## Dispersive wave blue-shift in supercontinuum generation

Optics Express, Vol. 14, Issue 25, pp. 11997-12007 (2006)

http://dx.doi.org/10.1364/OE.14.011997

Acrobat PDF (303 KB)

### Abstract

We numerically study dispersive wave emission during femtosecond-pumped supercontinuum generation in photonic crystal fibres. We show that dispersive waves are primarily generated over a short region of high temporal compression. Despite the apparent complexity of the pump pulse in this region, we show that the dynamics of dispersive wave generation are dominated by a single fundamental soliton. However, any straightforward application of the theory that is thought to describe the blue emission, considerably underestimates the frequency shift. We show that in fact the red-shift of the soliton, caused by spectral recoil from the growing dispersive wave, causes an additional blue-shift of the resonant frequency which is in good agreement with full simulations.

© 2006 Optical Society of America

## 1. Introduction

2. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901 (2001). [CrossRef] [PubMed]

*µ*m) photonic crystal fibers [3

3. J. Ranka, R. Windeler, and A. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **25**, 25–27 (2000). [CrossRef]

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

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

6. P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A **41**, 426–439 (1990). [CrossRef] [PubMed]

7. A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, “Direct Link betweenMicrowave and Optical Frequencies with a 300 THz Femtosecond Laser Comb,” Phys. Rev. Lett. **84**, 5102–5105 (2000). [CrossRef] [PubMed]

8. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber,” Opt. Lett. **26**, 608–610 (2001). [CrossRef]

9. H. N. Paulsen, K. M. Hilligsøe, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source,” Opt. Lett. **28**, 1123–1125 (2003). [CrossRef] [PubMed]

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

6. P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A **41**, 426–439 (1990). [CrossRef] [PubMed]

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

6. P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A **41**, 426–439 (1990). [CrossRef] [PubMed]

13. J. N. Elgin, T. Brabec, and S. Kelly, “A perturbative theory of soliton propagation in the presence of 3rd-order dispersion,” Opt. Commun. **114**, 321–328 (1995). [CrossRef]

*ω*

_{R}is governed by phase matching between the soliton and the low amplitude linear waves; that is

*β*

_{S}(

*ω*

_{R})=

*β*(

*ω*

_{R}) where

*β*

_{S}is the soliton wavenumber and

*β*is the fiber’s linear dispersion curve. The implied assumption that the soliton wavenumber is independent of

*z*was approximately true in the original studies [5

**51**, 2602–2607 (1995). [CrossRef] [PubMed]

**41**, 426–439 (1990). [CrossRef] [PubMed]

13. J. N. Elgin, T. Brabec, and S. Kelly, “A perturbative theory of soliton propagation in the presence of 3rd-order dispersion,” Opt. Commun. **114**, 321–328 (1995). [CrossRef]

*β*

_{S}

*ω*

_{S}and

*P*

_{S}are the soliton’s centre frequency, and peak power, subscripts on

*β*denote frequency derivatives, and γ is the fiber’s nonlinear coefficient. The power-dependent term represents the nonlinear phase-shift and is often small compared to the other terms in Eq. (1). Equating

*β*

_{S}given in Eq. (1) with the linear wavenumber yields the well-known resonance condition

2. A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901 (2001). [CrossRef] [PubMed]

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

14. I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express **12**, 124–135 (2004). [CrossRef] [PubMed]

15. K. M. Hilligsøe, H. N. Paulsen, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Initial steps of supercontinuum generation in photonic crystal fibers,” J. Opt. Soc. Am. B **20**, 1887–1893 (2003). [CrossRef]

16. G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express **12**, 4614–4624 (2004). [CrossRef] [PubMed]

*et al.*[16

16. G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express **12**, 4614–4624 (2004). [CrossRef] [PubMed]

## 2. Preliminary simulation results

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

*A*(

*T, z*) is the time domain envelope,

*T*=

*t*-

*β*

_{1z}is retarded time,

*z*is axial propagation distance,

*β*

_{n}and

*γ*are the

*n*

^{th}order dispersion and nonlinear coefficients evaluated at the reference frequency

*ω*

_{0},

*τ*

_{shock}models the dispersion of the nonlinearity and

*R*(

*T*) is the usual silica Raman response [18]. Femtosecond-pumped SC generation occurs in millimetres or centimetres of fiber and losses are therefore ignored. The initial conditions were unchirped hyperbolic-secant pulses i.e.

