Abstract

We derive the fundamental equations describing nonlinear propagation in multi-mode fibers in the presence of random mode coupling within quasi-degenerate groups of modes. Our result generalizes the Manakov equation describing mode coupling between polarizations in single-mode fibers. Nonlinear compensation of the modal dispersion is predicted and tested via computer simulations.

© 2012 OSA

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References

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  1. A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express20, 11673–11678 (2012).
    [CrossRef] [PubMed]
  2. D. Gloge, “Weakly guiding fibers,” Appl. Opt.10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  3. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).
  4. A. Mecozzi, C. Antonelli, and M. Shtaif, “Soliton trapping in multimode fibers with random mode coupling,” arXiv:1207.6506v2 [physics.optics] (2012).
  5. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B25, 1645–1654 (2008).
    [CrossRef]
  6. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
    [CrossRef]
  7. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6 × 6 MIMO processing,” J. Lightwave Technol.30, 521–531 (2012).
    [CrossRef]
  8. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, and G. Charlet, “Mode division multiplexing of 2 × 100Gb/s channels using an LCOS based spatial modulator,” J. Lightwave Technol.30, 618–623 (2012).
    [CrossRef]
  9. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express20, 11718–11733 (2012).
    [CrossRef] [PubMed]
  10. V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys.53, 55–67 (1982).
  11. A. Mecozzi, C. Antonelli, and M. Shtaif, “Optical nonlinearity in multi-mode fibers with random mode coupling,” in Proceedings of ECOC 2012, Paper P.1.11 (2012).
  12. S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” arXiv:1207.6645v1 [physics.optics] (2012).
  13. P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost “Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in Proceedings of ECOC 2011, Paper Tu.5.7 (2011).
  14. M. N. Islam, C. D. Poole, and J. P. Gordon, “Soliton trapping in birefringent optical fibers,” Opt. Lett.14, 1011–1013 (1989).
    [CrossRef] [PubMed]

2012 (4)

2011 (1)

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost “Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in Proceedings of ECOC 2011, Paper Tu.5.7 (2011).

2008 (1)

1997 (1)

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

1989 (1)

1982 (1)

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys.53, 55–67 (1982).

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).

1971 (1)

Agrawal, G. P.

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” arXiv:1207.6645v1 [physics.optics] (2012).

Antonelli, C.

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express20, 11718–11733 (2012).
[CrossRef] [PubMed]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express20, 11673–11678 (2012).
[CrossRef] [PubMed]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Soliton trapping in multimode fibers with random mode coupling,” arXiv:1207.6506v2 [physics.optics] (2012).

A. Mecozzi, C. Antonelli, and M. Shtaif, “Optical nonlinearity in multi-mode fibers with random mode coupling,” in Proceedings of ECOC 2012, Paper P.1.11 (2012).

Astruc, M.

Bigo, S.

Bigot-Astruc, M.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost “Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in Proceedings of ECOC 2011, Paper Tu.5.7 (2011).

Boivin, D.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost “Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in Proceedings of ECOC 2011, Paper Tu.5.7 (2011).

Bolle, C.

Boutin, A.

Brindel, P.

Burrows, E. C.

Charlet, G.

Esmaeelpour, M.

Essiambre, R.

Essiambre, R. J.

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” arXiv:1207.6645v1 [physics.optics] (2012).

Gloge, D.

Gnauck, A. H.

Gordon, J. P.

Horak, P.

Islam, M. N.

Koebele, C.

Lingle, R.

Maerten, H.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost “Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in Proceedings of ECOC 2011, Paper Tu.5.7 (2011).

Makhan’kov, V. G.

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys.53, 55–67 (1982).

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

Mardoyan, H.

McCurdy, A. H.

Mecozzi, A.

A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express20, 11673–11678 (2012).
[CrossRef] [PubMed]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express20, 11718–11733 (2012).
[CrossRef] [PubMed]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Optical nonlinearity in multi-mode fibers with random mode coupling,” in Proceedings of ECOC 2012, Paper P.1.11 (2012).

