## Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets

Optics Express, Vol. 17, Issue 17, pp. 14948-14955 (2009)

http://dx.doi.org/10.1364/OE.17.014948

Acrobat PDF (823 KB)

### Abstract

We measure the spatiotemporal field of ultrashort pulses with complex spatiotemporal profiles using the linear-optical, interferometric pulse-measurement technique SEA TADPOLE. Accelerating and decelerating ultrashort, localized, nonspreading Bessel-X wavepackets were generated from a ~27 fs duration Ti:Sapphire oscillator pulse using a combination of an axicon and a convex or concave lens. The wavefields are measured with ~5 μm spatial and ~15 fs temporal resolutions. Our experimental results are in good agreement with theoretical calculations and numerical simulations.

© 2009 OSA

## 1. Introduction

2. P. Saari and K. Reivelt, “Evidence of *X*-Shaped propagation-invariant localized light waves,” Phys. Rev. Lett. **79**(21), 4135–4138 (1997). [CrossRef]

8. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. **34**(15), 2276–2278 (2009). [CrossRef] [PubMed]

9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901–213904 (2007). [CrossRef]

11. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**(5924), 229–232 (2009). [CrossRef] [PubMed]

12. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **63**(2), 026601–026611 (2001). [CrossRef] [PubMed]

15. M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. Di Trapani, “Finite-energy, accelerating Bessel pulses,” Opt. Express **16**(24), 19807–19811 (2008). [CrossRef] [PubMed]

15. M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. Di Trapani, “Finite-energy, accelerating Bessel pulses,” Opt. Express **16**(24), 19807–19811 (2008). [CrossRef] [PubMed]

2. P. Saari and K. Reivelt, “Evidence of *X*-Shaped propagation-invariant localized light waves,” Phys. Rev. Lett. **79**(21), 4135–4138 (1997). [CrossRef]

3. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. **22**(5), 310–312 (1997). [CrossRef] [PubMed]

6. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche´, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A **67**(6), 063820–063825 (2003). [CrossRef]

8. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. **34**(15), 2276–2278 (2009). [CrossRef] [PubMed]

13. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express **16**(18), 13663–13675 (2008). [CrossRef] [PubMed]

16. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express **14**(24), 11892–11900 (2006). [CrossRef] [PubMed]

*E*(

*x*,

*y*,

*z*,

*t*) of complicated ultrashort pulses. Briefly, this method involves sampling a small spatial region of the Bessel pulse with a single-mode optical fiber and then interfering this pulse with a reference pulse in a spectrometer to reconstruct

*E*(

*λ*) for that spatial point. Then to measure the spatial dependence of the field, we scan the fiber axially (in

*x*) throughout the cross section of the Bessel pulse, so that

*E*(

*λ*) is measured at each

*x*, yielding

*E*(

*λ,x*). This field can be Fourier transformed to the time domain to yield

*E*(

*t,x*). In order to measure the z (propagation direction) dependence of the spatiotemporal field, the axicon and lens are translated along the propagation direction to bring them nearer or further from the sampling point (the fiber). As demonstrated previously [8

8. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. **34**(15), 2276–2278 (2009). [CrossRef] [PubMed]

13. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express **16**(18), 13663–13675 (2008). [CrossRef] [PubMed]

16. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express **14**(24), 11892–11900 (2006). [CrossRef] [PubMed]

17. P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express **15**(16), 10219–10230 (2007). [CrossRef] [PubMed]

## 2. Theoretical description of accelerating Bessel pulses

*t*= 0. Thinking in terms of the Huygens-Fresnel principle, this will yield an expanding, semi-toroidal wave-field immediately behind the slit. As the pulse propagates further, the tube radius of the half torus becomes larger than the annular-slit radius

*R*, and at times

*t*>

*R*/

*c*the wave-field evolves like a spindle torus, i.e., different parts of the torus start to overlap. Of course, the wave-field is treatable as a mathematical surface only for infinitesimally short delta-like pulses in time. Real ultrashort pulses are at least several cycles long, and so yield an interference pattern in the overlap region (see insets of Fig. 1 ). The radial dependence of the field in the interference region is approximately a zero

^{th}-order Bessel function of the first kind.

*z-*axis and the angle between the normal of the torus surface and the

*z*-axis (

*θ*) decreases. For ultrashort pulses, this intersection region is small, and the angle

*θ*is approximately the same for all points within it at a given instant. Therefore the field in the intersection region is approximately equivalent to the center of a Bessel beam or the apex of a Bessel-X pulse (see also [12

12. Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **63**(2), 026601–026611 (2001). [CrossRef] [PubMed]

*θ*—also called the

*axicon angle*—the larger the spacing between the Bessel rings and the smaller the superluminal velocity of the pulse. Hence, an annular ring transforms an ultrashort pulse into a decelerating Bessel wave-packet propagating along the

*z*-axis. Of course, outside of the intersection region, where there is no interference to generate phase fronts that are perpendicular to the

*z*-axis or a Bessel profile, the phase and pulse fronts expand with a constant velocity

*c*and propagate in their normal directions.

