## Selective zoning of high harmonic emission using counter-propagating light

Optics Express, Vol. 1, Issue 5, pp. 114-125 (1997)

http://dx.doi.org/10.1364/OE.1.000114

Acrobat PDF (163 KB)

### Abstract

High harmonic production can be dramatically increased by utilizing an interaction region much longer than a coherence length. Counter-propagating light pulses can be used to disrupt the out-of-phase harmonic emission from selected zones in the focus so that the remaining emission builds constructively. Counter-propagating light creates a standing field modulation repeating over a half laser wavelength in which phase cancellations for harmonic emission occur. A simple power-law model is used to demonstrate how such pulses can be designed to counteract geometrical phase mismatches and improve emission for individual harmonics by more than two orders of magnitude.

© Optical Society of America

## 1. Introduction

^{1–21. L’Huillier and Ph. Balcou, “High-Order harmonic Generation in Rare Gases with a 1-ps 1053nm Laser,” Phys. Rev. Lett. 70, 774 (1993). [CrossRef] }The harmonics have become an attractive source for coherent vacuum ultraviolet radiation. However, in many cases the generated harmonics (in particular the highest orders) have so little energy per pulse as to make them difficult to register, and this limits the prospects of their usefulness. The efficiency of converting the laser light into the high-order harmonics under optimal conditions has reached about 10

^{-7}(25nm light)

^{33. Ditmire, K. Kulander, J. K. Crane, H. Nguyen, and M. D. Perry, “Calculation and Measurement of High-Order Harmonic Energy Yields in Helium,” J. Opt. Soc. of Am. B 13, 406.}and is often much less. Some improvements to the efficiency of converting the laser light into the high-order harmonics can be made by decreasing the laser pulse duration

^{44. Peatross, J. Zhou, I. Christov, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “High-Order Harmonic Generation with a 25 Femtosecond Laser Pulse,” in Proceedings of the NATO Advanced Research Workshop on Super Intense Laser-Atom Physics IV (Moscow, Russia1995) p. 455.}or by designing laser pulse temporal profiles to enhance the atomic response.

^{5–75. V. T. Platonenko, V. V. Strelkov, G. Ferrante, V. Miceli, and E. Fiordilino, “Control of the Spectral Width and Pulse Duration of a Single High-Order Harmonic,” Laser Phys. 6, p. 1164–1167 (1996).}The efficiency of converting laser light into high-order harmonics is seriously limited by several macroscopic mechanisms (other than the atomic response itself): 1. Geometrical phase mismatches arise from discrepancies between the diffraction rates for the laser and for individual harmonics; 2. The refractive index for laser light in the generating medium can differ from the refractive index for the harmonics (particularly severe when free electrons are present); 3. The intrinsic phase of harmonic emission can vary spatially via the atomic response to local laser intensity throughout the focus. All together, these phase mismatches can cause strong cancellations as harmonic light emerges from different locations in the laser focus.

^{2222. Shkolnikov, A. E. Kaplan, and A. Lago, “Phase-Matching Optimization of Large-Scale Nonlinear Frequency Upconversion in Neutral and Ionized Gases,” J. Opt. Soc. Am. B 13, 412 (1996). [CrossRef] }or angular tuning of crossing beams

^{2424. Birulin, V. T. Platonenko, and V. V. Strelkov, “High-Harmonic Generation in Interfering Waves,” JETP 83, 33 (1996).}are proposed to allow for phase matching in a positive dispersive medium (i.e., plasma).

13. Peatross, M. V. Fedorov, and K. C. Kulander, “Intensity-Dependent Phase-Matching Effects in Harmonic Generation,” J. Opt. Soc. Am. B **12**,863 (1995). [CrossRef]

13. Peatross, M. V. Fedorov, and K. C. Kulander, “Intensity-Dependent Phase-Matching Effects in Harmonic Generation,” J. Opt. Soc. Am. B **12**,863 (1995). [CrossRef]

^{2525. Peatross, J. Chaloupka, and D. D. Meyerhofer, “High-Order Harmonic Generation with an Annular Laser Beam,” Opt. Lett. 19, 942 (1994). [CrossRef] [PubMed] }When an annular beam is focused, the center fills in to produce a central peak similar to that of a usual laser focus, surrounded by faint rings which do not contribute to harmonic production. Much of the harmonic energy emerges close to the laser axis so that it can pass inside of the hole. The hole is necessary because of the lack of suitable materials able to transmit vacuum ultraviolet light.

