## Laser noise compression by filamentation at 400 nm in argon

Optics Express, Vol. 15, Issue 20, pp. 13295-13303 (2007)

http://dx.doi.org/10.1364/OE.15.013295

Acrobat PDF (3085 KB)

### Abstract

Filamentation is an efficient way to produce an intense and spectrally broad, but poorly stable, source for coherent control spectroscopy. We first described both theoretically and experimentally the filamentation and broadening of a 410 nm ultrashort laser pulse in Argon. By observing the theoretical and experimental spectral cross-correlation in the filament, we then show that the stability of the source can be improved. The Signal-to-Noise Ratio of the intensity inside the filament is increased up to 7 dB by its spectral filtering which provide a low noise broad spectrum source.

© 2007 Optical Society of America

## 1. Introduction

1. T. Brixner, N. H. Damrauer, P. Niklaus, and G. Gerber, Photoselective adaptive femtosecond quantum control in the liquid phase, Nature **414**, 57–60 (2001). [CrossRef] [PubMed]

2. J. M. Dela Cruz, I. Pastirk, V. V. Lozovoy, K. A. Walowicz, and M. Dantus, Multiphoton intrapulse interference 3: Probing microscopic chemical environments, J. Phys. Chem. A **108**, 53–58 (2004). [CrossRef]

3. F. Courvoisier, V. Boutou, V. Wood, A. Bartelt, M. Roth, H. Rabitz, and J. P. Wolf, Femtosecond laser pulses distinguish bacteria from background urban aerosols, Appl. Phys. Lett. **87**, 063901 (2005). [CrossRef]

4. V. V. Lozovoy and M. Dantus, Coherent control in femtochemistry, Chemphyschem **6**, 1970–2000 (2005). [CrossRef] [PubMed]

5. A.M. Weiner, Femtosecond pulse shaping using spatial light modulators, Rev. Sci. Instrum. **71**, 1929–1960 (2000). [CrossRef]

6. J. P. Ogilvie, D. Dbarre, X. Solinas, J. L. Martin, E. Beaurepaire, and M. Joffre, Use of coherent control for selective two-photon fluorescence microscopy in live organisms, Opt. Express **14** (2), 759–766 (2006). [CrossRef] [PubMed]

7. J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysy-rowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, White-light filaments for atmospheric analysis, Science **301**, 61–64 (2003). [CrossRef] [PubMed]

8. H. Wille, M. Rodriguez, J. Kasparian, D. Mondelain, J. Yu, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, Teramobile: A mobile femtosecond-terawatt laser and detection system, Eur. Phys. J: Appl. Phys. **20**, 183–190 (2002). [CrossRef]

9. S. Coudreau, D. Kaplan, and P. Tournois, Ultraviolet acousto-optic programmable dispersive filter laser pulse shaping in KDP, Opt. Lett. **12**, 1899–1901 (2006). [CrossRef]

10. M. Hacker, G. Stobrawa, R. Sauerbrey, T. Buckup, M. Motzkus, M. Wildenhain, and A. Gehner, Micromirror SLM for femtosecond pulse shapiing in the ultraviolet, Appl. Phys. B , **76**, 711–714 (2003). [CrossRef]

11. F. G. Omenetto, B. P. Luce, and A. J. Taylor, Genetic algorithm pulse shaping for optimum femtosecond propagation in optical fibers, J. Opt. Soc. Am. B , **16**, 2005–2009 (1999). [CrossRef]

## 2. Theoretical background

### 2.1. Basic equations

_{0}= 410

*nm*with cylindrical symmetry around the propagation axis

*z*, written as ℜ

*e*{

*ϵ*exp[

*i*(

*k*

_{0}

*z*-

*ω*

_{0}

*t*)]}, where

*ϵ*(

*r,t,z*) is assumed to be slowly varying in time and along

*z*and evolves according to the propagation equation derive in [22

22. M. Mlejnek, E. M. Wright, and J. V. Moloney, Femtosecond pulse propagation in argon: A pressure dependence study, Phys. Rev. E **58**, 4903–4910 (1998). [CrossRef]

