## Attosecond pulses with sophisticated spatio-spectral waveforms: spatio-spectral Airy and auto-focusing beams |

Optics Express, Vol. 19, Issue 22, pp. 21730-21738 (2011)

http://dx.doi.org/10.1364/OE.19.021730

Acrobat PDF (1621 KB)

### Abstract

We propose a scheme for producing attosecond pulses with sophisticated spatio-spectral waveforms. The profile of a seed attosecond pulse is modified and its central frequency is up-converted through interaction with an infrared pump pulse. The transverse profile of the infrared beam and a spatiotemporal shift between the seed and infrared pulses are used for manipulating the spatio-spectral waveform of the generated pulse beam. We present several examples of sophisticated isolated attosecond pulse beam generation, including spatio-spectral Airy beam that exhibits prismatic self-bending effect and a beam undergoing auto-focusing to a sub-micron spot without the need of a focusing lens or nonlinearity.

© 2011 OSA

## 1. Introduction

1. P. H. Bucksbaum, “The future of attosecond spectroscopy,” Science **317**, 766–769 (2007). [CrossRef] [PubMed]

2. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature **419**, 803–807 (2002). [CrossRef] [PubMed]

3. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature **414**, 509–513 (2001). [CrossRef] [PubMed]

4. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science **320**, 1614–1617 (2008). [CrossRef] [PubMed]

5. T. Pfeifer, L. Gallmann, M. J. Abel, P. M. Nagel, D. M. Neumark, and S. R. Leone, “Heterodyne mixing of laser fields for temporal gating of high-order harmonic generation,” Phys. Rev. Lett. **97**, 163901 (2006). [CrossRef] [PubMed]

6. F. Calegari, C. Vozzi, M. Negro, G. Sansone, F. Frassetto, L. Poletto, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Efficient continuum generation exceeding 200 eV by intense ultrashort two-color driver,” Opt. Lett. **34**, 3125–3127 (2009). [CrossRef] [PubMed]

7. P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett. **19**, 1870–1872 (1994). [CrossRef] [PubMed]

8. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science **314**, 443–446 (2006). [CrossRef] [PubMed]

9. M. J. Abel, T. Pfeifer, P. M. Nagel, W. Boutu, M. J. Bell, C. P. Steiner, D. M. Neumark, and S. R. Leone, “Isolated attosecond pulses from ionization gating of high-harmonic emission,” Chem. Phys. **366**, 9–14 (2009). [CrossRef]

10. A. S. Sandhu, E. Gagnon, A. Paul, I. Thomann, A. Lytle, T. Keep, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, “Generation of sub-optical-cycle, carrier-envelope-phase insensitive, extreme-uv pulses via nonlinear stabilization in a waveguide,” Phys. Rev. A **74**, 061803 (2006). [CrossRef]

11. F. Ferrari, F. Calegari, M. Lucchini, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “High-energy isolated attosecond pulses generated by above-saturation few-cycle fields,” Nat. Photonics **4**, 875–879 (2010). [CrossRef]

12. X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. **103**, 183901 (2009). [CrossRef] [PubMed]

13. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. **81**, 163–234 (2009). [CrossRef]

14. M. Forre, E. Mevel, and E. Constant, “Seeded attosecond-pulse generation in structured media: a road for attosecond optics,” Phys. Rev. A **83**, 021402 (2011). [CrossRef]

15. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

15. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**, 979–981 (2007). [CrossRef] [PubMed]

17. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express **16**, 12880–12891 (2008). [CrossRef] [PubMed]

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

19. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. **103**, 123902 (2009). [CrossRef] [PubMed]

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

20. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics **3**, 395–398 (2009). [CrossRef]

21. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

*U*∝

_{p}*Iλ*

^{2}, where

*I*and

*λ*are the laser intensity and wavelength, respectively [21

21. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

22. K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. **92**, 023003 (2004). [CrossRef] [PubMed]

24. K. L. Ishikawa, E. J. Takahashi, and K. Midorikawa, “Single-attosecond pulse generation using a seed harmonic pulse train,” Phys. Rev. A **75**, 021801 (2007). [CrossRef]

23. G. Gademann, F. Kelkensberg, W. K. Siu, P. Johnsson, M. B. Gaarde, K. J. Schafer, and M. J. J. Vrakking, “Attosecond control of electronion recollision in high harmonic generation,” New J. Phys. **13**, 033002 (2011). [CrossRef]

25. J. Biegert, A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, M. B. Gaarde, K. J. Schafer, and U. Keller, “Control of high-order harmonic emission using attosecond pulse trains,” J. Mod. Opt. **53**, 87–96 (2006). [CrossRef]

