Abstract

We show that light propagation in a group of degenerate modes of a multi-mode optical fiber in the presence of random mode coupling is described by a multi-component Manakov equation, thereby making multi-mode fibers the first reported physical system that admits true multi-component soliton solutions. The nonlinearity coefficient appearing in the equation is expressed rigorously in terms of the multi-mode fiber parameters.

© 2012 OSA

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  1. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  2. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  3. 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]
  4. 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]
  5. 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]
  6. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fibers with multiple mode transmission,” Opt. Express, (submitted).
    [PubMed]
  7. V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
    [CrossRef]
  8. A. Hasegawa , “Self-confinement of multimode optical pulse in a glass fiber,” Opt. Lett. 5, 416–417 (1980).
    [CrossRef] [PubMed]
  9. B. Crosignani and P. Di Porto, “Soliton propagation in multi-mode optical fibers,” Opt. Lett. 6, 329–331 (1981).
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  10. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
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  12. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multi-mode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654 (2008).
    [CrossRef]
  13. 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]
  14. P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.
  15. M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
    [CrossRef]
  16. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [CrossRef] [PubMed]
  17. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collisions,” Opt. Lett. 20, 2060–2062 (1995).
    [CrossRef] [PubMed]
  18. A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (2012).
    [CrossRef] [PubMed]

2012

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]

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]

A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (2012).
[CrossRef] [PubMed]

2008

2003

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

2000

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

1997

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]

1995

1991

1982

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[CrossRef]

1981

1980

1974

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1972

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1971

Agrawal, G. P.

G. P. Agrawal, Nonlinear fiber optics, 4th ed. (Elsevier Science & Technology, 2006).

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. Express, (submitted).
[PubMed]

Astruc, M.

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]

Bigo, S.

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]

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 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Boivin, D.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Bolle, C.

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]

Boutin, A.

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]

Brindel, P.

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]

Burrows, E. C.

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]

Charlet, G.

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]

Chen, H. H.

Crosignani, B.

Di Porto, P.

Esmaeelpour, M.

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]

Essiambre, R.

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]

Gloge, D.

Gnauck, A. H.

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]

Gordon, J. P.

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

L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton–soliton collisions,” Opt. Lett. 20, 2060–2062 (1995).
[CrossRef] [PubMed]

Hasegawa, A.

Heismann, F.

Horak, P.

Jakubowski, M. H.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Koebele, C.

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]

Kogelnik, H.

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

Lingle, R.

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]

Maerten, H.

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Makhan’kov, V. G.

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[CrossRef]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 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.

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]

Matera, F.

McCurdy, A. H.

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]

Mecozzi, A.

A. Mecozzi and F. Matera, “Polarization scattering by intra-channel collisions,” Opt. Express 20, 1213–1218 (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. Express, (submitted).
[PubMed]

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]

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]

Mollenauer, L. F.

Mumtaz, S.

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]

Pashaev, O. K.

V. G. Makhan’kov and O. K. Pashaev, “Nonlinear Schrödinger equation with noncompact isogroup,” Theor. Math. Phys. 53, 979–987 (1982).
[CrossRef]

Peckham, D. W.

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]

Poletti, F.

Provost, L.

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]

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Randel, S.

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]

Ryf, R.

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]

Salsi, M.

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]

Sears, S. M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Segev, M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shtaif, M.

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

Sierra, A.

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]

Sillard, P.

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]

P. Sillard, M. Bigot-Astruc, D. Boivin, H. Maerten, and L. Provost“Few-mode fiber for uncoupled mode-division multiplexing transmissions,” in 37th European Conference and Exhibition on Optical Communication (ECOC), ECOC Technical Digest (Optical Society of America, 2011), Paper Tu.5.7.

Soljacic, M.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Sperti, D.

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]

Squier, R.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Steiglitz, K.

M. Soljačić, K. Steiglitz, S. M. Sears, M. Segev, M. H. Jakubowski, and R. Squier, “Collisions of two solitons in an arbitrary number of coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 90, 254102 (2003).
[CrossRef]

Tran, P.

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]

Verluise, F.

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]

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]

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]

Winzer, P. J.

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]

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

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Opt.

J. Lightwave Technol

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]

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]

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Figures (2)

Fig. 1
Fig. 1

(a) Collision of two solitons initially in two orthogonal modes in the case of full coupling, and initial angular frequency separation Δω = 0.4/τ. The ratio of the correlation length to the soliton length is Lc/Ls ≃ 2.8 × 10−3. (b) Collision of two solitons initially in two orthogonal modes in the case of absence of coupling, and initial angular frequency separation Δω = 0.4/τ. The time axis is normalized to the soliton time τ, the spatial axis to the soliton length Ls and the vertical axis to the soliton peak power P0.

Fig. 2
Fig. 2

(a) Collision of two solitons initially in two orthogonal modes in the same condition of Fig. 1, with ratio of the correlation length to the soliton length of Lc/Ls ≃ 2.8 × 10−1. (b) Collision of two solitons initially in two orthogonal modes in the same condition of Fig. 1, with ratio of the correlation length to the soliton length of Lc/Ls ≃ 2.8.

Equations (5)

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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 ,
D j h k m ( 1 ) = d x d y ( F j * F m ) ( F h * F k ) N j N h N k N m , D j h k m ( 2 ) = d x d y ( F j * F h * ) ( F m F k ) N j N h N k N m , N n 2 = d x d y | F n | 2 n f ,
κ = j h k m C j h k m δ h k δ j m + δ h m δ j k 2 N ( 2 N + 1 ) .
E z = i B ( 0 ) E B ( 1 ) E t i B ( 2 ) 2 2 E t 2 + i γ κ | E | 2 E .
E z = β E t i β 2 2 E t 2 + i γ κ | E | 2 E .

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