Abstract

Temporal solitons propagating in the vicinity of a zero-dispersion wavelength in an optical fiber emit phase-matched resonant radiations (RRs) as a result of perturbations due to higher-order dispersion effects. These RRs propagate linearly and they usually rapidly spread out in time, thus having a very low peak power. Here, we show that the use of an engineered dispersion-varying optical fiber allows us to induce a completely new dynamics, in which a new physical mechanism—cascade of RRs—is discovered. It is explained by the fact that the RR is temporally recompressed thanks to the change of dispersion sign induced by the varying geometry along the fiber. In addition, we report the experimental evidence of physical processes that had remained unobserved experimentally so far, such as the emission of multiple RRs from a single soliton and the generation of a 500 nm continuum exclusively composed of polychromatic RRs.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2014 (1)

2013 (3)

2012 (2)

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

C. Milián, A. Ferrando, D. V. Skryabin, “Polychromatic Cherenkov radiation and supercontinuum in tapered optical fibers,” J. Opt. Soc. Am. B 29, 589–593 (2012).
[Crossref]

2011 (2)

M. Erkintalo, J. M. Dudley, G. Genty, “Pump-soliton nonlinear wave mixing in noise-driven fiber supercontinuum generation,” Opt. Lett. 36, 3870–3872 (2011).
[Crossref]

S. P. Stark, A. Podlipensky, P. St.J. Russell, “Soliton blueshift in tapered photonic crystal fibers,” Phys. Rev. Lett. 106, 083903 (2011).
[Crossref]

2010 (2)

2009 (1)

2006 (1)

J. M. Dudley, G. Genty, S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

2005 (2)

D. V. Skryabin, A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005).
[Crossref]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

2004 (3)

2003 (1)

D. V. Skryabin, F. Luan, J. C. Knight, P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

1998 (1)

S. Burtsev, R. Camassa, I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).
[Crossref]

1995 (1)

N. Akhmediev, M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

1987 (1)

P. Beaud, W. Hodel, B. Zysset, H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[Crossref]

1986 (1)

1972 (1)

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

Agrawal, G.

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

Akhmediev, N.

N. Akhmediev, M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

Arteaga-Sierra, F. R.

Bang, O.

Beaud, P.

P. Beaud, W. Hodel, B. Zysset, H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[Crossref]

Belgiorno, F.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Bendahmane, A.

Biancalana, F.

Burtsev, S.

S. Burtsev, R. Camassa, I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).
[Crossref]

Camassa, R.

S. Burtsev, R. Camassa, I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).
[Crossref]

Chapman, B. H.

Chen, H. H.

Chen, Z.

Coen, S.

J. M. Dudley, G. Genty, S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Conforti, M.

Dávila, A.

Dudley, J. M.

M. Erkintalo, J. M. Dudley, G. Genty, “Pump-soliton nonlinear wave mixing in noise-driven fiber supercontinuum generation,” Opt. Lett. 36, 3870–3872 (2011).
[Crossref]

J. M. Dudley, G. Genty, S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Efimov, A.

Erkintalo, M.

Faccio, D.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Ferrando, A.

Genty, G.

M. Erkintalo, J. M. Dudley, G. Genty, “Pump-soliton nonlinear wave mixing in noise-driven fiber supercontinuum generation,” Opt. Lett. 36, 3870–3872 (2011).
[Crossref]

J. M. Dudley, G. Genty, S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Hasegawa, A.

A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer-Verlag, 2002).

Hodel, W.

P. Beaud, W. Hodel, B. Zysset, H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[Crossref]

Joly, N.

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

A. Efimov, A. Taylor, F. Omenetto, A. Yulin, N. Joly, F. Biancalana, D. Skryabin, J. Knight, P. St.J. Russell, “Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: experiment and modelling,” Opt. Express 12, 6498–6507 (2004).
[Crossref]

Judge, A. C.

Karlsson, M.

N. Akhmediev, M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

Kehr, S. C.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Knight, J.

