## Tunable modulational instability sidebands via parametric resonance in periodically tapered optical fibers |

Optics Express, Vol. 20, Issue 22, pp. 25096-25110 (2012)

http://dx.doi.org/10.1364/OE.20.025096

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### Abstract

We analyze the modulation instability induced by periodic variations of group velocity dispersion and nonlinearity in optical fibers, which may be interpreted as an analogue of the well-known parametric resonance in mechanics. We derive accurate analytical estimates of resonant detuning, maximum gain and instability margins, significantly improving on previous literature on the subject. We also design a periodically tapered photonic crystal fiber, in order to achieve narrow instability sidebands at a detuning of 35 THz, above the Raman maximum gain peak of fused silica. The wide tunability of the resonant peaks by variations of the tapering period and depth will allow to implement sources of correlated photon pairs which are far-detuned from the input pump wavelength, with important applications in quantum optics.

© 2012 OSA

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(190.3100) Nonlinear optics : Instabilities and chaos

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 16, 2012

Revised Manuscript: September 28, 2012

Manuscript Accepted: September 29, 2012

Published: October 18, 2012

**Citation**

Andrea Armaroli and Fabio Biancalana, "Tunable modulational instability sidebands via parametric resonance in periodically tapered optical fibers," Opt. Express **20**, 25096-25110 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-25096

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### References

- V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics) (Springer, 1989).
- L. D. Landau and E. M. Lifshitz, Mechanics, Third Edition: Volume 1 (Course of Theoretical Physics) (Butterworth-Heinemann, 1976).
- V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1988). [CrossRef]
- H. W. Broer and C. Simó, “Resonance tongues in Hill’s equations: a geometric approach,” J. Diff. Equations166, 290–327 (2000). [CrossRef]
- 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). [CrossRef]
- V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett.3, 307–310 (1966).
- G. B. Whitham, “Non-linear dispersive waves,” Proc. R. Soc. Lond., Ser. A283, 238–261 (1965). [CrossRef]
- 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). [CrossRef]
- A. Hasegawa, “Observation of self-trapping instability of a plasma cyclotron wave in a computer experiment,” Phys. Rev. Lett.24, 1165–1168 (1970). [CrossRef]
- C. K. W. Tam, “Amplitude dispersion and nonlinear instability of whistlers,” Phys. Fluids12, 1028–1035 (1969). [CrossRef]
- 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. A64, 043606 (2001). [CrossRef]
- R. Lai and A. J. Sievers, “Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains,” Phys. Rev. B57, 3433–3443 (1998). [CrossRef]
- V. I. Karpman, “Self-modulation of Nonlinear Plane Waves in Dispersive Media,” JETP Lett.6, 277–279 (1967).
- K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett.56, 135–138 (1986). [CrossRef] [PubMed]
- N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” JETP69, 1089–1093 (1986).
- 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). [CrossRef] [PubMed]
- 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). [CrossRef]
- D. Y. Tang, W. S. Man, H. Tam, and M. Demokan, “Modulational instability in a fiber soliton ring laser induced by periodic dispersion variation,” Phys. Rev. A61, 023804 (2000). [CrossRef]
- N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management.” Opt. Lett.21, 570–572 (1996). [CrossRef] [PubMed]
- J. C. Bronski and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management.” Opt. Lett.21, 937–939 (1996). [CrossRef] [PubMed]
- 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). [CrossRef]
- S. Ambomo, C. M. Ngabireng, and P. Tchofo-Dinda, “Critical behavior with dramatic enhancement of modulational instability gain in fiber systems with periodic variation dispersion,” J. Opt. Soc. Am. B25, 425–433 (2008). [CrossRef]
- F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Rev. A220, 213–218 (1996).
- F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B14, 27–33 (1997). [CrossRef]
- F. K. Abdullaev and J. Garnier, “Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion.” Phys. Rev. E60, 1042–1050 (1999). [CrossRef]
- R. Bauer and L. Melnikov, “Multi-soliton fission and quasi-periodicity in a fiber with a periodically modulated core diameter,” Opt. Commun.115, 190–198 (1995). [CrossRef]
- D. E. Pelinovsky and J. Yang, “Parametric resonance and radiative decay of dispersion-managed solitons,” SIAM J. Appl. Math.64, 1360 (2004). [CrossRef]
- M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, L. Bigot, and A. Mussot, “Demonstration of modulation instability assisted by a periodic dispersion landscape in an optical fiber,” in “CLEO: Science and Innovations,” (Optical Society of America, 2012), p. CTh4B.7.
- M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, L. Bigot, and A. Mussot, Université Lille 1, Laboratoire PhLAM, IRCICA, 59655 Villeneuve d’Ascq, France (private communication, 2012).
- P. St.J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003). [CrossRef] [PubMed]
- J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. St.J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express13, 534–544 (2005). [CrossRef] [PubMed]
- C. Söller, B. Brecht, P. J. Mosley, L. Y. Zang, A. Podlipensky, N. Y. Joly, P. St.J. Russell, and C. Silberhorn, “Bridging visible and telecom wavelengths with a single-mode broadband photon pair source,” Phys. Rev. A81, 031801 (2010). [CrossRef]
- F. Verhulst, “Perturbation analysis of parametric resonance,” in Encyclopedia of Complexity and Systems Science, R. A. Meyers, ed. (SpringerNew York, 2009), pp. 6625–6639.
- B. Deconinck and J. N. Kutz, “Computing spectra of linear operators using the Floquet-Fourier-Hill method,” J. Comp. Phys.219, 296–321 (2006). [CrossRef]
- V. A. Labay, J. Bornemann, and A. Labay, “Matrix singular value decomposition for pole-free solutions of homogeneous matrix equations as applied to numerical modeling methods,” IEEE Microw. Guided Wave Lett.2, 49–51 (1992). [CrossRef]
- G. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2006).
- F. Biancalana, D. V. Skryabin, and P. St.J. Russell, “Four-wave mixing instabilities in photonic-crystal and tapered fibers,” Phys. Rev. E68, 046603 (2003). [CrossRef]
- “Comsol Multiphysics”, http://www.comsol.com .
- S. V. Afshar and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express17, 2298–2318 (2009). [CrossRef]

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