Abstract

In this paper, we introduce the concept of spatial and spectral control of nonlinear parametric sidebands in multimode optical fibers by tailoring their linear refractive index profile. In all cases, the pump experiences Kerr self-cleaning, leading to a bell-shaped beam profile. Geometric parametric instability, owing to quasi-phase matching from the dynamic grating generated via the Kerr effect by pump self-imaging, leads to frequency multicasting of beam self-cleaning across a wideband array of sidebands. Our experiments show that introducing a Gaussian dip into the refractive index profile of a graded index fiber permits us to dramatically change the spatial content of spectral sidebands into higher-order modes. This is due to the breaking of the oscillation synchronism among low-order and higher-order modes. Hence, the inter-modal four-wave mixing approach should be used to describe the sideband generation mechanism. Observations agree well with theoretical predictions based on a perturbative analysis and with full numerical solutions of the (3+1)D nonlinear Schrödinger equation.

© 2019 Optical Society of America

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References

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2018 (7)

S. Wabnitz, K. Krupa, V. Couderc, D. Modotto, A. Barthélémy, and G. Millot, “Nonlinear dynamics in multimode optical fibers,” Proc. SPIE 10540, 105402B (2018).
[Crossref]

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light beam compression from nonlinear mode coupling,” Phys. Rev. A 97, 043836 (2018).
[Crossref]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
[Crossref]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Supplementary information for adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
[Crossref]

O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97, 013841 (2018).
[Crossref]

C. Mas Arabí, A. Kudlinski, A. Mussot, and M. Conforti, “Geometric parametric instability in periodically modulated graded-index multimode fibers,” Phys. Rev. A 97, 023803 (2018).
[Crossref]

A. Bendahmane, K. Krupa, A. Tonello, D. Modotto, T. Sylvestre, V. Couderc, S. Wabnitz, and G. Millot, “Seeded intermodal four-wave mixing in a highly multimode fiber,” J. Opt. Soc. Am. B 35, 295–301 (2018).
[Crossref]

2017 (6)

2016 (6)

2015 (3)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

A. Picozzi, G. Millot, and S. Wabnitz, “Nonlinear virtues of multimode fibre,” Nat. Photonics 9, 289–291 (2015).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

2013 (1)

J. Cuevas, P. G. Kevrekidis, B. A. Malomed, P. Dyke, and R. G. Hulet, “Interactions of solitons with a Gaussian barrier: splitting and recombination in quasi-one-dimensional and three-dimensional settings,” New J. Phys. 15, 063006 (2013).
[Crossref]

2012 (1)

W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4, 1719 (2012).
[Crossref]

2003 (1)

1980 (1)

J. P. Meunier, J. Pigeon, and J. N. Massot, “Analyse perturbative des caractéristiques de propagation des fibres optiques à gradient d’indice quasi-parabolique,” Opt. Quantum Electron. 12, 41–49 (1980).
[Crossref]

1978 (1)

R. A. Sammut and A. K. Ghatak, “Perturbation theory of optical fibres with power-law core profile,” Opt. Quantum Electron. 10, 475–482 (1978).
[Crossref]

1977 (1)

E. Khular, A. Kumar, A. Ghatak, and B. Pal, “Effect of the refractive index dip on the propagation characteristics of step index and graded index fibers,” Opt. Commun. 23, 263–267 (1977).
[Crossref]

1962 (1)

J. D. Swalen and J. A. Ibers, “Potential function for the inversion of ammonia,” J. Chem. Phys. 36, 1914–1918 (1962).
[Crossref]

Arabi, C. M.

Atkins, P. W.

P. W. Atkins and R. Friedman, Molecular Quantum Mechanics (Oxford University, 2005).

Auguste, J. L.

Barthélémy, A.

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light beam compression from nonlinear mode coupling,” Phys. Rev. A 97, 043836 (2018).
[Crossref]

S. Wabnitz, K. Krupa, V. Couderc, D. Modotto, A. Barthélémy, and G. Millot, “Nonlinear dynamics in multimode optical fibers,” Proc. SPIE 10540, 105402B (2018).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25, 22219–22227 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthélémy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42, 1293–1296 (2017).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

E. Deliancourt, M. Fabert, A. Tonello, K. Krupa, A. Desfarges-Berthelemot, V. Kermene, G. Millot, A. Barthélémy, S. Wabnitz, and V. Couderc, “Kerr beam self-cleaning on the LP11 mode in graded-index multimode fiber,” arXiv:1807.00563 (2018).

Bendahmane, A.

