## Relativistic deflection of photoelectron trajectories in elliptically polarized laser fields

Optics Express, Vol. 2, Issue 7, pp. 271-276 (1998)

http://dx.doi.org/10.1364/OE.2.000271

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### Abstract

We present simple analytical formulas describing the evolution from circular to linear polarization of the relativistic deflection of photoelectron trajectories along the direction of laser light propagation, and discuss conditions of applicability of the model used in calculations.

© Optical Society of America

## 1. Introduction

1. 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]

1. 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]

6. D. P. Crawford and H. R. Reiss, “Stabilization in relativistic ionization with circularly polarized light;,” Phys. Rev. A **50**, 1844–1850 (1994). [CrossRef] [PubMed]

1. 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]

9. Y. I. Salamin and F. H. M. Faisal, “Ponderomotive scattering of electrons in intense laser fields,” Phys. Rev. A **55**, 3678–3683 (1997). [CrossRef]

10. J. N. Bardsley, B. M. Penetrante, and M. H. Mittleman, “Relativistic dynamics of electrons in intense laser fields,” Phys. Rev. A **40**, 3823–3835 (1997). [CrossRef]

11. F. V. Hartemann, S. N. Fochs, G. P. Le Sage, N. C. Luhmann Jr., J. G. Woodworth, M. D. Perry, Y. J. Chen, and A. K. Kerman, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field in focus,” Phys. Rev. E **51**, 4833–4843 (1995). [CrossRef]

13. S. P. Goreslavsky and N. B. Narozhny, “Ponderomotive scattering at relativistic laser intensities,” J. Nonlinear Opt. Phys. Mater. **4**, 799–815 (1995). [CrossRef]

*φ*=

*t*-

*z*/

*c*and

*ξ*is the ellipticity. The first two arguments in

*A*⃗ reflect a smooth dependence of the field amplitude

*F*(

*x*⃗,

*φ*) on the coordinates and time, which is in fact scaled by the size of the focus

*R*, the pulse duration

*τ*, and the diffraction length

*kR*

^{2}as

*F*(

*x*⃗

_{⊥}/

*R*,

*z*/

*kR*

^{2},

*φ*/

*τ*) [12,13

13. S. P. Goreslavsky and N. B. Narozhny, “Ponderomotive scattering at relativistic laser intensities,” J. Nonlinear Opt. Phys. Mater. **4**, 799–815 (1995). [CrossRef]

*φ*

_{0}the transition from the bound state occurs to the only continuum state

**∣**

*p*⃗

**⟩**labeled by the canonical momentum, for which the classical kinetic momentum (and the velocity) equals zero:

*π*⃗(

*φ*

_{0}) =

*p*⃗ -

*e*

*A*⃗

_{0}/

*c*= 0. Here

*A*⃗

_{0}=

*A*⃗(

*x*⃗

_{0},

*φ*

_{0};

*φ*

_{0}) is the vector potential at the position of the atom at the time of the transition. The velocity distribution at the moment of ionization [17] is neglected in such an approach, and so the released electron starts motion in the laser field with zero velocity. After ionization the electron does not interact with the parent ion. The probability of ionization at the field phase

*φ*

_{0}is governed by the tunneling factor exp(-2

*F*

_{a}/3

**∣**

*F*⃗

**(**

*φ*

_{0}

**∣**

**)**and is largest when the instantaneous value of the electric field

**∣**

*F*⃗(

*φ*

_{0}

**)∣**corresponding to the potential Eq. (1) is maximal.

## 2. Short pulses

*φ*=

*t*-

*z*/

*c*as the parameter, and with the initial condition of zero velocity at

*φ*

_{0}, the kinetic momentum and the energy for

*φ*>

*φ*

_{0}are equal to

*φ*→ ∞ in Eq. (2) gives the final momentum and energy. As the field goes to zero, the transverse momentum of a free electron outside the field is

*π*⃗

**(**+∞

**)**=

*e*

*A*⃗

_{0}/

*c*, and

*π*

_{z}

**(**+ ∞

**)**and

*π*

_{0}

**(**+ ∞

**)**are found according to Eq. (2).

*π*/

*ω*in the parameter

*φ*denoted below as

**⟨**…

**⟩**[12,13

13. S. P. Goreslavsky and N. B. Narozhny, “Ponderomotive scattering at relativistic laser intensities,” J. Nonlinear Opt. Phys. Mater. **4**, 799–815 (1995). [CrossRef]

**⟨**

*A*⃗

**⟩**= 0 and

**⟨(**

*e*

*A*⃗/

*c*

**)**

^{2}

**⟩**= 2

*mU*where the ponderomotive potential

*U*in an elliptically polarized field is

*I*is the laser intensity. In the field of a focused pulse the ponderomotive potential varies in space and time in the same way as the field amplitude does. The averaging in Eq. (2) reveals the drift motion after ionization. With the notation

*q*

_{i}=

**⟨**

*π*

_{i}

**⟩**one gets at

*φ*>

*φ*

_{0}:

*φ*→ ∞ when the wiggling motion dies out, the kinetic and the average momenta coincide:

*π*⃗

**(**+∞

**)**=

*q*⃗

**(**+∞

**)**and equally represent the momentum of a free electron after interaction with the laser. The limit

*φ*→ ∞ in Eqs. (2) and (4) gives the same result.

