## Nonlinear dynamics of femtosecond supercontinuum generation with feedback

Optics Express, Vol. 17, Issue 18, pp. 15827-15841 (2009)

http://dx.doi.org/10.1364/OE.17.015827

Acrobat PDF (2302 KB)

### Abstract

We numerically study the impact of feedback on supercontinuum generation within a microstructured fiber inside a ring resonator, synchronously pumped with femtosecond pulses. In certain parameter ranges we observe a steady-state oscillator-like operation mode of the system. Depending on pump power also period doubling up to chaos is shown by the system. Even with the inclusion of realistic pump noise as perturbation, the periodic behavior was still achievable in numerical modeling as well as in a first experimental verification.

© 2009 Optical Society of America

## 1. Introduction

1. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **78**, 25–27 (2000).
[CrossRef]

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

5. Z. Zhu and T. Brown, “Effect of frequency chirping on supercontinuum generation in photonic crystal fibers,” Opt. Express **12**, 689–694 (2004).
[CrossRef] [PubMed]

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

6. S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber,” Opt. Lett. **26**, 1356–1358 (2001).
[CrossRef]

7. 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]

8. B. Washburn, S. Ralph, and R. Windeler, “Ultrashort pulse propagation in air-silica microstructure fiber,” Opt. Express **10**, 575–580 (2002).
[PubMed]

18. P. M. Moselund, M. H. Frosz, C. L. Thomsen, and O. Bang, “Back-seeding of higher order gain processes in picosecond supercontinuum generation,” Opt. Express **16**, 11954–11968 (2008).
[CrossRef] [PubMed]

19. M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. **32**, 662–664 (2007).
[CrossRef] [PubMed]

20. G. Steinmeyer, A. Buchholz, M. Hänsel, M. Heuer, A. Schwache, and F. Mitschke, “Dynamical pulse shaping in a nonlinear resonator,” Phys. Rev. A **52**, 830–838 (1995).
[CrossRef] [PubMed]

22. P. Groß, K.-J. Boller, and M. E. Klein, “High-precision wavelength-flexible frequency division for metrology,” Phys. Rev. A **71**, 043824 (2005).
[CrossRef]

## 2. Numerical model

### 2.1. Propagation equation

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

*E*=

*E*(

*z, t*) is the pulse envelope, where t is the time in a frame of reference moving with the group velocity

*β*

^{-1}

_{1}. The values

*β*are the dispersion coefficients at the center frequency

_{k}*ω*

_{0},

*α*is a measure of linear fiber loss and

*γ*is the usual nonlinear parameter of the fiber defined as

*γ*=(

*ω*

_{0}

*n*

_{2})/(

*cA*

_{eff}), where

*n*

_{2}is the nonlinear refractive index,

*c*is the vacuum speed of light, and

*A*

_{eff}is the effective mode area. The response function

*R*(

*t*), which includes the instantaneous electronic and the delayed Raman contribution, can be expressed by

*R*(

*t*′)=(1-

*f*)δ(

_{R}*t*′+

*f*(

_{R}h_{R}*t*′), with

*f*=0.18 representing the fractional part of the delayed Raman effect. For

_{R}*h*(

_{R}*t*′) we used the analytic form of the Raman response function [23

23. 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]

*h*(

_{R}*t*′)=[(τ

^{2}

_{1}+τ

^{2}

_{2})/(τ

_{1}τ

^{2}

_{2})]exp(-

*t*′/τ

_{2}) sin(

*t*′/τ

_{1}) with τ

_{1}=12.2 fs and τ

_{2}=32 fs. Equation (1) was solved numerically and stepwise along the fiber by using a split-step Fourier method [24]. In order to resolve the temporal and spectral features, which occur during the SC generation process, we used an array size of 2

^{16}points and the temporal resolution was set to 0.5 fs, corresponding to a spectral resolution of approximately 30GHz. Due to the usually large nonlinear parameter of MSF and resulting rapid spectral changes with propagation, a small step size in propagation direction of 10 µm was required, which was kept fixed after it had been verified to be sufficiently small.