*t*refers to the retarded time in the reference frame of the pulse

14. A. Couairon and A. Mysyrowicz, Femtosecond filamentation in transparent media, Phys. Rep. **441**, 47–189 (2007). [CrossRef]

^{K}corresponds to the coefficient of multiphoton absorption,

*K*being the minimal number of photons necessary to ionize Argon. This quantity is calculated as

*U*is the ionization potential of Argon (

_{i}*U*= 15.76 e

_{i}*V*[23

23. H. Ehrhardt, Hesselba. Kh, K. Jung, E. Schubert, and K. Willmann, Electron-impact ionization of argon -measurements of triple differential cross-sections, J. Phys. B **7**, 69–78 (1974). [CrossRef]

^{K}is expressed as β

^{K}=

*K*

*h*̄

*ω*where ρ

_{at}σ_{K}_{at}is the Argon density and σ

^{K}is the multiphoton ionization cross section. The dynamic of the electric field is coupled with the plasma density ρ because of the multiphoton ionization process. The ionization of Argon follows the equation [24

24. M. D. Feit and J. A. Fleck, Effect of refraction on spot-size dependence of laser-induced breakdown, Appl. Phys. Lett. **24**, 169–172 (1974). [CrossRef]

^{9}

*e*

^{-}.

*cm*

^{-3}. [25

25. A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, Infrared femtosecond light filaments in air: simulations and experiments, J. Opt. Soc. Am. B **19**, 1117–1131 (2002). [CrossRef]

*P*as

_{in}*z*axis, i.e. the diffraction and dispersion, we used a Crank-Nicholson scheme [26], more stable than the Euler method [27

27. A. Chiron, B. Lamouroux, R. Lange, J. F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysy-rowicz, Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases, Eur. Phys. J D **6**, 383–396 (1999). [CrossRef]

*w*

_{0}; 10Δt). As soon as the quadratic radius of the pulse is divided by a factor 2, the resolution of the meshgrid is increased by a factor 2. So, in the first part of the calculation, the radial resolution is only

*μm*and finally increases to 5.3

*μm*which is sufficient for resolving the plasma channel. In order to increase the resolution of the meshgrid, we used a cubic spline interpolation of the electric field (sampling error < 0.1% over the entire meshgrid). At the beginning of the propagation, a low resolution is sufficient because none of the nonlinear terms does not play a significant role in the propagation. Then the Kerr effect induces a plasma channel. As soon as plasma ionization becomes significant, the meshgrid has to resolve the plasma channel which has a FWHM width of a few microns. Hence the meshgrid resolution turns out to be relevant for any propagation step. This latter also changed during the propagation. It is initially fixed at 2 mm. Then because the nonlinear phase varies as |ϵ|

^{2}

*dz*, we control the relative error by decreasing the propagation step as a function of

*μm*during the filamentation process.

### 2.2. Results and discussion

_{0}= 410

*nm*Fourier Transformed limited Gaussian laser pulses with a pulse duration

*μJ*which corresponds to about 5

*P*at a pressure of 5 bar, which is the pressure used in the calculations below. We chose an input beam with an intensity FWHM diameter of 6

_{cr}*mm*and a focal length of 75

*cm*. Table I summarizes all the physical parameter values we used in our model. Some parameters are given as a function of

*p*which is the relative gas pressure.