*θ*is given by

*A*⃗

_{0}or, according to Eq. (4), the transverse drift momentum

*e*

*A*⃗

_{0}/

*c*that is an integral of the motion in short pulses.

*ωφ*

_{0}= ±

*π*/ 2 and directed along the major axis of the polarization ellipse (

*x*-axis). The vector potential at that time is along the smaller axis and hence the most probable initial drift momentum has only a

*y*-projection and is equal to

*ξeF*/

*ω*. That estimate of the value of the drift momentum is valid as well for circular polarization

*ξ*= 1. For very small ellipticities (in fact for

**4**, 799–815 (1995). [CrossRef]

*x*-axis and strictly at

*ωφ*

_{0}= ±

*π*/2 turns to zero. But the ionization occurs during the interval

*δφ*

_{0}around the maximum, which is easily estimated from the tunneling exponential factor as

*F*

_{0}

*δφ*

_{0}. Now combining these two estimates one can interpolate the value of the drift momentum acquired by the electron in the process of ionization as

*ξ*→ 1 and

*ξ*→ 0. As was mentioned above in laser fields satisfying the inequality

*F*

_{0}/

*F*

_{a}<< 1, the second term is essential only for small ellipticities. The condition

*F*

_{0}/

*F*

_{a}<< 1 is compatible with the definition of the relativistic intensity

*U*≈

*mc*

^{2}for ionization of multiple-charged ions with large ionization potentials

*I*∝

*Z*

^{2}. The atomic field

*F*

_{a}= (2

*I*)

^{3/2}is proportional to

*Z*

^{3}.

*U*

_{0}=

*U*(

*x*⃗

_{0},

*φ*

_{0}) is the ponderomotive potential at the time and at the place of ionization. At a fixed intensity the forward deflection is maximum for circular polarization and decreases monotonically with decreasing ellipticity. At

*ξ*→ 0 the momentum distribution flattens against the plane perpendicular to the direction of laser field propagation. The reason is that the longitudinal drift decreases faster with decreasing ellipticity than does the transverse drift. It is worth noting that variation of the ellipticity, in addition to the changes in the forward deflection, simultaneously leads to a strong rearrangement of the angular distribution in the polarization plane [15,16,18

18. S. J. McNaught, J.P. Knauer, and D. D. Meyerhofer, “Measurement of the initial condition of electrons ionized by a linearly polarized high-intensity laser,” Phys. Rev. Lett. **78**, 626–629 (1997). [CrossRef]

## 3. Long pulses

*φ*=

*φ*

_{0}in Eq (4). Secondly, the initial conditions are used to integrate the averaged equations of motion [12,13

**4**, 799–815 (1995). [CrossRef]

*U*=

*U*

**(**

*x*⃗

_{⊥}/

*R*,

*z*/

*kR*

^{2},

*φ*/

*τ*) are essential, only a numerical integration is possible. Some progress in derivation of an analytical solution is achieved if the longitudinal gradient of the ponderomotive potential is neglected in the averaged equation [7]. That procedure corresponds to the limit of an infinite diffraction length

*kR*

^{2}→ ∞ but

*R*remains constant. In other words the focal volume is considered as an infinite round cylinder so that

*U*=

*U*(

*x*⃗

_{⊥}/

*R*,

*φ*/

*τ*). In that simplified model of the focused pulse there exists an approximate integral of motion [12,13

**4**, 799–815 (1995). [CrossRef]

*m*

_{*}, is the electron effective mass in the light field. From Eqs. (8,9) in the limit

*φ*→ ∞ when the electron is in any case out of the field and the ponderomotive potential goes to zero, we get two equations for the final energy and momentum

*x*⃗

_{⊥}and hence

*q*⃗

_{⊥}is a constant equal to the initial value calculated from Eq. (4). Then Eq. (10) brings us back to the same results as the ones derived for short pulses from Eq. (2) or (4). New results are available in the opposite case of a long pulse when the ponderomotive potential does not depend on

*φ*and the average energy

*q*

_{0}and as well as

*q*

_{z}become integrals of the motion. Accounting for the initial values following from Eq. (4), we find the photoelectron momentum outside a long laser pulse to be

*ξ*= 0 and

*ξ*= 1 in Ref. [14].