### 2.2. Verification of algorithm for single shot SC generation

**78**, 1135–1184 (2006).
[CrossRef]

*γ*=0.095 (Wm)

^{-1}was extracted out of the data sheet of the MSF [25

25. Crystal Fibre, “NL-PM-750 data sheet,” http://www.crystal-fibre.com/datasheets/NL-PM-750.pdf.

^{th}order polynomial to the dispersion curve displayed also on the data sheet, the dispersion coefficients

*β*around 775 nm of the MSF were retrieved up to the 10

_{k}^{th}order. The initially injected pump pulse was numerically implemented as a hyperbolic secant field profile

*E*(0,

*t*)=√

*P*

_{0}·

*sech*((2log(1+√2)

*t*)/Δ

*t*)=

*E*

_{pump}, matching the temporal profile of the experimentally available, Fourier-limited pulses of the Titanium:Sapphire laser. The full width of the pulses at half maximum intensity (FWHM) was Δ

*t*=60 fs, and the peak power P0 was used as the variable parameter in our numerical investigations. With these parameters, a good agreement between experiment and simulation was observed. With the parameters given above, the GNLSE was numerically integrated for a number of different input pulse powers between 1mW and 18mW, and the results are shown in Fig. 1, where a) numerically calculated and b) experimentally measured spectra are displayed for varying effective average pump power. In both graphs, the spectral power is shown on a color-coded logarithmic density scale, and the effective average pump power increases from bottom to top, such that each single horizontal line within the graphs is representing a complete spectrum. We would like to note, that the effective average pump power inside the fiber was deduced from the fiber coupling efficiency, which was approximately 50%. As can be seen from Fig. 1, there is a very good agreement between simulation and experiment as all spectral features can be identified in both graphs. For example, the spectral feature evolving on the short-wavelength side (see the feature labeled ”1” in Fig. 1) as well as the Raman-shifted peak on the long-wavelength side (feature 2), or even the finer spectral structure in the middle part of both spectra (feature 3) show strong similarity. From this verification we concluded, that our numerical model and its implementation with the experimental parameters accomplish realistic predictions, such that we could proceed with first numerical investigations on SC generation with feedback and femtosecond pump pulses.

### 2.3. Implementation of the feedback loop

*E*

_{pump}denotes the electric pump pulse envelope which was adopted to a hyperbolic secant field profile, as before. The approach was straight forward: After integrating the GNLSE with the pump pulse field profile as the initial condition

*E*(

_{N}*z*=0,

*t*)|

*N*=0=

*E*

_{0}(0)=

*E*

_{pump}(

*N*=number of feedback iterations, and t was omitted for simplified presentation), the SC pulse envelope

*E*

_{0}(

*L*)=GNLSE(

*E*

_{0}(0)) (

*L*=fiber length) at the end of the fiber could be obtained. Then a certain fraction e of a few percent of the generated SC was superimposed onto the following pump pulse, by simply adding up the two arrays, resulting in

*E*

_{1}(0)=(

*ε*·

*E*

_{0}(

*L*)+

*E*

_{pump}). The GNLSE was integrated along the MSF length again, now with the modified input pulse array values, in order to get the next SC pulse

*E*

_{1}(

*L*)=GNLSE(

*E*

_{1}(0)). This procedure was repeated a few hundred times, by calculating the resulting recursive map

*ε*>45%), i.e. strong coupling, a reduced stability of the SC evolution and only a slow convergence was observed. The optimum fraction of the SC, to be fed back and superimposed onto the next pump pulse in order to achieve fast convergence, was identified to be

*ε*=30% of the electric field (or 9% of the power). In addition, from a practical point of view, a power efficiency of 9% can be realized easily in an experiment using standard broadband beam splitters for combining the SC pulse inside the feedback loop with a new undepleted pump pulse. All other parameters of the numerical simulation remained unchanged. In order to reach a steady state within the feedback loop, or to be able to identify patterns in the temporal spectral evolution, the numerical integration has to be repeated for typically several hundred times. Such an iterative numerical calculation would, in a simple case, take about two days on a normal personal computer. The results presented in this contribution correspond to 1.5 years of CPU time and were accomplished within a manageable time frame of a few weeks by grid-computing using the Condor software [26].

## 3. Results

### 3.1. Numerical analysis of the effect of feedback on supercontinuum generation

18. P. M. Moselund, M. H. Frosz, C. L. Thomsen, and O. Bang, “Back-seeding of higher order gain processes in picosecond supercontinuum generation,” Opt. Express **16**, 11954–11968 (2008).
[CrossRef] [PubMed]

*n*[27].

