## Time-frequency representations of high order harmonics

Optics Express, Vol. 8, Issue 10, pp. 529-536 (2001)

http://dx.doi.org/10.1364/OE.8.000529

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### Abstract

We calculate time-frequency representations (TFRs) of high-order short pulse harmonics generated in the interaction between neon atoms and an intense laser field, including macroscopic effects of propagation and phase matching in the non-linear medium. The phase structure of the harmonics is often complicated and the TFR can help to resolve the different components of this structure. The harmonic pulses exhibit an overall negative chirp, which can be attributed in part to the intensity dependence of the harmonic dipole phase. In some cases, the harmonic field separates in the time-frequency domain and clearly exhibits two different chirps. We also compute an experimental realization of a TFR (using Frequency Resolved Optical Gating, FROG) for a high harmonic. Due to the complicated time structure of the harmonics, the FROG trace is visually complex.

© Optical Society of America

## 1 Introduction

1. For a review, seeA. L’Huillier*et al*, “High-order harmonics: A coherent source in the XUV range,” J. of Nonl. Opt. Phys. and Mat. **4**, 647 (1995). [CrossRef]

2. M. Gisselbrecht*et al*, “Absolute photoionization cross sections of excited He states in the near-threshold region,” Phys. Rev. Lett. **82**, 4607 (1999). [CrossRef]

3. Z. Chang*et al*., “Generation of coherent soft X rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. **79**, 2967 (1997). [CrossRef]

4. C.-G. Wahlström*et al*., “High-order harmonic generation in rare gases with an intense short-pulse laser,” Phys. Rev. A **48**, 4709 (1993). [CrossRef] [PubMed]

3. Z. Chang*et al*., “Generation of coherent soft X rays at 2.7 nm using high harmonics,” Phys. Rev. Lett. **79**, 2967 (1997). [CrossRef]

5. R. Trebino*et al*, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**, 3277 (1997). [CrossRef]

6. B. Sheehy*et al*, “High Harmonic Generation at Long Wavelengths,” Phys. Rev. Lett. **83**, 5270 (1999). [CrossRef]

7. T. Sekikawa*et al*, “Pulse Compression of a High-Order Harmonic by Compensating the Atomic Dipole Phase”, Phys. Rev. Lett. **83**, 2564 (1999). [CrossRef]

8. M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A **52**, 4747 (1995). [CrossRef] [PubMed]

9. M. B. Gaarde*et al*, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A **59**, 1367 (1999). [CrossRef]

^{14}W/cm

^{2}and a pulse length of 150–200 fs. The 45th harmonic represents a typical harmonic which is deep in the plateau region of the harmonic spectrum, whereas the 89th harmonic is representative of the cutoff region. We first calculate spectograms for the single atom harmonic emission. For the 45th harmonic we also calculate the TFR for the fully phase matched harmonic pulse emitted from a macroscopic collection of atoms after propagation through a gas jet. Both the single atom and the propagated 45th harmonic exhibit an overall negative chirp. The propagated harmonic also has several (weaker) chirp components to it. We can spatially resolve the time-frequency behavior, and find that at different radii in the farfield profile, the coherence properties of the harmonic pulse are very different. We also calculate a polarization gate FROG trace (see for instance [5

5. R. Trebino*et al*, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**, 3277 (1997). [CrossRef]

## 2 Theoretical approach

*E*(

*t*) is a simultaneous representation of the temporal and spectral characteristics of the pulse. We define it here as:

*ω*is the frequency and

*τ*is a time delay which defines the position of the window or gate function

*W*(

*t*-

*τ*). For every

*τ, S*(

*ω*,

*τ*) represents the instantaneous spectrum of the pulse at the time specified by

*W*(

*t*-

*τ*). As the window function slides through all delays the TFR trace illustrates how the spectrum of the pulse changes with time. As the electric field

*E*(

*t*) we use either the single atom or the macroscopic harmonic pulse. The window function is a Gaussian with a full width at half maximum (FWHM) duration which is approximately 65% of the length of the harmonic pulse.

*q*th harmonic is found by multiplying the spectrum with a window function centered around

*qω*

_{0}, where

*ω*

_{0}is the driving frequency, and Fourier transforming into the time domain.