*tg*

*θ*

_{lin}/

*tg*

*θ*

_{cir}=

**√**2 [7].

## 4. Conditions of applicability

*π*

_{z}=

*π*

_{z}

**(**∞

**)**, Δ

*π*

_{0}=

*π*

_{0}

**(**∞

**)**-

*mc*

^{2}and so on, and remembering that

*π*⃗

**(**∞

**)**=

*q*⃗

**(**∞

**)**, we have from Eq. (10)

10. J. N. Bardsley, B. M. Penetrante, and M. H. Mittleman, “Relativistic dynamics of electrons in intense laser fields,” Phys. Rev. A **40**, 3823–3835 (1997). [CrossRef]

*π*

_{z}= 2

*k*

_{z}but its energy does not change Δ

*π*

_{0}= 0. If the calculations leading to Eq. (10) are repeated with arbitrary initial value

*π*_ ≠

*mc*, the result Δ

*π*

_{z}= Δ

**(**

**)**/2

*π*_ is different from Eq. (14).

*π*_ is the consequence of the

**(**

*t*-

*z*/

*c*

**)**-dependence specific to a plane wave field. Averaging of Eq. (15) over oscillations turns it into Eq. (8). The quantum theory sheds some light on the conservation of

*π*_. From that point of view an arbitrary plane wave propagating in the positive direction of the

*z*-axis consists of photons with different frequencies but strictly parallel wave vectors

*k*

_{z}=

*ω*/

*c*> 0. Matrix elements corresponding to absorption and/or emission of several photons of the wave relate the initial electron state with the energy

*π*

_{0}and the momentum

*π*

_{z}to the virtual states characterized by the energy

*π̃*

_{0}and the momentum

*π̃*

_{z}equal to

*π̃*

_{0}-

*cπ̃*

_{z}=

*π*

_{0}-

*cπ*

_{0}. That selection rule for transitions to the virtual states originates due to the specific dispersion relation for photons with parallel wave vectors. Real processes with energy-momentum conservation as in Eq. (16) are forbidden.

*ω*and wave vector

*k*⃗ =

**(**

*k*⃗

_{⊥},

*k*

_{z}

**)**has the form

*k*

_{z}≈

*ω*/

*c*, as in the plane wave field. Accounting for the term quadratic in

*k*

_{⊥}in the decomposition of the exact formula

## Acknowledgments

## References and links

1. | 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. |

2. | D. D. Meyerhofer, J. P. Knauer, S. J. McNaught, and C. I. Moore, “Mass shift effects during high-intensity laser-electron interactions,” J. Opt. Soc. Am. B |

3. | D. D. Meyerhofer, “Laser-electron scattering at relativistic intensities,” in |

4. | H. R. Reiss, “Relativistic strong-field ionization,” J. Opt. Soc. Am. B |

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

6. | D. P. Crawford and H. R. Reiss, “Stabilization in relativistic ionization with circularly polarized light;,” Phys. Rev. A |

7. | P. B. Corkum, N. H. Burnett, and F. Brunel, “Multiphoton ionization at large ponderomotive potentials,” in |

8. | S. P. Goreslavskii, M. V. Fedorov, and A. A. Kil’pio, “Relativistic drift of an electron under the influence of a short intense laser pulse,” Laser Phys. |

9. | Y. I. Salamin and F. H. M. Faisal, “Ponderomotive scattering of electrons in intense laser fields,” Phys. Rev. A |

10. | J. N. Bardsley, B. M. Penetrante, and M. H. Mittleman, “Relativistic dynamics of electrons in intense laser fields,” Phys. Rev. A |

11. | F. V. Hartemann, S. N. Fochs, G. P. Le Sage, N. C. Luhmann Jr., J. G. Woodworth, M. D. Perry, Y. J. Chen, and A. K. Kerman, “Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field in focus,” Phys. Rev. E |

12. | S. P. Goreslavskii, N. B. Narozhny, O. V. Shcherbachev, and V. P. Yakovlev, “The dynamics and radiation of a relativistic electron in the field of an intense, focused laser pulse,” Laser Phys. |

13. | S. P. Goreslavsky and N. B. Narozhny, “Ponderomotive scattering at relativistic laser intensities,” J. Nonlinear Opt. Phys. Mater. |

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

15. | S. P. Goreslavskii and S. V. Popruzhenko, “Momentum distribution of photoelectrons in strong low-frequency elliptically polarized laser field,” Laser Phys. |

16. | S. P. Goreslavskii and S. V. Popruzhenko, “Differential photoelectron distributions in a strong elliptically polarized low-frequency laser field,” Zh. Eksp. Teor. Fiz. |