## 2. Microscopic phase disruption in counter-propagating pulses

_{1}and E

_{2}which propagate in opposite directions. We will assume that E

_{1}is the larger of the fields and write their sum as a single field having the form of the stronger plane wave, although with a standing intensity and phase modulation:

_{2}≪ E

_{1}), the total field reduces to that of the stronger plane wave (i.e., E

_{t}(z)→ E

_{1}; ϕ(z)→ 0). Nevertheless, even a relatively weak field E

_{2}can cause significant standing modulations. The fringe visibility for the standing intensity modulation is given by

_{2}is only a tenth as strong as E

_{1}(i.e., 100 times less intense), the standing intensity pattern has a fringe visibility of 0.20. In this case, I

_{max}is 49% more intense than I

_{min}. Fig. 2 shows the standing intensity profile for this case. By intensity, we do not refer to net energy flow (i.e., Poynting vector), but rather to the square of the combined field amplitude (i.e., ε

_{o}c

^{th}order. As seen in Fig. 2, both the standing intensity modulation and the standing phase modulation are periodic over a half laser wavelength.

_{2}= 0). To perform this calculation, we invoke a simple model: The strength of the q

^{th}harmonic is assumed to follow the laser field strength raised to the p

^{th}power. That is, the field emission from individual atoms goes as

_{2}is zero. By effective emission, we mean the apparent response of the atoms in the interval as though the phase matching had not been disrupted. We therefore define a microscopic phase-mismatch factor as

_{t}(z) is even and ϕ(z) is odd over the interval of integration. Although Eq. (7) is defined in terms of the power law model used in the present work, the definition can be generalized to accommodate other models including results of numerical simulations.

## 3. Geometrical phase mismatches

_{q}~

^{88. L’Huillier, K. J. Schafer, and K. C. Kulander, “Theoretical Aspects of Intense Field Harmonic Generation,” J. Phys. B: At. Mol. Opt. Phys. 24, 3315 (1991). [CrossRef] }Similarly, the phase matching calculation for the arbitrary power-law model used in the present work (i.e., E

_{q}~

^{1212. Peatross and D. D. Meyerhofer, “Intensity-Dependent Phase Effects in High-Order Harmonic Generation,” Phys. Rev. A 52, 3976–3987 (1995). [CrossRef] [PubMed] }The integral which describes the field of individual harmonics at a screen far from the laser focus is

_{o}is the laser Rayleigh range, and θ is the angle from the axis to a point on the screen. Note that 2f

^{#}θ is a parameter which compares the angle of the harmonic emission to that of the laser beam. We have included the microscopic phase-mismatch factor ξ(T, z) which is identically one in the absence of counter-propagating pulses. T is a time parameter that refers to a specific point on the temporal envelope of the generating pulse which moves through the focus. In Eq. (8), we have neglected proportionality factors since we are interested only in changes in harmonic production when counter-propagating pulses are used.

^{st}harmonic with p chosen to be 5. The calculation is for that light which hits the screen on axis (i.e., the figure shows the integrand of Eq. (8) with θ = 0, N = 1, and ξ = 1). The phase changes by about 25π over the range -z

_{o}< z < z

_{o}. Fig. 4(b) shows the real part of the emission to emphasize the strong effect of the varying phase. The curve in Fig. 4(b) may be interpreted as the electric field contributions from different positions in the focus arriving simultaneously and interfering at the center of the distant screen. Since these contributions arrive at the screen together, it is apparent that they are produced at different laboratory times as a single temporal point T of the laser pulse moves through the focus. When the contributions are summed (i.e., the integral of Eq. (8) is taken), the strongly varying phase takes a serious toll on the overall harmonic emission. To counteract this effect, counter-propagating pulses can be chosen with appropriate timing and durations so that the microscopic factor ξ(z,T) tends towards zero in regions with undesirable phase. Alternatively, the gas distribution can be restricted to a coherence length (phase interval of π) to minimize phase cancellations, as is the usual practice, but this makes only limited use of the focal region.

