## Relativistic signatures in laser-assisted scattering at high field intensities

Optics Express, Vol. 2, Issue 7, pp. 262-267 (1998)

http://dx.doi.org/10.1364/OE.2.000262

Acrobat PDF (478 KB)

### Abstract

The complete Dirac-Volkov relativistic treatment of the first Born limit of laser-assisted potential scattering of electrons within a circularly polarized laser field has been compared to the nonrelativistic Bunkin-Fedorov approach. The dependence of the quiver energy on the electron four-momentum in an ultrastrong laser field leads to different energy transfer cross sections depending on the scattering geometry with respect to the laser propagation direction. Visible differences between the relativistic and non-relativistic differential cross sections for small-angle scattering occur already for 10^{16} W/cm^{2} intensity of near infrared wavelength and moderate electron initial energies.

© Optical Society of America

^{16}W/cm

^{2}– 10

^{19}W/cm

^{2}. Recent exciting experiments give clear evidence of several relativistic effects [3

3. D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berrigde, W. M. Bugg, K. Shmakov, A. W. Weidemann, C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D.A. Reis, and W. Ragg, “Positron Production in Multiphoton Light-by-Light Scattering”, Phys. Rev. Lett. **79**, 1626–1629 (1997). [CrossRef]

4. C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, W. Ragg, D. L. Burke, R. C. Field, G. Horton-Smith, A. C. Odian, J. E. Spencer, D. Walz, S. C. Berrigde, W. M. Bugg, K. Shmakov, and A. W. Weidemann, “Observation of Nonlinear Effects in Compton Scattering”, Phys. Rev. Lett. **76**, 3116–3119 (1996). [CrossRef] [PubMed]

5. P. A. Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, 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. **76**, 1832–1835 (1996). [CrossRef] [PubMed]

6. P. Monot, T. Auguste, P. Gibbon, F. Jakober, G. Mainfray, A. Dulieu, M. Louis-Jacquet, G. Malka, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling of a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. **74**, 2953–2956 (1995). [CrossRef] [PubMed]

7. C. I. Moore, J. P. Knauer, and D. D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. **74**, 2439–2442 (1995). [CrossRef] [PubMed]

8. L. S. Brown and T. W. B. Kibble, “Interaction of Intense Laser Beams with Electrons”, Phys. Rev. **133**, A705 (1964). [CrossRef]

9. H. R. Reiss, “Absorption of Light by Light”, J. Math. Phys. **3**, 59–67 (1962). [CrossRef]

13. J. H. Eberly, “Proposed Experiment for Observation of Nonlinear Compton Wavelength Shift”, Phys. Rev. Lett. **15**, 91–93 (1965). [CrossRef]

15. J. H. Eberly and H. R. Reiss, “Electron Self-Energy in Intense Plane-Wave Field”, Phys. Rev. **145**, 1035 (1966). [CrossRef]

16. H. R. Reiss and J. H. Eberly, “Green’s function in intense field electrodynamics”, Phys. Rev. **151**, 1058–1066 (1966). [CrossRef]

20. C. Szymanowski, V. Véniard, R. Taïeb, A. Maquet, and C. H. Keitel, “Mott scattering in strong laser fields”, Phys. Rev. A **56**3846–3859 (1997). [CrossRef]

*c*-corrections at relatively low laser intensity have been realized [21].

*p*(outside the field and genuine quantum number) inside a circularly polarized plane wave with vector potential

^{μ}*A*and wavevector

^{μ}*k*propagating along the

^{μ}*e*̂

_{z}direction read in Feynman slash notation [22,23

23. D. M. Volkov, “Uber eine Klasse von Lösungen der Diracschen Gleichung”, Z. f. Phys. **94**,250–260 (1935). [CrossRef]

*u*represents a free bispinor normalized by

*u*̄

*u*=

*u*

^{*}

*γ*

^{0}

*u*= 2

*c*

^{2}in the volume

*V*. The physical significance of

*q*= (

^{μ}*Q*/

*c*,

*q*⃗) is the averaged four-momentum (dressed momentum) of the particle inside the laser field:

*q*=

^{μ}*p*-

^{μ}*k*[

^{μ}*A*

^{2}/2(

*kp*)

*c*

^{2}].

