Abstract

In this review we present recent theoretical and experimental progress on modulation instability and parametric amplification processes in dispersion oscillating fibers. These optical fibers are characterized by longitudinal periodic variations of their outer diameter engineered over the meter-long scale, which provides an additional degree of freedom to the system and leads to the generation of multiple MI sideband pairs. The main results published in single-pass configurations and in passive cavities are summarized in this review.

© 2017 Optical Society of America

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2017 (4)

C. G. L. Tiofack, S. Coulibaly, M. Taki, S. De Bièvre, and G. Dujardin, “Periodic modulations controlling Kuznetsov-Ma soliton formation in nonlinear Schrödinger equations,” Phys. Lett. A 381, 1999–2003 (2017).
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F. Copie, M. Conforti, A. Kudlinski, A. Mussot, F. Biancalana, and S. Trillo, “Instabilities in passive dispersion oscillating fiber ring cavities,” Eur. Phys. J. D 71, 133 (2017).
[Crossref]

F. Copie, M. Conforti, A. Kudlinski, S. Trillo, and A. Mussot, “Dynamics of Turing and Faraday instabilities in a longitudinally modulated fiber-ring cavity,” Opt. Lett. 42, 435–438 (2017).
[Crossref]

F. Copie, M. Conforti, A. Kudlinski, S. Trillo, and A. Mussot, “Modulation instability in the weak dispersion regime of a dispersion modulated passive fiber-ring cavity,” Opt. Express 25, 11283–11296 (2017).
[Crossref]

2016 (10)

M. Conforti, F. Copie, A. Mussot, A. Kudlinski, and S. Trillo, “Parametric instabilities in modulated fiber ring cavities,” Opt. Lett. 41, 5027–5030 (2016).
[Crossref]

U. Al Khawaja, S. M. Al-Marzoug, H. Bahlouli, and F. K. Abdullaev, “Modulational instability of the Kuznetsov-Ma breather in optical fibers with constant and periodic dispersion,” Commun. Nonlinear Sci. Numer. Simul. 32, 1–9 (2016).
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A. M. Perego, N. Tarasov, D. V. Churkin, S. K. Turitsyn, and K. Staliunas, “Pattern generation by dissipative parametric instability,” Phys. Rev. Lett. 116, 028701 (2016).
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N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
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P. Suret, R. E. Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
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B. Wetzel, D. Bongiovanni, M. Kues, Y. Hu, Z. Chen, S. Trillo, J. M. Dudley, S. Wabnitz, and R. Morandotti, “Experimental generation of Riemann waves in optics: a route to shock wave control,” Phys. Rev. Lett. 117, 073902 (2016).
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M. Conforti, A. Mussot, A. Kudlinski, S. Rota Nodari, G. Dujardin, S. De Biévre, A. Armaroli, and S. Trillo, “Heteroclinic structure of parametric resonance in the nonlinear Schrodinger equation,” Phys. Rev. Lett. 117, 013901 (2016).
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M. Conforti, S. Trillo, A. Kudlinski, and A. Mussot, “Multiple QPM resonant radiations induced by MI in dispersion oscillating fibers,” IEEE Photon. Technol. Lett. 28, 740–743 (2016).
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F. Copie, M. Conforti, A. Kudlinski, A. Mussot, and S. Trillo, “Competing Turing and Faraday instabilities in longitudinally modulated passive resonators,” Phys. Rev. Lett. 116, 143901 (2016).
[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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2015 (8)