*N*=5 pump soliton. The other parameters are given in Table 1. The fiber is modeled on Crystal Fibre NL-2.0-740, a typical high air-fill hexagonal photonic crystal fiber designed for nonlinear experiments at 800 nm. Dispersion coefficients were measured using white-light interferometry [19

19. Q. H. Ye, C. Xu, X. Liu, W. H. Knox, M. F. Yan, R. S. Windeler, and B. Eggleton, “Dispersion measurement of tapered air-silica microstructure fiber by white-light interferometry,” Appl. Opt. **41**, 4467–4470 (2002). [CrossRef] [PubMed]

*τ*

_{shock}=1/

*ω*

_{0}, ignoring the frequency-dependent effective area which has been shown to have negligible effect for this parameter regime [20

20. B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B **81**, 337–342 (2005). [CrossRef]

*z*=36 mm, indicated by the vertical lines in Figs. 1(b) and 1(d).

## 3. Soliton wavenumber description

*P*

_{S}, thus finding the peak power required to achieve the observed nonlinear phase-shift. In Fig. 3(b), this is shown to greatly exceed the actual peak power

*P*

_{max}obtained from the simulations. Thus we conclude that no conceivable straightforward application of Eq. (2) is consistent with the numerical results. In the remainder of this paper we thus investigate why this is so, and how this can be improved.

## 4. Conditions during dispersive wave generation

*z*. The results are shown in Fig. 4(a) for several

*N*, and the discontinuous, sudden nature of dispersive wave generation is apparent. We defined the “dispersive wave generation region” (DWGR) as the

*z*values over which the dispersive wave energy grows from 20 % to 80 % of its final value. We tested a number of alternative definitions for the dispersive wave energy and DWGR and found that our conclusions were unaffected. For

*N*=5, the DWGR is illustrated by the white horizontal lines in Fig. 1, whilst the start and finish are shown for 3<

*N*<10 in Fig. 4(b). Also shown is the point

*z*=

*z*

_{max}, defined as the

*z*-coordinate where the pump pulse reaches its maximum temporal intensity

*P*

_{max}. For the range of

*N*considered here,

*z*

_{max}lies within the DWGR, which is 2–8 mm in length. Furthermore, both the DWGR and

*z*

_{max}show good agreement with an approximate expression for the soliton fission distance

*z*

_{fiss}=

*L*

_{D}/N, where

*L*

_{D}=

*β*

_{2}| is the quadratic dispersion length [4

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

*z*=

*z*

_{max}=36 mm. Several temporally overlapping structures are evident, including the beginnings of the dispersive wave around 650 nm and a well defined pulse which goes on to become the most intense ejected soliton.

## 5. Nonlinear wavenumber

*ω′*=

*ω*-

*ω*

_{0}, so that

*β*

_{NL}(

*ω, z*) and

*α*

_{NL}(

*ω, z*) are the

*z*-dependent nonlinear wavenumber and loss respectively. Equating coefficients we find

*ω′*. Using the pre-computed solution

*A*(

*ω, z*), the nonlinear wavenumber can be evaluated as desired. Clearly, at low intensities linear effects dominate and

*β*

_{NL}≈

*β*, whilst for a fundamental soliton at the carrier frequency experiencing quadratic dispersion and self-phase modulation only, Eq. (5) reduces to Eq. (1). The

*N*-soliton pump pulses are

*N*

^{2}times more intense, and therefore would “compensate” for a correspondingly larger amount of quadratic dispersion, if it were present. At