*p*⃗ and

*q*⃗ have different longitudinal but equal transverse momentum projections with respect to

*k*⃗. This implies that angles might change due to field envelope effects [24,25]. For scattering off a Yukawa-type potential the matrix element for the transition ∣

*q*⟩ → ∣

*q*′⟩ is

26. C. H. Keitel, C. Szymanowski, P. L. Knight, and A. Maquet, “Radiative reaction in ultra-intense laser-atom interaction”, J. Phys. B **31**, L75–L83 (1998). [CrossRef]

27. F. V. Hartemann and A. K. Kerman, “Classical Theory of Nonlinear Compton Scattering”, Phys. Rev. Lett. **76**, 624–627 (1996). [CrossRef] [PubMed]

*J*a sum over Fourier components whose order are associated to the net number of exchanged photons is obtained which leads to well-known Fourier transform integrals of the central potential. The appropriate expression for the square of the

_{n}*δ*-function is given by the usual procedure [22]. This leads to the expression for the differential cross section into the solid angle

*o*:

*q*⃗′∣ is fixed through the condition

*q*=

^{μ}q_{μ}*q′*

^{μ}*q*′

_{μ}. The differential cross section

*dσ*(

*n*)/

*do*for each net n-photon process for unpolarized projectiles is [20

20. C. Szymanowski, V. Véniard, R. Taïeb, A. Maquet, and C. H. Keitel, “Mott scattering in strong laser fields”, Phys. Rev. A **56**3846–3859 (1997). [CrossRef]

*v*̃

^{μ}denotes (

*v*

^{0}, -

*v*⃗

^{i}),

*ζ*=

*A*{(

*p*/

_{x}*c*(

*kp*) -

*p*′

_{x}/

*c*(

*kp*′)]

^{2}+ [

*p*/

_{y}*c*(

*kp*) -

*p*′

_{y}/

*c*(

*kp*′)]

^{2}}

^{1/2},

*ϕ*

_{0}= arccos{

*A*[

*p*/(

_{x}*kp*) -

*p*′

_{x}/(

*kp*′)]/

*cζ*} and Å ≡

*A*(0, cos

*ϕ*

_{0}, sin

*ϕ*

_{0}, 0). The influence of the spin-laser interaction on the degree of polarization of initially polarized electrons has also been studied [28].

*ζ*and its nonrelativistic limit. The physical meaning of

*ζ*is the maximal attainable transient energy difference per photon energy during a collision [20

20. C. Szymanowski, V. Véniard, R. Taïeb, A. Maquet, and C. H. Keitel, “Mott scattering in strong laser fields”, Phys. Rev. A **56**3846–3859 (1997). [CrossRef]

*ω*= 0.043 a.u., corresponding to the Neodymium laser angular frequency (1.17eV photon energy). For illustrating the main features of our results we will concentrate here on the case of small-angle scattering, which dominates the total cross section.

*q*⃗,

*q*⃗′) = 0.6mrad are shown for an intensity of 3.5 × 10

^{16}W/cm

^{2}and a moderate initial electron energy of

*W*

_{kin}≃ 2.7keV . The red curve gives the envelope of the result according to Eq. (4) for an incoming electron with momentum parallel to the laser propagation direction

*k*⃗, while the yellow curve denotes the case of antiparallel propagation. The black line sketches the nonrelativistic result virtually independent from the direction of

*k*⃗. A significant dependence on geometry of the amount of photon energy transfer is noticed.

*ζ*, meaning to the change in the energy dressing difference. The simulations clearly show that the nonrelativistic limit 1/

*ω*of the four-product 1/(

*kp*) is not any longer a good approximation in this laser intensity regime. In fact, for a kinetic energy of

*W*

_{kin}=

*E*-

*c*

^{2}≃ 2.7keV the differences amount to about +11% in propagation direction parallel to

*k*⃗ and about -9% in direction antiparallel to

*k*⃗. In the full Dirac-Volkov treatment, four-vector products of this form come additionally into play due to the Volkov prefactor (∝

*A*) containing

*γ*-matrices. However, these terms of the cross section describing the spin-laser interaction do not contribute significantly at intensities considered here.