11. Ph. Antoine*et al*, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A **53**, 1725 (1995). [CrossRef]

*S*(

_{p}*ω*,

*τ*) is calculated for each radial point in the farfield harmonic profile

*E*(

*r*,

*t*) and integrated over the radial coordinate:

## 3 Results

^{14}W/cm

^{2}. The driving field has a wavelength of 810 nm. The TFR traces the instantaneous value of the harmonic frequency (shown in units of the driving frequency,

*ω*

_{0}, relative to a zero point at the harmonic frequency), which changes almost linearly with time from a positive to a negative value. Both harmonics exhibit negative linear chirps, and the chirp of the 45th harmonic is much larger than that of the 89th harmonic. In the following we first concentrate on the 45th harmonic. The FWHM duration of the harmonic pulse can be found from the TFR trace to be approximately 25 fs. The Fourier transform limited FWHM bandwidth of a 25 fs pulse is 18 THz, which corresponds to 0.05

*ω*

_{0}. The presence of the negative chirp broadens the spectral profile to 0.3ω0. Both the time profile and the spectrum are somewhat asymmetric, with more weight in the latter part of the pulse and thereby - via the negative chirp - on the low frequency side of the spectrum. The time on the x-axis is really a measure of the delay between the harmonic pulse and the window function. In this case it also measures the time profile of the pulse relative to

*t*=0, since the position of the peak of the window function is well known. In an experiment, an absolute time scale can often not be defined.

8. M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A **52**, 4747 (1995). [CrossRef] [PubMed]

12. Ph. Balcou*et al*, “Quantum-path analysis and phase matching of high-order frequency mixing processes in strong laser fields,” J. Phys. B **32**, 2973 (1999). [CrossRef]

*i*is linearly proportional to the intensity with a proportionality constant -

*α*. These phases each give rise to a negative chirp. Since the driving laser pulse intensity is approximately quadratic in time close to its peak, these negative chirps are approximately linear during the time when the harmonic is generated. The chirp rate

_{i}*β*for each of these negative linear chirps is proportional to

_{i}*α*, the peak intensity

_{i}*I*

_{0}, and 1/

*T*

^{2}, where

*T*is the pulse length of the driving pulse [9

9. M. B. Gaarde*et al*, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A **59**, 1367 (1999). [CrossRef]

^{2}envelope, which gives the following expression for

*β*:

_{i}12. Ph. Balcou*et al*, “Quantum-path analysis and phase matching of high-order frequency mixing processes in strong laser fields,” J. Phys. B **32**, 2973 (1999). [CrossRef]

9. M. B. Gaarde*et al*, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A **59**, 1367 (1999). [CrossRef]

*ω*

_{0}/fs. This corresponds to a proportionality constant

*α*of 40×10

^{-14}cm

^{2}/W. This finding is typical for harmonics in neon calculated within the SAE approximation. The semi-classical model predicts the quantum path with α≈25×10

^{-14}cm

^{2}/W to be by far the most dominant [12

12. Ph. Balcou*et al*, “Quantum-path analysis and phase matching of high-order frequency mixing processes in strong laser fields,” J. Phys. B **32**, 2973 (1999). [CrossRef]

*α*≈13×10

^{-14}cm

^{2}/W, dominates the cutoff harmonics. The time-frequency behavior of these harmonics is in general simpler than the plateau harmonics, due to the absence of interference between different quantum path contributions.

^{14}W/cm

^{2}and a pulse length of 150 fs. The laser focus is long compared to the gas jet, with a confocal parameter of 5 mm.

*ω*

_{0}/fs and is the cause of the large spectral bandwidth of 0.4

*ω*

_{0}. Through Eq.(3) and using

*α*=40×10

^{-14}cm

^{2}/W, this corresponds to an “average” peak intensity experienced by the atoms which is between 3 and 4×10

^{14}W/cm

^{2}. This reflects the fact that due to bad phase matching conditions, the radiation that exits the medium often does not originate from the center of the laser focus [13

13. P. Salières*et al*, “Studies of the spatial and temporal coherence of high order harmonics,” Adv. At. Mol. Opt. Phys. **41**, 83 (1999). [CrossRef]

5. R. Trebino*et al*, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**, 3277 (1997). [CrossRef]

*E*(

*t*-

*τ*)|

^{2}of the harmonic pulse is used as the window function. Since the time-structure of the harmonic pulse is very complicated, as demonstrated in Fig.4, the FROG trace looks very non-intuitive and does not indicate the negative chirp found above. The full information about the time-frequency behavior of the radially integrated harmonic pulse can of course be obtained from the FROG trace using a phase retrieval algorithm.

