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

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    [CrossRef]
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2010 (1)

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]

2009 (1)

2008 (1)

2006 (1)

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]

2005 (1)

2004 (1)

D. E. Pelinovsky and J. Yang, “Parametric resonance and radiative decay of dispersion-managed solitons,” SIAM J. Appl. Math.64, 1360 (2004).
[CrossRef]

2003 (3)

P. St.J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
[CrossRef] [PubMed]

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]

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]

2001 (1)

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]

2000 (2)

H. W. Broer and C. Simó, “Resonance tongues in Hill’s equations: a geometric approach,” J. Diff. Equations166, 290–327 (2000).
[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]

1999 (1)

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]

1998 (1)

R. Lai and A. J. Sievers, “Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains,” Phys. Rev. B57, 3433–3443 (1998).
[CrossRef]

1997 (1)

1996 (3)

1995 (2)

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]

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]

1993 (1)

1992 (1)

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]

1986 (2)

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).

1970 (1)

A. Hasegawa, “Observation of self-trapping instability of a plasma cyclotron wave in a computer experiment,” Phys. Rev. Lett.24, 1165–1168 (1970).
[CrossRef]

1969 (1)

C. K. W. Tam, “Amplitude dispersion and nonlinear instability of whistlers,” Phys. Fluids12, 1028–1035 (1969).
[CrossRef]

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).
[CrossRef]

1967 (2)

V. I. Karpman, “Self-modulation of Nonlinear Plane Waves in Dispersive Media,” JETP Lett.6, 277–279 (1967).

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]

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett.3, 307–310 (1966).

1965 (1)

G. B. Whitham, “Non-linear dispersive waves,” Proc. R. Soc. Lond., Ser. A283, 238–261 (1965).
[CrossRef]

Abdullaev, F. K.

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]

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]

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, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Rev. A220, 213–218 (1996).

Afshar, S. V.

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2006).

Akhmediev, N. N.

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” JETP69, 1089–1093 (1986).

Ambomo, S.

Arnold, V. I.

V. I. Arnold, A. Weinstein, and K. Vogtmann, Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics) (Springer, 1989).

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, 1988).
[CrossRef]

Baizakov, B. B.

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]

Bauer, R.

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]

Benjamin, T. B.

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]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett.3, 307–310 (1966).

Biancalana, F.

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]

Bigot, L.

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).

Bischoff, S.

Bornemann, J.

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]

Bouwmans, G.

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).

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.

Brecht, B.

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]

Broer, H. W.

H. W. Broer and C. Simó, “Resonance tongues in Hill’s equations: a geometric approach,” J. Diff. Equations166, 290–327 (2000).
[CrossRef]

Bronski, J. C.

Darmanyan, S. A.

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]

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, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Rev. A220, 213–218 (1996).

Deconinck, B.

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]

Demokan, M.

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]

Doran, N. J.

Droques, M.

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).

Duligall, J.

Feir, J. E.

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]

Fulconis, J.

Futami, F.

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]

Garnier, J.

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]

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett.56, 135–138 (1986).
[CrossRef] [PubMed]

A. Hasegawa, “Observation of self-trapping instability of a plasma cyclotron wave in a computer experiment,” Phys. Rev. Lett.24, 1165–1168 (1970).
[CrossRef]

Joly, N. Y.

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]

Kaneko, S.

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]

Karpman, V. I.

V. I. Karpman, “Self-modulation of Nonlinear Plane Waves in Dispersive Media,” JETP Lett.6, 277–279 (1967).

Kikuchi, K.

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]

Kobyakov, A.

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).

Konotop, V. V.

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]

Korneev, V. I.

N. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrödinger equation,” JETP69, 1089–1093 (1986).

Kudlinski, A.

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).

Kumar, A.

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]

Kutz, J. N.

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]

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]

Labay, A.

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]

Labay, V. A.

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]

Labruyere, A.

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]

Lai, R.

R. Lai and A. J. Sievers, “Modulational instability of nonlinear spin waves in easy-axis antiferromagnetic chains,” Phys. Rev. B57, 3433–3443 (1998).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Mechanics, Third Edition: Volume 1 (Course of Theoretical Physics) (Butterworth-Heinemann, 1976).

Lederer, F.

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).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Mechanics, Third Edition: Volume 1 (Course of Theoretical Physics) (Butterworth-Heinemann, 1976).

Lorattanasane, C.