*q*⃗,

*q*⃗′) = 0.6mrad and for a relativistically high intensity of about 10

^{18}W/cm

^{2}are plotted. The incoming electron energy is again about 2.7keV, and the incoming electron momentum is oriented parallely to the laser propagation direction. The red curve denotes the full Dirac-Volkov result according to equation 4. The black line corresponds to a simplified treatment for spinless relativistic particles, while the yellow line sketches the outcome of the nonrelativistic calculation. The discrepancy between the relativistic and nonrelativistic treatments amount now to a factor 2 in the occurence of the cutoff. The maximal energy transfer is fixed by the condition

*n*≃

*ζ*when the Bessel functions reach asymptotically the Airy function Ai. One can verify in classical terms that

*ζ*corresponds to the maximal attainable transient energy difference per photon energy of the electron travelling in the laser field. Nevertheless in comparison to the spinless particle case the complete Dirac-Volkov calculation predicts a higher cross section due to the spin-laser interaction pushing the electron in the laser propagation direction. This effect is mediated by the additional terms in Eq. 4 proportional to powers of A. In comparison, the blue line denotes the contribution arising only from the first part of the Volkov prefactor (∝ 1) which is largely dominating in the nonrelativistic and moderately relativistic intensity regime.

## References

1. | see, for instance, “High-field interactions and short-wavelength generation”, edited by H. Milch-berg and R. Freeman, part 1 of special issue of J. Opt. Soc. Am. |

2. | see, for instance, “High-field interactions and short-wavelength generarion”, edited by H. Milch-berg and R. Freeman, part 2 of special issue of J. Opt. Soc. Am. |

3. | D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berrigde, W. M. Bugg, K. Shmakov, A. W. Weidemann, C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D.A. Reis, and W. Ragg, “Positron Production in Multiphoton Light-by-Light Scattering”, Phys. Rev. Lett. |

4. | C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, W. Ragg, D. L. Burke, R. C. Field, G. Horton-Smith, A. C. Odian, J. E. Spencer, D. Walz, S. C. Berrigde, W. M. Bugg, K. Shmakov, and A. W. Weidemann, “Observation of Nonlinear Effects in Compton Scattering”, Phys. Rev. Lett. |

5. | P. A. Norreys, M. Zepf, S. Moustaizis, A. P. Fews, J. Zhang, P. Lee, 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. |

6. | P. Monot, T. Auguste, P. Gibbon, F. Jakober, G. Mainfray, A. Dulieu, M. Louis-Jacquet, G. Malka, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling of a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. |

7. | C. I. Moore, J. P. Knauer, and D. D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. |

8. | L. S. Brown and T. W. B. Kibble, “Interaction of Intense Laser Beams with Electrons”, Phys. Rev. |

9. | H. R. Reiss, “Absorption of Light by Light”, J. Math. Phys. |

10. | I. I. Gol’dman, “Intensity Effects in Compton Scattering”, Zh. Éksp. Teor. Fiz. |

11. | A. I. Nikishov and V. I. Ritus, “Quantum Processes in the field of a plane electromagnetic wave and in a constant field I”, Zh. Éksp. Teor. Fiz. |

12. | A. I. Nikishov and V. I. Ritus, “Quantum Processes in the field of a plane electromagnetic wave and in a constant field II” Zh. Éksp. Teor. Fiz. |

13. | J. H. Eberly, “Proposed Experiment for Observation of Nonlinear Compton Wavelength Shift”, Phys. Rev. Lett. |

14. | T. W. B. Kibble, “Mutual Refraction of Electrons and Photons”, Phys. Rev. |

15. | J. H. Eberly and H. R. Reiss, “Electron Self-Energy in Intense Plane-Wave Field”, Phys. Rev. |

16. | H. R. Reiss and J. H. Eberly, “Green’s function in intense field electrodynamics”, Phys. Rev. |

17. | J. H. Eberly, “Interaction of very intense light with free electrons”, Prog. Opt. |

