Abstract

Soliton propagation is modeled in a tapered photonic crystal fiber for various taper profiles with the purpose of optimizing the soliton self-frequency shift (SSFS) in such geometries. An optimal degree of tapering is found to exist for tapers with an axially uniform waist. In the case of axially nonuniform waists, an additional enhancement of the SSFS is achieved by varying the taper waist diameter along its length in a carefully designed fashion in order to present an optimal level of group-velocity dispersion to the soliton at each point, thus avoiding the spectral recoil due to the emission of dispersive waves. In doing so, the increased nonlinearity and dispersion engineering afforded by the reduction of the core size are exploited while circumventing the limitation imposed on the soliton redshift by the associated shortening of the red zero-dispersion wavelength.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2009 (2)

2008 (3)

2006 (1)

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

2005 (1)

2004 (3)

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748-751 (2004).

F. Biancalana, D. V. Skryabin, and 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]

S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864-2869 (2004).
[CrossRef] [PubMed]

2003 (2)

O. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61-68 (2003).
[CrossRef]

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

2001 (1)

1999 (2)

M. E. Fermann, A. Galvanauskas, M. L. Stock, K. K. Wong, D. Harter, and L. Goldberg, “Ultrawide tunable Er soliton fiber laser amplified in Yb-doped fiber,” Opt. Lett. 24, 1428-1430 (1999).
[CrossRef]

N. Nishizawa and T. Goto, “Compact system of wavelength-tunable femtosecond soliton pulse generation using optical fibers,” IEEE Photon. Technol. Lett. 11, 325-327 (1999).
[CrossRef]

1995 (1)

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

1994 (1)

1989 (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

1986 (2)

1975 (1)

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964-967 (1975).
[CrossRef]

1972 (1)

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62-64 (1972).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Akhmediev, N.

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

Biancalana, F.

F. Biancalana, D. V. Skryabin, and 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]

Birks, T.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Chai, L.

Chandalia, J. K.

Chen, Z.

Cherlow, J.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964-967 (1975).
[CrossRef]

Coen, S.

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

Dudley, J. M.

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

Efimov, A.

Eggleton, B. J.

Fang, X.-H.

Fateev, N. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748-751 (2004).

Fermann, M. E.

Galvanauskas, A.

Garcia, H.

Genty, G.

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

Goldberg, L.

Gordon, J. P.

Goto, T.

N. Nishizawa and T. Goto, “Compact system of wavelength-tunable femtosecond soliton pulse generation using optical fibers,” IEEE Photon. Technol. Lett. 11, 325-327 (1999).
[CrossRef]

Harter, D.

Hellwarth, R.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964-967 (1975).
[CrossRef]

Herrmann, J.

Holzlohner, R.

Hu, M.-L.

Ippen, E. P.

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62-64 (1972).
[CrossRef]

Johnson, A. M.

Karlsson, M.

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

Knight, J. C.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Knox, W. H.

Kobtsev, S. M.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748-751 (2004).

Kosinski, S. G.

Kukarin, S. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748-751 (2004).

Lee, J. H.

J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Sel. Top. Quantum Electron. 14, 713-723 (2008).
[CrossRef]

Leon-Saval, S.

Li, Y.-F.

Liu, B.-W.

Liu, X.

J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Sel. Top. Quantum Electron. 14, 713-723 (2008).
[CrossRef]

X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in a short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358-360 (2001).
[CrossRef]

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Luo, J.

Mason, M.

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

Nazarkin, A.

Nishizawa, N.

N. Nishizawa and T. Goto, “Compact system of wavelength-tunable femtosecond soliton pulse generation using optical fibers,” IEEE Photon. Technol. Lett. 11, 325-327 (1999).
[CrossRef]

Oguama, F. A.

Russel, P. S. J.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Russell, P. S. J.

Sinkin, O.

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and 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]

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Smirnov, S. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748-751 (2004).

Stock, M. L.

