRef]

Eyal, A.

Fatome, J.

Ferrario, M.

Fisher, R. A.

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.

Jackson, K. R.

Jauslin, H. R.

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 (2010).
[CrossRef] [PubMed]

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] [PubMed]

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]

Kozlov, V. V.

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. , in press (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] [PubMed]

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 (2010).
[CrossRef] [PubMed]

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] [PubMed]

Manakov, S. V.

A. Degasperis, S. V. Manakov, and P. M. Santini, “On the initial-boundary value problems for soliton equations,” JETP Lett. 74, 481–485 (2001).
[CrossRef]

Marazzi, L.

Marks, B. S.

Martelli, P.

Martijn de Sterke, C.

Martinelli, M.

Menyuk, C. R.

C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. 24, 2806–2826 (2006).
[CrossRef]

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(1996).
[CrossRef]

Mikhailov, A. V.

A. V. Mikhailov and S. Wabnitz, “Polarization dynamics of counterpropagating beams in optical fibers,” Opt. Lett. 15, 1055–1057 (1990).
[CrossRef] [PubMed]

V. E. Zakharov and A. V. Mikhailov, “Polarization domains in nonlinear optics,” Pis’ma Zh. Eksp. Teor. Fiz., 45, 279–282 (1987) [JETP Lett. 45, 349–352 (1987)].

Millot, G.

Morin, P.

Picozzi, A.

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 (2010).
[CrossRef] [PubMed]

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] [PubMed]

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]

Pitois, S.

Robert, B. D.

Santini, P. M.

A. Degasperis, S. V. Manakov, and P. M. Santini, “On the initial-boundary value problems for soliton equations,” JETP Lett. 74, 481–485 (2001).
[CrossRef]

Sugny, D.

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 (2010).
[CrossRef] [PubMed]

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] [PubMed]

Thèvenaz, L.

Tur, M.

Wabnitz, S.

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. , in press (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] [PubMed]

S. Wabnitz, “Cross polarization modulation domain wall solitons for WDM signals in birefringent optical fibers,” IEEE Photonics Technol. Lett. 21, 875–877 (2009).
[CrossRef]

S. Wabnitz, “Chiral polarization solitons in elliptically birefringent spun optical fibers,” Opt. Lett. 34, 908–910 (2009).
[CrossRef] [PubMed]

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]

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
[CrossRef]

S. Wabnitz and B. Daino, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289–293(1993).
[CrossRef]

A. V. Mikhailov and S. Wabnitz, “Polarization dynamics of counterpropagating beams in optical fibers,” Opt. Lett. 15, 1055–1057 (1990).
[CrossRef] [PubMed]

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(1996).
[CrossRef]

Zadok, A.

Zakharov, V. E.

V. E. Zakharov and A. V. Mikhailov, “Polarization domains in nonlinear optics,” Pis’ma Zh. Eksp. Teor. Fiz., 45, 279–282 (1987) [JETP Lett. 45, 349–352 (1987)].

Zilka, E.

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 Photonics Technol. Lett. (1)

S. Wabnitz, “Cross polarization modulation domain wall solitons for WDM signals in birefringent optical fibers,” IEEE Photonics Technol. Lett. 21, 875–877 (2009).
[CrossRef]

J. Lightwave Technol. (2)

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(1996).
[CrossRef]

C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. 24, 2806–2826 (2006).
[CrossRef]

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

JETP Lett. (1)

A. Degasperis, S. V. Manakov, and P. M. Santini, “On the initial-boundary value problems for soliton equations,” JETP Lett. 74, 481–485 (2001).
[CrossRef]

Lett. Math. Phys. (1)

V. V. Kozlov and S. Wabnitz, “Instability of optical solitons in the boundary value problem for a medium of finite extension,” Lett. Math. Phys. , in press (2010).
[CrossRef]

Opt. Express (5)

Opt. Lett. (5)

Phys. Lett. A (1)

S. Wabnitz and B. Daino, “Polarization domains and instabilities in nonlinear optical fibers,” Phys. Lett. A 182, 289–293(1993).
[CrossRef]

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. (2)

S. Pitois, G. Millot, and S. Wabnitz, “Polarization domain wall solitons with counterpropagating laser beams,” Phys. Rev. Lett. 81, 1409–1412 (1998).
[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] [PubMed]

Pis’ma Zh. Eksp. Teor. Fiz., 45, 279–282 (1987) (1)

V. E. Zakharov and A. V. Mikhailov, “Polarization domains in nonlinear optics,” Pis’ma Zh. Eksp. Teor. Fiz., 45, 279–282 (1987) [JETP Lett. 45, 349–352 (1987)].

