## Coherence of subsequent supercontinuum pulses generated in tapered fibers in the femtosecond regime

Optics Express, Vol. 15, Issue 5, pp. 2732-2741 (2007)

http://dx.doi.org/10.1364/OE.15.002732

Acrobat PDF (357 KB)

### Abstract

We measure the degree of coherence of supercontinua generated in tapered fibers by subsequent fs pulses. By means of interference experiments we study its dependence on the input pulse duration and power. We also present numerical simulations that allow us to explain the experimental observations which show a decreasing degree of coherence with increasing input power. We attribute this loss of coherence to phase noise due to the cross-phase modulation by several solitons with randomly varying parameters due to quantum noise.

© 2007 Optical Society of America

## 1. Introduction

## 2. Experimental results

17. J. Teipel, K. Franke, D. Türke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, “Characteristics of supercontin-uum generationin tapered fibers using femtosecond laser pulses, ” Appl. Phys. B **77**, 245–251 (2003). [CrossRef]

2. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers, ” Opt. Lett . **25**, 1415–1417 (2000). [CrossRef]

15. F. Lu and W. H. Knox, “Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers, ” Opt. Express **12**, 347–353 (2004). [CrossRef] [PubMed]

18. J. Stenger and H. R. Telle, “Kerr-lens mode-locked lasers for optical frequency measurements, ” in Laser Frequency Stabilization, Standards, Measurement, and Applications, John L. Hall and Jun Ye, eds., Proc. SPIE **4269**, 72–76 (2001). [CrossRef]

*V*. As the degradation of the contrast is not only affected by the phase noise but also by the amplitude noise, the latter has to be measured independently for different frequency bands by monitoring a train of pulses using a fast photodiode. A simple simulation of the dependence of the visibility on the amplitude noise allows to calculate the phase noise part separately as follows: We assume a perfectly stable phase for a certain frequency component ω, but we allow a Gaussian distributed amplitude noise of the e-field:

*P*〉 denotes the average power and Δ

*P*the deviation from 〈

*P*〉. We choose a Gaussian distribution due to the fact that it serves in the case of high photon numbers as an approximation for a Poisson distribution, which describes the photon statistic of coherent laser light. Since the detector integrates over a large number of subsequent pulses with individual amplitudes, the measured power reads as

*P*) describes the Gaussian distribution with the standard deviation σ of the amplitude noise. σ is determined by evaluating the standard deviation of the normalized amplitudes of the electric field recorded by the photodiode.

*V*(σ) denotes the fraction in the previous expression, with

*x*=Δ

*P*for better readability. The numerically calculated decrease of the amplitude

*V*(σ) of the visibility fringes given by Eq. 3 [see also Fig. 2(b)] in dependence on the width of the Gaussian distribution is shown in Fig. 3. It is remarkable that even a completely random amplitude of the electric field (rectangu-lar distribution with σ=1) leads to a decrease of the visibility of only 12 %.

*phase noise*.

20. J. W. Nicholson and M. F. Yan, “Cross-coherence measurements of supercontinua generated in highly-nonlinear, dispersion shifted fiber at 1550 nm, ” Opt. Express **12**, 679–688 (2004). [CrossRef] [PubMed]

## 3. Numerical model and results

21. P. D. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations, ” J. Opt. Soc. Am. B **18**, 139–152 (2001). [CrossRef]

3. A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, ” Phys. Rev. Lett . **87**, 203901 (2001). [CrossRef] [PubMed]

*E*(

*z*,ω) is the field strength characterizing the evolution of the pulse during propagation along the

*z*coordinate. The transverse distribution of the field is determined by the spatial structure of the fundamental mode, which satisfies a corresponding Helmholtz eigenvalue equation with the wavenumber β(ω), and

*n*is the characteristic group refractive index. For tapered fibers, the transverse mode distribution and the wavenumber including both bulk and waveguide dispersion can be found by an analytical solution of the Helmholtz equation; the difference to standard fibers is here only in the smaller core radius and the large index contrast between the fused silica core and the air surrounding. The quantity

_{g}*P*(

_{NL}*z*,ω) is the Fourier transform of the nonlinear polarization

*n*¯ =

*E*

_{0}/(

*h*¯ω

_{0}) is the number of photons in the pulse,

*E*

_{0}being the pulse energy. The quantity which characterizes the coherence of the supercontinuum is defined as