^{-1}z/z

_{o}in Eq. (8) is sufficient since it is by far the most rapidly varying phase term. The derivative of this phase term is

_{o}must be chosen depending on the selected position z such that

_{P}will be shorter than the coherence length L

_{C}whenever we have

^{1010. Ph. Balcou and A. L’Huillier, “Phase-Matching Effects in Strong-Field Harmonic Generation,” Phys. Rev. A 47, 1447–1459 (1993). [CrossRef] [PubMed] ,1414. Salieres, A. L’Huillier, and M. Lewenstein, “Coherence Control of High-Order Harmonics,” Phys. Rev. Lett. 74, 3776 (1995). [CrossRef] [PubMed] }since the reports refer to experiments made with fixed f-number and higher-than necessary intensity in the focus.

## 4. Simulation of harmonic production in a focus with counter-propagating light

_{o}to +z

_{o}as a function of θ (blue). The gas density was taken to be uniform over this region, as might be the case for a gas cell. The plot shows intensity (not field) at the screen for the instant associated with T=0 (the laboratory time when the peak of the laser pulse hits the screen). Other values for T yield similar pictures (assuming a main pulse which is shorter than individual counter-propagating peaks).

_{o}, a geometrical coherence length. As is evident, restriction of the gas dimension represents a substantial improvement over the case shown by the green line. However, the use of counter-propagating light increases the intensity over the narrow jet case by a factor of 40. Moreover, slightly off axis the intensity has been increased by many orders of magnitude. To compare over all conversion efficiency, it is necessary to sum the intensity over the area on the screen:

## 5. Summary and discussion

## References and links

1. | L’Huillier and Ph. Balcou, “High-Order harmonic Generation in Rare Gases with a 1-ps 1053nm Laser,” Phys. Rev. Lett. |

2. | Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, “Enhanced High-Harmonic Generation Using 25 Femtosecond Laser Pulses,” Phys. Rev. Lett. |

3. | Ditmire, K. Kulander, J. K. Crane, H. Nguyen, and M. D. Perry, “Calculation and Measurement of High-Order Harmonic Energy Yields in Helium,” J. Opt. Soc. of Am. B |

4. | Peatross, J. Zhou, I. Christov, A. Rundquist, M. M. Murnane, and H. C. Kapteyn, “High-Order Harmonic Generation with a 25 Femtosecond Laser Pulse,” in |

5. | V. T. Platonenko, V. V. Strelkov, G. Ferrante, V. Miceli, and E. Fiordilino, “Control of the Spectral Width and Pulse Duration of a Single High-Order Harmonic,” Laser Phys. |

6. | Fiordilino and V. Miceli, “Laser Pulse Shape Effects in Harmonic Generation from a Two-Level Atom,” J. Mod. Opt. |

7. | Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, “Quantum Control of Wave Packet Evolution with Tailored Femtosecond Pulses,” Phys. Rev. Lett. |

8. | L’Huillier, K. J. Schafer, and K. C. Kulander, “Theoretical Aspects of Intense Field Harmonic Generation,” J. Phys. B: At. Mol. Opt. Phys. |

9. | L’Huillier, Ph. Balcou, S. Candel, K. J. Schafer, and K. C. Kulander, “Calculations of High-Order Harmonic-Generation Processes in Xenon at 1064 nm,” Phys. Rev. A |

10. | Ph. Balcou and A. L’Huillier, “Phase-Matching Effects in Strong-Field Harmonic Generation,” Phys. Rev. A |

11. | Peatross and D. D. Meyerhofer, “The Angular Distribution of High-Order Harmonics Emitted from Rare Gases at Low Density,” Phys. Rev. A |

12. | Peatross and D. D. Meyerhofer, “Intensity-Dependent Phase Effects in High-Order Harmonic Generation,” Phys. Rev. A |

13. | Peatross, M. V. Fedorov, and K. C. Kulander, “Intensity-Dependent Phase-Matching Effects in Harmonic Generation,” J. Opt. Soc. Am. B |

14. | Salieres, A. L’Huillier, and M. Lewenstein, “Coherence Control of High-Order Harmonics,” Phys. Rev. Lett. |