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Figures (5)

Fig. 1
Fig. 1

Components of the (normalized) mean Stokes vector of the output signal beam as a function of the relative power of the pump beam: S 1 + (black squares), S 2 + (red circles), and S 3 + (green triangles) for six input SOPs of the pump beam located near the poles of the Poincaré sphere: (a) ( 0.99 , 0.01 , 0.14 ) , (b) ( 0.99 , 0.01 , 0.14 ) , (c) ( 0.01 , 0.99 , 0.14 ) , (d) ( 0.01 , 0.99 , 0.14 ) , (e) ( 0.01 , 0.01 , 0.9999 ) , (f) ( 0.01 , 0.01 , 0.9999 ) . The Stokes parameters of the signal and pump beam are normalized with respect to S 0 + ( z , t ) and S 0 ( z , t ) , respectively.

Fig. 2
Fig. 2

DOP D of the output signal beam as a function of the relative pump beam power for six input SOPs of the pump beam: (a) ( 0.99 , 0.01 , 0.14 ) (black squares), ( 0.01 , 0.99 , 0.14 ) (red circles), ( 0.01 , 0.01 , 0.9999 ) ) (green triangles); (b) ( 0.99 , 0.01 , 0.14 ) (black squares), ( 0.01 , 0.99 , 0.14 ) (red circles), ( 0.01 , 0.01 , 0.9999 ) (green triangles).

Fig. 3
Fig. 3

N = 110 points on the Poincaré sphere illustrating the output SOPs of the signal beam corresponding to an input distribution of N = 110 points uniformly distributed over the Poincaré sphere. Here, the input SOP of the pump beam is ( 0.01 , 0.01 , 0.9999 ) . The pump power is S 0 = 5.5 S 0 + .

Fig. 4
Fig. 4

DOP of the signal as the function of the distance propagated inside the fiber. The initial SOP of the pump is ( 0.01 , 0.01 , 0.9999 ) .

Fig. 5
Fig. 5

DOP of the pump at z = 0 as a function of the relative pump power. The initial SOP of the pump is ( 0.99 , 0.01 , 0.14 ) (black solid curve), ( 0.01 , 0.99 , 0.14 ) (red dashed curve), ( 0.01 , 0.01 , 0.9999 ) (green dotted curve).

Equations (35)

