Abstract

Multiple-scale analysis is used to study linear wave propagation in a rapidly-spun fiber and its predictions are shown to be consistent with results obtained by other methods. Subsequently, multiple-scale analysis is used to derive a generalized Schroedinger equation for nonlinear wave propagation in a rapidly-spun fiber. The consequences of this equation for pulse propagation and four-wave mixing are discussed briefly.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. Barlow , J. J. Ramskov-Hansen , and D. N. Payne , “ Birefringence and polarization-mode dispersion in spun single-mode fibers , ” Appl. Opt.   20 , 2962 – 2968 ( 1981 ).
    [CrossRef] [PubMed]
  2. C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves , ” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
    [CrossRef]
  3. J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
    [CrossRef]
  4. S. Radic and C. J. McKinstrie , “ Optical amplification and signal processing in highly-nonlinear optical fiber , ” IEICE Trans. Electron.   E88C , 859 – 869 ( 2005 ).
    [CrossRef]
  5. R. M. Jopson and R. E. Tench , “ Polarisation-independent phase conjugation of lightwave signals , ” Electron. Lett.   29 , 2216 – 2217 ( 1993 ).
    [CrossRef]
  6. K. Inoue , “ Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies , ” J. Lightwave Technol.   12 , 1916 – 1920 ( 1994 ).
    [CrossRef]
  7. C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence , ” Opt. Express   12 , 2033 – 2055 ( 2004 ).
    [CrossRef] [PubMed]
  8. T. Tanemura , K. Katoh , and K. Kikuchi , “ Polarization-insensitive asymmetric four-wave mixing using circularly polarized pumps in a twisted fiber , ” Opt. Express   13 , 7497 – 7505 ( 2005 ).
    [CrossRef] [PubMed]
  9. C. J. McKinstrie , J. D. Harvey , S. Radic , and M. G. Raymer , “ Translation of quantum states by four-wave mixing in fibers , ” Opt. Express   13 , 9131 – 9142 ( 2005 ).
    [CrossRef] [PubMed]
  10. J. P. Gordon and H. Kogelnik , “ PMD fundamentals: Polarization mode dispersion in optical fibers , ” Proc. Nat. Acad. Sci.   97 , 4541 – 4550 ( 2000 ).
    [CrossRef] [PubMed]
  11. W. A. Shurcliff , Polarized Light ( Harvard University Press, 1962 ).
  12. P. McIntyre and A. W. Snyder , “ Light propagation in twisted anisotropic media: Application to photoreceptors , ” J. Opt. Soc. Am.   68 , 149 – 157 ( 1978 ).
    [CrossRef] [PubMed]
  13. A. Galtarossa , P. Griggio , L. Palmieri , and A. Pizzinat , “ First- and second-order PMD statistical properties of constantly spun randomly birefringent fibers , ” J. Lightwave Technol.   22 , 1127 – 1136 ( 2004 ).
    [CrossRef]
  14. A. H. Nayfeh , Introduction to Perturbation Techniques ( Wiley, 1981 ).
  15. P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength , ” Phys. Rev.   137 , A801 – A818 ( 1965 ).
    [CrossRef]
  16. C. R. Menyuk , “ Nonlinear pulse propagation in birefringent optical fibers , ” IEEE J. Quantum Electron.   23 , 174 – 176 ( 1987 ).
    [CrossRef]
  17. P. K. A. Wai , C. R. Menyuk , and H. H. Chen , “ Stability of solitons in randomly varying birefringent fibers , ” Opt. Lett.   16 , 1231 – 1233 ( 1991 ).
    [CrossRef] [PubMed]
  18. S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
    [CrossRef]
  19. U. Leonhardt , Measuring the Quantum State of Light ( Cambridge University Press, 1997 ).