*θ*as the pulse propagates and hence an accelerating pulse.

*θ*to the

*z*axis. In this case, the field can be described using the known expression for the field of a Bessel-X pulse, which is a broadband wave-packet of monochromatic Bessel beams [1–3

3. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. **22**(5), 310–312 (1997). [CrossRef] [PubMed]

5. I. Alexeev, K. Y. Kim, and H. M. Milchberg, “Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse,” Phys. Rev. Lett. **88**(7), 073901–073904 (2002). [CrossRef] [PubMed]

7. F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express **17**(12), 9804–9809 (2009). [CrossRef] [PubMed]

15. M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. Di Trapani, “Finite-energy, accelerating Bessel pulses,” Opt. Express **16**(24), 19807–19811 (2008). [CrossRef] [PubMed]

*ρ*,

*z*, and

*t*are the spatial (cylindrical) and temporal coordinates, and

*G*(

*ω–ω*) is the (Gaussian-like) spectrum of the pulse having a central frequency

_{0}*ω*

_{0}. However, unlike the case of the Bessel-X pulse, here the axicon angle depends on the propagation distance

*z*from the lens with the focal length

*f*as

*θ*(

*z*)

*=*arctan[|

*f*(

*f*-

*z*)

^{−1}| tan

*θ*

_{a}], where

*θ*

_{a}is the axicon angle without the lens. Because the group velocity of the wave-packet along the

*z*direction is given by v

_{g}

*= c*/cos(

*θ*) [1–3

3. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. **22**(5), 310–312 (1997). [CrossRef] [PubMed]

5. I. Alexeev, K. Y. Kim, and H. M. Milchberg, “Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse,” Phys. Rev. Lett. **88**(7), 073901–073904 (2002). [CrossRef] [PubMed]

7. F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express **17**(12), 9804–9809 (2009). [CrossRef] [PubMed]

**16**(24), 19807–19811 (2008). [CrossRef] [PubMed]

*f*is positive and decelerate if

*f*is negative. The approximations made in this approach are valid as long as the pulse duration

*τ*is much shorter than its characteristic time of flight given by

*f/c*. Considering our experimental parameters, which are given below, the phase fronts at the intersection (apex) region deviate from those of conical waves by less than 10

^{−5}of the wavelength, which is negligible.

*r*=

_{a}*|f|*tan(

*θ*) is the radius of the ring along which the integration is carried out by the polar coordinate

_{a}*ϕ*of the source points and the origin of the

*z*axis is in the plane of the ring. One advantage to this approach is that we can take into account aberrations in the lens and axicon. For example, chromatic aberrations can be modeled using a frequency-dependent ring radius

*r*(

_{a}*ω*). Also, this expression can be used for numerical calculations of the field under the conditions in which the previous approach is not valid, i.e., also outside of the apex region.

## 3. Experimental results

_{0}= 805 nm. The spot size of the laser beam was 4 mm (FWHM). A fused-silica axicon with an apex angle of 176° was used, which transforms plane wave pulses at

*λ*

_{0}= 805 nm into conical wave pulses (Bessel-X pulses) with

*θ*= 0.92°. We used lenses with focal lengths of + 153 mm and −152 mm. For convenience in the actual set-up, the axicon was mounted behind the lens in a lens tube, i.e., in reverse order of Fig. 1. So the two components effectively constituted a single thin phase element, whose transmission function does not depend on the ordering of components. However, the small distance between them (a few mm) was taken into account in our simulations.

_{a}*x*. For more details about the SEA TADPOLE device that we used, see reference [13

13. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express **16**(18), 13663–13675 (2008). [CrossRef] [PubMed]

*z*positions and for the decelerating Bessel pulse at nine positions. In all cases, we measured the complete spatiotemporal intensity and phase, but we show only the spatiotemporal intensities here, as this information is more interesting. Three of these measurements for each are shown in Fig. 2 and Fig. 4 . For comparison, numerical simulations were performed using Eq. (1) with the experimental parameters, and as seen in the figures, the two are in very good agreement.