*z*= 0

*m*,

*z*= 75

*cm*,

*z*= 78

*cm*,

*z*= 80

*cm*,

*z*= 84

*cm*and

*z*= 99

*cm*). The vertical axis represents the radial distance (in

*μm*), whereas the horizontal axis represents the retarded time (in

*fs*). When the intensity peak reaches a value of about 6 10

^{13}

*W*.cm

^{-2}, the plasma contribution becomes higher than that of the Kerr effect for the trailing edge because the ionization threshold is reached. Therefore the trailing edge is defocused (see

*z*= 80

*cm*). The plasma density subsequently decreases so that the trailing part of the pulse focuses again when the power is high enough (see

*z*= 84

*cm*). In such a case, the pulse is divided into two subpulses (Fig. 2(a)). For even longer propagation distance, the mechanism is rather complex because it involves pulse splitting and self-phase modulation modified by ionization. Consequently, the resulting electric field intensity is divided in several subpulses (Fig. 1,

*z*= 99

*cm*).

^{13}

*W*.

*cm*

^{-2}because the generated electron density has a threshold like response, which saturates self-focusing locally and limits the peak intensity inside the filament by defocusing the beam. Concerning the plasma density, it reaches a value of about 10

^{17–18}

*e*

^{-}

*cm*

^{-3}. This value is consistent with previous calculations with other wavelengths in Argon or in the air (248 nm, 586 nm and 810 nm) [31

31. A. Couairon and L. Berge, Light filaments in air for ultraviolet and infrared wavelengths, Phys. Rev. Lett. **88**, 135003 (2002). [CrossRef] [PubMed]

32. S. Champeaux and L. Berge, Femtosecond pulse compression in pressure-gas cells filled with argon, Phys. Rev. E **68**, 066603 (2003). [CrossRef]

*μm*(see Fig. 3, Right, inset) but this calculation does not take into account the photon bath which feeds the filament core. Therefore, it is preferable to consider the quadratic radius calculated as

*μm*.

33. T. Brixner, N. H. Damrauer, B. Kiefer, and G. Gerber, Liquid-phase adaptive femtosecond quantum control: Removing intrinsic intensity dependencies, J. Chem. Phys. **118**, 3692–3701 (2003). [CrossRef]

*fs*

^{2}is observed on the electric field.

*μm*instead of 100

*μm*at 800 nm) but it implies a less efficient white-light generation. However, 400 nm filament generates a sufficiently broad spectrum and its shape can be tailored with simple parameters such as chirp and gas pressure. Moreover, the filament is intrinsically energetic which is favorable for remote sensing applications. However, the filamentation process is highly nonlinear, which means that small changes in the input beam induce strong variations of the filament spectrum. This instability is the main drawback of using filament as a usefull coherent source for closed-loop optimization experiments. However, the statistical properties of filamentation can be used to restrict, up to a certain point, this instability. Recent studies [34, 35] indeed show that correlations exist between specific wavelengths pairs. These correlations are positive if the wavelengths are created simultaneously, negative if one wavelength is used to create the other. The variance of the intensity σ

^{2}(

*I*) = <

*I*

^{2}> - <

*I*>

^{2}(where <

*X*> denotes the average of X over a representative sample) can be written as

*I*(

_{i}) the average of

*I*(λ

_{i},

*z*) over the propagation calculated as:

## 3. Experiments

### 3.1. Experimental setup

*e*

^{-2}level). A 1 mm BBO crystal was used for frequency doubling the fundamental wavelength. Adequate filters and dichroic mirrors allow to reject the residual of the fundamental. The beam energy at 410 nm could be varied up to 1 mJ. As depicted in Fig. 8, the 410 nm beam was focused by a 75 cm lens inside a 1m length cell containing Argon. The Argon pressure was varied from 1 to 8 bar. Depending on the pressure, the filament formed about the center of the cell was typically a few centimeters long. At the cell exit, as shown in Fig. 8(a), after about 50 cm of free propagation outside the Argon cell, the beam was scattered on a neutral target, and the light was collected with a fiber and injected into a spectrometer (Ocean Optics HR2000) providing 0.6 nm resolution between 380 and 450 nm. 1000 spectra were recorded and used to compute the intensity cross-correlation maps across the spectrum. The cross-correlation between two wavelengths λ