27. E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. **99**, 053904 (2007). [CrossRef] [PubMed]

28. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A **49**, 2117–2132 (1994). [CrossRef] [PubMed]

29. M. B. Gaarde, F. Salin, E. Constant, P. Balcou, K. J. Schafer, K. C. Kulander, and A. L’Huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A **59**, 1367–137 (1999). [CrossRef]

30. M. B. Gaarde and K. J. Schafer, “Space-time considerations in the phase locking of high harmonics,” Phys. Rev. Lett. **89**, 213901 (2002). [CrossRef] [PubMed]

31. N. Dudovich, J. L. Tate, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and M. B. Gaarde, “Subcycle spatial mapping of recollision dynamics,” Phys. Rev. A **80**, 011806 (2009). [CrossRef]

## 2. Scheme for producing sophisticated attosecond pulses

*eV*) (Fig. 1a). Here

*λ*

^{IR}= 2

*μm*,

*λ*

^{atto}= 30

*nm*and

*c*is the speed of light. The temporal envelopes,

*f*

^{IR}(

*t*) and

*f*

^{EUV}(

*t*–

*τ*) are Gaussians with full width at half maximum (FWHM) of 50

*fs*and 250

*as*, respectively, and peak intensities of 7.4 · 10

^{14}

*W*·

*cm*

^{−2}and 10

^{12}

*W*·

*cm*

^{−2}, respectively. The parameter

*τ*denotes the time delay between the pulses which in the first example is set to −3.06

*fs*(+15° after a peak of the mid-IR field). This time delay was chosen so electrons that are ionized by the attosecond pulse would emit the highest photon-energy upon recombination (the cut-off radiation). The motion of the electron under the influence of the mid-IR and EUV fields is calculated by solving the one dimensional time dependent Schrödinger equation for the potential

*V*(

*x*,

*t*) =

*V*

_{bind}(

*x*) +

*x*(

*E*+

^{IR}*E*). The ion binding potential is modeled by

^{EUV}*eV*(

*a*

_{0}is Bohrs radius) [32

32. J. Javanainen, J. H. Eberly, and Q. Su, “Numerical simulations of multiphoton ionization and above-threshold electron spectra,” Phys. Rev. A **38**, 3430–344 (1988). [CrossRef] [PubMed]

33. H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A **81**, 063813 (2010). [CrossRef]

^{−7}) whereas ionization is enhanced by the EUV field (Fig. 1a). The high-order radiation is calculated by the acceleration expectation value, using Ernfest theorem

*ψ*is the electronic wavefunction. As shown in Fig. 1b, the radiated spectral field highly resembles an Airy function,

*α*= 1600

*fs*

^{−3}is the chirp quadratic coefficient and

*ω*

_{cut off}= 930

*eV*is the cut-off frequency as predicted by the classical model [21

21. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

34. M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Staraceet, “Analytic formulae for high harmonic generation,” J. Phys. B **42**, 035601 (2009). [CrossRef]

*t*

_{cut-off}corresponds to the emission time of the cut-off radiation, and the amplitude,

*E*

_{0}, is time-independent because the seed EUV pulse populates uniformly the quantum trajectories that recombine around the cut-off time. Notably, the spectral field,

*Ẽ*(

*ω*), is real (Fig. 1b) because the emitted field in time domain obeys

*E*(

*t*–

*t*

_{cut-off}) =

*E*

^{*}(

*t*

_{cut-off}–

*t*) in the vicinity of

*t*

_{cut-off}.

*eV*). Figure 1c shows that the pulse-width increases when more lobs are included. This results from the

*π*phase jump between adjacent lobs of the Airy field. Moreover, as the number of spectral lobs increases, the pulse temporal shape becomes squarish (flat-top) (Fig. 1d). Finally, figures 1e and 1f show the pulse shapes for additional +0.1 and −0.1 femtoseconds time-delays, respectively (+5° and −5° shift in the phase of the mid-IR wave). For the large bandwidth case, the pulse shape exhibit a triangle (leading or trailing edges) pulses. The asymmetry emerges from uneven population of the quantum trajectories set to recombine before and after

*t*

_{cut-off}. The square and triangle pulses demonstrate a simple form for attosecond pulse shaping. More complicated pulses can be produced by using complex structures of ionizing pulses.