A. Bendahmane, K. Krupa, A. Tonello, D. Modotto, T. Sylvestre, V. Couderc, S. Wabnitz, and G. Millot, “Seeded intermodal four-wave mixing in a highly multimode fiber,” J. Opt. Soc. Am. B 35, 295–301 (2018).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25, 22219–22227 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthélémy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42, 1293–1296 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
[Crossref]

K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guenard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, “Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber,” Opt. Lett. 41, 5785–5788 (2016).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Caravaca-Aguirre, A. M.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
[Crossref]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Supplementary information for adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
[Crossref]

Christodoulides, D.

Christodoulides, D. N.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358, 94–97 (2017).
[Crossref]

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10, 771–776 (2016).
[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41, 3675–3678 (2016).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

Conforti, M.

C. Mas Arabí, A. Kudlinski, A. Mussot, and M. Conforti, “Geometric parametric instability in periodically modulated graded-index multimode fibers,” Phys. Rev. A 97, 023803 (2018).
[Crossref]

M. Conforti, C. M. Arabi, A. Mussot, and A. Kudlinski, “Fast and accurate modeling of nonlinear pulse propagation in graded-index multimode fibers,” Opt. Lett. 42, 4004–4007 (2017).
[Crossref]

Correa, R. A.

Couderc, V.

S. Wabnitz, K. Krupa, V. Couderc, D. Modotto, A. Barthélémy, and G. Millot, “Nonlinear dynamics in multimode optical fibers,” Proc. SPIE 10540, 105402B (2018).
[Crossref]

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light beam compression from nonlinear mode coupling,” Phys. Rev. A 97, 043836 (2018).
[Crossref]

A. Bendahmane, K. Krupa, A. Tonello, D. Modotto, T. Sylvestre, V. Couderc, S. Wabnitz, and G. Millot, “Seeded intermodal four-wave mixing in a highly multimode fiber,” J. Opt. Soc. Am. B 35, 295–301 (2018).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25, 22219–22227 (2017).
[Crossref]

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthélémy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42, 1293–1296 (2017).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
[Crossref]

K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guenard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, “Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber,” Opt. Lett. 41, 5785–5788 (2016).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

E. Deliancourt, M. Fabert, A. Tonello, K. Krupa, A. Desfarges-Berthelemot, V. Kermene, G. Millot, A. Barthélémy, S. Wabnitz, and V. Couderc, “Kerr beam self-cleaning on the LP11 mode in graded-index multimode fiber,” arXiv:1807.00563 (2018).

Cuevas, J.

J. Cuevas, P. G. Kevrekidis, B. A. Malomed, P. Dyke, and R. G. Hulet, “Interactions of solitons with a Gaussian barrier: splitting and recombination in quasi-one-dimensional and three-dimensional settings,” New J. Phys. 15, 063006 (2013).
[Crossref]

Deliancourt, E.

E. Deliancourt, M. Fabert, A. Tonello, K. Krupa, A. Desfarges-Berthelemot, V. Kermene, G. Millot, A. Barthélémy, S. Wabnitz, and V. Couderc, “Kerr beam self-cleaning on the LP11 mode in graded-index multimode fiber,” arXiv:1807.00563 (2018).

Desfarges-Berthelemot, A.

Dupiol, R.

Dyke, P.

J. Cuevas, P. G. Kevrekidis, B. A. Malomed, P. Dyke, and R. G. Hulet, “Interactions of solitons with a Gaussian barrier: splitting and recombination in quasi-one-dimensional and three-dimensional settings,” New J. Phys. 15, 063006 (2013).
[Crossref]

Eftekhar, M. A.

Eznaveh, Z. S.

Fabert, M.

R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthélémy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42, 1293–1296 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
[Crossref]

K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25, 22219–22227 (2017).
[Crossref]

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[Crossref]

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R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
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K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light beam compression from nonlinear mode coupling,” Phys. Rev. A 97, 043836 (2018).
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S. Wabnitz, K. Krupa, V. Couderc, D. Modotto, A. Barthélémy, and G. Millot, “Nonlinear dynamics in multimode optical fibers,” Proc. SPIE 10540, 105402B (2018).
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A. Bendahmane, K. Krupa, A. Tonello, D. Modotto, T. Sylvestre, V. Couderc, S. Wabnitz, and G. Millot, “Seeded intermodal four-wave mixing in a highly multimode fiber,” J. Opt. Soc. Am. B 35, 295–301 (2018).
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C. Mas Arabí, A. Kudlinski, A. Mussot, and M. Conforti, “Geometric parametric instability in periodically modulated graded-index multimode fibers,” Phys. Rev. A 97, 023803 (2018).
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K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
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K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
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[Crossref]

A. Bendahmane, K. Krupa, A. Tonello, D. Modotto, T. Sylvestre, V. Couderc, S. Wabnitz, and G. Millot, “Seeded intermodal four-wave mixing in a highly multimode fiber,” J. Opt. Soc. Am. B 35, 295–301 (2018).
[Crossref]

