Abstract

Parametric amplification or four-wave mixing in high-birefringence optical fibers may be exploited to implement a novel type of nonlinear polarizer. Such a device leads to the simultaneous amplification, frequency conversion, and repolarization of both signal and idler waves along one of the principal birefringence axes of the fiber, independently of the pump, signal, and idler input state of polarization, power, and frequency detuning. We discuss the conditions for the observation of polarization attraction in fiber optics parametric amplifiers operating with a pump in either the normal or the anomalous dispersion regime.

© 2012 Optical Society of America

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References

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  1. M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.
  2. R. Noe, H. Heidrich, and D. Hoffmann, “Endless polarization control systems for coherent optics,” J. Lightwave Technol. 6, 1199–1208 (1988).
    [CrossRef]
  3. B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
    [CrossRef]
  4. A. Zadok, E. Zilka, A. Eyal, L. Thevenaz, and M. Tur, “Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers,” Opt. Express 16, 21692–21707 (2008).
    [CrossRef]
  5. M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express 17, 947–955 (2009).
    [CrossRef]
  6. V. V. Kozlov, Javier Nuño, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35, 3970–3972 (2010).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. S. Pitois, J. Fatome, and G. Millot, “Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths,” Opt. Express 16, 6646–6651 (2008).
    [CrossRef]
  12. J. Fatome, S. Pitois, P. Morin, and G. Millot, “Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications,” Opt. Express 18, 15311–15317 (2010).
    [CrossRef]
  13. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. 38, 2018–2021 (1988).
    [CrossRef]
  14. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. 42, 682–685 (1990).
    [CrossRef]
  15. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
    [CrossRef]
  16. E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
    [CrossRef]
  17. J. F. L. Freitas, C. J. S. de Matos, M. B. Costa e Silva, and A. S. L. Gomes, “Impact of phase modulation and parametric gain on signal polarization in an anomalously dispersive optical fiber,” J. Opt. Soc. Am. B 24, 1469–1474 (2007).
    [CrossRef]
  18. 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]
  19. 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]
  20. V. V. Kozlov, J. Nun¯o, J. D. Ania-Castañón, and S. Wabnitz, “Multi-channel Raman polarizer with suppressed relative intensity noise for WDM transmission lines,” submitted to Opt. Lett. (2012).

2012

2011

2010

2009

2008

2007

2005

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

1996

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

1990

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. 42, 682–685 (1990).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

1988

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. 38, 2018–2021 (1988).
[CrossRef]

R. Noe, H. Heidrich, and D. Hoffmann, “Endless polarization control systems for coherent optics,” J. Lightwave Technol. 6, 1199–1208 (1988).
[CrossRef]

Ania-Castañón, J. D.

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

V. V. Kozlov, J. Nun¯o, J. D. Ania-Castañón, and S. Wabnitz, “Multi-channel Raman polarizer with suppressed relative intensity noise for WDM transmission lines,” submitted to Opt. Lett. (2012).

Bayer, S.

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

Bilbault, J. M.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

Cirigliano, M.

Costa e Silva, M. B.

de Matos, C. J. S.

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Dumas, D.

M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.

Eyal, A.

Fatome, J.

Ferrario, M.

Freitas, J. F. L.

Gomes, A. S. L.

Griesser, H.

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

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]

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

Harvey, J. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Heidrich, H.

R. Noe, H. Heidrich, and D. Hoffmann, “Endless polarization control systems for coherent optics,” J. Lightwave Technol. 6, 1199–1208 (1988).
[CrossRef]

Hoffmann, D.

R. Noe, H. Heidrich, and D. Hoffmann, “Endless polarization control systems for coherent optics,” J. Lightwave Technol. 6, 1199–1208 (1988).
[CrossRef]

Jauslin, H. R.

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]

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Koch, B.

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

Kozlov, V. V.

Leonhardt, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Marazzi, L.

Martelli, P.

Martinelli, M.

Millot, G.

Mirvoda, V.

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

Morin, P.

Noe, R.

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

R. Noe, H. Heidrich, and D. Hoffmann, “Endless polarization control systems for coherent optics,” J. Lightwave Technol. 6, 1199–1208 (1988).
[CrossRef]

Nun¯o, J.

Nuño, Javier

OSullivan, M.

M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.

Picozzi, A.

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.

Reimer, M.

M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.

Remoissenet, M.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. 42, 682–685 (1990).
[CrossRef]

Seve, E.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

Soliman, G.

M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.

Tchofo Dinda, P.

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

Thevenaz, L.

Tur, M.

Turitsyn, K.

Wabnitz, S.

Wernz, H.

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

Yevick, D.

M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.

Zadok, A.

Zilka, E.

Europhys. Lett.

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 Photon. Technol. Lett.