C. Finot, A. Sysoliatin, and S. Wabnitz, “Nonlinear parametric resonances in quasiperiodic dispersion oscillating fibers,” Opt. Commun. 348, 24–30 (2015).
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S. R. Nodari, M. Conforti, G. Dujardin, A. Kudlinski, A. Mussot, S. Trillo, and S. De Bièvre, “Modulational instability in dispersion-kicked optical fibers,” Phys. Rev. A 92, 013810 (2015).
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M. Conforti, S. Trillo, A. Mussot, and A. Kudlinski, “Parametric excitation of multiple resonant radiations from localized wavepackets,” Sci. Rep. 5, 9433 (2015).
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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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F. Feng, P. Morin, Y. K. Chembo, A. Sysoliatin, S. Wabnitz, and C. Finot, “Experimental demonstration of spectral sideband splitting in strongly dispersion oscillating fibers,” Opt. Lett. 40, 455–458 (2015).
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F. Copie, A. Kudlinski, M. Conforti, G. Martinelli, and A. Mussot, “Modulation instability in amplitude modulated dispersion oscillating fibers,” Opt. Express 23, 3869–3875 (2015).
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C. Fourcade-Dutin, Q. Bassery, D. Bigourd, A. Bendahmane, A. Kudlinski, M. Douay, and A. Mussot, “12  THz flat gain fiber optical parametric amplifiers with dispersion varying fibers,” Opt. Express 23, 10103–10110 (2015).
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C. Finot and S. Wabnitz, “Influence of the pump shape on the modulation instability process induced in a dispersion-oscillating fiber,” J. Opt. Soc. Am. B 32, 892–899 (2015).
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2014 (7)

X. Wang, D. Bigourd, A. Kudlinski, K. K. Y. Wong, M. Douay, L. Bigot, A. Lerouge, Y. Quiquempois, and A. Mussot, “Correlation between multiple modulation instability side lobes in dispersion oscillating fiber,” Opt. Lett. 39, 1881–1884 (2014).
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M. Conforti, A. Mussot, A. Kudlinski, and S. Trillo, “Modulational instability in dispersion oscillating fiber ring cavities,” Opt. Lett. 39, 4200–4203 (2014).
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A. Armaroli and F. Biancalana, “Suppression and splitting of modulational instability sidebands in periodically tapered optical fibers because of fourth-order dispersion,” Opt. Lett. 39, 4804–4807 (2014).
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M. Conforti, F. Baronio, and S. Trillo, “Resonant radiation shed by dispersive shock waves,” Phys. Rev. A 89, 013807 (2014).
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F. Feng, J. Fatome, A. Sysoliatin, Y. Chembo, S. Wabnitz, and C. Finot, “Wavelength conversion and temporal compression of pulse train using dispersion oscillating fibre,” Electron. Lett. 50, 768–770 (2014).
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C. Finot, F. Feng, Y. Chembo, and S. Wabnitz, “Gain sideband splitting in dispersion oscillating fibers,” Opt. Fiber Technol. 20, 513–519 (2014).
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A. Mussot, A. Kudlinski, M. Droques, P. Szriftgiser, and N. Akhmediev, “Fermi-Pasta-Ulam recurrence in nonlinear fiber optics: the role of reversible and irreversible losses,” Phys. Rev. X 4, 011054 (2014).
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2013 (11)

M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, “Dynamics of the modulation instability spectrum in optical fibers with oscillating dispersion,” Phys. Rev. A 87, 013813 (2013).
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A. Armaroli and F. Biancalana, “Vector modulational instability induced by parametric resonance in periodically tapered highly birefringent optical fibers,” Phys. Rev. A 87, 063848 (2013).
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M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7, 868–874 (2013).
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K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7, 102–112 (2013).
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K. Staliunas, C. Hang, and V. V. Konotop, “Parametric patterns in optical fiber ring nonlinear resonators,” Phys. Rev. A 88, 023846 (2013).
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K. E. Webb, Y. Q. Xu, M. Erkintalo, and S. G. Murdoch, “Generalized dispersive wave emission in nonlinear fiber optics,” Opt. Lett. 38, 151–153 (2013).
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T. Godin, B. Wetzel, T. Sylvestre, L. Larger, A. Kudlinski, A. Mussot, A. Ben Salem, M. Zghal, G. Genty, F. Dias, and J. M. Dudley, “Real time noise and wavelength correlations in octave-spanning supercontinuum generation,” Opt. Express 21, 18452–18460 (2013).
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M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, A. Mussot, A. Armaroli, and F. Biancalana, “Fourth-order dispersion mediated modulation instability in dispersion oscillating fibers,” Opt. Lett. 38, 3464–3467 (2013).
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M. Conforti and S. Trillo, “Dispersive wave emission from wave breaking,” Opt. Lett. 38, 3815–3818 (2013).
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C. Finot, J. Fatome, A. Sysoliatin, A. Kosolapov, and S. Wabnitz, “Competing four-wave mixing processes in dispersion oscillating telecom fiber,” Opt. Lett. 38, 5361–5364 (2013).
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B. Sévigny, O. Vanvincq, C. Valentin, N. Chen, Y. Quiquempois, and G. Bouwmans, “Four-wave mixing stability in hybrid photonic crystal fibers with two zero-dispersion wavelengths,” Opt. Express 21, 30859–30873 (2013).
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2012 (10)