*nm*which is much smaller than the transverse variation scales of the incoming laser fields (several microns). Subsequently, we simulated the propagation of the produced pulse beam along the propagation axis, z, by solving the linear (refractive index is 1), paraxial, and absorption-free Helmholtz equation for each narrow spectral component (intensity is too weak for nonlinear effects and propagation distance is much smaller than the absorption length). Coherent super position of the spectral components gives the space-time dependence of the pulse at propagation distance z:

*x*) profiles of Gaussians with FWHM of 10

*μm*and 150

*μm*respectively. A lateral shift of

*x*

_{0}= 63.7

*μm*between the peaks of the pulses was introduced such that the EUV pulse resides on an approximately linear mid-IR intensity slope (zero 2

*derivative) (inset of Fig. 2a). Consequently, in this geometry the local cut-off frequency also grows linearly with*

^{nd}*x*. Thus, the spectral Airy of Fig. 1 becomes

*x*-dependent:

*ω*

_{0}is the cut-off at

*x*

_{0}and

*β*is the cut-off lateral slope. Indeed, the calculated intensity exhibits spatial and spectral Airy profile (Fig. 2a). In this pulse, each spectral component forms a spatial Airy beam (Fig. 2b) that propagates along a curved path (Fig. 2c). The dependences of the curvature, initial propagation angle and initial position of the peak on the photon-energy (Fig. 2d) disperse the spectral components in space. The consequence of this prismatic effect is shown in figure 3. Figures 3(a–c) show spatiotemporal profiles after 5

*mm*propagation, formed by a 10

*eV*band pass filter centered at 608

*eV*, 578

*eV*and 548

*eV*, respectively. As shown, each spectral region undergoes a different lateral shift. As a result, a narrow slit that is located downstream can be used as a tunable spectral filter of the attosecond pulse (Fig. 3d). A linear dependence of the spectral intensity vs. the slit central-position is shown in Fig. 3e. Figure 3 shows that spatio-spectral Airy pulse beam may be used for tunable manipulation and control of the spectrum of attosecond pulses or for spectrometer-less attosecond spectroscopy, where different segments of a sample are irradiated by an attosecond pulse of varying spectrum.

*nm*) source pulse is 10

*μm*wide and centered at the mid-IR (2

*μm*) trough (inset of Fig. 4a). The mid-IR intensity is given by

*A*= 1.4,

_{w}*A*= 0.4,

_{n}*σ*= 90

_{w}*μm*,

*σ*= 30

_{w}*μm*, and

*I*is the mid-IR intensity previously discussed. Figures 4a and 4b show the intensity and phase of the radiation, clearly exhibiting the parabolic profiles. The parabolic phase front leads to focusing of the produced pulse after 5

^{IR}*mm*. Figure 4c shows the propagation and focusing of the 929

*eV*photon-energy spectral component. The intricate focusing spot details and the extended Rayleigh range result from the

*π*phase jumps at the beam margins (see inset in Fig. 4b). Figure 4d shows the spatio-temporal profile of the focused pulse at the focal point. The pulse consists of a 250

*eV*spectral bandwidth that is centered at 929

*eV*and exhibit a sub-micron attosecond focal spot. The focusing distance and width can be manipulated by the profile of the mid-IR beam.

35. H. Kapteyn, O. Cohen, I. Christov, and M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science **317**, 775–778 (2007). [CrossRef] [PubMed]

36. O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. **99**, 053902 (2007). [CrossRef] [PubMed]

37. P. Sidorenko, M. Kozlov, A. Bahabad, T. Popmintchev, M. Murnane, H. Kapteyn, and O. Cohen, “Sawtooth grating-assisted phase-matching,” Opt. Express **18**, 22686–22692 (2010). [CrossRef] [PubMed]

38. T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A **106**, 10516–10521 (2009). [CrossRef] [PubMed]

38. T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A **106**, 10516–10521 (2009). [CrossRef] [PubMed]

## 3. Conclusions

## Acknowledgments

## References and links

1. | P. H. Bucksbaum, “The future of attosecond spectroscopy,” Science |

2. | M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature |

3. | M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature |

4. | E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science |

5. | T. Pfeifer, L. Gallmann, M. J. Abel, P. M. Nagel, D. M. Neumark, and S. R. Leone, “Heterodyne mixing of laser fields for temporal gating of high-order harmonic generation,” Phys. Rev. Lett. |

6. | F. Calegari, C. Vozzi, M. Negro, G. Sansone, F. Frassetto, L. Poletto, P. Villoresi, M. Nisoli, S. De Silvestri, and S. Stagira, “Efficient continuum generation exceeding 200 eV by intense ultrashort two-color driver,” Opt. Lett. |