14. M. Bellini*et al*, “Temporal coherence of ultrashort high-order harmonic pulses,” Phys. Rev. Lett. **81**, 297 (1998). [CrossRef]

*et al*, “Spatiotemporal separation of high harmonic radiation into two quantum path components,” Phys. Rev. A **59**, 1367 (1999). [CrossRef]

^{-14}cm

^{2}/W in the semi-classical model [12

*et al*, “Quantum-path analysis and phase matching of high-order frequency mixing processes in strong laser fields,” J. Phys. B **32**, 2973 (1999). [CrossRef]

*et al*, “Quantum-path analysis and phase matching of high-order frequency mixing processes in strong laser fields,” J. Phys. B **32**, 2973 (1999). [CrossRef]

14. M. Bellini*et al*, “Temporal coherence of ultrashort high-order harmonic pulses,” Phys. Rev. Lett. **81**, 297 (1998). [CrossRef]

15. C. Lyngå*et al*, “Studies of the temporal coherence of high-order harmonics,” Phys. Rev. A **60**, 4823 (1999). [CrossRef]

## 4 Summary

*separates*in the spatial domain. Very close to the axis of propagation we find harmonic radiation which has only a very small chirp and therefore a narrow spectrum. Far from the propagation axis we find radiation that has a well defined linear chirp, and which could therefore possibly be recompressed to a very short duration.

## References and links

1. | For a review, seeA. L’Huillier |

2. | M. Gisselbrecht |

3. | Z. Chang |

4. | C.-G. Wahlström |

5. | R. Trebino |

6. | B. Sheehy |

7. | T. Sekikawa |

8. | M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A |

9. | M. B. Gaarde |

10. | K. C. Kulander, K. J. Schafer, and J. L. Krause, in |

11. | Ph. Antoine |

12. | Ph. Balcou |

13. | P. Salières |

14. | M. Bellini |

15. | C. Lyngå |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 19, 2001

Published: May 7, 2001

**Citation**

Mette Gaarde, "Time-frequency representations of high order harmonics," Opt. Express **8**, 529-536 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-10-529

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### References

- For a review, see A. L'Huillier et al, "High-order harmonics: A coherent source in the XUV range," J. of Nonl. Opt. Phys. and Mat. 4, 647 (1995). [CrossRef]
- M. Gisselbrecht et al, "Absolute photoionization cross sections of excited He states in the near-threshold region," Phys. Rev. Lett. 82, 4607 (1999). [CrossRef]
- Z. Chang et al., "Generation of coherent soft X rays at 2.7 nm using high harmonics," Phys. Rev. Lett. 79, 2967 (1997). ; R. Bartels et al., Nature 406, 164 (2000). [CrossRef]
- C.-G. Wahlstr�m et al., "High-order harmonic generation in rare gases with an intense short-pulse laser," Phys. Rev. A 48, 4709 (1993). [CrossRef] [PubMed]
- R. Trebino et al, "Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating," Rev. Sci. Instrum. 68, 3277 (1997). [CrossRef]
- B. Sheehy et al, "High Harmonic Generation at Long Wavelengths," Phys. Rev. Lett. 83, 5270 (1999). [CrossRef]
- T. Sekikawa et al, "Pulse Compression of a High-Order Harmonic by Compensating the Atomic Dipole Phase," Phys. Rev. Lett. 83, 2564 (1999). [CrossRef]
- M. Lewenstein, P. Sali�res, and A. L'Huillier, "Phase of the atomic polarization in high-order harmonic generation," Phys. Rev. A 52, 4747 (1995). [CrossRef] [PubMed]
- M. B. Gaarde et al, "Spatiotemporal separation of high harmonic radiation into two quantum path components," Phys. Rev. A 59, 1367 (1999). [CrossRef]
- K. C. Kulander, K. J. Schafer, and J. L. Krause, in Atoms in Intense Radiation Fields, Ed. M. Gavrila (Academic Press, New York, 1992).
- Ph. Antoine et al, "Theory of high-order harmonic generation by an elliptically polarized laser field," Phys. Rev. A 53, 1725 (1995). [CrossRef]
- Ph. Balcou et al, "Quantum-path analysis and phase matching of high-order frequency mixing processes in strong laser fields," J. Phys. B 32, 2973 (1999). [CrossRef]
- P. Sali�res et al, "Studies of the spatial and temporal coherence of high order harmonics," Adv. At. Mol. Opt. Phys. 41, 83 (1999). [CrossRef]
- M. Bellini et al, "Temporal coherence of ultrashort high-order harmonic pulses," Phys. Rev. Lett. 81, 297 (1998). [CrossRef]
- C. Lyng� et al, "Studies of the temporal coherence of high-order harmonics," Phys. Rev. A 60, 4823 (1999). [CrossRef]

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