_{1}and λ

_{2}was calculated as

*V*(

*x*) the variance of variable

*x*and

*n*the photon number (or the intensity) at the wavelength

_{i}*λ*. Hence, the cross-correlation measured experimentally (whereas it was not the case theoretically) is a statistical property of the filamentation. In a second experimental arrangement, depicted on Fig. 8(b), the continuum generated by the filament in argon was further dispersed by a diffraction grating (order -1 blazed, efficiency 75 % at 410 nm). The setup was used to observe the noise reduction induced by self phase modulation on the central wavelength. For that purpose, after the dispersion of the beam on the grating, an iris was placed in the Fourier plane of two cylindrical lenses in order to filter out the edge frequencies of the continuum. The remaining part of the beam was then detected on a photomultiplier tube (PMT) connected to a boxcar amplifier. Another PMT, located before the argon filled cell, was dedicated to measure simultaneously the laser input fluctuations. Acquisitions of about 2000 laser shots have been performed for each pressure and/or each position and size of the iris in order to retrieve histograms of the signal prior and after nonlinear propagation in the argon filled cell. Optimal parameters maximizing the Signal-to-Noise Ratio were determined from these histograms.

_{i}### 3.2. Results and discussions

22. M. Mlejnek, E. M. Wright, and J. V. Moloney, Femtosecond pulse propagation in argon: A pressure dependence study, Phys. Rev. E **58**, 4903–4910 (1998). [CrossRef]

_{0}= 410

*nm*. At higher pressures, the critical power decreases and therefore lowers the threshold, yielding to a larger broadening of the input pulses. As expected from χ

^{3}induced broadening processes, two photons from the pump at λ

_{0}are annihilated to produce one photon at λ

_{1}and its conjugated photon at λ

_{2}(through the conservation of the energy,

_{1}= λ

_{2}, whereas negative correlations form a dark cross centered on λ

_{0}. Up to 3 bar, the increase of the positive correlations area reflects further broadening of the input pulse and the creation of conjugated wavelengths. At higher pressure (for example, 5 bar on Fig. 9), cascaded events are responsible for the occurrence of new negative correlations (additional dark stripes). As described previously in the calculations section, when the spectrum is broadened enough, one or both conjugated wavelengths can be involved in a secondary event involving the third order polarization, which partially destroy the previously formed correlations.

*S*

_{1}) and after (

*S*

_{2}) (with spectral filtering [406–414 nm]) propagation through the cell (argon pressure 2 bar). As compared to the incoming beam, filamentation yields a SNR increase of about 7 dB. This noise reduction disappears if spectral filtering is applied only to one side of the spectrum as SPM rejects the fluctuations to the two edges of the spectrum.

*S*

_{2}) versus the intensities on the channel before filamentation (

*S*

_{1}). In Fig. 11, such plots are given as a function of the gas pressure inside the cell. For pressure lower than 2.5 bar,

*S*

_{2}is an affine function of

*S*

_{1}(

*S*

_{2}≅ 0.9

*S*

_{1}). The broadening does not take place, nothing is filtered out. Up to 2.5 bar, the evolution of

*S*

_{2}is no longer proportional to

*S*

_{1}but a plateau appears for a range of input intensities (between 1.5 and 2.5 arb. unit.). Three intensity regimes can be defined, labeled from I to III. In interval I, the incident intensity is too low and consequently SPM generation is not efficient enough to induce noise reduction. Interval II corresponds to the range of input intensities for which noise reduction is optimal. Although the input intensity

*S*

_{1}increases by a factor 2, the spatially filtered output intensity

*S*

_{2}remains stable. The fluctuations of

*S*

_{1}versus

*S*

_{2},

## 4. Conclusions

## Acknowledgments

## References and links

1. | T. Brixner, N. H. Damrauer, P. Niklaus, and G. Gerber, Photoselective adaptive femtosecond quantum control in the liquid phase, Nature |