*c*. The origin of our time axis can be considered as the location of the reference pulse if it propagated along the axis

*z*with the Bessel pulses. So, if the Bessel pulse were traveling at the speed of light, then, for each value of

*z*, its spatiotemporal intensity would be centered at the same time origin

*t*= 0 which is emphasized with the white bar in the figures. But it is easy to see in Figs. 2 and 3 that this is not the case. The superluminal group velocity and the pulse’s acceleration or deceleration are both apparent from the z-dependent shifts of the pulses relative the origin

*t*= 0. The time shifts were compared to theoretically calculated shift function and we found a good agreement, see Fig. 4.

*J*

_{0}. Thanks to the many measurable fringes we could accurately determine the mean radial wavelength Λ

*=*

_{B}*λ*

_{0}/sin

*θ*of the Bessel profile.

_{g}=

*c*/cos

*θ*, the distance-dependent group velocity can be found using the following equation: Equation (3) was used to estimate the group velocity of the measured pulses at various propagation distances, and these results are shown in Fig. 5 along with the theoretical values. The experimental values are in good agreement with our theoretical predictions, except for two points at

*z =*32 mm and 52 mm for the decelerating pulses. This discrepancy is likely due to the imperfect surface of the axicon as discussed below.

*Without*a lens, the propagation depth over which the Bessel-X pulses last, starts, in principle, at the tip of axicon and ends at

*z*≈

*w*/tan

*θ*, where

_{a}*w*is the radius of the input beam or aperture. Imperfections in our axicon reduce the distance over which the Bessel-X pulse maintains its perfect ring profile (let us call it the Bessel zone). At values of

*z*less than 50 mm, the profile distortions were caused by the slightly spherical tip of the axicon (see, e. g., [18

18. S. Akturk, B. Zhou, B. Pasquiou, M. Franco, and A. Mysyrowicz, “Intensity distribution around the focal regions of real axicons,” Opt. Commun. **281**(17), 4240–4244 (2008). [CrossRef]

*z*≈130 mm [8

**34**(15), 2276–2278 (2009). [CrossRef] [PubMed]

*z*

_{max}≈

*w*/tan

*θ*). At values of

_{a}*z >*130 mm, these deviations distort the phase-fronts of the interfering plane wave constituents of the conical wave to an extent that affects the shape of the central spot of the Bessel profile, but it does not noticeably affect the pulse's group velocity as established in [8

**34**(15), 2276–2278 (2009). [CrossRef] [PubMed]

19. D. Abdollahpour, P. Panagiotopoulos, M. Turconi, O. Jedrkiewicz, D. Faccio, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Long spatio-temporally stationary filaments in air using short pulse UV laser Bessel beams,” Opt. Express **17**(7), 5052–5057 (2009). [CrossRef] [PubMed]

20. J.-M. Manceau, A. Averchi, F. Bonaretti, D. Faccio, P. Di Trapani, A. Couairon, and S. Tzortzakis, “Terahertz pulse emission optimization from tailored femtosecond laser pulse filamentation in air,” Opt. Lett. **34**(14), 2165–2167 (2009). [CrossRef] [PubMed]

*z*≈30 mm to 80 mm. With the negative lens (decelerating Bessel pulses) the Bessel zone is greatly lengthened, and in principle to tens of kilometers starting at

*z ≈*70 mm. Therefore the two points at

*z =*32 mm and 52 mm in Fig. 5 deviate (as clearly seen in the right plot with expanded ordinate scale) from the theoretical curve because they are outside of the Bessel zone. Although we could not measure the decelerating pulse kilometers from the axicon, the Bessel ring pattern was observable by eye on the lab wall ~10 m after the axicon, and, at this point, the radial wavelength Λ

*had increased to about 1 mm.*

_{B}*e*of the maximum decreased by a factor of 1.6, from 23.0 μm to 14.8 μm, after 40 mm of propagation from z = 32 mm to 72 mm inside the Bessel zone. For the decelerating pulse, the spot size instead increased by a factor of 1.4, from 39.1 μm to 56.0 μm after 10 cm of propagation from

*z =*72 mm to 172 mm. This represents a much larger Rayleigh range than that of a Gaussian beam, which would only be 0.2 mm if the waist diameter were 14.8 μm or 1.5 mm if it were 39.1 μm.

## 5. Conclusions

*E*(

*x*,

*t*,

*z*) of ultrashort accelerating and decelerating Bessel pulses with micron spatial resolution and femtosecond temporal resolution using SEA TADPOLE. The field after a lens and axicon was described and modeled theoretically, and we used this model to analyze our experimental results. The features in the measured spatiotemporal profiles, including the ring spacings and the central spot sizes, were found to be in good agreement with our theoretical calculations and numerical simulations. We also measured the group velocities of the pulses along the propagation direction and observed their acceleration and deceleration. The accelerating Bessel pulse’s speed went from 1.0002

*c*after 3.2 cm of propagation to 1.0009

*c*at 7.2 cm and the decelerating Bessel pulse had a speed of 1.00007

*c*after 5.2 cm and 1.00003

*c*after 17.2 cm of propagation. The measured group velocities were also in good agreement with our theoretical calculations.