J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express 20, 9227–9242 (2012).
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B. P.-P. Kuo, J. M. Fini, L. Grüner-Nielsen, and S. Radic, “Dispersion-stabilized highly-nonlinear fiber for wideband parametric mixer synthesis,” Opt. Express 20, 18611–18619 (2012).
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A. Armaroli and F. Biancalana, “Tunable modulational instability sidebands via parametric resonance in periodically tapered optical fibers,” Opt. Express 20, 25096–25110 (2012).
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M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, “Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape,” Opt. Lett. 37, 4832–4834 (2012).
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A. Mussot, A. Kudlinski, R. Habert, I. Dahman, G. Mélin, L. Galkovsky, A. Fleureau, S. Lempereur, L. Lago, D. Bigourd, T. Sylvestre, M. W. Lee, and E. Hugonnot, “20  THz-bandwidth continuous-wave fiber optical parametric amplifier operating at 1  μm using a dispersion-stabilized photonic crystal fiber,” Opt. Express 20, 28906–28911 (2012).
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B. Wetzel, A. Stefani, L. Larger, P. A. Lacourt, J. M. Merolla, T. Sylvestre, A. Kudlinski, A. Mussot, G. Genty, F. Dias, and J. M. Dudley, “Real-time full bandwidth measurement of spectral noise in supercontinuum generation,” Sci. Rep. 2, 882 (2012).
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D. R. Solli, G. Herink, B. Jalali, and C. Ropersand, “Fluctuations and correlations in modulation instability,” Nat. Photonics 6, 463–468 (2012).
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S. K. Turitsyn, B. G. Bale, and M. P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521, 135–203 (2012).
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B. Kibler, J. Fatome, C. Finot, G. Millot, G. Genty, B. Wetzel, N. Akhmediev, F. Dias, and J. M. Dudley, “Observation of Kuznetsov-Ma soliton dynamics in optical fibre,” Sci. Rep. 2, 463 (2012).
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S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. 18, 670–680 (2012).
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2010 (3)

2009 (1)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Phys. D 238, 540–548 (2009).
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2008 (1)

2007 (3)

2006 (2)

2005 (2)

2004 (2)

2003 (5)

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. Knight, W. J. Wadsworth, and P. St.J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28, 2225–2227 (2003).
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S. Atkins, A. Rosen, A. Bekker, and B. Fischer, “Evolution of localization in frequency for modulated light pulses in a recirculating fiber loop,” Opt. Lett. 28, 2228–2230 (2003).
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A. Kumar, A. Labruyere, and P. Tchofo Dinda, “Modulational instability in fiber systems with periodic loss compensation and dispersion management,” Opt. Commun. 219, 221–232 (2003).
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P. St.J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
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S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226, 415–422 (2003).
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2002 (3)

F. Consolandi, C. De Angelis, A.-D. Capobianco, G. Nalesso, and A. Tonello, “Parametric gain in fiber systems with periodic dispersion management,” Opt. Commun. 208, 309–320 (2002).
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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).
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G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental study of the reversible behavior of modulational instability in optical fibers,” J. Opt. Soc. Am. B 19, 477–486 (2002).
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2001 (4)

K. Inoue and T. Mukai, “Signal wavelength dependence of gain saturation in a fiber optical parametric amplifier,” Opt. Lett. 26, 10–12 (2001).
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S. Coen and M. Haelterman, “Continuous-wave ultrahigh-repetition-rate pulse-train generation through modulational instability in a passive fiber cavity,” Opt. Lett. 26, 39–41 (2001).
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G. Van Simaeys, P. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001).
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F. K. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose-Einstein condensates,” Phys. Rev. A 64, 043606 (2001).
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1999 (5)