2. | J. M. Dela Cruz, I. Pastirk, V. V. Lozovoy, K. A. Walowicz, and M. Dantus, Multiphoton intrapulse interference 3: Probing microscopic chemical environments, J. Phys. Chem. A |

3. | F. Courvoisier, V. Boutou, V. Wood, A. Bartelt, M. Roth, H. Rabitz, and J. P. Wolf, Femtosecond laser pulses distinguish bacteria from background urban aerosols, Appl. Phys. Lett. |

4. | V. V. Lozovoy and M. Dantus, Coherent control in femtochemistry, Chemphyschem |

5. | A.M. Weiner, Femtosecond pulse shaping using spatial light modulators, Rev. Sci. Instrum. |

6. | J. P. Ogilvie, D. Dbarre, X. Solinas, J. L. Martin, E. Beaurepaire, and M. Joffre, Use of coherent control for selective two-photon fluorescence microscopy in live organisms, Opt. Express |

7. | J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysy-rowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, White-light filaments for atmospheric analysis, Science |

8. | H. Wille, M. Rodriguez, J. Kasparian, D. Mondelain, J. Yu, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, and L. Woste, Teramobile: A mobile femtosecond-terawatt laser and detection system, Eur. Phys. J: Appl. Phys. |

9. | S. Coudreau, D. Kaplan, and P. Tournois, Ultraviolet acousto-optic programmable dispersive filter laser pulse shaping in KDP, Opt. Lett. |

10. | M. Hacker, G. Stobrawa, R. Sauerbrey, T. Buckup, M. Motzkus, M. Wildenhain, and A. Gehner, Micromirror SLM for femtosecond pulse shapiing in the ultraviolet, Appl. Phys. B , |

11. | F. G. Omenetto, B. P. Luce, and A. J. Taylor, Genetic algorithm pulse shaping for optimum femtosecond propagation in optical fibers, J. Opt. Soc. Am. B , |

12. | Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, C. Spielman, and F. Krausz, Generation of Intense Diffraction-Limited White Light and 4-fs Pulses, |

13. | C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation, Appl. Phys. B |

14. | A. Couairon and A. Mysyrowicz, Femtosecond filamentation in transparent media, Phys. Rep. |

15. | G. Mejean, J. Kasparian, J. Yu, S. Frey, E. Salmon, R. Ackermann, J. P. Wolf, L. Berge, and S. Skupin, Uv-supercontinuum generated by femtosecond pulse filamentation in air: Meter-range experiments versus numerical simulations, Appl. Phys. B , |

16. | S. L. Chin, F. Thberge, and W. Liu, Filamentation nonlinear optics, Appl. Phys. B |

17. | S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, Observation of optical soliton photon-number squeezing, Phys. Rev. Lett. |

18. | S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, Observation of multimode quantum correlations in fiber optical solitons, Phys. Rev. Lett. |

19. | L. Boivin, F. X. Kartner, and H. A. Haus, Analytical solution to the quantum-field theory of self-phase modulation with a finite response-time, Phys. Rev. Lett. |

20. | T. Opatrny, N. Korolkova, and G. Leuchs, Mode structure and photon number correlations in squeezed quantum pulses, Phys. Rev. A |

21. | E. Schmidt, L. Knoll, and D. G. Welsch, Cumulant expansion for studying damped quantum solitons, Phys. Rev. A |

22. | M. Mlejnek, E. M. Wright, and J. V. Moloney, Femtosecond pulse propagation in argon: A pressure dependence study, Phys. Rev. E |

23. | H. Ehrhardt, Hesselba. Kh, K. Jung, E. Schubert, and K. Willmann, Electron-impact ionization of argon -measurements of triple differential cross-sections, J. Phys. B |

24. | M. D. Feit and J. A. Fleck, Effect of refraction on spot-size dependence of laser-induced breakdown, Appl. Phys. Lett. |

25. | A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, and A. Mysyrowicz, Infrared femtosecond light filaments in air: simulations and experiments, J. Opt. Soc. Am. B |

26. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipies, Cambridge University Press, Cambridge, Numerical Recipies, 1989. |

27. | A. Chiron, B. Lamouroux, R. Lange, J. F. Ripoche, M. Franco, B. Prade, G. Bonnaud, G. Riazuelo, and A. Mysy-rowicz, Numerical simulations of the nonlinear propagation of femtosecond optical pulses in gases, Eur. Phys. J D |

28. | E. R. Peck and D. J. Fischer, Dispersion of argon, J. Opt. Soc. Am. |

29. | V. Mizrahi and D. P. Shelton, Dispersion of nonlinear susceptibilities of Ar, |

30. | Y. P. Raizer, Plasma physics, Springer, Berlin, Plasma Physics (1994). |

31. | A. Couairon and L. Berge, Light filaments in air for ultraviolet and infrared wavelengths, Phys. Rev. Lett. |

32. | S. Champeaux and L. Berge, Femtosecond pulse compression in pressure-gas cells filled with argon, Phys. Rev. E |

33. | T. Brixner, N. H. Damrauer, B. Kiefer, and G. Gerber, Liquid-phase adaptive femtosecond quantum control: Removing intrinsic intensity dependencies, J. Chem. Phys. |

34. | P. Bejot, J. Kasparian, E. Salmon, R. Ackermann, N. Gisin, and J. P. Wolf, Laser noise reduction in air, Appl. Phys. Lett. |

35. | P. Bejot, J. Kasparian, E. Salmon, R. Ackermann, and J. P. Wolf, Spectral correlation and noise reduction in laser filaments, Appl. Phys. B |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(190.3270) Nonlinear optics : Kerr effect

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 17, 2007

Revised Manuscript: August 23, 2007

Manuscript Accepted: August 29, 2007

Published: September 28, 2007

**Citation**

Pierre Béjot, Christophe Bonnet, Véronique Boutou, and Jean-Pierre Wolf, "Laser noise compression by filamentation at 400 nm in argon," Opt. Express **15**, 13295-13303 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13295