*z*=0, this leads to

21. C.-M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B **19**, 1961–1967 (2002). [CrossRef]

*z*

_{max}. The linear and nonlinear wavenumbers intersect at

*ν*=478 THz, close to the dispersive wave frequency of

*ν*=476 THz. By contrast, the soliton wavenumber intersects at

*ν*=458 THz, underestimating the blue-shift. For plotting, the transformation

*β*

_{NL}(

*ω*)′=

*β*

_{NL}(

*ω*)-

*β*(

*ω*

_{0})-

*β*

_{1}(

*ω*

_{0})

*ω′*(defined similarly for

*β*and

*β*

_{S}) is applied. This removes the linear phase and group velocities (equivalent to the transformation from the lab to the retarded frame) to make the curvature evident. Also, we plot the spectra and wavenumbers against frequency rather than wavelength as that is the natural domain of the latter. Figure 5(c) shows

*β*

_{NL}for various pump powers, and the blue-shift of the intersection point with increasing power is evident. The anomalous behavior of the

*N*=7 curve around 450 THz is due the rapid change in phase that occurs when self-phase modulation drives the complex spectral amplitude

*A*(

*ω, z*) through or close to zero. In general,

*β*

_{NL}is ill-defined for any frequency at which

*A*(

*ω, z*) is close to zero. In Fig. 5(d) we present the resonant wavelength obtained from

*β*

_{NL}for the full range of

*N*, showing good agreement with the simulation data. The average and maximum differences between the two curves are 7 nm and 18 nm respectively.

*β*

_{NL}is shown in the animation attached to Fig. 6(a). The initial condition, as expected from Eq. (6), is roughly parabolic, but after only 5 mm the nonlinear wavenumber has become nearly straight. After soliton fission has occurred, the linear behavior is lost as complex inter-soliton interactions occur. Two simple interpretations are evident from the animation. First for

*z*>50 mm,

*β*

_{NL}≈

*β*for

*ν*>470 THz, showing that nonlinear interactions with the dispersive waves have effectively ceased. Second, the straight section of

*β*

_{NL}at low

*n*for

*z*>50 mm corresponds to the emergence of a spectrally distinct fundamental soliton, which gradually red-shifts due to Raman scattering.

*β*

_{NL}around

*ν*=477 THz is not caused by the influence of the local linear wavenumber, we simulated SC in a hypothetical fibre with

*β*

_{3}=0 and

*β*

_{4}=4.459×10

^{-4}ps

^{4}/km, chosen to give the same dispersive wave resonance wavelength as before, i.e. 659 nm. The other parameters were identical. The resulting dispersion curve is symmetric about the pump frequency (in the retarded frame), and the fourth order dispersion creates a second resonance at 1052 nm. Over 3<

*N*<10, we found no power-dependent blue-shift of the dispersive wave. Therefore, the blue-shift is not caused merely by the upturn of the linear dispersion curve at high

*ν*, or some property of the higher order soliton evolution.

## 6. Spectral recoil of dominant fundamental soliton

*β*

_{NL}from an approximately straight line occur around the pump frequency (

*ν*=370 THz), where the spectral intensity is low since SPM is driving the field through or nearby the origin causing a rapid change in phase. The linear dependence of

*β*

_{NL}on frequency is a signature of fundamental soliton propagation, and we attribute it to the dominant effect of the most intense fundamental soliton which forms part of the multisoliton initial state [11

11. Y. Kodama and A. Hasegawa, “Nonlinear Pulse-Propagation In A Monomode Dielectric Guide,” IEEE J. Quantum Electron. **23**, 510–524 (1987). [CrossRef]

*β*

_{NL}in Fig. 5(c) corresponds to a red-shift of the soliton, resulting in a blue-shift of the intersection point with the linear wavenumber curve. This analysis is consistent with previous work, which attributed pump red-shift, in a general sense, to Raman scattering and third-order dispersion [15