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

ξ S + = S + × J s S + + S + × J x S ,
η S = S × J s S + S × J x S + .
± i U p z + i β ( ω p ) U p t + Δ B ( ω p ) U p + γ p p [ 2 3 ( U p * · U p ) U p + 1 3 ( U p · U p ) U p * ] + 2 3 γ p s [ ( U s * · U s ) U p + ( U s · U p ) U s * + ( U p · U s * ) U s ] = 0 .
M ( z ) = ( cos θ 2 sin θ 2 sin θ 2 cos θ 2 ) .
Δ B ¯ ( ω p , s ) = ( Δ β ( ω p , s ) i 2 g θ ± i 2 g θ Δ β ( ω p , s ) ) .
T p ( z ) = ( a 1 a 2 a 2 * a 1 * ) , T s ( z ) = ( b 1 b 2 b 2 * b 1 * ) ,
± i z T p , s + Δ B ¯ ( ω p , s ) T p , s = 0.
± i Φ p z + i β ( ω p ) Φ p t + 1 3 γ [ 2 ( Φ p * · Φ p ) Φ p + N ^ s p m ] + 2 3 γ [ ( Φ s * · Φ s ) Φ p + N ^ x p m ] = 0 .
N s 1 = | Φ p 1 | 2 Φ p 1 + u 3 2 ( 2 | Φ p 2 | 2 | Φ p 1 | 2 ) Φ p 1 u 3 u 6 * ( 2 | Φ p 1 | 2 | Φ p 2 | 2 ) Φ p 2 u 3 u 6 Φ p 1 2 Φ p 2 * u 6 * 2 Φ p 2 2 Φ p 1 * ,
N s 2 = | Φ p 2 | 2 Φ p 2 + u 3 2 ( 2 | Φ p 1 | 2 | Φ p 2 | 2 ) Φ p 2 + u 3 u 6 ( 2 | Φ p 2 | 2 | Φ p 1 | 2 ) Φ p 1 + u 3 u 6 * Φ p 2 2 Φ p 1 * u 6 2 Φ p 1 2 Φ p 2 * ,
N x 1 = ( | u 13 | 2 + | u 10 | 2 ) | Φ s 1 | 2 Φ p 1 u 9 u 13 Φ s 1 Φ p 1 Φ s 2 * + u 10 * u 14 * Φ s 1 * Φ p 1 Φ s 2 + ( u 10 u 14 * u 9 u 13 * ) | Φ s 1 | 2 Φ p 2 + u 9 2 Φ s 1 Φ p 2 Φ s 2 * + u 14 * 2 Φ s 1 * Φ p 2 Φ s 2 u 9 * u 13 * Φ s 2 Φ p 1 Φ s 1 * + u 14 u 10 Φ s 2 * Φ p 1 Φ s 1 + ( | u 9 | 2 + | u 14 | 2 ) | Φ s 2 | 2 Φ p 1 u 13 * 2 Φ s 2 Φ p 2 Φ s 1 * u 10 2 Φ s 2 * Φ p 2 Φ s 1 ( u 9 u 13 * u 10 u 14 * ) | Φ s 2 | 2 Φ p 2 ,
N x 2 = ( | u 13 | 2 + | u 10 | 2 ) | Φ s 2 | 2 Φ p 2 + u 9 * u 13 * Φ s 2 Φ p 2 Φ s 1 * u 10 u 14 Φ s 2 * Φ p 2 Φ s 1 + ( u 10 * u 14 + u 9 * u 13 * ) | Φ s 2 | 2 Φ p 1 + u 9 * 2 Φ s 2 Φ p 1 Φ s 1 * + u 14 2 Φ s 2 * Φ p 1 Φ s 1 + u 9 u 13 Φ s 1 Φ p 2 Φ s 2 * u 14 * u 10 Φ s 1 * Φ p 2 Φ s 2 + ( | u 9 | 2 + | u 14 | 2 ) | Φ s 1 | 2 Φ p 2 u 13 2 Φ s 1 Φ p 1 Φ s 2 * u 10 * 2 Φ s 1 * Φ p 1 Φ s 2 ( u 9 * u 13 + u 10 * u 14 ) | Φ s 1 | 2 Φ p 1 .
( S 1 S 2 S 3 ) = ( S 2 S 1 0 ) g θ + ( 0 2 Δ β ( ω p ) S 3 ± 2 Δ β ( ω p ) S 2 ) .
G = 1 2 L c [ S 2 2 2 S 1 2 + S 1 2 2 S 2 2 + 2 θ 2 2 S 1 S 2 2 S 1 S 2 + 2 S 2 2 θ S 1 2 S 1 2 θ S 2 ] 2 Δ β ( ω p ) S 3 S 2 ± 2 Δ β ( ω p ) S 2 S 3 .
z S 1 2 = 2 L c 1 ( S 1 2 S 2 2 ) ,
z S 2 2 = 2 L c 1 ( S 1 2 S 2 2 ) 4 Δ β ( ω p ) S 2 S 3 ,
z S 3 2 = ± 4 Δ β ( ω p ) S 2 S 3 ,
z S 2 S 3 = L c 1 S 2 S 3 ± 2 Δ β ( ω p ) ( S 2 2 S 3 2 ) .
( S 1 S 2 S 3 S 4 ) = ( S 2 S 1 0 0 ) g θ + ( Δ ± ( ) S 4 Δ ± ( + ) S 3 Δ ± ( + ) S 2 Δ ± ( ) S 1 ) ,
G = 1 L c [ S 2 2 2 S 1 2 + S 1 2 2 S 2 2 + 2 θ 2 2 S 1 S 2 2 S 1 S 2 + 2 S 2 2 θ S 1 2 S 1 2 θ S 2 S 1 S 1 S 2 S 2 ] Δ ± ( ) S 1 S 4 + Δ ± ( ) S 4 S 1 Δ ± ( + ) S 3 S 2 + Δ ± ( + ) S 2 S 3 ,
z S 1 2 = 2 L c 1 ( S 1 2 S 2 2 ) + 2 Δ ± ( ) S 1 S 4 ,
z S 2 2 = 2 L c 1 ( S 1 2 S 2 2 ) 2 Δ ± ( + ) S 2 S 3 ,
z S 3 2 = 2 Δ ± ( + ) S 2 S 3 ,
z S 4 2 = 2 Δ ± ( ) S 1 S 4 ,
z S 2 S 3 = L c 1 S 2 S 3 + Δ ± ( + ) ( S 2 2 S 3 2 ) ,
z S 1 S 4 = L c 1 S 1 S 4 + Δ ± ( ) ( S 4 2 S 1 2 ) .
S i + ( z = 0 , t ) = S i + ( z , t = 0 ) ,
S i ( z = L , t ) = S i ( z , t = 0 ) ,
S i + ( L ) = 1 N j = 1 N [ S i + ( L ) ] j ,
θ 0 = arccos ( S 3 S 1 + 2 + S 2 + 2 + S 3 + 2 ) ,
ϕ 0 = arctan 2 ( S 2 + , S 1 + ) .
a r c tan 2 ( x , y ) = { arctan ( y / x ) , x > 0 , π + arctan ( y / x ) , y 0 , x < 0 π + arctan ( y / x ) , y < 0 , x < 0 .
S ¯ 1 = S 0 + sin θ 0 cos ϕ 0 ,
S ¯ 2 = S 0 + sin θ 0 sin ϕ 0 ,
S ¯ 3 = S 0 + cos θ 0 .

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