15. | Wahlstrom, J. Larsson, A. Persson, T. Starczewski, and S. Svanberg, “High-Order Harmonic Generation in Rare Gases with an Intense Short-Pulse Laser,” Phys. Rev. A. |

16. | Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, and A. L’Huillier, “Influence of Atomic Density in High-Order Harmonic Generation,” J. Opt. Soc. Am. B |

17. | Lynga, A. L’Huillier, and C.-G. Wahlstrom, “High-Order Harmonic Generation in Molecular Gases,” J. Phys. B |

18. | T. Ditmire, J. W. G. Tisch, D. J. Fraser, J. P. Marangos, N. Hay, M. H. R. Hutchinson, T. Donnelly, R. W. Falcone, and M. D. Perry, “High-Order Harmonic Generation in Large Molecules and Atomic Clusters,” in |

19. | Wahlstrom, S. Borgstrom, J. Larsson, and S.-G. Pettersson, “High-Order Harmonic Generation in Laser-Produced Ions Using a Near-Infrared Laser,” Phys. Rev. A |

20. | Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, M. Bakarezos, C. N. Danson, A. Dyson, P. Gibbon, P. Loukakos, D. Neely, F. N. Walsh, J. S. Wark, and A. E. Dangor, “Efficient Extreme UV Harmonics Generated from Picosecond Laser Pulse Interactions with Solid Targets,” Phys. Rev. Lett. |

21. | H. M. Milchberg, C. G. Durfee, and T. J. McIlrath, “High-Order Frequency Conversion in the Plasma Waveguide,” Phys. Rev. Lett. |

22. | Shkolnikov, A. E. Kaplan, and A. Lago, “Phase-Matching Optimization of Large-Scale Nonlinear Frequency Upconversion in Neutral and Ionized Gases,” J. Opt. Soc. Am. B |

23. | Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. |

24. | Birulin, V. T. Platonenko, and V. V. Strelkov, “High-Harmonic Generation in Interfering Waves,” JETP |

25. | Peatross, J. Chaloupka, and D. D. Meyerhofer, “High-Order Harmonic Generation with an Annular Laser Beam,” Opt. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(190.4160) Nonlinear optics : Multiharmonic generation

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 13, 1997

Revised Manuscript: August 4, 1997

Published: September 1, 1997

**Citation**

Justin Peatross, Sergei Voronov, and I. Prokopovich, "Selective zoning of high harmonic emission using counter-propagating light," Opt. Express **1**, 114-125 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-5-114