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### References

- T. Brixner, N. H. Damrauer, P. Niklaus, G. Gerber, Photoselective adaptive femtosecond quantum control in the liquid phase, Nature 414, 57–60 (2001). [CrossRef] [PubMed]
- J. M. Dela Cruz, I. Pastirk, V. V. Lozovoy, K. A. Walowicz, M. Dantus, Multiphoton intrapulse interference 3: Probing microscopic chemical environments, J. Phys. Chem. A 108, 53–58 (2004). [CrossRef]
- F. Courvoisier, V. Boutou, V. Wood, A. Bartelt, M. Roth, H. Rabitz, J. P. Wolf, Femtosecond laser pulses distinguish bacteria from background urban aerosols, Appl. Phys. Lett. 87, 063901 (2005). [CrossRef]
- V. V. Lozovoy, M. Dantus, Coherent control in femtochemistry, Chemphyschem 6, 1970–2000 (2005). [CrossRef] [PubMed]
- A.M. Weiner, Femtosecond pulse shaping using spatial light modulators, Rev. Sci. Instrum. 71, 1929–1960 (2000). [CrossRef]
- J. P. Ogilvie, D. Dbarre, X. Solinas, J. L. Martin, E. Beaurepaire, M. Joffre, Use of coherent control for selective two-photon fluorescence microscopy in live organisms, Opt. Express 14 (2), 759–766 (2006). [CrossRef] [PubMed]
- J. Kasparian, M. Rodriguez, G. Mejean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y. B. Andre, A. Mysy-rowicz, R. Sauerbrey, J. P. Wolf, L. Woste, White-light filaments for atmospheric analysis, Science 301, 61–64 (2003). [CrossRef] [PubMed]
- H. Wille, M. Rodriguez, J. Kasparian, D. Mondelain, J. Yu, A. Mysyrowicz, R. Sauerbrey, J. P. Wolf, L. Woste, Teramobile: A mobile femtosecond-terawatt laser and detection system, Eur. Phys. J: Appl. Phys. 20, 183–190 (2002). [CrossRef]
- S. Coudreau, D. Kaplan, P. Tournois, Ultraviolet acousto-optic programmable dispersive filter laser pulse shaping in KDP, Opt. Lett. 12, 1899–1901 (2006). [CrossRef]
- M. Hacker, G. Stobrawa, R. Sauerbrey, T. Buckup, M. Motzkus, M. Wildenhain, A. Gehner, Micromirror SLM for femtosecond pulse shapiing in the ultraviolet, Appl. Phys. B, 76, 711–714 (2003). [CrossRef]
- F. G. Omenetto, B. P. Luce, A. J. Taylor, Genetic algorithm pulse shaping for optimum femtosecond propagation in optical fibers, J. Opt. Soc. Am. B, 16, 2005–2009 (1999). [CrossRef]
- Z. Cheng, G. Tempea, T. Brabec, K. Ferencz, C. Spielman, F. Krausz, Generation of Intense Diffraction-Limited White Light and 4-fs Pulses, Lasers and Electro-Optics Europe, 1998. 1998 CLEO/Europe. Conference on
- C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, U. Keller, Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation, Appl. Phys. B 79, 673–677 (2004) [CrossRef]
- A. Couairon, A. Mysyrowicz, Femtosecond filamentation in transparent media, Phys. Rep. 441, 47–189 (2007). [CrossRef]
- G. Mejean, J. Kasparian, J. Yu, S. Frey, E. Salmon, R. Ackermann, J. P. Wolf, L. Berge, S. Skupin, Uv-supercontinuum generated by femtosecond pulse filamentation in air: Meter-range experiments versus numerical simulations, Appl. Phys. B, 82, 341–345 (2006). [CrossRef]
- S. L. Chin, F. Thberge, W. Liu, Filamentation nonlinear optics, Appl. Phys. B 86, 477–483 (2007). [CrossRef]
- S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, T. Mukai, Observation of optical soliton photon-number squeezing, Phys. Rev. Lett. 77, 3775–3778 (1996). [CrossRef] [PubMed]
- S. Spalter, N. Korolkova, F. Konig, A. Sizmann, G. Leuchs, Observation of multimode quantum correlations in fiber optical solitons, Phys. Rev. Lett. 81, 786–789 (1998). [CrossRef]
- L. Boivin, F. X. Kartner, H. A. Haus, Analytical solution to the quantum-field theory of self-phase modulation with a finite response-time, Phys. Rev. Lett. 73, 240–243 (1994). [CrossRef] [PubMed]
- T. Opatrny, N. Korolkova, G. Leuchs, Mode structure and photon number correlations in squeezed quantum pulses, Phys. Rev. A 66, 053813 (2002). [CrossRef]
- E. Schmidt, L. Knoll, D. G. Welsch, Cumulant expansion for studying damped quantum solitons, Phys. Rev. A 59, 2442–2457 (1999). [CrossRef]
- M. Mlejnek, E. M. Wright, J. V. Moloney, Femtosecond pulse propagation in argon: A pressure dependence study, Phys. Rev. E 58, 4903–4910 (1998). [CrossRef]
- H. Ehrhardt, Hesselba. Kh, K. Jung, E. Schubert, K. Willmann, Electron-impact ionization of argon -measurements of triple differential cross-sections, J. Phys. B 7, 69–78 (1974). [CrossRef]
- M. D. Feit, J. A. Fleck, Effect of refraction on spot-size dependence of laser-induced breakdown, Appl. Phys. Lett. 24, 169–172 (1974). [CrossRef]
- A. Couairon, S. Tzortzakis, L. Berge, M. Franco, B. Prade, A. Mysyrowicz, Infrared femtosecond light filaments in air: simulations and experiments, J. Opt. Soc. Am. B 19, 1117–1131 (2002). [CrossRef]
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