2005 (3)

2004 (2)

2002 (2)

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves , ” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[CrossRef]

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

2000 (1)

J. P. Gordon and H. Kogelnik , “ PMD fundamentals: Polarization mode dispersion in optical fibers , ” Proc. Nat. Acad. Sci.   97 , 4541 – 4550 ( 2000 ).
[CrossRef] [PubMed]

1994 (1)

K. Inoue , “ Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies , ” J. Lightwave Technol.   12 , 1916 – 1920 ( 1994 ).
[CrossRef]

1993 (1)

R. M. Jopson and R. E. Tench , “ Polarisation-independent phase conjugation of lightwave signals , ” Electron. Lett.   29 , 2216 – 2217 ( 1993 ).
[CrossRef]

1992 (1)

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[CrossRef]

1991 (1)

1987 (1)

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

1981 (1)

1978 (1)

1965 (1)

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength , ” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[CrossRef]

Andrekson, P. A.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

Barlow, A. J.

Bergano, N. S.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[CrossRef]

Chen, H. H.

Chraplyvy, A. R.

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves , ” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[CrossRef]

Evangelides, S. G.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[CrossRef]

Galtarossa, A.

Gordon, J. P.

J. P. Gordon and H. Kogelnik , “ PMD fundamentals: Polarization mode dispersion in optical fibers , ” Proc. Nat. Acad. Sci.   97 , 4541 – 4550 ( 2000 ).
[CrossRef] [PubMed]

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[CrossRef]

Griggio, P.

Hansryd, J.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

Harvey, J. D.

Hedekvist, P. O.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

Inoue, K.

K. Inoue , “ Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies , ” J. Lightwave Technol.   12 , 1916 – 1920 ( 1994 ).
[CrossRef]

Jopson, R. M.

C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence , ” Opt. Express   12 , 2033 – 2055 ( 2004 ).
[CrossRef] [PubMed]

R. M. Jopson and R. E. Tench , “ Polarisation-independent phase conjugation of lightwave signals , ” Electron. Lett.   29 , 2216 – 2217 ( 1993 ).
[CrossRef]

Kanaev, A. V.

Katoh, K.

Kikuchi, K.

Kogelnik, H.

C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence , ” Opt. Express   12 , 2033 – 2055 ( 2004 ).
[CrossRef] [PubMed]

J. P. Gordon and H. Kogelnik , “ PMD fundamentals: Polarization mode dispersion in optical fibers , ” Proc. Nat. Acad. Sci.   97 , 4541 – 4550 ( 2000 ).
[CrossRef] [PubMed]

Leonhardt, U.

U. Leonhardt , Measuring the Quantum State of Light ( Cambridge University Press, 1997 ).

Li, J.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

Maker, P. D.

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength , ” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[CrossRef]

McIntyre, P.

McKinstrie, C. J.

S. Radic and C. J. McKinstrie , “ Optical amplification and signal processing in highly-nonlinear optical fiber , ” IEICE Trans. Electron.   E88C , 859 – 869 ( 2005 ).
[CrossRef]

C. J. McKinstrie , J. D. Harvey , S. Radic , and M. G. Raymer , “ Translation of quantum states by four-wave mixing in fibers , ” Opt. Express   13 , 9131 – 9142 ( 2005 ).
[CrossRef] [PubMed]

C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence , ” Opt. Express   12 , 2033 – 2055 ( 2004 ).
[CrossRef] [PubMed]

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves , ” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[CrossRef]

Menyuk, C. R.

P. K. A. Wai , C. R. Menyuk , and H. H. Chen , “ Stability of solitons in randomly varying birefringent fibers , ” Opt. Lett.   16 , 1231 – 1233 ( 1991 ).
[CrossRef] [PubMed]

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

Mollenauer, L. F.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[CrossRef]

Nayfeh, A. H.

A. H. Nayfeh , Introduction to Perturbation Techniques ( Wiley, 1981 ).

Palmieri, L.

Payne, D. N.

Pizzinat, A.

Radic, S.