*L*is the length of the fiber. The quantity

*g*(ω) directly corresponds to the visibility

*g*=

*V*=(

*I*

_{max}-

*I*

_{min})/(

*I*

_{max}+

*I*

_{min}) measured in the interference experiment. To characterize the coherence of the radiation, we used also the average coherence

*g*¯ calculated by using the spectral power as weight. The propagation equation (4) was solved by the split-step Fourier method with nonlinear steps performed by the fourth-order Runge-Kutta method. To model the SC coherence of a pulse train as observed in the experiment, the numerical results were smoothed as functions of ω to reproduce experimental conditions. The values of the material parameters for fused-silica fibers are

*n*

^{2}=3×10

^{-16}cm

^{2}/W,

*f*= 0.18,

*T*= 70 fs, and τ

_{R}_{R}=30 fs. For the calculation of wavenumber β(ω) we have used the Sellmeyer coefficients of fused silica [22], and a fiber diameter of 2.1 μm. These values yield a zero-dispersion wavelength of 730 nm and a GVD-parameter at the input wavelength of 775 nm of -11.2 fs

^{2}/mm.

*g*¯ of 0.93. For the higher power of 5.9 kW and the same pulse duration [Fig. 5(b)], the average coherence drops to 0.15 which is within the uncertainty due to the finite number of the noise realizations. The coherence has a peak around the input wavelength but is quite low in other parts of the spectrum. Note that coherence is observed for 2.5 kW and disappears for 5.9 kW input power except for the peak at the input wavelength, in good agreement with the experimental results in Fig. 4. The widths of the spectrum correspond also to the experimental values in both cases. Note that we simulated pulse propagation over a shorter distance than the experimental length of the fiber of around 30 cm, however this is justified by the saturation of the spectral broadening and coherence after about 4-cm propagation.

## 4. Discussion

9. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber, ” Phys. Rev. Lett . **90**, 113904 (2003). [CrossRef] [PubMed]

3. A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, ” Phys. Rev. Lett . **87**, 203901 (2001). [CrossRef] [PubMed]

*f*(

_{i}*z*,

*t*) by a certain soliton with amplitude

*A*after the fission of the input pulse can be described by the soliton perturbation theory. Using the ansatz

_{i}*E*(

*z*,

*t*)=∑

*(*

_{i}A_{i}*z*,

*t*)

*e*

*+∑*

^{iωit-ikiz}*i fi*(

*z*,

*t*)e

^{iω´it-iḱiz}the non-solitonic component

*fi*(

*z*,

*t*) is described by the following linear propagation equation:

## 5. Conclusion

## 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 800nm, ” Opt. Lett . |

2. | T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers, ” Opt. Lett . |

3. | A. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, ” Phys. Rev. Lett . |

4. | J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J. Russell, and G. Korn, “Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers, ” Phys. Rev. Lett . |

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

6. | B. Schenkel, R. Paschotta, and U. Keller, “Pulse compression with supercontinuum generation in microstructure fibers, ” J. Opt. Soc. Am. B |

7. | R. Holzwarth, Th. Udem, T. W. Haänsch, J. C. Knight, W. J. Wadsworth, and p. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy, ” Phys. Rev. Lett . |

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

9. | K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber, ” Phys. Rev. Lett . |

10. | T. M. Fortier, Ye J., 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 . |

11. | N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, “Noise amplification during supercontinuum generation in microstructure fiber, ” Opt. Lett . |

12. | A. L. Gaeta, “Nonlinear propagation and continuum generation in microstructured optical fibers, ” Opt. Lett . |

13. | M. Bellini and T.W. Haänsch, “Phase-locked white-light continuum pulses: toward a universal optical frequency-comb synthesizer, ” Opt. Lett . |

14. | X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express |

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

16. | J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, and S. T. Cundiff, “Excess noise generation during spectral broadening in a microstructured fiber, ” Appl. Phys. B |

17. | J. Teipel, K. Franke, D. Türke, F. Warken, D. Meiser, M. Leuschner, and H. Giessen, “Characteristics of supercontin-uum generationin tapered fibers using femtosecond laser pulses, ” Appl. Phys. B |