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]

Man, W. S.

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]

Martinelli, G.

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).

Matera, F.

Mecozzi, A.

Melnikov, L.

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]

Monro, T. M.

Mosley, P. J.

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]

Mussot, A.

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).

Ngabireng, C. M.

Pelinovsky, D. E.

D. E. Pelinovsky and J. Yang, “Parametric resonance and radiative decay of dispersion-managed solitons,” SIAM J. Appl. Math.64, 1360 (2004).
[CrossRef]

Podlipensky, A.

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[CrossRef]

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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. A64, 043606 (2001).
[CrossRef]

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[CrossRef]

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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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P. St.J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
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Figures (7)

Fig. 1
Fig. 1

Resonant tongues in the (δ, h) plane for the parametric resonance instability of a NLS with varying dispersion and nonlinearity, for (a) the first and (b) the second resonant bands. GVD is normal (s0 = +1) and α = 10, n1 = −s1 = 1 are chosen in order to obtain the maximum gain and bandwidthm, see Eqs. (13) and (16). The colormap provides the values of gain computed by means of an ODE solver. The solid green lines are the analytic predictions of Eq. (17) and the blue dotted lines with markers are the band edges computed numerically by employing the Hill determinant method (see in the text). The insets show the maximum gain vs. h (blue dotted line with markers) and in (a) also the curve obtained from Eq. (13) (green solid). Notice that since we reported in the text only the perturbation results of the first PR tongue, in (b) we show only numerical results. The vertical dashed line in the inset of (b) highlights the threshold behavior of g2 vs. h.

Fig. 2
Fig. 2

Same as in Fig. 1, but in the anomalous GVD regime (s0 = −1). In this case n1 = + s1 = 1.

Fig. 3
Fig. 3

Characterization of the first three PR peaks (m = 1, 2, 3) as a function of α. Normal GVD (s0 = +1), s1 = −1, n1 = 1 and h = 0.5. We plot in (a) the resonant frequency calculated in Eq. (10) and obtained from the ODE solution as the point which maximizes the instability gain, which is shown, in logarithmic scale, in (b), which includes also the solution of Eq. (13), for m = 1. Finally in (c) the instability bandwidth calculated numerically by means of the Hill determinant method and analytically for m = 1, according to Eq. (16). In every panel the same convention is used, i.e. solid lines represent analytical results, while dotted lines with markers are obtained by numerical calculations: specifically the blue dotted line with crosses is associated to m = 1, the green dotted line with stars to m = 2 and the red dotted line with circles to m = 3. The solid lines use the same color convention.

Fig. 4
Fig. 4

Same of Fig. 3 in the case of anomalous GVD (s0 = −1) and s1 = +1.

Fig. 5
Fig. 5

Numerical evolution of the NLS equation: (a) comparison of input (blue dashed line) and output (black solid) spectra. GVD is normal and the total propagation length is z = 38, α = 10, h = 0.5 and n1 = −s1 = 1. The m = 1, 2, 3 PR peaks as well as the first two mixing products of m = 1 are highlighted by arrows. (b) Amplification of the first three peaks: extracted from the spectrum (solid line) and predicted by the linearized analysis, i.e. exponential growth with gain gm (dashed line). (c) The detail of the amplification process on a shorter scale: we use a different set of parameters, α = 5 and h = 0.9 (with n1 = −s1 = 1 as above) in order to have higher gain and a larger period. Blue solid line represents the evolution of the 1st PR peak spectral component, the dashed green line the solution of the averaged equations, Eq. (11). The amplification-deamplification process is apparent and agrees with the prediction of the averaged linear equation.

Fig. 6
Fig. 6

Main parameters of a PCF as a function of pitch Λ at wavelength λ0 = 1064 nm. The fiber is made of pure silica and the filling fraction d/Λ = 0.4. The red dashed lines in (a,b) show the range spanned by the parameters when tapering the fiber.

Fig. 7
Fig. 7

Output spectrum after the propagation in a periodically tapered PCF. All fiber parameters are listed in Table 3. We detect the two Raman gain bands at Δf = ±13 THz, Stokes (red-detuned) and anti-Stokes (blue-detuned), the former exhibits as expected a larger gain. We also label the main two PR instability peak pairs which correspond respectively to the design requirement of Δf = 35 THz and to the additional phase matching allowed for by FOD.