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K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Spatial beam self-cleaning in multimode fibres,” Nat. Photonics 11, 237–241 (2017).
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R. Dupiol, A. Bendahmane, K. Krupa, A. Tonello, M. Fabert, B. Kibler, T. Sylvestre, A. Barthélémy, V. Couderc, S. Wabnitz, and G. Millot, “Far-detuned cascaded intermodal four-wave mixing in a multimode fiber,” Opt. Lett. 42, 1293–1296 (2017).
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R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
[Crossref]

K. Krupa, C. Louot, V. Couderc, M. Fabert, R. Guenard, B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A. Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, “Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber,” Opt. Lett. 41, 5785–5788 (2016).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

E. Deliancourt, M. Fabert, A. Tonello, K. Krupa, A. Desfarges-Berthelemot, V. Kermene, G. Millot, A. Barthélémy, S. Wabnitz, and V. Couderc, “Kerr beam self-cleaning on the LP11 mode in graded-index multimode fiber,” arXiv:1807.00563 (2018).

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O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, “Coexistence of collapse and stable spatiotemporal solitons in multimode fibers,” Phys. Rev. A 97, 013841 (2018).
[Crossref]

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O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Supplementary information for adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
[Crossref]

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
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S. Wabnitz, K. Krupa, V. Couderc, D. Modotto, A. Barthélémy, and G. Millot, “Nonlinear dynamics in multimode optical fibers,” Proc. SPIE 10540, 105402B (2018).
[Crossref]

K. Krupa, A. Tonello, V. Couderc, A. Barthélémy, G. Millot, D. Modotto, and S. Wabnitz, “Spatiotemporal light beam compression from nonlinear mode coupling,” Phys. Rev. A 97, 043836 (2018).
[Crossref]

A. Bendahmane, K. Krupa, A. Tonello, D. Modotto, T. Sylvestre, V. Couderc, S. Wabnitz, and G. Millot, “Seeded intermodal four-wave mixing in a highly multimode fiber,” J. Opt. Soc. Am. B 35, 295–301 (2018).
[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Nonlinear beam self-cleaning in a coupled cavity composite laser based on multimode fiber,” Opt. Express 25, 22219–22227 (2017).
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[Crossref]

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[Crossref]

R. Guenard, K. Krupa, R. Dupiol, M. Fabert, A. Bendahmane, V. Kermene, A. Desfarges-Berthelemot, J. L. Auguste, A. Tonello, A. Barthélémy, G. Millot, S. Wabnitz, and V. Couderc, “Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber,” Opt. Express 25, 4783–4792 (2017).
[Crossref]

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[Crossref]

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[Crossref]

A. Picozzi, G. Millot, and S. Wabnitz, “Nonlinear virtues of multimode fibre,” Nat. Photonics 9, 289–291 (2015).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
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E. Deliancourt, M. Fabert, A. Tonello, K. Krupa, A. Desfarges-Berthelemot, V. Kermene, G. Millot, A. Barthélémy, S. Wabnitz, and V. Couderc, “Kerr beam self-cleaning on the LP11 mode in graded-index multimode fiber,” arXiv:1807.00563 (2018).

Wagner, K.

O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Supplementary information for adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
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O. Tzang, A. M. Caravaca-Aguirre, K. Wagner, and R. Piestun, “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres,” Nat. Photonics 12, 368–374 (2018).
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[Crossref]

Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41, 3675–3678 (2016).
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L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
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W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4, 1719 (2012).
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Wright, L. G.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal mode-locking in multimode fiber lasers,” Science 358, 94–97 (2017).
[Crossref]

L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10, 771–776 (2016).
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G. Lopez-Galmiche, Z. S. Eznaveh, M. A. Eftekhar, J. A. Lopez, L. G. Wright, F. Wise, D. Christodoulides, and R. A. Correa, “Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm,” Opt. Lett. 41, 2553–2556 (2016).
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Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41, 3675–3678 (2016).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
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[Crossref]

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L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

A. Picozzi, G. Millot, and S. Wabnitz, “Nonlinear virtues of multimode fibre,” Nat. Photonics 9, 289–291 (2015).
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L. G. Wright, Z. Liu, D. A. Nolan, M.-J. Li, D. N. Christodoulides, and F. W. Wise, “Self-organized instability in graded-index multimode fibres,” Nat. Photonics 10, 771–776 (2016).
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Figures (12)