15. K. M. Hilligsøe, H. N. Paulsen, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Initial steps of supercontinuum generation in photonic crystal fibers,” J. Opt. Soc. Am. B **20**, 1887–1893 (2003). [CrossRef]

*z*

_{max}with and without the non-instantaneous Raman response included in the simulation. The difference is minor, and it therefore not surprising that the wavelengths of dispersive wave emission are nearly identical, as shown in Fig. 7(b). Despite this, we found that energy transfer to the dispersive wave was significantly reduced by Raman scattering, consistent with previous numerical results by Dudley

*et al.*[4

**78**, 1135–1184 (2006). [CrossRef]

*β*

_{3}is considered here. In addition to dispersive wave emission, third-order dispersion causes a spectral recoil of fundamental solitons away from the zero-dispersion wavelength in order to conserve the spectral “centre ofmass” [5

**51**, 2602–2607 (1995). [CrossRef] [PubMed]

13. J. N. Elgin, T. Brabec, and S. Kelly, “A perturbative theory of soliton propagation in the presence of 3rd-order dispersion,” Opt. Commun. **114**, 321–328 (1995). [CrossRef]

*β*

_{3}=0 case mentioned earlier — the two resonances spaced equidistantly (in frequency space) on either side of the pump cancel out any spectral recoil.

*ω*

_{R}and

*ω*

_{S}as unknowns. The fraction of energy

*f*

_{R}transferred to the dispersive wave is extracted fromthe simulations. We therefore calculate a pair of pump soliton and dispersive wave frequencies which are both phase matched and conserve the spectral centre of mass at the pump frequency. As expected, the recoil of the pump soliton increases with

*N*. Furthermore, the calculated resonant frequency agrees to within 15 nm with the observed dispersive wave emission for

*N*≤9. Whilst a detailed study of the energy transferred to the dispersive wave is beyond the scope of this work, a rough

*a priori*estimate of this quantity would enable Eq. (7) to be used in conjunction with Eq. (2) as an improved prediction of the dispersive wave wavelength.

*N*, the blue-shift is underestimated, which we attribute to the complicating influence of cross phase modulation and four-wave mixing between the solitonic and dispersive wave components [16

16. G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express **12**, 4614–4624 (2004). [CrossRef] [PubMed]

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

23. A. Efimov, A. Yulin, D. Skryabin, J. Knight, N. Joly, F. Omenetto, A. Taylor, and P. St. J. Russell, “Interaction of an Optical Soliton with a Dispersive Wave,” Phys. Rev. Lett. **95**, 213902 (2005). [CrossRef] [PubMed]

24. A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B **21**, 18–23 (2004). [CrossRef]

*z*=18 mm, a spectrally distinct dispersive wave is evident at 585 nm, as indicated by a black dashed line. However, the complex series of interactions that occur with subsequent propagation have the net effect of blue-shifting the peak an additional 10 nm to 575 nm, indicated by the second line. The interactions also introduce fine structure into the dispersive wave spectrum, suggesting that a “clean”, near-transform limited dispersive wave pulse can be produced by truncating the fibre just after the DWGR.

25. D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton Self-Frequency Shift Cancellation in Photonic Crystal Fibers,” Science **301**, 1705–1708 (2003). [CrossRef] [PubMed]

26. 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]

## 7. Summary

## References and links

1. | R. R. Alfano, |

2. | A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. |

3. | J. Ranka, R. Windeler, and A. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. |

4. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

5. | N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A |

6. | P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A |

7. | A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch, “Direct Link betweenMicrowave and Optical Frequencies with a 300 THz Femtosecond Laser Comb,” Phys. Rev. Lett. |

8. | I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber,” Opt. Lett. |

9. | H. N. Paulsen, K. M. Hilligsøe, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source,” Opt. Lett. |

10. | J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. |

11. | Y. Kodama and A. Hasegawa, “Nonlinear Pulse-Propagation In A Monomode Dielectric Guide,” IEEE J. Quantum Electron. |