F. K. Abdullaev and J. Garnier, “Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion,” Phys. Rev. E 60, 1042–1050 (1999).
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S. Coen, M. Haelterman, P. Emplit, L. Delage, L. M. Simohamed, and F. Reynaud, “Bistable switching induced by modulational instability in a normally dispersive all-fibre ring cavity,” J. Opt. B 1, 36–42 (1999).
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M. E. Marhic and F. S. Yang, “High-nonlinearity fiber optical parametric amplifier with periodic dispersion compensation,” J. Lightwave Technol. 17, 210–215 (1999).
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S. Coen and M. Haelterman, “Competition between modulational instability and switching in optical bistability,” Opt. Lett. 24, 80–82 (1999).
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B. Fischer, A. Rosen, and S. Fishman, “Localization in frequency for periodically kicked light propagation in a dispersive single-mode fiber,” Opt. Lett. 24, 1463–1465 (1999).
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1998 (2)

M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B 15, 2269–2275 (1998).
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R. Lai and A. J. Sievers, “Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains,” Phys. Rev. B 57, 3433–3443 (1998).
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1997 (5)

1996 (5)

I. R. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21, 327–329 (1996).
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N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management,” Opt. Lett. 21, 570–572 (1996).
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J. C. Bronski and J. Nathan Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. 21, 937–939 (1996).
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N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electron. Lett. 32, 54–55 (1996).
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F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
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1995 (3)

K. Kikuchi, C. Lorattanasane, F. Futami, and S. Kaneko, “Observation of quasi-phase matched four-wave mixing assisted by periodic power variation in a long-distance optical amplifier chain,” IEEE Photon. Technol. Lett. 7, 1378–1380 (1995).
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S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20, 866–868 (1995).
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C. De Angelis, M. Santagiustina, and S. Trillo, “Four-photon homoclinic instabilities in nonlinear highly birefringent media,” Phys. Rev. A 51, 774–791 (1995).
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1994 (1)

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
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1993 (2)

F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. 18, 1499–1501 (1993).
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N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
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1992 (2)

M. Haelterman, S. Trillo, and S. Wabnitz, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett. 17, 745–747 (1992).
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M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Opt. Commun. 91, 401–407 (1992).
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1991 (8)

G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
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S. V. Chernikov and P. V. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. Am. B 8, 1633–1641 (1991).
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S. Trillo and S. Wabnitz, “Dynamics of the nonlinear modulational instability in optical fibers,” Opt. Lett. 16, 986–988 (1991).
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S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
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S. Trillo and S. Wabnitz, “Nonlinear modulation of coupled waves in birefringent optical fibers,” Phys. Lett. A 159, 252–256 (1991).
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E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
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V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
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P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. 27, 2347–2355 (1991).
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1987 (2)

K. Tajima, “Compensation of soliton broadening in nonlinear optical fibers with loss,” Opt. Lett. 12, 54–56 (1987).
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L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
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1986 (2)

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear schrödinger equation,” Teor. Mat. Fiz. 69, 1089–1093 (1986).
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K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56, 135–138 (1986).
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1984 (1)

1979 (1)

B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep. 52, 263–379 (1979).
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1968 (1)

T. Taniuti and H. Washimi, “Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma,” Phys. Rev. Lett. 21, 209–212 (1968).
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1967 (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water part 1. Theory,” J. Fluid Mech. 27, 417–430 (1967).
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U. Al Khawaja, S. M. Al-Marzoug, H. Bahlouli, and F. K. Abdullaev, “Modulational instability of the Kuznetsov-Ma breather in optical fibers with constant and periodic dispersion,” Commun. Nonlinear Sci. Numer. Simul. 32, 1–9 (2016).
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F. K. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, “Nonlinear excitations in arrays of Bose-Einstein condensates,” Phys. Rev. A 64, 043606 (2001).
[Crossref]

F. K. Abdullaev and J. Garnier, “Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion,” Phys. Rev. E 60, 1042–1050 (1999).
[Crossref]

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sorensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27–33 (1997).
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F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
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F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
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J. M. Dudley and J. Taylor, Supercontinuum Generation in Optical Fibers, 1st ed. (Cambridge University, 2010).

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

C. Denz, S. Flach, and Y. S. Kivshar, eds., Nonlinearities in Periodic Structures and Metamaterials, Springer Series in Optical Sciences (Springer, 2010), Vol. 150.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, 1st ed. (Cambridge University, 2007).