Knight, J. C.

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

D. V. Skryabin, F. Luan, J. C. Knight, P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

König, F.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Kudlinski, A.

Kuklewicz, C. E.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Lee, Y. C.

Leonhardt, U.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Martijn de Sterke, C.

Matsumoto, M.

A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer-Verlag, 2002).

McLenaghan, J.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Menyuk, C. R.

Milián, C.

Murdoch, S. G.

Mussot, A.

Omenetto, F.

Omenetto, F. G.

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

Podlipensky, A.

S. P. Stark, A. Podlipensky, P. St.J. Russell, “Soliton blueshift in tapered photonic crystal fibers,” Phys. Rev. Lett. 106, 083903 (2011).
[Crossref]

Popov, S. V.

Rohr, S.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Rubino, E.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Russell, P. St.J.

S. P. Stark, A. Podlipensky, P. St.J. Russell, “Soliton blueshift in tapered photonic crystal fibers,” Phys. Rev. Lett. 106, 083903 (2011).
[Crossref]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

A. Efimov, A. Taylor, F. Omenetto, A. Yulin, N. Joly, F. Biancalana, D. Skryabin, J. Knight, P. St.J. Russell, “Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: experiment and modelling,” Opt. Express 12, 6498–6507 (2004).
[Crossref]

A. V. Yulin, D. V. Skryabin, P. St.J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004).
[Crossref]

D. V. Skryabin, F. Luan, J. C. Knight, P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Shabat, A. B.

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

Skryabin, D.

Skryabin, D. V.

C. Milián, A. Ferrando, D. V. Skryabin, “Polychromatic Cherenkov radiation and supercontinuum in tapered optical fibers,” J. Opt. Soc. Am. B 29, 589–593 (2012).
[Crossref]

D. V. Skryabin, A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005).
[Crossref]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

F. Biancalana, D. V. Skryabin, A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

A. V. Yulin, D. V. Skryabin, P. St.J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004).
[Crossref]

D. V. Skryabin, F. Luan, J. C. Knight, P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Stark, S. P.

S. P. Stark, A. Podlipensky, P. St.J. Russell, “Soliton blueshift in tapered photonic crystal fibers,” Phys. Rev. Lett. 106, 083903 (2011).
[Crossref]

Taylor, A.

Taylor, A. J.

Z. Chen, A. J. Taylor, A. Efimov, “Coherent mid-infrared broadband continuum generation in non-uniform ZBLAN fiber taper,” Opt. Express 17, 5852–5860 (2009).
[Crossref]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

Timofeyev, I.

S. Burtsev, R. Camassa, I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).
[Crossref]

Torres-Cisneros, M.

Torres-Gómez, I.

Townsend, D.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

Travers, J. C.

Trillo, S.

Vanvincq, O.

Wai, P. K. A.

Webb, K. E.

Weber, H.

P. Beaud, W. Hodel, B. Zysset, H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[Crossref]

Xu, Y. Q.

Yang, J.

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, 1st ed. (SIAM, 2012).

Yulin, A.

Yulin, A. V.

D. V. Skryabin, A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005).
[Crossref]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

F. Biancalana, D. V. Skryabin, A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

A. V. Yulin, D. V. Skryabin, P. St.J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004).
[Crossref]

Zakharov, V. E.

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

Zysset, B.

P. Beaud, W. Hodel, B. Zysset, H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[Crossref]

IEEE J. Quantum Electron. (1)

P. Beaud, W. Hodel, B. Zysset, H. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1938–1946 (1987).
[Crossref]

J. Comput. Phys. (1)

S. Burtsev, R. Camassa, I. Timofeyev, “Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems,” J. Comput. Phys. 147, 166–186 (1998).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Express (4)

Opt. Lett. (6)

Phys. Rev. A (1)

N. Akhmediev, M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

Phys. Rev. E (2)

D. V. Skryabin, A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005).
[Crossref]

F. Biancalana, D. V. Skryabin, A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).
[Crossref]

Phys. Rev. Lett. (3)

