## Energy distribution of relativistic electrons in the tunneling ionization of atoms by super-intense laser radiation

Optics Express, Vol. 2, Issue 7, pp. 268-270 (1998)

http://dx.doi.org/10.1364/OE.2.000268

Acrobat PDF (117 KB)

### Abstract

We studied theoretically the influence of relativistic effects on the energy distribution of electrons in the tunneling ionization of atoms by a field of linearly polarized super-intense laser radiation. It was shown that the energy distribution of ejected electrons is determined by relativistic law though the electron kinetic energy can be less than its rest energy. The relativistic probability of ionization along the field strength decreases exponentially with the electron kinetic energy, but more quickly than in the non-relativistic case.

© Optical Society of America

1. N.B. Delone and V.P. Krainov, *Atoms in Strong Light Fields* (Springer, Berlin-Heidelberg1985). [CrossRef]

*E*is the binding energy of the initial atomic state, and

_{i}*E*(

_{f}*t*) is the energy of the final continuum state taking into account the field of laser radiation. Here and below, the atomic system of units is used, as a rule, where

*e*=

*m*=

_{e}*ħ*= 1 and the speed of light is

*c*= 137.02 . Finally,

*E*(

_{f}*t*) = -

*E*.

_{i}*ħω*<<

*E*. Eq. (1) is correct as well as in the relativistic theory, since the classical action

_{i}*S*=

*P*is relativistic invariant quantity, and the coordinate part of this action influences only the pre-exponential factor in the ionization rate, Eq. (1). We neglect this factor both in the non-relativistic energy distribution of ejected electrons [2

_{i}dx^{i}2. N.B. Delone and V.P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation”, J. Opt. Soc. Am. B **8**, 1207–1211 (1991). [CrossRef]

*E*(

_{f}*t*) of a free electron in the field of laser radiation.

*γ*is [3

3. N.B. Delone and V.P. Krainov, *Multiphoton Processes in Atoms* (Springer, Berlin-Heidelberg1994). [CrossRef]

4. H.R. Reiss, “Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes”, Prog. Quantum Electron. **16**, 1–71 (1992). [CrossRef]

*F*and

*ω*are the amplitude and the frequency of laser field, respectively. We studied the influence of relativistic effects on the angular distribution of ejected electrons in the field of laser radiation in Refs. [5

5. V.P. Krainov and S.P. Roshchupkin, “Relativistic effects in the angular distribution of ejected electronds in tunneling ionization of atoms by strong electromagnetic field”, J. Opt. Soc. Am. B **9**, 1231–1233 (1992). [CrossRef]

*relativistic*energy distribution of ejected electrons in the field of linearly polarized low-frequency laser radiation. We restrict ourselves by the case of electron ejection along the electric field strength vector only, since the most part of electrons are emitted along this direction in a strong laser field. Then the classical relativistic energy

*E*(

_{f}*t*) of an electron in laser field is given by well-known expression [7]

*p*=

*p*(

*t*) is the electron momentum for arbitrary time moment

*t*, and

*p*

_{0}=

*p*(0) is the initial value of this momentum for

*t*= 0. The value of

*p*is determined from the cubic equation [7]:

*ω*

_{0}is the non-relativistic total tunneling ionization rate which can be calculated, foe example, in the frames of ADK theory [8].

2. N.B. Delone and V.P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation”, J. Opt. Soc. Am. B **8**, 1207–1211 (1991). [CrossRef]

9. P.B. Corkum, N.H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit”, Phys. Rev. Lett. **62**, 1259–1262 (1989). [CrossRef] [PubMed]

*E*exponentially decreases with

_{e}*E*. The non-relativistic width of this distribution is

_{e}*E*<

_{e}*c*

^{2}since

*γ*

^{2}<< 1.

*γ*= 0.1 the conditions

*c*

^{2}>

*E*>

_{e}*γ*

^{2}

*c*

^{2}are required for 500 keV >

*E*> 5 keV. According to (9) the value of Δ

_{e}*E*(

_{e}*rel*) is equal to 20 keV for radiation of CO

_{2}- laser.

_{2}- laser pulse with the intensity exceeding 10

^{15}W/cm

^{2}produces

*relativistic*distribution of ejected electrons with keV-energies along the polarization axis for most kinds of atoms, though the kinetic energies of these electrons are less than their rest energy

*mc*

^{2}. The results of this work can be used in the analysis of experimental data of Rochester group [10].

## References

1. | N.B. Delone and V.P. Krainov, |

2. | N.B. Delone and V.P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation”, J. Opt. Soc. Am. B |

3. | N.B. Delone and V.P. Krainov, |

4. | H.R. Reiss, “Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes”, Prog. Quantum Electron. |

5. | V.P. Krainov and S.P. Roshchupkin, “Relativistic effects in the angular distribution of ejected electronds in tunneling ionization of atoms by strong electromagnetic field”, J. Opt. Soc. Am. B |

6. | V.P. Krainov and B. Shokri, “Angular distribution of relativistic electrons in the tunneling ionization of atoms by an ac field”, Laser Phys. |

7. | L.D. Landau and E.M. Lifshitz, |

8. | M.V. Ammosov, N.B. Delone, and V.P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field”, Sov. Phys. JETP |

9. | P.B. Corkum, N.H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit”, Phys. Rev. Lett. |

10. | B. Buerke, J.P. Knauer, S.J. McNaught, and D.D. Meyerhofer, “Precision tests of laser-tunneling ionization models”: in |

**OCIS Codes**

(270.6620) Quantum optics : Strong-field processes

**ToC Category:**

Focus Issue: Relativistic effects in strong eectromagnetic fields

**History**

Original Manuscript: December 1, 1997

Published: March 30, 1998

**Citation**

V. Krainov, "Energy distribution of relativistic electrons in the tunneling ionization of
atoms by super-intense laser radiation," Opt. Express **2**, 268-270 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-7-268

Sort: Journal | Reset

### References

- N.B. Delone and V.P. Krainov, Atoms in Strong Light Fields (Springer, Berlin-Heidelberg 1985). [CrossRef]
- N.B. Delone and V.P. Krainov, "Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation", J. Opt. Soc. Am. B 8, 1207-1211 (1991). [CrossRef]
- N.B. Delone and V.P. Krainov, Multiphoton Processes in Atoms (Springer, Berlin-Heidelberg 1994). [CrossRef]
- H.R. Reiss, "Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes", Prog. Quantum Electron. 16, 1-71 (1992). [CrossRef]
- V.P. Krainov and S.P. Roshchupkin, "Relativistic eects in the angular distribution of ejected electronds in tunneling ionization of atoms by strong electromagnetic field", J. Opt. Soc. Am. B 9, 1231-1233 (1992). [CrossRef]
- V.P. Krainov and B. Shokri, "Angular distribution of relativistic electrons in the tunneling ionization of atoms by an ac field", Laser Phys. 5, 793-796 (1995).
- L.D. Landau and E.M. Lifshitz, Field Theory (Oxford, Pergamon 1977).
- M.V. Ammosov, N.B. Delone and V.P. Krainov, "Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field", Sov. Phys. JETP 64, 1191-1194 (1986).
- P.B. Corkum, N.H. Burnett and F. Brunel, "Above-threshold ionization in the long-wavelength limit", Phys. Rev. Lett. 62, 1259-1262 (1989). [CrossRef] [PubMed]
- B. Buerke, J.P. Knauer, S.J. McNaught and D.D. Meyerhofer, "Precision tests of laser-tunneling ionization models": in Applications of High Field and Short Wavelength Sources VII, OSA Technical Digest Series 7, 75-76 (1997).

## 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.