B. A. Malomed, Soliton Management in Periodic Systems, 1st ed. (Springer, 2005).

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Figures (21)

Figure 1
Figure 1 (a) Group-velocity dispersion evolution along the fiber. Gain spectra at the output of (b) a uniform fiber by pumping in the anomalous dispersion, (c) a uniform fiber by pumping in the normal dispersion, (d) a dispersion oscillating fiber by pumping in the anomalous dispersion, and (e) a dispersion oscillating fiber by pumping in the normal dispersion. Color and line styles are the same for (a)–(e) figures. (f) Detuning of the quasi-phase-matched MI sidelobes versus sideband order m: solid line from Eq. (4) and circles from numerical simulations shown in (e). Parameters: β¯2=±1×10−27  s2/m, βM=0.9×10−27  s2/m, γ=7.5/W/km, PP=20  W, and Z=5  m. Numerical simulations are inspired from the method described in [65] where signal and idler characteristics at the output of a fiber made of several uniform pieces of fibers are evaluated by means of a matrix product, each matrix depending on the uniform fiber piece characteristics. We applied this method to DOFs by sampling the fiber with a very short step to obtain a concatenation of very short uniform fiber pieces. The advantage of doing so compared to NLSE integration is that it is extremely rapid to calculate the gain curve at the output of DOFs. Note that it is possible to include linear losses and well as Raman effect, as reported in [66]. We will refer to as “segmentation method” in the rest of the paper.

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Figure 2
Figure 2 2D plots of the output gain spectrum in a dispersion oscillating fiber with a sinusoidal modulation versus (a) the amplitude of modulation of the group-velocity dispersion and (b) the average value of the group-velocity dispersion. Parameters: similar to those used in Fig. 1, β¯2 is equal to 1×10−27  s2/m in (a) and βM is equal to 0.9×10−27  s2/m in (b).

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Figure 3
Figure 3 (a) Group-velocity dispersion evolution along the fiber length. (b) Corresponding output spectra. Inset: close-up of the first sidebands. Similar parameters as those used in Fig. 1 except the modulation formats.

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Figure 4
Figure 4 (a) GVD evolution versus wavelength for different pitch values, ΛREF=3.4  μm (dashed black lines), ΛMAX=4.76  μm (solid blue line), and Λmin=2.04  μm (solid red line). The gray area delimits a variation of ±7% of the pitch around the reference value. (b)–(e) Evolutions of β2,3,4 and γ versus the normalized value of the pitch [(Λ−ΛREF)/ΛREF].

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Figure 5
Figure 5 (a) Outer diameter evolution with ±7% variations, targeted curve (solid line), measurements (crosses). (b) Calculated GVD at 1065.7 nm.

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Figure 6
Figure 6 Relative variation of the GVD [β2Max−β2minβ¯2] and of the nonlinear coefficient [γMax−γminγ¯] as a function of the pump.

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Figure 7
Figure 7 Sketch illustrating, in the complex plane, the effect of the periodic modulation terms fZ(z) and gZ(z) on the eigenvalues of the linearized Floquet map (13). Black dots correspond to the unperturbed eigenvalues lying on the unit circle (dashed line). Colored dots show the new position of the eigenvalues after switching on the modulations, leading to a stable regime when k≠πmZ (left sketch) and an unstable one when k=πmZ (right sketch, red dots).

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Figure 8
Figure 8 Color level plot of the parametric g(ω) for a fiber with periodic GVD modulation β¯2=1×10−27  s2/m, βa,b=1±βM, La=Lb=Z/2=2.5  m, γ=7.5/W/km, P=20  W for (a) and βM=1×0.9−27  s2/m. Dashed black curves indicate the parametric resonance frequencies from Eq. (16), whereas magenta curves indicate the quasi-phase-matching frequencies from Eq. (4).

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Figure 9
Figure 9 Gain calculated from three-wave model Eq. (22) (blue circles), from Floquet theory (black curve), “segmentation” (red curve), and NLSE (green dots). Parameters correspond to the case studied in [45]: L=120  m β¯2=1.2×10−27  s2/m, βM=1.5×10−27  s2/m, γ=7/W/Km, P=20  W.