S. P. Stark, A. Podlipensky, P. St.J. Russell, “Soliton blueshift in tapered photonic crystal fibers,” Phys. Rev. Lett. 106, 083903 (2011).
[Crossref]

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett. 108, 253901 (2012).
[Crossref]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, P. St.J. Russell, “Interaction of an optical soliton with a dispersive wave,” Phys. Rev. Lett. 95, 213902 (2005).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Science (1)

D. V. Skryabin, F. Luan, J. C. Knight, P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003).
[Crossref]

Sov. Phys. JETP (1)

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

Other (3)

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, 1st ed. (SIAM, 2012).

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

A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer-Verlag, 2002).

Supplementary Material (1)

» Media 1: AVI (3170 KB)     

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Figures (8)

Fig. 1.
Fig. 1. (a) Left axis: evolution of the outer diameter of the dispersion-varying PCF versus length measured during the draw process (markers). Right axis: simulated evolution of the second ZDW versus fiber length (solid line). (b) Simulated dispersion curves for the maximum (blue line) and minimum (red line) fiber diameters.
Fig. 2.
Fig. 2. Experimental setup. fs, femtosecond; ISO, optical isolator; L, lens; HWP, half-wave plates; P, polarizer; OSA, optical spectrum analyzer.
Fig. 3.
Fig. 3. (a), (d) Output spectrum after 20 m for a pump peak power of 75 W in (a) experiments and (d) simulations; (b), (e) output spectrum after 12 m for a pump peak power of 75 W in (b) experiments and (e) simulations. Red and blue dashed lines represent, respectively, the minimum and maximum ZDW. (c), (f) Dynamics of the spectrum formation versus fiber length in (c) experiments and (f) simulations. The white solid line represents the second ZDW, and the horizontal dashed line corresponds to a fiber length of 12 m. Black and red dots represent phase-matched wavelengths of RR obtained with Eq. (1) from the soliton when it reaches the second ZDW and from the first generated RR when it crosses the ZDW, respectively.
Fig. 4.
Fig. 4. (a) Simulated time domain evolution (in linear scale) versus fiber length corresponding to Fig. 3(f); (b) evolution of the peak power of RR1 with fiber length. Horizontal dotted lines depict locations where the RR1 experiences a change in dispersion sign (N, normal dispersion; A, anomalous dispersion). (α)–(ϵ) Simulated temporal profiles of RR1 at fiber lengths of 18, 16, 14.3, 12.1, and 11 m, respectively.
Fig. 5.
Fig. 5. Numerical spectrograms for fiber lengths of 10.4 and 10.6 m for the profiles displayed in (a) [(b) and (c)] (Media 1) and (d) [(e) and (f)]. Horizontal white-dashed lines depict the second ZDW. (g) Comparison between spectra obtained at 10.6 m in the profile of plots (a) (black line) and (d) (red line).
Fig. 6.
Fig. 6. (a), (c) Output spectrum after 20 m for a pump peak power of 110 W in (a) experiments and (c) simulations; (b), (d) dynamics of the spectrum formation versus fiber length in (b) experiments and (d) simulations. The white line represents the second ZDW. Red and blue dashed lines in (a) and (c) represent, respectively, the minimum and maximum ZDW.
Fig. 7.
Fig. 7. Close-up on the RR emission observed around 9 m in Fig. 6(d). The white line represents the second ZDW. The black line represents the phase-matching relation given by Eq. (1).
Fig. 8.
Fig. 8. (a) Experimental output spectrum after 20 m for a pump peak power of 380 W, (b) measured dynamics of the spectrum formation versus fiber length, (c) simulated output spectrum, and (d) simulated dynamics of the spectrum formation. The white line represents the second ZDW. Red and blue dashed lines in (a) and (c) represent, respectively, the minimum and maximum ZDW.

Equations (1)

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β22Ω2+β36Ω3=γP2,

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