Stolen, R. H.

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62-64 (1972).
[CrossRef]

Taylor, A. J.

Taylor, J. R.

Tong, W.

Travers, J. C.

Tynes, A. R.

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62-64 (1972).
[CrossRef]

van Howe, J.

J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Sel. Top. Quantum Electron. 14, 713-723 (2008).
[CrossRef]

Voronin, A. A.

Wadsworth, W.

Wang, C.-Y.

Windeler, R. S.

Wong, K. K.

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Xu, C.

J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Sel. Top. Quantum Electron. 14, 713-723 (2008).
[CrossRef]

X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in a short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358-360 (2001).
[CrossRef]

Yang, T.-T.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964-967 (1975).
[CrossRef]

Yulin, A. V.

F. Biancalana, D. V. Skryabin, and 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]

Zheltikov, A. M.

Zweck, J.

Appl. Phys. Lett. (1)

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguide,” Appl. Phys. Lett. 20, 62-64 (1972).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. H. Lee, J. van Howe, C. Xu, and X. Liu, “Soliton self-frequency shift: experimental demonstrations and applications,” IEEE J. Sel. Top. Quantum Electron. 14, 713-723 (2008).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

N. Nishizawa and T. Goto, “Compact system of wavelength-tunable femtosecond soliton pulse generation using optical fibers,” IEEE Photon. Technol. Lett. 11, 325-327 (1999).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Laser Phys. (1)

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Generation of self-frequency-shifted solitons in tapered fibers in the presence of femtosecond pumping,” Laser Phys. 14, 748-751 (2004).

Opt. Express (3)

Opt. Lett. (7)

Phys. Rev. A (1)

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

Phys. Rev. B (1)

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964-967 (1975).
[CrossRef]

Phys. Rev. E (1)

F. Biancalana, D. V. Skryabin, and 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]

Rev. Mod. Phys. (1)

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

Science (1)

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Document NL-15-670.pdf may be obtained at http://www.nktphotonics.com/side5328.html.

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of the effects of tapering on the SSFS. (a) In the untapered fiber the relatively low nonlinearity and high dispersion lead to a low rate of redshift. (b) Tapering the fiber to a uniform waist enhances the shift rate through a higher nonlinearity but is accompanied by a shortening of the second ZDW. The soliton reaches a point where the dispersion in the taper facilitates the emission of low-amplitude waves in the normally dispersive region with a resultant halt of the redshift. (c) If the waist diameter of the taper is varied such that the soliton experiences the maximum possible nonlinearity and a level of dispersion that minimizes the loss of energy to DWs, the soliton may achieve a redshift above that realized in cases (a) and (b).

Fig. 2
Fig. 2

(a) Raman gain spectrum α R for bulk silica overlaid with the convolution function C for three values of T s . The areas under the latter curves are equal. As indicated, the vertical axis scale corresponds to the dimensionless units of the gain spectrum. The curves drawn with broken lines, referring to the convolution function, have been scaled to convenient but arbitrary units. (b) The functions R, R s , and R l ; the values of R at the example pulse widths used in panel (a) are indicated with squares.

Fig. 3
Fig. 3

(a) GVD and (b) the nonlinear parameter for the NL-15-670 fiber from Crystal Fibre, using an assumed value of n 2 = 1.81 × 10 20 m 2 W [22] for a range of fiber diameters. Also shown is (c) the pulse width and (d) a representation of the SSFS rate, calculated with Eqs. (15, 7), respectively, for an initial soliton of energy E 0 = 100 pJ at λ 0 = 2 π c ω 0 = 850 nm . In each panel results are shown for three ratios of the tapered to the untapered fiber diameter: 1.0 (solid curves), 0.8 (dashed) and 0.6 (dotted–dashed). The dotted vertical lines indicate the long-wavelength edge of the anomalous dispersion region for the three diameter scalings.