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Figure 10
Figure 10 Evolution of (a) the gain from Eqs. (22) and (24) and (b) the relative phase θ(z) from Eq. (23) of the first amplified frequency (m=1) for a sinusoidally modulated optical fiber whose parameters can be found in [45]. Analytical results are represented in red lines and numerical simulations in solid black lines. The green area corresponds to amplification and the red one to deamplification over one period. [Figure 2] reprinted with permission from Droques et al., Phys. Rev. A, 87, 013813 (2013) [45]. Copyright 2013 by the American Physical Society.

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Figure 11
Figure 11 Schemes representing different fiber optic systems in which one of the longitudinal parameters is modulated. (a) A telecommunication system with a periodic modulation of the power [37], (b) a recirculating loop aiming at mimicking a telecommunication system [61], (c) a birefringent fiber with a periodic modulation of the effective index by applying a mechanical stress [76], (d) a PCF with a periodic modulation of the outer diameter during the drawing process [49], (e) a passive fiber ring cavity made of different fibers, and (e) [77], and a graded-index multimode fiber with a periodic variation of the field due to self-imaging [78].

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Figure 12
Figure 12 Experimental (dashed blue lines) and numerical (solid red line) output spectra. Vertical dashed lines represent quasi-phase-matched frequencies obtained from Eq. (4) for m=1 to 10.

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Figure 13
Figure 13 (a) Average spectrum. (b)–(d) Scattergram between Q1−Q1*, Q2−Q2*, and Q1−Q2, respectively. (e) Energy correlation map. β¯2=0.7×10−27  s2/m, β2A=+1.8×10−27  s2/m, β3=0.52×10−40  s2/m, β4=−1.1×10−55  s2/m, γ=5.5/W/km, PP=39  W, α=1  dB/km, Z=10  m, and L=68  m.

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Figure 14
Figure 14 (a) Quasi-phase-matching curves calculated from Eq. (26) with and without the β4 term (blue solid and red dashed lines, respectively) as a function of average GVD. (b) QPM curves calculated from Eq. (26) (solid line) and measurement of MI sideband frequencies done by tuning the pump wavelength (markers). Reprinted with permission from [50]. Copyright 2013 Optical Society of America.

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Figure 15
Figure 15 (a) Gain spectrum in amplitude modulated fibers as a function of the Z1/Z2 ratio (Z1 is fixed and equal to 7.5 m, Z2 varies from ∞ to Z1). The dotted lines represent the predicted positions of the sidelobes maximum. The external diameter and dispersion profile of (b) the reference oscillating fiber, and (c) the amplitude modulated fiber. (d) Experimental spectra out of the reference fiber (blue dotted line) and the amplitude modulated fiber (solid red line). The red arrows point out the predicted positions of the first two sidelobes in the amplitude modulated fiber. Reprinted with permission from [53]. Copyright 2015 Optical Society of America.

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Figure 16
Figure 16 (a) Outer diameter evolution versus fiber length. Inset: snapshot on one peak. (b) Spectra recorded at the fiber output and (c) numerical simulations. [Figures 5 and 7] reprinted with permission from Nodari et al., Phys. Rev. A, 92, 013810 (2015) [51]. Copyright 2015 by the American Physical Society.

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Figure 17
Figure 17 (a) Gain curves, experimental results (circles), numerical simulation (solid line). Parameters: pump power 26 W, signal power 10 μW, fiber length 120 m, period of modulation 10 m, amplitude of modulation 2×10−27  s2/m, and linear loss 8 dB/km. (b) Optimal longitudinal evolution of the group-velocity dispersion to obtain the flat gain curve shown in (c). (c) Gain curve at the output of fiber with dispersion shown in (b) for 30 W pump power.

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Figure 18
Figure 18 Contour plot of experimental spectra versus input pump power. The fiber was sinusoidally modulated with a period of 20 m and the average dispersion value is 0.74  ps2/km. Reprinted with permission from [60]. Copyright 2013 Optical Society of America.

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Figure 19
Figure 19 (a) Evolution of extracted pump (blue line) and sideband power (red line) fractions from numerical simulations of the NLSE Eq. (1) compared with those from the average model (dashed lines), (Eq. (10) of [52]). (b) Output sideband fraction η versus frequency detuning ω from NLSE numerical integration at fixed z (black solid line; red solid line is the maximum achievable conversion along z) with superimposed small-signal gain (blue). Inset: optimal frequency ωc versus input signal fraction η0. [Figures 4 and 5] reprinted with permission from Conforti et al., Phys. Rev. Lett., 117, 013901 (2016) [52]. Copyright 2016 by the American Physical Society.