12. | J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. |

13. | J. N. Elgin, T. Brabec, and S. Kelly, “A perturbative theory of soliton propagation in the presence of 3rd-order dispersion,” Opt. Commun. |

14. | I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express |

15. | K. M. Hilligsøe, H. N. Paulsen, J. Thogersen, S. R. Keiding, and J. J. Larsen, “Initial steps of supercontinuum generation in photonic crystal fibers,” J. Opt. Soc. Am. B |

16. | G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express |

17. | K. Blow and D Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. |

18. | G. P. Agrawal, |

19. | Q. H. Ye, C. Xu, X. Liu, W. H. Knox, M. F. Yan, R. S. Windeler, and B. Eggleton, “Dispersion measurement of tapered air-silica microstructure fiber by white-light interferometry,” Appl. Opt. |

20. | B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B |

21. | C.-M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B |

22. | D. Skryabin and A. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E |

23. | A. Efimov, A. Yulin, D. Skryabin, J. Knight, N. Joly, F. Omenetto, A. Taylor, and P. St. J. Russell, “Interaction of an Optical Soliton with a Dispersive Wave,” Phys. Rev. Lett. |

24. | A. Peleg, M. Chertkov, and I. Gabitov, “Inelastic interchannel collisions of pulses in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B |

25. | D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton Self-Frequency Shift Cancellation in Photonic Crystal Fibers,” Science |

26. | 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 |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(060.7140) Fiber optics and optical communications : Ultrafast processes in fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 2, 2006

Revised Manuscript: December 3, 2006

Manuscript Accepted: December 5, 2006

Published: December 11, 2006

**Citation**

Dane R. Austin, C. Martijn de Sterke, Benjamin J. Eggleton, and Thomas G. Brown, "Dispersive wave blue-shift in supercontinuum generation," Opt. Express **14**, 11997-12007 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11997

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### References

- R. R. Alfano, The supercontinuum laser source: fundamentals with updated references, 2nd ed. (Springer, New York, 2006).
- A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]
- J. Ranka, R. Windeler, and A. Stentz, "Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 25, 25-27 (2000). [CrossRef]
- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006). [CrossRef]
- N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995). [CrossRef] [PubMed]
- P. K. A. Wai, H. H. Chen, and Y. C. Lee, "Radiations by "solitons" at the zero group-dispersion wavelength of single-mode optical fibers," Phys. Rev. A 41, 426-439 (1990). [CrossRef] [PubMed]
- A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link betweenMicrowave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000). [CrossRef] [PubMed]
- I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001). [CrossRef]
- H. N. Paulsen, K. M. Hilligsøe, J. Thogersen, S. R. Keiding, and J. J. Larsen, "Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source," Opt. Lett. 28, 1123-1125 (2003). [CrossRef] [PubMed]
- J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, "Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers," Phys. Rev. Lett. 88, 173901 (2002). [CrossRef] [PubMed]
- Y. Kodama and A. Hasegawa, "Nonlinear Pulse-Propagation In A Monomode Dielectric Guide," IEEE J. Quantum Electron. 23, 510-524 (1987). [CrossRef]
- J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662-664 (1986). [CrossRef] [PubMed]
- J. N. Elgin, T. Brabec, and S. Kelly, "A perturbative theory of soliton propagation in the presence of 3rd-order dispersion," Opt. Commun. 114, 321-328 (1995). [CrossRef]
- I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, "Dispersive wave generation by solitons in microstructured optical fibers," Opt. Express 12, 124-135 (2004). [CrossRef] [PubMed]
- K. M. Hilligsøe, H. N. Paulsen, J. Thogersen, S. R. Keiding, and J. J. Larsen, "Initial steps of supercontinuum generation in photonic crystal fibers," J. Opt. Soc. Am. B 20, 1887-1893 (2003). [CrossRef]
- G. Genty, M. Lehtonen, and H. Ludvigsen, "Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses," Opt. Express 12, 4614-4624 (2004). [CrossRef] [PubMed]
- K. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989). [CrossRef]
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