C. J. McKinstrie , J. D. Harvey , S. Radic , and M. G. Raymer , “ Translation of quantum states by four-wave mixing in fibers , ” Opt. Express   13 , 9131 – 9142 ( 2005 ).
[CrossRef] [PubMed]

S. Radic and C. J. McKinstrie , “ Optical amplification and signal processing in highly-nonlinear optical fiber , ” IEICE Trans. Electron.   E88C , 859 – 869 ( 2005 ).
[CrossRef]

C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence , ” Opt. Express   12 , 2033 – 2055 ( 2004 ).
[CrossRef] [PubMed]

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves , ” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[CrossRef]

Ramskov-Hansen, J. J.

Raymer, M. G.

Shurcliff, W. A.

W. A. Shurcliff , Polarized Light ( Harvard University Press, 1962 ).

Snyder, A. W.

Tanemura, T.

Tench, R. E.

R. M. Jopson and R. E. Tench , “ Polarisation-independent phase conjugation of lightwave signals , ” Electron. Lett.   29 , 2216 – 2217 ( 1993 ).
[CrossRef]

Terhune, R. W.

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength , ” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[CrossRef]

Wai, P. K. A.

Westlund, M.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

R. M. Jopson and R. E. Tench , “ Polarisation-independent phase conjugation of lightwave signals , ” Electron. Lett.   29 , 2216 – 2217 ( 1993 ).
[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 J. Sel. Top. Quantum Electron. (2)

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves , ” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[CrossRef]

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications , ” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[CrossRef]

IEICE Trans. Electron. (1)

S. Radic and C. J. McKinstrie , “ Optical amplification and signal processing in highly-nonlinear optical fiber , ” IEICE Trans. Electron.   E88C , 859 – 869 ( 2005 ).
[CrossRef]

J. Lightwave Technol. (3)

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons , ” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[CrossRef]

K. Inoue , “ Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies , ” J. Lightwave Technol.   12 , 1916 – 1920 ( 1994 ).
[CrossRef]

A. Galtarossa , P. Griggio , L. Palmieri , and A. Pizzinat , “ First- and second-order PMD statistical properties of constantly spun randomly birefringent fibers , ” J. Lightwave Technol.   22 , 1127 – 1136 ( 2004 ).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. (1)

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength , ” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[CrossRef]

Proc. Nat. Acad. Sci. (1)

J. P. Gordon and H. Kogelnik , “ PMD fundamentals: Polarization mode dispersion in optical fibers , ” Proc. Nat. Acad. Sci.   97 , 4541 – 4550 ( 2000 ).
[CrossRef] [PubMed]

Other (3)

W. A. Shurcliff , Polarized Light ( Harvard University Press, 1962 ).

A. H. Nayfeh , Introduction to Perturbation Techniques ( Wiley, 1981 ).

U. Leonhardt , Measuring the Quantum State of Light ( Cambridge University Press, 1997 ).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Laboratory-frame rotation angle plotted as a function of distance for a birefringence-to-spin ratio of 0.15. The red curve represents the exact formula (32), whereas the blue curve represents the approximate formula (35).

Fig. 2.
Fig. 2.

Rotating-frame Stokes components plotted as functions of distance for a birefringence-to-spin ratio of 0.15. The red, green and blue curves represent the 1, 2 and 3 components, respectively.

Fig. 3.
Fig. 3.

Trajectory of the rotating-frame Stokes vector for a birefringence-to-spin ratio of 0.15.

Fig. 4.
Fig. 4.

Laboratory-frame Stokes components plotted as functions of distance for a birefringence-to-spin ratio of 0.15. The red, green and blue curves represent the 1, 2 and 3 components, respectively.

Fig. 5.
Fig. 5.

Trajectory of the laboratory-frame Stokes vector for a birefringence-to-spin ratio of 0.15.