Fig. 1.
Fig. 1. Iso-intensity surfaces identifying the local variation of the refractive index induced by the Kerr effect and obtained for δn=0 (a) and δn=0.004 (b).
Fig. 2.
Fig. 2. Numerical simulation of sideband generation for an input intensity of 10GW/cm2, pulse duration of 9 ps, input beam diameter of 40 μm, and propagation distance of 0.16 m. First (a) and second (b) anti-Stokes sidebands obtained for δn=0 (green), δn=0.002 (orange), and δn=0.004 (violet). The lower frames show the corresponding beam shapes integrated over a 10 THz bandwidth, for the two extreme cases of δn=0 (green frame) and δn=0.004 (violet frame).
Fig. 3.
Fig. 3. (a) Beam diameter versus input (guided) peak power (Ppp) measured at the FWHMI, emerging from a 10-m-long GRIN dip MMF. The blue fitting curve is a guide for the eye. Insets: output near-field patterns for different Ppp. (b) Measured index profile of the dip fiber (red filled curve). Parabolic profile used in the simulations (black curve) and Gaussian approximation of the dip (yellow curve).
Fig. 4.
Fig. 4. Experimental output two-dimensional near-field shapes (normalized intensity in the linear scale) as a function of input guided power Ppp measured at the pump wavelength of 1064 nm in 10-m-long GRIN dip MMF. Panels (a) and (b) show the results obtained for slightly different input conditions. Asterisk (*): results for the Ppp at which frequency conversion into sidebands was also observed.
Fig. 5.
Fig. 5. Experimental spectra obtained from a 10-m-long GRIN MMF without (top, red curve) and with (bottom, blue curve) a central dip in the refractive index profile. The input guided power was Ppp=36kW. The vertical dashed lines indicate the analytically calculated sideband frequencies. The blue spectrum was down-shifted by 50 dB for better visualization.
Fig. 6.
Fig. 6. Experimental output two-dimensional near-field shapes (normalized intensity in linear scale) of a series of selected spectral components measured from 10-m-long GRIN MMF with a dip in their index profile at Ppp=36kW including the first four-orders anti-Stokes parametric sidebands (upper panel). Asterisk (*): near-field shape at the pump wavelength (1064 nm) in the linear regime with Ppp=18W.
Fig. 7.
Fig. 7. Measured refractive index profiles of two types of GRIN fibers: (a) fiber with a nearly parabolic profile—the black curve shows the parabolic profile used in the numerical simulations; (b) fiber with a parabolic profile with a dip on the top—the black curves and the yellow curve represent the parabolic profiles and the Gaussian approximation of the dip, respectively, as used in the numerical simulations.
Fig. 8.
Fig. 8. Low-order mode profiles for a GRIN fiber with a Gaussian dip with δn=4×103. Since these modes have intensity profiles similar to the linearly polarized modes of a step-index fiber, we adopted the same numbering.
Fig. 9.
Fig. 9. Calculated shift of the sideband positions upon barrier depth.
Fig. 10.
Fig. 10. Experimental evolutions of two-dimensional near-field shape at 1064 nm (normalized intensity in linear scale) (a) and spectrum (b) as a function of input guided power Ppp measured at the output of a 10-m-long GRIN MMF with perturbed index profile.
Fig. 11.
Fig. 11. Supercontinuum generation in the GRIN fiber with dip: fiber length of 10 m and average power of 36 kW. We used a series of 10-nm-wide bandpass filters with center wavelengths of 550, 650, 750, 900, 1000, 1200, 1300, 1500, and 1600, and a 3-nm-wide bandpass filter at 1064 nm.
Fig. 12.
Fig. 12. Experimental output two-dimensional near-field shapes (normalized intensity in linear scale) as a function of input guided power Ppp measured at the pump wavelength of (a) 1064 nm and (b) 532 nm in a 30-m-long GRIN dip MMF. Asterisk (*): results for the Ppp at which frequency conversion into sidebands was also observed.

Equations (14)

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δβδnωcS|ψ|2er2/Γ2dS,
n¯(r)=n012Δr2R2,
Ezi12k02E+iκ22Et2+ik0Δr2R2Eik0δnn0Eer2/Γ2=iγ(1fR)|E|2E.
2in0ω0cA˜zn02ω02c2A˜+2A˜=ω2c2n2A˜ω2c2ε0f˜NL.
n(r)=n¯(r)+δner2/Γ2,
iAz=12k02A+k0Δr2R2A,
iAz=12k02A+V0(r)A=H(0)A.
iAz=HA=(H(0)+H1)A,
βϵ=ψ*(H1ϵ)ψdxdy=ψ*(er2/Γ2)ψdxdy.
H(0)ψ0,n=β0,nψ0,n,
(H(0)β0,n)ψ0,n(1)=(β0,n(1)H(1))ψ0,n.
β0,n(1)=ψ0,n*(H(1))ψ0,ndxdy.
ψn(1)=ψ0,n+hh|H(1)|nβ0,nβ0,hψ0,h.
βn+2βn=β0,n+2β0,n+{ψ0,n+2*(H(1))ψ0,n+2dxdyψ0,n*(H(1))ψ0,ndxdy}.