*n*phase space, with the used array size

*n*, as from each complex-valued array element a phase as well as an amplitude can be derived. The results presented here, however, did not depend on the individual number of array element considered. For the results presented in the following, the spectral phase at the wavelength of 775 nm was considered only as an example. In order to show that for a different wavelength the bifurcation points occurred at the same pump power and that the same system behavior within the same intervals was developed, in Fig. 3 b the spectral phase at a wavelength of 750 nm (gray dots) is included. We have verified, that the system displayed the same qualitative behavior within all the other spectral components by varying the bifurcation parameter (effective average input power), no matter which phase, or even amplitude or spectral intensity was regarded. The system behavior could also be deduced from the total pulse energy, which represents a measurable observable for an experimental verification.

### 3.2. Nonlinear dynamics with dynamical dispersion compensation

*E*(λ) is the Fourier-transformed electric field after the

_{N}*N*

^{th}iteration. The upper index p has to be set to 1, if the variability between consecutive spectra should be determined, and has to set to a value of two, if a spectrum is not to be compared to the one directly following, but to the one

*after that*, and so on. If

*p*equals 1 the value of

*V*

^{1}

_{N}is an indication for the average spectral change of two consecutive spectra. The more consecutive spectra differ from each other, the closer

*V*

^{1}

_{N}will be to the maximum value of one, and if the spectral evolution converged towards a steady state, the variability should reach zero.

*V*

^{1}

_{N}has been calculated for the SC with feedback shown in Fig. 4 a, and the result is displayed in Fig. 4 b on a linear scale as a function of the number of iterations. From this graph, it can be seen, that

*V*

^{1}

_{N}showed a rather high value during the transition period, but decreased rapidly. By replotting

*V*

^{1}

_{N}on logarithmic scale (Fig. 4 c) one can see, that the variability decreased exponentially and that after approximately 115 iterations the variability

*V*

^{1}

_{N}reached a value in the order of 10

^{-24}, which corresponded to the round-off errors of the simulation. This indicated that the system reached steady state.

*n*-dimensional phase space is shown, which displays amplitude and phase evolution of a single spectral component, namely at 790 nm, since for this single spectral component the behavior in the transition region could easily be analyzed on a first view. Additionally, this cross section is the maximum fraction of phase space, which can be displayed reasonably, and which at the same time contains all important information, as qualitatively the same behavior was observed and verified for all spectral components. In Fig. 5 a the spectral component at 790 nm is displayed with the spectral power density (|

*E*|

^{2}) in radial direction and the spectral phase (φ) in azimuthal direction for a pump power range between 8.75mW and 9.35mW, including the transition from steady state to period-2 cycle. The color indicates, after which number of iterations the respective coordinate was reached. By increasing the pump power the system switched abruptly into a period-2 cycle, which is presented by two new fixed points with different amplitudes and phases (outer two fixed points in Fig. 5 a). Fig. 5 b shows the evolution of the spectral component at 790 nm in a magnification, from left to right with the pump power increasing from 8.75mW to 9.1mW in steps of 0.05mW. Here, the system was approaching the respective fixed point on a spiral trajectory and from the color gradient it can be seen that the convergence was the slower, the higher the power and the closer the system was to the bifurcation point. This observation agrees well with general analysis of nonlinear dynamics, which states that any periodic attractor looses its stability when approaching a bifurcation point [28].

*V*

^{1}

_{N}was calculated as well as

*V*

^{2}

_{N}and

*V*

^{2}

_{N+1}, and these values are plotted as a function of the feedback iteration number in Fig. 6 b. As expected,

*V*

^{1}

_{N}did not decrease at all, but remained at a constant value, close to one. On the other hand,

*V*

^{2}

_{N}and

*V*

^{2}

_{N}+1, which are measures for the similarity of the spectra obtained after even and odd numbered iterations, respectively, both did decrease exponentially, reaching the numerical noise floor after approximately 100 iterations. This is a clear evidence, that the system was showing two different fixed spectra, which were alternating after each iteration. We have thus observed period doubling, in analogy to classic (mechanic or electric) nonlinear oscillators for the first time within femtosecond SC generation. At a pump power of 10.3mW we observed a quadrupled period as shown in Fig. 6 c. This spectral evolution exhibited once again a transient oscillation regime of about 15 iterations. But then, the generated spectra did repeat only after every fourth one. For a clear verification,