Equations (106)

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

A ( z ) = T ( z ) A ( 0 ) .
A ( z ) = U ( z ) A ( 0 ) .
U = [ μ ν ν * μ * ] ,
μ = cos ( ϕ 2 ) + i sin ( ϕ 2 ) cos ( 2 ϕ 1 ) ,
ν = i sin ( ϕ 2 ) + sin ( 2 ϕ 1 ) ,
μ = cos ( ϕ 3 ) ,
ν = sin ( ϕ 3 ) ,
μ = cos ( ϕ 2 ) cos ( ϕ 3 ) + i sin ( ϕ 2 ) cos ( 2 ϕ 1 ϕ 3 ) ,
ν = cos ( ϕ 2 ) sin ( ϕ 3 ) + i sin ( ϕ 2 ) sin ( 2 ϕ 1 ϕ 3 ) .
U = λ + E + E + + λ E E .
E + = [ cos ( θ 1 2 ) exp ( i θ 2 2 ) , sin ( θ 1 2 ) , exp ( i θ 2 2 ) ] t ,
E = [ sin ( θ 1 2 ) exp ( i θ 2 2 ) , cos ( θ 1 2 ) , exp ( i θ 2 2 ) ] t ,
μ = cos ( θ 3 2 ) i cos ( θ 1 ) sin ( θ 3 2 ) ,
ν = sin ( θ 1 ) sin ( θ 3 2 ) exp ( i θ 2 ) .
A 1 = A x 2 A y 2 ,
A 2 = A x A y * + A x * A y ,
A 3 = i ( A x A y * A x * A y ) .
E ± = ± ( cos θ 1 , sin θ 1 cos θ 2 , sin θ 1 sin θ 2 ) .
A r ( z ) = R ( z ) A l ( z ) ,
R ( z ) = [ cos ( ρz ) sin ( ρz ) sin ( ρz ) cos ( ρz ) ] .
d z A r = L r A r ,
L r = [ ρ ρ ] ,
d z U r = L r U r ,
μ r ( z ) = cos ( kz ) + i ( δ k ) sin ( kz ) ,
ν r ( z ) = ( ρ k ) sin ( kz ) ,
U l ( z ) = R ( z ) U r ( z ) .
μ l ( z ) = cos ( ρz ) cos ( kz ) + ( ρ k ) sin ( ρz ) sin ( kz ) + i ( δ k ) cos ( ρz ) sin ( kz ) ,
ν l ( z ) = ( ρ k ) cos ( ρz ) sin ( kz ) sin ( ρz ) cos ( kz ) + i ( δ k ) sin ( ρz ) sin ( kz ) .
2 ϕ 1 ( z ) = tan 1 [ ρ tan ( kz ) k ] + ,
ϕ 2 ( z ) = ( 1 ) n sin 1 [ δ sin ( kz ) k ] ,
ϕ 3 ( z ) = tan 1 [ ρ tan ( kz ) k ] .
ϕ 3 ( z ) = tan 1 [ ρ tan ( kz ) k ] ρz .
2 ϕ 1 ( z ) ρz ,
ϕ 2 ( z ) ( δ k ) sin ( kz ) ,
ϕ 3 ( z ) ( δ 2 ρ 2 k 3 ) [ kz sin ( 2 kz ) 2 ] ,
L l ( z ) = d z U l ( z ) U l ( z ) ,
L l ( z ) = R ( z ) L r R ( z ) R ( z ) d z R ( z ) ,
L l ( z ) = [ cos ( 2 ρz ) sin ( 2 ρz ) sin ( 2 ρz ) cos ( 2 ρz ) ] .