Fig. 4
Fig. 4

Left axes, final λ s (circles) and the second ZDW in the waist (dashed line). Right axes, E a E f (diamonds). The data shown relate to the propagation of an initial soliton with E 0 = 100 pJ along a 1 m taper waist for various uniform diameter scalings, simulated by using Eq. (1). The simulations corresponding to the data points lying between the vertical dotted lines are depicted in more detail in Fig. 5.

Fig. 5
Fig. 5

Spectral evolution of an initial ideal fundamental soliton with E 0 = 100 pJ for three uniform scalings of the the fiber waist diameter relative to the untapered fiber: (a) 1.00, (b) 0.69, (c) 0.68, and (d) 0.67. The white dashed lines indicate the position of the second ZDW in each case, while the black dashed curves show the evolution of the soliton center frequency predicted by Eq. (7).

Fig. 6
Fig. 6

Spectral evolution of an initial ideal fundamental soliton with E 0 = 100 pJ for tapers with axially nonuniform waists designed with a minimum waist scaling of 0.60 and a GVD threshold of (a) 70 , (b) 60 , and (c) 50 ps 2 km . Also shown (d) is the spectral evolution of a soliton with E 0 = 110 pJ propagating in the taper from (a). The white dashed curves indicate the position of the second ZDW in each case, while in (a)–(c) the black dashed curve shows the evolution of the soliton center frequency predicted by Eq. (7).

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

z A ( z , T ) = i m 2 i m β m ( z ) T m m ! A ( z , T ) + i m 0 i m γ m ( z ) T m m ! A ( z , T ) G ( T T ) | A ( z , T ) | 2 d T .
γ ( z , ω ) = ω n ( ω ) n 2 ( ω ) c n eff 2 ( z , ω ) A eff ( z , ω ) ,
G ( T ) = δ ( T ) + 2 f R 3 ( 1 f R ) g R ( T ) ,
z A ( z , T ) = i β 2 T 2 2 A ( z , T ) + i γ 0 A ( z , T ) | A ( z , T ) | 2 ,
A ( z , T ) = ( | β 2 | T s 2 γ 0 ) 1 2 sech ( T T s ) exp ( i | β 2 | 2 T s 2 z ) ,
T s = 2 | β 2 | γ 0 E .
d ω ¯ d z = | β 2 ( ω ¯ ) | T s 3 ( ω ¯ ) R ( ω ¯ ) ,
ω ¯ = ω | A ̃ ( ω ) | 2 d ω | A ̃ ( ω ) | 2 d ω .
R = α R ( ω ) C ( ω ) d ω 2 π ,
C ( ω ) = f R 1 f R π 2 T s 4 6 ω 3 sinh 2 ( π T s ω 2 )
α R ( ω ) | ω α R | ω = 0 ω ,
R R l = f R 1 f R Γ ( 5 ) ζ ( 4 ) 3 π 4 T s ,
C ( ω ) f R 1 f R 2 T s 2 ω 3 ,
R R s = f R 1 f R 2 T s 2 3 α R ( ω ) ω d ω 2 π .
T s ( ω ¯ ) = 2 | β 2 ( ω ¯ ) | γ ( ω ¯ ) E ( ω ¯ ) ,
E ( ω ¯ ) E 0 γ ( ω ¯ ) γ 0 ,
z | A ̃ ( z , ω ) | 2 γ ( ω ) d ω 2 π = 0 ,
z | A ̃ ( z , ω ) | 2 γ ( z , ω ) d ω 2 π + | A ̃ ( z , ω ) | 2 γ ( z , ω ) z ln ( γ ( z , ω ) ) = 0 ,
E ( ω ¯ ) E 0 γ ( z 0 , ω ¯ ) γ 0 ( z 0 ) .
6 | β 2 ( ω ¯ ) | 2 γ ( ω ¯ ) E ( ω ¯ ) | β 3 ( ω ¯ ) | Δ Ω min ,
| β 2 ( ω ¯ ) | | β 2 | min ,

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