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Figure 20
Figure 20 Experimental (red curve) and numerical (blue curve) output spectra for (a) 5 W of pump power and (c) 8 W of pump power. (b) QPM relation. Vertical black lines correspond to the MI sidelobe positions determined by the graphical solutions of Eq. (26), and (d) QPM relation for a soliton. The vertical green lines correspond to the RR positions determined by the graphical solution of Eq. (2) of [57] while black dashed lines to the one of Eq. (26). © 2016 IEEE. Reprinted, with permission, from Conforti et al., IEEE Photonics. Technol. Lett., 28, 740–743 (2016) [57].

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Figure 21
Figure 21 (a) Bistable response of the cavity for the operating normalized detuning Δ=δ/α=4 (δ=π/4.5  rad; loss coefficient α=0.157); the hatched region is unstable (inaccessible). (b) Pseudo-color level plot of the gain spectrum as a function of the intracavity power calculated from Floquet analysis [114]. (c), (d) Comparison of experimental spectra (solid blue), spectra obtained from numerical integrations of the periodic LLE (dashed red), and analytical estimates (vertical line). (c) Pin=3.16  W; (d) Pin=1.97  W. [Figures 3 and 4] reprinted with permission from Copie et al., Phys. Rev. Lett., 116, 143901 (2016) [77]. Copyright 2016 by the American Physical Society.

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Equations (28)

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i∂u∂z−β2(z)2∂2u∂t2+γ|u|2u=0.
β2(z)=β¯2+βM×sin(2πz/Z),
β2ω2+2γP=0.
β¯2ω2+2γP=2mπ/Z,
dFiber=dPreformVPreformVFiber,
i∂u∂z−β2(z)2∂2u∂t2+γ|u|2u≃0.
i∂u∂z−β22∂2u∂t2+γ(z)|u|2u≃0.
i∂u∂z−β2(z)2∂2u∂t2+γ(z)|u|2u=0,
β2(z)=β¯2+βMfZ(z),γ(z)=γ¯+γMgZ(z),
i∂v∂z−β2(z)2∂2v∂t2+γ(z)P(v+v*)=0.
{∂q∂z−β2(z)2∂2p∂t2=0,∂p∂z+β2(z)2∂2q∂t2−2γ(z)Pq=0.
{∂q^∂z+β2(z)2ω2p^=0,∂p^∂z−β2(z)2ω2q^−2γ(z)Pq^=0.
g(ω)=1Zln(max{|λ+|,|λ−|}),
ϕav=(cos(kavZ)−β¯2ω22kav sin(kavZ)2kβ¯2ω2 sin(kavZ)cos(kavZ)),
kav=β¯22ω2(β¯22ω2+2γ¯P).
λav±=exp(±ikavZ).
ωm=2β¯2((γ¯P)2+(mπZ)2−γ¯P),
kav(ωm)=mπZ=mkg2
β¯2ωm2+2γ¯P=2πmZ.
Ψ=ϕaϕb,
λ±=Δ2±Δ24−1,
Δ=2 cos(kaLa)cos(kbLb)−σ sin(kaLa)sin(kbLb)
g(ω,z)=2γPp∑q=−∞q=+∞Jq(βMω22π/Z)sin[(β2¯ω2+2γPp−q2πZ)z+Kq],
θ(ω,z)=[β2¯ω2+2γPp]z+βMω22π/Z[1−cos(2πz/Z)]+θ(ω,0).
g(ωm,z)=2γPp|Jm(βMωm22π/Z)|,
ρ=⟨E(f1)E(f2)⟩−⟨E(f1)⟩⟨E(f2)⟩(⟨E(f1)2⟩−⟨E(f1)⟩2)(⟨E(f2)2⟩−⟨E(f2)⟩2),
β¯2ω2+β¯412ω4+2γP=2mπ/Z,
−Δk1ω+β2¯ω2/2+β3¯ω3/6+β4¯ω4/24−γPS/2=2mπ/Z,

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