U l ( z ) = [ cos ( δ 2 z 2 ρ ) sin ( δ 2 z 2 ρ ) sin ( δ 2 z 2 ρ ) cos ( δ 2 z 2 ρ ) ] .
d z U l = L l U l ,
L l = [ 0 δ 2 2 ρ δ 2 2 ρ 0 ]
μ ( ζ ) cos [ ( 1 + ε 2 2 ) ζ ] + sin [ ( 1 + ε 2 2 ) ζ ] ,
ν ( ζ ) sin [ ( 1 + ε 2 2 ) ζ ] .
DA = LA ,
L = [ 1 1 ] .
( D 0 + ε 2 D 2 ) ( A 0 + ε A 1 + ε 2 A 2 ) ( L 0 + ε L 1 ) ( A 0 + ε A 1 + ε 2 A 2 ) .
( D 0 L 0 ) A 0 = 0 ,
L 0 = [ 0 1 1 0 ] .
A 0 ( ζ 0 , ζ 2 ) = A + ( ζ 0 , ζ 2 ) E + + A ( ζ 0 , ζ 2 ) E ,
A ± ( ζ 0 , ζ 2 ) = A ¯ ± ( ζ 2 ) exp ( ± i ζ 0 ) .
( D 0 L 0 ) A 1 = L 1 A 0 ,
L l = [ i 0 0 i ] .
A 1 ( ζ 0 , ζ 2 ) = B + ( ζ 0 , ζ 2 ) E + + B ( ζ 0 , ζ 2 ) E ,
( D 0 i ) B ± = i A ¯ exp ( i ζ 0 ) .
B ± ( ζ 0 , ζ 2 ) = i A ¯ ( ζ 2 ) sin ( ζ 0 )
( D 0 L 0 ) A 2 = D 2 A 0 + L 1 A 1 .
A 2 ( ζ 0 , ζ 2 ) = C + ( ζ 0 , ζ 2 ) E + + C ( ζ 0 , ζ 2 ) E
( D 0 i ) C ± = D 2 A ¯ ± exp ( ± i ζ 0 ) A ¯ ± sin ( ζ 0 ) .
D 2 A ¯ ± = ± i A ¯ ± 2 .
A ¯ ± ( ζ 2 ) = A ¯ ± ( 0 ) exp ( ± i ζ 2 2 ) .
A ( z ) { A ¯ + ( 0 ) exp [ i ( 1 + ε 2 2 ) z ] + A ¯ ( 0 ) sin ( z ) } E +
{ A ¯ + ( 0 ) sin ( z ) + A ¯ ( 0 ) exp [ i ( 1 + ε 2 2 ) z ] } E .
A ( z ) { A ξ ( 0 ) [ cos ( kz ) + sin ( kz ) ] + A η ( 0 ) sin ( kz ) } E ξ
{ A ξ ( 0 ) sin ( kz ) + A η ( 0 ) [ cos ( kz ) sin ( kz ) ] } E η ,
z A ξ = A ξ + ρ A η β 1 ξ τ A ξ i β 2 ξ ττ 2 A ξ 2 + N ξ ( A ξ , A η ) .
N ξ = ( A ξ 2 A ξ + 2 A η 2 A ξ 3 + A ξ * A η 2 3 ) ,
( D 0 + ε 2 D 2 ) ( A 0 + ε A 1 + ε 2 A 2 ) ( L 0 + ε L 1 + ε 2 L 2 ) ( A 0 + ε A 1 + ε 2 A 2 ) + ε 2 N 2 ( A 0 ) .
I = [ 1 0 0 1 ] , J = [ 1 0 0 1 ] , K = [ 0 1 1 0 ] ,
L 2 a = I ( i β 2 a ττ 2 2 ) ,
L 2 d = J ( β 1 d τ + i β 2 d ττ 2 2 ) ,
N 2 = [ 2 ( A A ) A + ( A t A ) A * ] 3 .
A 0 ( z 0 , z 2 ) = A + ( z 0 , z 2 ) E + + A ( z 0 , z 2 ) E ,
A ± ( z 0 , z 2 ) = A ¯ ± ( z 2 ) exp ( ± z 0 ) ,