*V*

^{2}

_{N}and

*V*

^{2}

_{N}+1 were calculated, but also

*V*

^{4}

_{N},

*V*

^{4}

_{N+1},

*V*

^{4}

_{N+2}, and

*V*

^{4}

_{N+3}, which represent the similarity of spectra obtained after every fourth iteration, each with a different initial spectrum. In Fig. 6 d these curves are plotted as a function of the number of iterations. As can be seen,

*V*

^{2}

_{N}and

*V*

^{2}

_{N}+1 did not decrease, but remained close to one. However,

*V*

^{4}

_{N},

*V*

^{4}

_{N+1},

*V*

^{4}

_{N+2}, and

*V*

^{4}

_{N+3}all decreased exponentially to the numerical noise level, which they reached after approximately 950 iterations. This time span was somewhat longer than the previously investigated period-2 cycle (see Fig. 6 b), however, the observation of

*V*

^{4}

_{N}reaching the noise level is a proof for period quadrupling in the SC feedback system.

### 3.3. Simulation of the feedback system without dispersion compensation

*V*

^{1}

_{N}stayed at a high level, whereas the variability

*V*

^{2}

_{N}and

*V*

^{2}

_{N+1}decreased to the numerical noise level close to zero, which is clear evidence that also without DDC period doubling was observed.

### 3.4. Stability of a fixed state in the presence of pump power noise and quantum noise

*E*

_{pump}was randomly varied within a Gaussian distribution with a standard deviation (rms) of 0.5%, realistically modeling technical amplitude noise of the available pump laser. In addition, the effect of input quantum noise was included, with the phenomenological approach of adding one photon with random phase to every spectral mode [2

**78**, 1135–1184 (2006).
[CrossRef]

*V*

^{2}

_{N}and

*V*

^{2}

_{N+1}(see Fig. 9 d).

## 4. First experimental verification

## 5. Summary and conclusion

## Acknowledgements

## References and links

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

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

3. | M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett. |

4. | X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. of Opt. A |

5. | Z. Zhu and T. Brown, “Effect of frequency chirping on supercontinuum generation in photonic crystal fibers,” Opt. Express |

6. | S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “White-light supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber,” Opt. Lett. |

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

8. | B. Washburn, S. Ralph, and R. Windeler, “Ultrashort pulse propagation in air-silica microstructure fiber,” Opt. Express |

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

10. | 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. |

11. | A. D. Aguirre, N. Nishizawa, J. G. Fujimoto, W. Seitz, M. Lederer, and D. Kopf, “Continuum generation in a novel photonic crystal fiber for ultrahigh resolution optical coherence tomography at 800 nm and 1300 nm,” Opt. Express |

12. | D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science |

13. | T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature |

14. | J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. |

15. | T. M. Fortier, J. Ye, S. T. Cundiff, and R. S. Windeler, “Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase,” Opt. Lett. |

16. | F. Lu and W. H. Knox, “Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers,” Opt. Express |

17. | G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B |

18. | P. M. Moselund, M. H. Frosz, C. L. Thomsen, and O. Bang, “Back-seeding of higher order gain processes in picosecond supercontinuum generation,” Opt. Express |

19. | M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G. Vladimirov, and M. Taki, “Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities,” Opt. Lett. |

20. | G. Steinmeyer, A. Buchholz, M. Hänsel, M. Heuer, A. Schwache, and F. Mitschke, “Dynamical pulse shaping in a nonlinear resonator,” Phys. Rev. A |

21. | A. E. Siegman, |

22. | P. Groß, K.-J. Boller, and M. E. Klein, “High-precision wavelength-flexible frequency division for metrology,” Phys. Rev. A |

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

24. | G. P. Agrawal, |

25. | Crystal Fibre, “NL-PM-750 data sheet,” http://www.crystal-fibre.com/datasheets/NL-PM-750.pdf. |

26. | M. J. Litzkow, M. Livny, and M. W. Mutka, “Condor - a hunter of idle workstations,” in |

27. | S. H. Strogatz and R. F. Fox, |

28. | A. Scott, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(320.7140) Ultrafast optics : Ultrafast processes in fibers

(320.6629) Ultrafast optics : Supercontinuum generation

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: April 27, 2009

Revised Manuscript: July 1, 2009

Manuscript Accepted: July 29, 2009

Published: August 21, 2009

**Citation**

Michael Kues, Nicoletta Brauckmann, Till Walbaum, Petra Groß, and Carsten Fallnich, "Nonlinear dynamics of femtosecond supercontinuum generation with feedback," Opt. Express **17**, 15827-15841 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15827