A. Mecozzi, C. Antonelli, and M. Shtaif, “Soliton trapping in multimode fibers with random mode coupling,” arXiv:1207.6506v2 [physics.optics] (2012).

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

Mumtaz, S.

Pashaev, O. K.

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys.53, 55–67 (1982).

Peckham, D. W.

Poletti, F.

Poole, C. D.

Provost, L.

Randel, S.

Ryf, R.

Salsi, M.

Shtaif, M.

A. Mecozzi, C. Antonelli, and M. Shtaif, “Nonlinear propagation in multi-mode fibers in the strong coupling regime,” Opt. Express20, 11673–11678 (2012).
[CrossRef] [PubMed]

C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express20, 11718–11733 (2012).
[CrossRef] [PubMed]

A. Mecozzi, C. Antonelli, and M. Shtaif, “Optical nonlinearity in multi-mode fibers with random mode coupling,” in Proceedings of ECOC 2012, Paper P.1.11 (2012).

A. Mecozzi, C. Antonelli, and M. Shtaif, “Soliton trapping in multimode fibers with random mode coupling,” arXiv:1207.6506v2 [physics.optics] (2012).

Sierra, A.

Sillard, P.

Sperti, D.

Tran, P.

Verluise, F.

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

Winzer, P. J.

Appl. Opt. (1)

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Opt. Lett. (1)

Sov. Phys. JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).

Theor. Math. Phys. (1)

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys.53, 55–67 (1982).

Other (4)

A. Mecozzi, C. Antonelli, and M. Shtaif, “Optical nonlinearity in multi-mode fibers with random mode coupling,” in Proceedings of ECOC 2012, Paper P.1.11 (2012).

S. Mumtaz, R. J. Essiambre, and G. P. Agrawal, “Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations,” arXiv:1207.6645v1 [physics.optics] (2012).

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost “Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in Proceedings of ECOC 2011, Paper Tu.5.7 (2011).

A. Mecozzi, C. Antonelli, and M. Shtaif, “Soliton trapping in multimode fibers with random mode coupling,” arXiv:1207.6506v2 [physics.optics] (2012).

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Figures (1)

Fig. 1
Fig. 1

(a) Three-dimensional plot of the power envelope of the two modes obtained from Eq. (1). The fact that solitary evolution is demonstrated validates the coupled ME, on the basis of which solitary evolution was predicted. (b) Timing difference TaTb, center frequencies (denoted by ωa and ωb), and pulse-widths τa and τb as a function of the propagation distance z. The black-dashed curves represent the solution of Eq. (1) and the solid-red by the solution of the coupled Manakov equations (2)(4). These two curves are hardly differentiable in the resolution of the figure. The horizontal dashed-dotted lines show the analytical steady state values obtained in [4].

Equations (8)

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

E z = i B ( 0 ) E B ( 1 ) E t i B ( 2 ) 2 2 E t 2 + i γ j h k m C j h k m E h * E k E m e ^ j ,
E a z = i β a E a β a E a t i β a 2 2 E a t 2 + i γ ( κ a a | E a | 2 + κ a b | E b | 2 ) E a ,
E b z = i β b E b β b E b t i β b 2 2 E b t 2 + i γ ( κ b a | E a | 2 + κ b b | E b | 2 ) E b ,
κ u v = k , m j u h v C j h k m δ h k δ j m + δ h m δ j k ( 2 N u ) ( 2 N v + δ v u ) ,
k , m j u h v C j h k m E h * E k E m e ^ j κ u v | E v | 2 E u .
κ u v = k , m j u h v C j h k m [ E h * E k E m E j * ] | E u | 2 | E v | 2 ,
E u = a u sech ( t T u τ u ) exp ( i ω u t )
[ X h * X k X m X j * ] [ | X u | 2 | X v | 2 ] = δ h k δ j m + δ h m δ j k ( 2 N u ) ( 2 N v + δ v u ) ,

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