## Acknowledgements

## References and links

1. | H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., |

2. | P. Saari and K. Reivelt, “Evidence of |

3. | H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. |

4. | K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | I. Alexeev, K. Y. Kim, and H. M. Milchberg, “Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse,” Phys. Rev. Lett. |

6. | R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche´, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A |

7. | F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express |

8. | P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. |

9. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

10. | P. Saari, “Laterally accelerating airy pulses,” Opt. Express |

11. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

12. | Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

13. | P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express |

14. | P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, Measuring the spatio-temporal field of diffracting ultrashort pulses,” |

15. | M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. Di Trapani, “Finite-energy, accelerating Bessel pulses,” Opt. Express |

16. | P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express |

17. | P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express |

18. | S. Akturk, B. Zhou, B. Pasquiou, M. Franco, and A. Mysyrowicz, “Intensity distribution around the focal regions of real axicons,” Opt. Commun. |

19. | D. Abdollahpour, P. Panagiotopoulos, M. Turconi, O. Jedrkiewicz, D. Faccio, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Long spatio-temporally stationary filaments in air using short pulse UV laser Bessel beams,” Opt. Express |

20. | J.-M. Manceau, A. Averchi, F. Bonaretti, D. Faccio, P. Di Trapani, A. Couairon, and S. Tzortzakis, “Terahertz pulse emission optimization from tailored femtosecond laser pulse filamentation in air,” Opt. Lett. |

**OCIS Codes**

(320.0320) Ultrafast optics : Ultrafast optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 7, 2009

Revised Manuscript: August 4, 2009

Manuscript Accepted: August 4, 2009

Published: August 7, 2009

**Citation**

Heli Valtna-Lukner, Pamela Bowlan, Madis Lõhmus, Peeter Piksarv, Rick Trebino, and Peeter Saari, "Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets," Opt. Express **17**, 14948-14955 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14948

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### References

- H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and Applications (New Jersey: John Wiley & Sons Ltd, 2008).
- P. Saari and K. Reivelt, “Evidence of X-Shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997). [CrossRef]
- H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22(5), 310–312 (1997). [CrossRef] [PubMed]
- K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056611 (2002). [CrossRef]
- I. Alexeev, K. Y. Kim, and H. M. Milchberg, “Measurement of the superluminal group velocity of an ultrashort Bessel beam pulse,” Phys. Rev. Lett. 88(7), 073901–073904 (2002). [CrossRef] [PubMed]
- R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche´, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67(6), 063820–063825 (2003). [CrossRef]
- F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor,” Opt. Express 17(12), 9804–9809 (2009). [CrossRef] [PubMed]
- P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34(15), 2276–2278 (2009). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901–213904 (2007). [CrossRef]
- P. Saari, “Laterally accelerating airy pulses,” Opt. Express 16(14), 10303–10308 (2008). [CrossRef] [PubMed]
- P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]
- Z. L. Horváth and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(2), 026601–026611 (2001). [CrossRef] [PubMed]
- P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16(18), 13663–13675 (2008). [CrossRef] [PubMed]
- P. Bowlan, M. Lohmus, P. Piksarv, H. Valtna-Lukner, P. Saari, and R. Trebino, Measuring the spatio-temporal field of diffracting ultrashort pulses,” arXiv:0905.4381 (2009).
- M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. Di Trapani, “Finite-energy, accelerating Bessel pulses,” Opt. Express 16(24), 19807–19811 (2008). [CrossRef] [PubMed]
- P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14(24), 11892–11900 (2006). [CrossRef] [PubMed]
- P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15(16), 10219–10230 (2007). [CrossRef] [PubMed]
- S. Akturk, B. Zhou, B. Pasquiou, M. Franco, and A. Mysyrowicz, “Intensity distribution around the focal regions of real axicons,” Opt. Commun. 281(17), 4240–4244 (2008). [CrossRef]
- D. Abdollahpour, P. Panagiotopoulos, M. Turconi, O. Jedrkiewicz, D. Faccio, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Long spatio-temporally stationary filaments in air using short pulse UV laser Bessel beams,” Opt. Express 17(7), 5052–5057 (2009). [CrossRef] [PubMed]
- J.-M. Manceau, A. Averchi, F. Bonaretti, D. Faccio, P. Di Trapani, A. Couairon, and S. Tzortzakis, “Terahertz pulse emission optimization from tailored femtosecond laser pulse filamentation in air,” Opt. Lett. 34(14), 2165–2167 (2009). [CrossRef] [PubMed]

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