18. | F. V. Bunkin and M. V. Fedorov, “Bremsstrahlung in a strong radiation field”, Zh. Éksp. Teor. Fiz. |

19. | M. M. Denisov and M. V. Fedorov, “Bremsstrahlung effect on relativistic electrons in a strong radiation field”, Zh. Éksp. Teor. Fiz. |

20. | C. Szymanowski, V. Véniard, R. Taïeb, A. Maquet, and C. H. Keitel, “Mott scattering in strong laser fields”, Phys. Rev. A |

21. | V. P. Krainov and S. P. Roshupkin, “Relativistic effects in the angular distribution of ejected electrons in tunneling ionization of atoms by strong electromagnetic fields”, J. Opt. Soc. Am. |

22. | V. B. Berestetzkiĭ, E. M. Lifshitz, and L. P. Pitaevskiĭ, |

23. | D. M. Volkov, “Uber eine Klasse von Lösungen der Diracschen Gleichung”, Z. f. Phys. |

24. | S. P. Goreslavskiĭ, “The BSI model and relativistic ponderomotive scattering”, Laser Phys. |

25. | N. B. Narozhnyi and M. S. Fofanov, “Creation of a pair by a photon colliding with a short focused laser pulse”, Laser Phys. |

26. | C. H. Keitel, C. Szymanowski, P. L. Knight, and A. Maquet, “Radiative reaction in ultra-intense laser-atom interaction”, J. Phys. B |

27. | F. V. Hartemann and A. K. Kerman, “Classical Theory of Nonlinear Compton Scattering”, Phys. Rev. Lett. |

28. | C. Szymanowski, R. Taïeb, and A. Maquet, “Laser-assisted scattering of polarized electrons at high field intensities”, Laser Phys. |

**OCIS Codes**

(020.2070) Atomic and molecular physics : Effects of collisions

(020.4180) Atomic and molecular physics : Multiphoton processes

(020.5580) Atomic and molecular physics : Quantum electrodynamics

**ToC Category:**

Focus Issue: Relativistic effects in strong eectromagnetic fields

**History**

Original Manuscript: October 28, 1997

Published: March 30, 1998

**Citation**

Carsten Szymanowski and Alfred Maquet, "Relativistic signatures in
laser-assisted scattering at high field intensities," Opt. Express **2**, 262-267 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-7-262