7. | P. B. Corkum, N. H. Burnett, and M. Y. Ivanov, “Subfemtosecond pulses,” Opt. Lett. |

8. | G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science |

9. | M. J. Abel, T. Pfeifer, P. M. Nagel, W. Boutu, M. J. Bell, C. P. Steiner, D. M. Neumark, and S. R. Leone, “Isolated attosecond pulses from ionization gating of high-harmonic emission,” Chem. Phys. |

10. | A. S. Sandhu, E. Gagnon, A. Paul, I. Thomann, A. Lytle, T. Keep, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, “Generation of sub-optical-cycle, carrier-envelope-phase insensitive, extreme-uv pulses via nonlinear stabilization in a waveguide,” Phys. Rev. A |

11. | F. Ferrari, F. Calegari, M. Lucchini, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “High-energy isolated attosecond pulses generated by above-saturation few-cycle fields,” Nat. Photonics |

12. | X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett. |

13. | F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. |

14. | M. Forre, E. Mevel, and E. Constant, “Seeded attosecond-pulse generation in structured media: a road for attosecond optics,” Phys. Rev. A |

15. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

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

17. | J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

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

19. | P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. |

20. | T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics |

21. | P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. |

22. | K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. |

23. | G. Gademann, F. Kelkensberg, W. K. Siu, P. Johnsson, M. B. Gaarde, K. J. Schafer, and M. J. J. Vrakking, “Attosecond control of electronion recollision in high harmonic generation,” New J. Phys. |

24. | K. L. Ishikawa, E. J. Takahashi, and K. Midorikawa, “Single-attosecond pulse generation using a seed harmonic pulse train,” Phys. Rev. A |

25. | J. Biegert, A. Heinrich, C. P. Hauri, W. Kornelis, P. Schlup, M. P. Anscombe, M. B. Gaarde, K. J. Schafer, and U. Keller, “Control of high-order harmonic emission using attosecond pulse trains,” J. Mod. Opt. |

26. | A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B |

27. | E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. |

28. | M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A |

29. | M. B. Gaarde, F. Salin, E. Constant, P. Balcou, K. J. Schafer, K. C. Kulander, and A. L’Huillier, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A |

30. | M. B. Gaarde and K. J. Schafer, “Space-time considerations in the phase locking of high harmonics,” Phys. Rev. Lett. |

31. | N. Dudovich, J. L. Tate, Y. Mairesse, D. M. Villeneuve, P. B. Corkum, and M. B. Gaarde, “Subcycle spatial mapping of recollision dynamics,” Phys. Rev. A |

32. | J. Javanainen, J. H. Eberly, and Q. Su, “Numerical simulations of multiphoton ionization and above-threshold electron spectra,” Phys. Rev. A |

33. | H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A |

34. | M. V. Frolov, N. L. Manakov, T. S. Sarantseva, and A. F. Staraceet, “Analytic formulae for high harmonic generation,” J. Phys. B |

35. | H. Kapteyn, O. Cohen, I. Christov, and M. Murnane, “Harnessing attosecond science in the quest for coherent X-rays,” Science |

36. | O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. |

37. | P. Sidorenko, M. Kozlov, A. Bahabad, T. Popmintchev, M. Murnane, H. Kapteyn, and O. Cohen, “Sawtooth grating-assisted phase-matching,” Opt. Express |

38. | T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A |

**OCIS Codes**

(190.7220) Nonlinear optics : Upconversion

(320.7120) Ultrafast optics : Ultrafast phenomena

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: August 31, 2011

Revised Manuscript: September 25, 2011

Manuscript Accepted: September 25, 2011

Published: October 19, 2011

**Citation**

Ofer Kfir, Maxim Kozlov, Avner Fleischer, and Oren Cohen, "Attosecond pulses with sophisticated spatio-spectral waveforms: spatio-spectral Airy and auto-focusing beams," Opt. Express **19**, 21730-21738 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21730

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### References

- P. H. Bucksbaum, “The future of attosecond spectroscopy,” Science317, 766–769 (2007). [CrossRef] [PubMed]
- M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature419, 803–807 (2002). [CrossRef] [PubMed]
- M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature414, 509–513 (2001). [CrossRef] [PubMed]
- E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science320, 1614–1617 (2008). [CrossRef] [PubMed]
- T. Pfeifer, L. Gallmann, M. J. Abel, P. M. Nagel, D. M. Neumark, and S. R. Leone, “Heterodyne mixing of laser fields for temporal gating of high-order harmonic generation,” Phys. Rev. Lett.97, 163901 (2006). [CrossRef] [PubMed]
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