Tables (3)

Tables Icon

Table 1 Values of resonant frequencies and bandwidth for α = 10. Comparison between data extracted from NLS evolution and the results of linearized model (solved both as phase matching condition and Hill equation). The resonant frequency is estimated at any order analytically, while numerical values are reported for bandwidth.

Tables Icon

Table 2 Same as in Table 1, but for α = 5.

Tables Icon

Table 3 PCF parameters used in the generalized NLS model.

Equations (24)

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i z A 1 2 s ( z ) t 2 A + n ( z ) | A | 2 A = 0 .
i z a 1 2 s ( z ) t 2 a + n ( z ) P 0 ( a + a * ) = 0 .
a ( z , t ) = a A ( z ) e i δ t + a S ( z ) e i δ t
i | ψ ˙ = H s ( z ) | ψ , H s ( z ) ( s ( z ) δ 2 2 n ( z ) P 0 ) σ ^ z in ( z ) P 0 σ ^ y
s ( z ) = s 0 + s ˜ ( z ) = s 0 + h s 1 cos α z , n ( z ) = n 0 + n ˜ ( z ) = n 0 + h n 1 cos α z ,
s 0 δ m 2 + 2 n 0 P 0 = m α
R = 1 2 ( 1 1 1 1 )
H pq = R H s R 1 = ( 0 c 1 ( z ) c 2 ( z ) 0 )
ϕ ¨ 1 , 2 c ˙ 1 , 2 c 1 , 2 ϕ ˙ 1 , 2 + c 1 c 2 ϕ 1 , 2 = 0 ,
ϕ 1 , 2 = exp ( 1 2 0 z c ˙ ( z ) 1 , 2 c ( z ) 1 , 2 d z ) ϕ ˜ 1 , 2 = c 1 , 2 ϕ ˜ 1 , 2 .
ϕ ˜ ¨ 1 , 2 + { c 1 c 2 + 1 2 c ¨ 1 , 2 c 1 , 2 3 4 [ c ˙ 1 , 2 c 1 , 2 ] 2 } ϕ ˜ 1 , 2 = 0
i | ϕ ˙ = H pq ( z ) | ϕ
δ m = 1 | s 0 | 2 n 0 P 0 [ s 0 + | s 0 | 1 + ( m α 2 n 0 P 0 ) 2 ]
ϕ 1 = A ( z ) cos ω 0 z + B ( z ) sin ω 0 z , ϕ 2 = i ω 0 c 1 0 [ A ( z ) sin ω 0 z B ( z ) cos ω 0 z ] ,
( A ˙ B ˙ ) = α ( ρ 2 ρ 1 ) ω 0 2 2 π ( α 2 4 ω 0 2 ) ( 1 + cos ( 4 π ω 0 α ) sin ( 4 π ω 0 α ) sin ( 4 π ω 0 α ) 1 cos ( 4 π ω 0 α ) ) ( A B )
g 1 = δ 1 2 P 0 2 α h | s 0 n 1 s 1 n 0 |
1 2 δ 2 = d 0 + h d 1 + h 2 d 2 , | ϕ ( z ) = | ϕ 0 ( z ) + h | ϕ 1 ( z ) + h 2 | ϕ 2 ( z ) + , | ϕ i = ( ϕ 1 i , ϕ 2 i ) T .
ϕ ¨ 10 + ω 0 2 ϕ 10 = 0
ϕ ¨ 11 + ω 0 2 ϕ 11 = [ d 1 s 0 ( c 1 0 + c 2 0 ) + ( c 2 0 c 1 0 ) d 0 s 1 2 + c 1 0 n 1 P 0 ] cos ω 0 z + [ ( 3 c 2 0 + c 1 0 ) d 0 s 1 2 + c 1 0 n 1 P 0 ] cos 3 ω 0 z .
d 1 = 1 2 s 0 ( d 0 s 0 n 0 P 0 ) d 0 P 0 ( s 0 n 1 n 0 s 1 ) .
δ 2 2 = δ 1 2 2 h d 1
i | ϕ ˙ = H gnls ( z ) | ϕ , H gnls H 0 + H ˜ ( z ) .
1 2 s ( z ) δ 2 1 2 s ( z ) δ 2 1 24 β 4 n δ 4 ,
Λ = Λ 0 + Λ 1 cos α ˜ z , β 2 = β 2 0 + β 2 1 cos α ˜ z , γ = γ 0 + γ 1 cos α ˜ z ,

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