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### References

- see, for instance, "High-field interactions and short-wavelength generation", edited by H. Milchberg and R. Freeman, part 1 of special issue of J. Opt. Soc. Am. B13, 51-222 (1996).
- see, for instance, "High-field interactions and short-wavelength generarion", edited by H. Milchberg and R. Freeman, part 2 of special issue of J. Opt. Soc. Am. B13, 314-468 (1996).
- D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berrigde, W. M. Bugg, K. Shmakov, A. W. Weidemann, C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D.A. Reis, and W. Ragg, "Positron Production in Multiphoton Light-by-Light Scattering", Phys. Rev. Lett. 79, 1626-1629 (1997). [CrossRef]
- C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, W. Ragg, D. L. Burke, R. C. Field, G. Horton-Smith, A. C. Odian, J. E. Spencer, D. Walz, S. C. Berrigde, W. M. Bugg, K. Shmakov, and A. W. Weidemann, "Observation of Nonlinear Effects in Compton Scattering", Phys. Rev. Lett. 76, 3116-3119 (1996). [CrossRef] [PubMed]
- P. A. 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. 76, 1832-1835 (1996). [CrossRef] [PubMed]
- P. Monot, T. Auguste, P. Gibbon, F. Jakober, G. Mainfray, A. Dulieu, M. Louis-Jacquet, G. Malka, and J. L. Miquel, "Experimental Demonstration of Relativistic Self-Channeling of a Multiterawatt Laser Pulse in an Underdense Plasma", Phys. Rev. Lett. 74, 2953-2956 (1995). [CrossRef] [PubMed]
- C. I. Moore, J. P. Knauer, and D. D. Meyerhofer, "Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons", Phys. Rev. Lett. 74, 2439-2442 (1995). [CrossRef] [PubMed]
- L. S. Brown and T. W. B. Kibble, "Interaction of Intense Laser Beams with Electrons", Phys. Rev. 133, A705 (1964). [CrossRef]
- H. R. Reiss, "Absorption of Light by Light", J. Math. Phys. 3, 59-67 (1962). [CrossRef]
- I. I. Gol'dman, "Intensity Effects in Compton Scattering", Zh. Eksp. Teor. Fiz. 46, 1412-1417 (1964) [Sov. Phys. JETP 19, 954-957 (1964)].
- A. I. Nikishov and V. I. Ritus, "Quantum Processes in the field of a plane electromagnetic wave and in a constantfieldI",Zh. Eksp. Teor. Fiz. 46, 776-796 (1964) [Sov. Phys. JETP 19, 529-541 (1964)].
- A. I. Nikishov and V. I. Ritus, "Quantum Processes in the field of a plane electromagnetic wave and in a constant field II" Zh. Eksp. Teor. Fiz. 46, 1768-1781 (1964) [Sov. Phys. JETP 19, 1191-1199 (1964)].
- J. H. Eberly, "Proposed Experiment for Observation of Nonlinear Compton Wavelength Shift", Phys. Rev. Lett. 15, 91-93 (1965). [CrossRef]
- T. W. B. Kibble, "Mutual Refraction of Electrons and Photons", Phys. Rev. 150, 1060 - 1069 (1966). [CrossRef]
- J. H. Eberly and H. R. Reiss, "Electron Self-Energy in Intense Plane-Wave Field", Phys. Rev. 145, 1035 (1966). [CrossRef]
- H. R. Reiss and J. H. Eberly, "Green's function in intense field electrodynamics", Phys. Rev. 151, 1058-1066 (1966). [CrossRef]
- J. H. Eberly, "Interaction of very intense light with free electrons", Prog. Opt. 7, 361-415, edited by E. Wolf, (Noth-Holland, Amsterdam, 1969).
- F. V. Bunkin and M. V. Fedorov, " Bremsstrahlung in a strong radiation field", Zh. Eksp. Teor. Fiz. 49, 1215-1221 (1965) [Sov. Phys. JETP 22, 844-847 (1966)].
- M. M. Denisov and M. V. Fedorov, "Bremsstrahlung effect on relativistic electrons in a strong radiation field", Zh. Eksp. Teor. Fiz. 53, 1340-1348 (1967) [Sov. Phys. JETP 26 779-783 (1968)].
- C. Szymanowski, V. Veniard, R. Taieb, A. Maquet, and C. H. Keitel, "Mott scattering in strong laser fields", Phys. Rev. A 56 3846-3859 (1997). [CrossRef]
- V. P. Krainov and S. P. Roshupkin, "Relativistic effects in the angular distribution of ejected electrons in tunneling ionization of atoms by strong electromagnetic fields", J. Opt. Soc. Am. B9, 1231-1233 (1992).
- V. B. Berestetzkii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd Edition (Pergamon Press, Oxford, 1982).
- D. M. Volkov, "Uber eine Klasse von L" osungen der Diracschen Gleichung", Z. f. Phys. 94, 250-260 (1935). [CrossRef]
- S. P. Goreslavskii, "The BSI model and relativistic ponderomotive scattering", Laser Phys. 6, 74-78 (1996).
- N. B. Narozhnyi and M. S. Fofanov, "Creation of a pair by a photon colliding with a short focused laser pulse", Laser Phys. 7, 141-149 (1997).
- C. H. Keitel, C. Szymanowski, P. L. Knight, and A. Maquet, "Radiative reaction in ultra-intense laser-atom interaction", J. Phys. B 31, L75-L83 (1998). [CrossRef]
- F. V. Hartemann and A. K. Kerman, "Classical Theory of Nonlinear Compton Scattering", Phys. Rev. Lett. 76, 624-627 (1996). [CrossRef] [PubMed]
- C. Szymanowski, R. Taieb, and A. Maquet, "Laser-assisted scattering of polarized electrons at high field intensities", Laser Phys. 8, 102-106 (1998).

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