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### References

- J.K. Ranka, R. S. Windeler, and A. J. Stentz, "Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 78, 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]
- M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, "Supercontinuum generation in a highly birefringent microstructured fiber," Appl. Phys. Lett. 82, 2197-2199 (2003). [CrossRef]
- Q1. X. Fu, L. Qian, S. Wen, and D. Fan, "Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre," J. of Opt. A 6, 1012-1016 (2004). [CrossRef]
- Z. Zhu and T. Brown, "Effect of frequency chirping on supercontinuum generation in photonic crystal fibers," Opt. Express 12, 689-694 (2004). [CrossRef] [PubMed]
- S. Coen, A.H. L. Chau, R. Leonhardt, J.D. Harvey, J.C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Whitelight supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber," Opt. Lett. 26, 1356-1358 (2001). [CrossRef]
- 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]
- B. Washburn, S. Ralph, and R. Windeler, "Ultrashort pulse propagation in air-silica microstructure fiber," Opt. Express 10, 575-580 (2002). [PubMed]
- H.N. Paulsen, K.M. Hilligsøe, J. Thøgersen, SR. Keiding, and J. J. Larsen, "Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source," Opt. Lett. 27, 1123-1125 (2003). [CrossRef]
- I. Hartl, X.D. Li, C. Chudoba, R.K. Ghanta, T.H. Ko, J.G. Fujimoto, J.K. Ranka, and R. S. Windeler, "Ultrahighresolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001). [CrossRef]
- A.D. Aguirre, N. Nishizawa, J.G. Fujimoto, W. Seitz, M. Lederer, and D. Kopf, "Continuum generation in a novel photonic crystal fiber for ultrahigh resolution optical coherence tomography at 800 nm and 1300 nm," Opt. Express 14, 1145-1160 (2006). [CrossRef] [PubMed]
- D. J. Jones, S.A. Diddams, J.K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000). [CrossRef] [PubMed]
- T. Udem, R. Holzwarth, and T.W. H¨ansch, "Optical frequency metrology," Nature 416, 233-237 (2002). [CrossRef] [PubMed]
- J.M. Dudley and S. Coen, "Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers," Opt. Lett. 27, 1180-1182 (2002). [CrossRef]
- T.M. Fortier, J. Ye, S. T. Cundiff, and R. S. Windeler, "Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase," Opt. Lett. 27, 445-447 (2002). [CrossRef]
- F. Lu and W.H. Knox, "Generation of a broadband continuum with high spectral coherence in tapered singlemode optical fibers," Opt. Express 12, 347-353 (2004). [CrossRef] [PubMed]
- G. Genty, J.M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009). [CrossRef]
- P.M. Moselund, M.H. Frosz, C. L. Thomsen, and O. Bang, "Back-seeding of higher order gain processes in picosecond supercontinuum generation," Opt. Express 16, 11954-11968 (2008). [CrossRef] [PubMed]
- M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A.G. Vladimirov, and M. Taki, "Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities," Opt. Lett. 32, 662-664 (2007). [CrossRef] [PubMed]
- G. Steinmeyer, A. Buchholz, M. H¨ansel, M. Heuer, A. Schwache, and F. Mitschke, "Dynamical pulse shaping in a nonlinear resonator," Phys. Rev. A 52, 830-838 (1995). [CrossRef] [PubMed]
- A. E. Siegman, Lasers, (University Science Books, 1986).
- P. Groß, K.-J. Boller, and M. E. Klein, "High-precision wavelength-flexible frequency division for metrology," Phys. Rev. A 71, 043824 (2005). [CrossRef]
- 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]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
- Crystal Fibre, "NL-PM-750 data sheet," http://www.crystal-fibre.com/datasheets/NL-PM-750.pdf.
- M. J. Litzkow, M. Livny, and M.W. Mutka, "Condor - a hunter of idle workstations," in Proc. 8th Int. Conf. on Distributed Computing Systems (IEEE Computer Society Press, 1988), pp. 104-111.
- S.H. Strogatz and R. F. Fox, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Physics Today, 1995).
- A. Scott, Encyclopedia of Nonlinear Science (Routledge Taylor and Francis Group, 2005).

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