Sort: Journal | Reset

### References

- LHuillier and Ph. Balcou, "High-Order harmonic Generation in Rare Gases with a 1-ps 1053nm Laser," Phys. Rev. Lett. 70, 774 (1993). [CrossRef]
- Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, Enhanced High-Harmonic Generation Using 25 Femtosecond Laser Pulses, Phys. Rev. Lett. 76, 752 (1996). [CrossRef] [PubMed]
- Ditmire, K. Kulander, J. K. Crane, H. Nguyen, M. D. Perry, "Calculation and Measurement of High-Order Harmonic Energy Yields in Helium," J. Opt. Soc. of Am. B 13, 406.
- Peatross, J. Zhou, I. Christov, A. Rundquist, M. M. Murnane, H. C. Kapteyn, High-Order Harmonic Generation with a 25 Femtosecond Laser Pulse, in Proceedings of the NATO Advanced Research Workshop on Super Intense Laser-Atom Physics IV (Moscow, Russia 1995) p. 455.
- V. T. Platonenko, V. V. Strelkov, G. Ferrante, V. Miceli, E. Fiordilino, "Control of the Spectral Width and Pulse Duration of a Single High-Order Harmonic," Laser Phys. 6, p. 1164-1167 (1996).
- Fiordilino and V. Miceli, "Laser Pulse Shape Effects in Harmonic Generation from a Two-Level Atom," J. Mod. Opt. 41, 1415-1426 (1994). [CrossRef]
- Kohler, V. V. Yakovlev, Jianwei Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Yijing Yan, "Quantum Control of Wave Packet Evolution with Tailored Femtosecond Pulses," Phys. Rev. Lett. 74, 3360-3363 (1995). [CrossRef] [PubMed]
- LHuillier, K. J. Schafer, and K. C. Kulander, "Theoretical Aspects of Intense Field Harmonic Generation," J. Phys. B: At. Mol. Opt. Phys. 24, 3315 (1991). [CrossRef]
- L'Huillier, Ph. Balcou, S. Candel, K. J. Schafer, and K. C. Kulander, "Calculations of High-Order Harmonic-Generation Processes in Xenon at 1064 nm," Phys. Rev. A 46, 2778-2790 (1992). [CrossRef]
- Ph. Balcou and A. L'Huillier, "Phase-Matching Effects in Strong-Field Harmonic Generation," Phys. Rev. A 47, 1447-1459 (1993). [CrossRef] [PubMed]
- Peatross and D. D. Meyerhofer, "The Angular Distribution of High-Order Harmonics Emitted from Rare Gases at Low Density," Phys. Rev. A 51, R906 (1995). [CrossRef] [PubMed]
- Peatross and D. D. Meyerhofer, "Intensity-Dependent Phase Effects in High-Order Harmonic Generation," Phys. Rev. A 52, 3976-3987 (1995). [CrossRef] [PubMed]
- Peatross, M. V. Fedorov, K. C. Kulander, "Intensity-Dependent Phase-Matching Effects in Harmonic Generation," J. Opt. Soc. Am. B 12,863 (1995). [CrossRef]
- Salieres, A. L'Huillier, and M. Lewenstein, "Coherence Control of High-Order Harmonics," Phys. Rev. Lett. 74, 3776 (1995). [CrossRef] [PubMed]
- Wahlstrom, J. Larsson, A. Persson, T. Starczewski, and S. Svanberg, "High-Order Harmonic Generation in Rare Gases with an Intense Short-Pulse Laser," Phys. Rev. A. 48, 4709 (1993). [CrossRef] [PubMed]
- Altucci, T. Starczewski, E. Mevel, C.-G. Wahlstrom, B. Carre, A. L`Huillier, "Influence of Atomic Density in High-Order Harmonic Generation," J. Opt. Soc. Am. B 13, 148 (1996). [CrossRef]
- Lynga, A. L'Huillier, C.-G. Wahlstrom, "High-Order Harmonic Generation in Molecular Gases," J. Phys. B 29, 3293 (1996). [CrossRef]
- T. Ditmire, J. W. G. Tisch, D. J. Fraser, J. P. Marangos, N. Hay, M. H. R. Hutchinson, T. Donnelly, R. W. Falcone, M. D. Perry, "High-Order Harmonic Generation in Large Molecules and Atomic Clusters," in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1996) p. 544.
- Wahlstrom, S. Borgstrom, J. Larsson, and S.-G. Pettersson, "High-Order Harmonic Generation in Laser-Produced Ions Using a Near-Infrared Laser," Phys. Rev. A 51, 585 (1995). [CrossRef] [PubMed]
- Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, M. Bakarezos, C. N. Danson, A. Dyson, P. Gibbon, P. Loukakos, D. Neely, F. N. Walsh, J. S. Wark, A. E. Dangor, "Efficient Extreme UV Harmonics Generated from Picosecond Laser Pulse Interactions with Solid Targets," Phys. Rev. Lett. 76, 1832 (1996). [CrossRef] [PubMed]
- H. M. Milchberg, C. G. Durfee, T. J. McIlrath, High-Order Frequency Conversion in the Plasma Waveguide, Phys. Rev. Lett. 75, 2494 (1995). [CrossRef] [PubMed]
- Shkolnikov, A. E. Kaplan, and A. Lago, "Phase-Matching Optimization of Large-Scale Nonlinear Frequency Upconversion in Neutral and Ionized Gases," J. Opt. Soc. Am. B 13, 412 (1996). [CrossRef]
- Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 127, 1918 (1962). [CrossRef]
- Birulin, V. T. Platonenko, and V. V. Strelkov, "High-Harmonic Generation in Interfering Waves," JETP 83, 33 (1996).
- Peatross, J. Chaloupka, and D. D. Meyerhofer, "High-Order Harmonic Generation with an Annular Laser Beam," Opt. Lett. 19, 942 (1994). [CrossRef] [PubMed]

## 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.