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]

B. Koch, R. Noe, V. Mirvoda, H. Griesser, S. Bayer, and H. Wernz, “Record 59  krad/s polarization tracking in 112  Gb/s 640-km PDM-RZ-DQPSK transmission,” IEEE Photon. Technol. Lett. 22, 1407–1409 (2010).
[CrossRef]

J. Lightwave Technol.

R. Noe, H. Heidrich, and D. Hoffmann, “Endless polarization control systems for coherent optics,” J. Lightwave Technol. 6, 1199–1208 (1988).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. 38, 2018–2021 (1988).
[CrossRef]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. 42, 682–685 (1990).
[CrossRef]

Phys. Rev. A

E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J. M. Bilbault, and M. Haelterman, “Modulational instability and critical regime in a highly birefringent fiber,” Phys. Rev. A 54, 3519 (1996).
[CrossRef]

Other

V. V. Kozlov, J. Nun¯o, J. D. Ania-Castañón, and S. Wabnitz, “Multi-channel Raman polarizer with suppressed relative intensity noise for WDM transmission lines,” submitted to Opt. Lett. (2012).

M. Reimer, D. Dumas, G. Soliman, D. Yevick, and M. OSullivan, “Polarization evolution in dispersion compensation modules,” presented at the Optical Fiber Communications/National Fiber Optic Engineers Conference (OFC/NFOEC), San Diego, CA, USA, March2009, paper OWD4.

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Figures (9)

Fig. 1.
Fig. 1.

Gain g in the normal dispersion regime [in (a),  P tot = 10 W ; in (b),  P tot = 5 W ]; the circles in (a) represent the approximated gain of Eq. (A6); α p = 5 ° , 25°, 45°, 65°, 85°.

Fig. 2.
Fig. 2.

Angle α i in the normal dispersion regime [in (a),  P tot = 10 W ; in (b),  P tot = 5 W ); α p = 5 ° , 25°, 45°, 65°, 85°.

Fig. 3.
Fig. 3.

Sideband gain g in the anomalous dispersion regime ( P tot = 10 W ). Black solid line for band B 1 , L , red dash-dot line for B 1 , H ; α p = 5 ° , 25°, 45°, 50° 65°, 85°. The inset provides a zoom for the band B 1 , H in the frequency range 1.6 THz Δ ν 1.9 THz .

Fig. 4.
Fig. 4.

Angle α i in the anomalous dispersion regime ( P tot = 10 W ). Black solid line for band B 1 , L , red dash-dot line for B 1 , H ; α p = 5 ° , 25°, 40°, 50° 65°, 85°. The inset provides a zoom for the band B 1 , H in the frequency range 1.6 THz Δ ν 1.9 THz .

Fig. 5.
Fig. 5.

Sideband gain g in the normal dispersion regime ( P tot = 100 W ); α p = 5 ° , 25°, 45°, 65°, 85°.

Fig. 6.
Fig. 6.

Angle α i in the normal dispersion case discussed in Section 3 ( P tot = 100 W ); α p = 5 ° , 25°, 45°, 65°, 85°. The pump-independent QLPA is not reached as α i sweeps from nearly 25° to 85° depending on the detuning frequency Δ ν and on the input pump angle α p .

Fig. 7.
Fig. 7.

(a) Fixed input unit Stokes vector of the pump S⃗ p ( 0 ) = [ S 1 = 1 , S 2 = 0 , S 3 = 0 ] (empty triangle) and 225 input unit Stokes vectors of the sidebands S⃗ i , n ( 0 ) = S⃗ s , n ( 0 ) (filled circles) are shown on the Poincaré sphere ( 1 n 225 ); (b) output unit Stokes vectors of the idler S⃗ i , n ( L ) (filled circles) after fiber propagation (fiber length L = 100 m )

Fig. 8.
Fig. 8.

Output unit Stokes vectors of the idler S⃗ i , n ( L ) (filled circles) after fiber propagation (fiber length L = 100 m ). The input idler is null; the input unit Stokes vector of the pump and of the signal are the same as in Fig. 7(a).

Fig. 9.
Fig. 9.

(a) Fixed input unit Stokes vectors of the sidebands S⃗ i , n ( 0 ) = S⃗ s , n ( 0 ) = [ 1 , 0 , 0 ] (filled circle) and 225 input unit Stokes vectors of the pump (empty triangles) are shown on the Poincaré sphere; (b) output unit Stokes vectors of the idler S⃗ i , n ( L ) (filled circles) after fiber propagation (fiber length L = 300 m ).