18. | J. Stenger and H. R. Telle, “Kerr-lens mode-locked lasers for optical frequency measurements, ” in Laser Frequency Stabilization, Standards, Measurement, and Applications, John L. Hall and Jun Ye, eds., Proc. SPIE |

19. | L. Mandel and E. Wolf, |

20. | J. W. Nicholson and M. F. Yan, “Cross-coherence measurements of supercontinua generated in highly-nonlinear, dispersion shifted fiber at 1550 nm, ” Opt. Express |

21. | P. D. Drummond and J. F. Corney, “Quantum noise in optical fibers. I. Stochastic equations, ” J. Opt. Soc. Am. B |

22. | M. Bass (ed.), |

23. | A. Husakou and J. Herrmann, in preparation. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(320.7140) Ultrafast optics : Ultrafast processes in fibers

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: December 22, 2006

Revised Manuscript: February 23, 2007

Manuscript Accepted: February 26, 2007

Published: March 5, 2007

**Citation**

D. Türke, S. Pricking, A. Husakou, J. Teipel, J. Herrmann, and H. Giessen, "Coherence of subsequent
supercontinuum pulses generated in
tapered fibers in the femtosecond regime," Opt. Express **15**, 2732-2741 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2732

Sort: Year | Journal | Reset

### References

- J. K. Ranka, R. S. Windeler, A. J. Stentz, "Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm," Opt. Lett. 25, 25-27 (2000). [CrossRef]
- T. A. Birks, W. J. Wadsworth, P. St. J. Russell, "Supercontinuum generation in tapered fibers," Opt. Lett. 25,1415-1417 (2000). [CrossRef]
- A. Husakou, J. Herrmann, "Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers," Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]
- J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002). [CrossRef] [PubMed]
- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006). [CrossRef]
- B. Schenkel, R. Paschotta, U. Keller, "Pulse compression with supercontinuum generation in microstructure fibers," J. Opt. Soc. Am. B 22, 687-693 (2005). [CrossRef]
- R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000). [CrossRef] [PubMed]
- J. M. Dudley, S. Coen, "Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers," Opt. Lett. 27, 1180-1182 (2002). [CrossRef]
- K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003). [CrossRef] [PubMed]
- T. M. Fortier, J. Ye, S. T. Cundiff, 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]
- N. R. Newbury, B. R. Washburn, K. L. Corwin, R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003). [CrossRef] [PubMed]
- A. L. Gaeta, "Nonlinear propagation and continuum generation in microstructured optical fibers," Opt. Lett. 27, 924-926 (2002). [CrossRef]
- M. Bellini, T. W. H¨ansch, "Phase-locked white-light continuum pulses: toward a universal optical frequency-comb synthesizer," Opt. Lett. 25, 1049-1051 (2000). [CrossRef]
- X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, R. S. Windeler, "Experimental studies of the coherence of microstructure-fiber supercontinuum," Opt. Express 11, 2697-2703 (2003). [CrossRef] [PubMed]
- F. Lu, W. H. Knox, "Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers," Opt. Express 12, 347-353 (2004). [CrossRef] [PubMed]
- J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003). [CrossRef]
- J. Teipel, K. Franke, D. T¨urke, F.Warken, D. Meiser, M. Leuschner, H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77,245-251 (2003). [CrossRef]
- J. Stenger and H. R. Telle, "Kerr-lens mode-locked lasers for optical frequency measurements," in Laser Frequency Stabilization, Standards, Measurement, and Applications, John L. Hall and Jun Ye, eds., Proc. SPIE 4269, 72-76 (2001). [CrossRef]
- L. Mandel, E. Wolf, Optical coherence and quantum optics (Cambridge University Press, Cambridge, 1995).
- J. W. Nicholson, M. F. Yan, "Cross-coherence measurements of supercontinua generated in highly-nonlinear, dispersion shifted fiber at 1550 nm," Opt. Express 12, 679-688 (2004). [CrossRef] [PubMed]
- P. D. Drummond, J. F. Corney, "Quantum noise in optical fibers. I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001). [CrossRef]
- M. Bass (ed.), Handbook of Optics (McGraw-Hill, New York, 1995).
- A. Husakou and J. Herrmann, in preparation.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.