A 1 ( z 0 , z 2 ) = B + ( z 0 , z 2 ) E + + B ( z 0 , z 2 ) E ,
B ± ( z 0 , z 2 ) = i A ¯ ( z 2 ) δ sin ( ρ z 0 ) ρ .
( D 0 L 0 ) A 2 = D 2 A 0 + L 1 A 1 + L 2 A 0 + N 2 ( A 0 )
L 1 A 1 = [ δ 2 sin ( ρ z 0 ) ρ ] ( A ¯ + E + + A ¯ E ) .
L 2 a A 0 = ( i β 2 a ττ 2 A + 2 ) E + ( i β 2 a ττ 2 A 2 ) E .
L 2 d A 0 = ( β 1 d τ A + i β 2 d ττ 2 A - 2 ) E +
( β 1 d τ A + + i β 2 d ττ 2 A + 2 ) E .
N 2 = i ( 2 γ 3 ) ( A + 2 + 2 A - 2 ) A + E +
+ i ( 2 γ 3 ) ( 2 A + 2 + A 2 ) A E .
z 2 A ¯ ± = ± i ( δ 2 2 ρ ) A ¯ ± i β 2 a ττ 2 A ¯ ± 2
+ i ( 2 γ 3 ) ( A ¯ ± 2 + 2 A ¯ 2 ) A ¯ ± .
z A ± = ± i ( ρ + δ 2 2 ρ ) A ± i β 2 a ττ 2 A ± 2
+ i ( 2 γ 3 ) ( A ± 2 + 2 A 2 ) A ± .
z A ξ = ( ρ + δ 2 2 ρ ) A η i β 2 a ττ 2 A ξ 2
+ ( A ξ 2 A ξ + 2 A η 2 A ξ 3 + A ξ * A η 2 3 ) ,
z A = LA + i β a ( i τ ) A + [ 2 ( A A ) A + ( A t A ) ] A * 3 ,
z B = i β a ( i τ ) B + [ 2 ( B B ) B + ( B t B ) B * ] 3 ,
μ = cos ( ϕ 2 ) exp [ i ( ϕ 3 + ϕ 1 ) ] ,
ν = sin ( ϕ 2 ) exp [ i ( ϕ 3 ϕ 1 ) ] ,
ϕ 1 = tan 1 [ δ tan ( kz ) k ] 2 ,
ϕ 2 = sin 1 [ ρ sin ( kz ) k ] 2 ,
B ± ( ζ 0 , ζ 2 ) = ± [ B ¯ ± ( ζ 2 ) exp ( ± i ζ 0 ) A ¯ ( ζ 2 ) exp ( i ζ 0 ) ] 2 ,
( D 0 i ) C ± = D 2 A ¯ ± exp ( ± i ζ 0 ) i [ B ¯ exp ( i ζ 0 ) A ¯ ± exp ( ± i ζ 0 ) ] 2
D 2 A ¯ ± = ± i A ¯ ± 2 .
A ¯ ± ( ζ 2 ) = A ¯ ± ( 0 ) exp ( ± i ζ 2 2 ) .
C ± ( ζ 0 , ζ 2 ) = [ C ¯ ± ( ζ 2 ) exp ( ± i ζ 0 ) B ¯ ( ζ 2 ) exp ( i ζ 0 ) ] 4 ,
( D 0 L 0 ) A 3 = D 2 A 1 + L 1 A 2 .
A 3 ( ζ 0 , ζ 2 ) = D + ( ζ 0 , ζ 2 ) E + + D ( ζ 0 , ζ 2 ) E
( D 0 i ) D ± = D 2 [ B ¯ exp ( ± i ζ 0 ) A ¯ exp ( i ζ 0 ) ] 2
i [ C ¯ exp ( i ζ 0 ) B ¯ ± exp ( ± i ζ 0 ) ] 4 .
D 2 B ¯ ± = ± i B ¯ ± 2 .
B ¯ ± ( ζ 2 ) = A ¯ ( 0 ) exp ( ± i ζ 2 2 ) .
B ± ( ζ 0 , ζ 2 ) = i A ( 0 ) sin ( ζ 0 + ζ 2 2 ) ,

Metrics