17. | S. P. Goreslavskii and S. V. Popruzhenko, “Photoelectron velocity distribution at the time of ionization by elliptically polarized laser field,” Laser Phys. |

18. | S. J. McNaught, J.P. Knauer, and D. D. Meyerhofer, “Measurement of the initial condition of electrons ionized by a linearly polarized high-intensity laser,” Phys. Rev. Lett. |

19. | S. J. McNaught, J. P. Knauer, and D. D. Meyerhofer, “Photoelectron drift momentum in the long-pulse tunneling limit for an elliptically polarized laser,” Laser Phys. |

**OCIS Codes**

(190.4180) Nonlinear optics : Multiphoton processes

(260.3230) Physical optics : Ionization

(350.5720) Other areas of optics : Relativity

**ToC Category:**

Focus Issue: Relativistic effects in strong eectromagnetic fields

**History**

Original Manuscript: November 5, 1997

Published: March 30, 1998

**Citation**

S. P. Goreslavsky and Sergei Popruzhenko, "Relativistic deflection of photoelectron trajectories in elliptically polarized laser fields," Opt. Express **2**, 271-276 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-7-271

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### References

- 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]
- D. D. Meyerhofer, J. P. Knauer, S. J. McNaught and C. I. Moore, Mass shift effects during high-intensity laser-electron interactions, J. Opt. Soc. Am. B 13, 113-117 (1996). [CrossRef]
- D. D. Meyerhofer, Laser-electron scattering at relativistic intensities, in Super - Intense Laser - Atom Physics IV, H. G. Muller and M. V. Fedorov, eds. (Kluwer Academic Publishers, Netherlands, 1996). [CrossRef]
- H. R. Reiss, "Relativistic strong-field ionization," J. Opt. Soc. Am. B 7, 574-586 (1990). [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]
- D. P. Crawford and H. R. Reiss, "Stabilization in relativistic ionization with circularly polarized light," Phys. Rev. A 50, 1844-1850 (1994). [CrossRef] [PubMed]
- P. B. Corkum, N. H. Burnett and F. Brunel, Multiphoton ionization at large ponderomotive potentials, in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic Press, New York, 1992).
- S. P. Goreslavskii, M. V. Fedorov and A. A. Kilpio, Relativistic drift of an electron under the influence of a short intense laser pulse, Laser Phys. 5, 1020 -1028 (1995).
- Y. I. Salamin, F. H. M. Faisal, Ponderomotive scattering of electrons in intense laser fields, Phys. Rev. A 55, 3678-3683 (1997). [CrossRef]
- J. N. Bardsley, B. M. Penetrante, M. H. Mittleman, Relativistic dynamics of electrons in intense laser fields, Phys. Rev. A 40, 3823-3835 (1997). [CrossRef]
- F. V. Hartemann, S. N. Fochs, G. P. Le Sage, N. C. Luhmann, Jr., J. G. Woodworth, M. D. Perry, Y. J. Chen, A. K. Kerman, Nonlinear ponderomotive scattering of relativistic electrons by an intense laser field in focus, Phys. Rev. E 51, 4833-4843 (1995). [CrossRef]
- S. P. Goreslavskii, N. B. Narozhny, O. V. Shcherbachev, and V. P. Yakovlev, The dynamics and radiation of a relativistic electron in the field of an intense, focused laser pulse, Laser Phys. 3, 421-434, (1993).
- S. P. Goreslavsky and N. B. Narozhny, Ponderomotive scattering at relativistic laser intensities, J. Nonlinear Opt. Phys. Mater. 4, 799 - 815 (1995). [CrossRef]
- S. P. Goreslavskii, The BSI model and relativistic ponderomotive scattering, Laser Phys. 6, 74-78 (1996).
- S. P. Goreslavskii and S. V. Popruzhenko, Momentum distribution of photoelectrons in strong low- frequency elliptically polarized laser field, Laser Phys. 6, 780-784 (1996).
- S. P. Goreslavskii and S. V. Popruzhenko, Differential photoelectron distributions in a strong elliptically polarized low-frequency laser field, Zh. Eksp. Teor. Fiz. 110, 1200-1215 (1996) (JETP 83, 661-669).
- S. P. Goreslavskii and S. V. Popruzhenko, Photoelectron velocity distribution at the time of ionization by elliptically polarized laser field, Laser Phys. 7, 700-705 (1997).
- S. J. McNaught, J.P. Knauer, and D. D. Meyerhofer, Measurement of the initial condition of electrons ionized by a linearly polarized high-intensity laser, Phys. Rev. Lett. 78, 626-629 (1997). [CrossRef]
- S. J. McNaught, J. P. Knauer, and D. D. Meyerhofer, Photoelectron drift momentum in the long-pulse tunneling limit for an elliptically polarized laser, Laser Phys. 7, 712-718 (1996).

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