Tables (1)

Tables Icon

Table 1. The Processes X ( 1 , 2 , 3 , 4 ) are Listed With Their Corresponding Optimal Detuning Frequency Δ ν X i that Leads the Maximum Amplification of the Sidebands, the Attraction of the Sidebands Towards the Birefringence Axes, and the Dependence of the Gain on the Pump Polarization ( Stands for Directly Proportional) a

Equations (16)

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

A x z + 1 v x p A x t + β 2 2 2 A x t 2 = i γ ( | A x | 2 A x + 2 3 | A y | 2 A x ) , A y z + 1 v y p A y t + β 2 2 2 A y t 2 = i γ ( | A y | 2 A y + 2 3 | A x | 2 A y ) .
β 2 = 2 k j p ω 2 | ω = ω p , ( j = { x , y } ) ,
A j ( z , t ) = A j p ( z ) + A j s ( z ) exp ( i Δ k j s z + i 2 π Δ ν t ) + A j i ( z ) exp ( i Δ k j i z i 2 π Δ ν t ) ( j = { x , y } ) ,
A x p ( z ) = P exp [ i ϕ x p ( z ) ] , A y p ( z ) = Q exp [ i ϕ y p ( z ) ] , ϕ x p ( z ) = ψ x p ( 0 ) + γ ( P + ( 2 / 3 ) Q ) z , ϕ y p ( z ) = ψ y p ( 0 ) + γ ( Q + ( 2 / 3 ) P ) z ,
d A x s d z = i γ ( 2 | A x p | 2 + 2 3 | A y p | 2 ) A x s + i γ 2 3 A x p 2 A x i * exp [ i ( 2 k x p k x i k x s ) ] X 1 + i γ 2 3 A x p A y p * A y s + i γ 2 3 A x p A y p A y i * exp [ i ( k x p k x s k y i + k y p ) ] X 3 , d A x i d z = i γ 2 3 A x p 2 A x s * exp [ i ( 2 k x p k x i k x s ) ] X 1 + i γ ( 2 | A x p | 2 + 2 3 | A y p | 2 ) A x i + i γ 2 3 A x p A y p A y s * exp [ i ( k x p k x i k y s + k y p ) ] X 4 + i γ 2 3 A x p A y p * A y i , d A y s d z = i γ ( 2 | A y p | 2 + 2 3 | A x p | 2 ) A y s + i γ 2 3 A y p 2 A y i * exp [ i ( 2 k y p k y i k y s ) ] X 2 + i γ 2 3 A y p A x p * A x s + i γ 2 3 A y p A x p A x i * exp [ i ( k x p k y s k x i + k y p ) ] X 4 , d A y i d z = i γ 2 3 A y p 2 A y s * exp [ i ( 2 k y p k y i k y s ) ] X 2 + i γ ( 2 | A y p | 2 + 2 3 | A x p | 2 ) A y i + i γ 2 3 A x p A y p A x s * exp [ i ( k y p k y i k x s + k x p ) ] X 3 + i γ 2 3 A y p A x p * A x i .
P tot < P c = Δ n 2 4 γ c 2 β 2
M = [ v / 2 + K / 2 + γ P γ P ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q γ P v / 2 K / 2 γ P ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q v / 2 + K / 2 + γ Q γ Q ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q γ Q v / 2 K / 2 γ Q ] ,
A j v ( z ) = | A ˜ j v | exp [ i ϕ j v ( z ) + i r z ] exp ( g z ) ( j = { x , y } ; v = { i , s } ) ,
β 2 ( 2 π Δ ν X 4 ) 2 ( Δ n / c ) ( 2 π Δ ν X 4 ) + γ P tot = 0.
v ( Δ ν ) / 2 + K ( Δ ν ) / 2 + γ P / 2 + γ Q / 2 = v ( Δ ν X 4 ) / 2 + K ( Δ ν X 4 ) / 2 + P / 2 + Q / 2 = 0 π ν Δ n / c + 4 π 2 ν Δ ν X 4 β 2 = π ν Δ n / c + 4 π 2 ν Δ ν X 4 β 2 .
v ( Δ ν ) / 2 + K ( Δ ν ) / 2 + γ P = π ν Δ n / c + 4 π 2 ν Δ ν X 4 β 2 + γ ( P Q ) / 2 ,
v ( Δ ν ) / 2 K ( Δ ν ) / 2 γ Q = + π ν Δ n / c 4 π 2 ν Δ ν X 4 β 2 + γ ( P Q ) / 2.
M ˜ = [ 0 0 0 0 0 v / 2 K / 2 0 0 0 0 v / 2 + K / 2 0 0 0 0 0 ] , M = [ M ( 1 , 1 ) γ P ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q γ P γ P ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q γ Q γ Q ( 2 / 3 ) γ P Q ( 2 / 3 ) γ P Q γ Q M ( 4 , 4 ) ] .
[ λ < v 01 , M v 01 > < v 01 , M v 02 > < v 02 , M v 01 > λ < v 02 , M v 02 > ] .
g app = [ ( 4 / 9 ) γ 2 P Q ( Δ ν Δ ν X 4 ) 2 ( 4 π 2 β 2 Δ ν X 4 π Δ n / c ) 2 ] 1 / 2 .
g app 2 = 4 a c b 2 , a = 2 ( γ P tot K + K 2 + v 2 ) , b = 4 v K γ ( P Q ) , c = 2 γ P tot ( K 3 K v 2 ) + ( v 2 K 2 ) 2 ( 20 / 9 ) γ 2 P Q K 2 .

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