## Temporal and spatial characterization of harmonics structures of relativistic nonlinear Thomson scattering

Optics Express, Vol. 11, Issue 4, pp. 309-316 (2003)

http://dx.doi.org/10.1364/OE.11.000309

Acrobat PDF (4283 KB)

### Abstract

The harmonics of the scattering of a femtosecond intense laser pulse by an electron has been numerically investigated. The harmonic spectrum shows interesting red shifts and parasitic lines in the blue sides of harmonic lines. The red shift of the lines is found to be caused by the dilation of laser oscillation experienced by an electron due to its relativistic drift motion along the direction of a driving laser propagation and the parasitic lines come from the variation of the laser intensity. The angular distribution of each higher harmonic line shows double peak patterns in the forward direction. The backward scattering has its own distinct pattern: line-shaped nodes perpendicular to the laser electric field, the number of which is the harmonic order number minus one. As the harmonic order increases, the primary peaks of higher harmonics move from the backward to the forward direction of the laser propagation. In the time domain, each radiation pulse in the case of a linearly-polarized laser pulse has a double peak structure due to the disappearance of the acceleration during the half cycle of an electron’s oscillation.

© 2002 Optical Society of America

## 1. Introduction

1. M. D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers,” Science **264**, 917 (1994) [CrossRef] [PubMed]

2. G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-Intensity Lasers: Physics of the Extreme on a Tabletop,” Phys. Today **51**, 22 (1998). [CrossRef]

3. P. Sprangle, A. Ting, E. Esarey, and A. Fisher, “Tunable, short pulse hard x-rays from a compact laser synchrotron source,” J. Appl. Phys. **72**, 5032 (1992). [CrossRef]

4. E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson Scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E **48**, 3003 (1993). [CrossRef]

5. F. V. Hartemann, “High-intensity scattering processes of relativistic electrons in vacuum,” Phys. Plasmas **5**, 2037 (1998). [CrossRef]

6. Y. Ueshima, Y. Kishimoto, A. Sasaki, and T. Tajima, “Laser Larmor X-ray radiation from low-Z matter,” Laser and Particle Beams **17**, 45 (1999). [CrossRef]

7. A. E. Kaplan and P. L. Shkolnikov, “Lasetron: A Proposed Source of Powerful Nuclear-Time-Scale Electromagnetic Bursts,” Phys. Rev. Lett. **88**, 074801 (2002). [CrossRef] [PubMed]

9. D. Kim, S. H. Son, J. H. Kim, C. Toth, and C. P. J. Barty, “Gain characteristics of inner-shell photoionization-pumped L23M1 transition in Ca,” Phys. Rev. A **63**, 023806 (2001). [CrossRef]

*I*the laser intensity in W/cm

^{2}.

## 2. Results and Discussions

^{17}W/cm

^{2}and 10

^{18}W/cm

^{2}. The generation of an ultrashort x-ray pulse using higher intensities are presented in Ref. [20

20. K. Lee, Y. H. Cha, M. S. Shin, B. H. Kim, and D. Kim, “Relativistic nonlinear Thomson scattering as attosecond x-ray source”, in print in Phys. Rev. E **67**2003. [CrossRef]

^{18}W/cm

^{2}amounts to 0.1 µm, which is much smaller compared with a typical beam size of 10 um in diameter in real experiments. Hence the relative variation of laser intensity during the quivering motion of an electron can be neglected. The estimation of the ponderomotive force reveals that the ponderomotive force is about three orders of magnitude smaller than the Lorentz force. Thus the plane wave approximation is adopted for this calculation.

^{17}W/cm

^{2}and 10

^{18}W/cm

^{2}. Harmonic structures are clearly seen. The normalized vector potentials of these intensities correspond to 0.21 and 0.68, respectively. At first, one can see that for a case of 10

^{17}W/cm

^{2}, the line shapes of the harmonic lines are broadened to the red side compared with the unshifted lines (the vertical lines). As the laser intensity increases, the harmonic lines shift more toward the red side and get broader. Also, small peaks between the harmonic lines can be noticed for a laser intensity of 10

^{18}W/cm

^{2}. The radiations in the direction of laser propagation (θ = 0 - 90°, θ is the polar angle measured from the +z axis) and the backward direction (θ = 90 - 180°) for the case of 10

^{18}W/cm

^{2}are separately plotted in Fig. 1(b), which shows clearly different harmonic spectral lines. The energy of the fundamental harmonic radiation,

*E*with respect to the laser intensity and the direction from an electron initially at rest changes as [4

_{s}4. E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson Scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E **48**, 3003 (1993). [CrossRef]

13. E. S. Sarachik and G. T. Schappert, “Classical Theory of the Scattering of Intense Laser Radiation by Free Electrons,” Phys. Rev. D **1**, 2738 (1970). [CrossRef]

*E*is the fundamental photon energy of a laser, 1.55 eV for 800 nm laser. Below, we will show that such a red shift can be explained by the drift motion of an electron due to β⃗ ×

_{o}*B*⃗ force which makes the electron feel a dilated laser oscillation period. β is the electron’s velocity divided by the speed of light and

*B*⃗ the magnetic field of a driving laser pulse.

^{18}W/cm

^{2}is plotted in Fig. 2(a). An interesting feature in the spectrum is the low-intensity parasitic lines to the blue side of harmonic lines. This is caused by the variation of the laser intensity during a laser pulse. As seen from Eq. (2), the shift of a harmonic line increases as the laser intensity increases, and the principal line is generated during the peak laser intensity with a larger shift, while the parasitic lines are generated during the lower laser intensities with a less shift. The rapid variation of a laser intensity manifests itself in different ways in high harmonics generation by neutral atoms under a 25 fs TW laser pulse that the harmonic generation is enhanced and the degree of chirp on the rising edge of a pulse determines the direction as well as the amount of the shifts of harmonic lines [21

21. J. Zhou, J. Peatross, M. M. Murnane, and H. C. Kapteyn, “Enhanced High-Harmonic Generation Using 25 fs Laser Pulses”, Phys. Rev. Lett. **76**, 752 (1996). [CrossRef] [PubMed]

4. E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson Scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E **48**, 3003 (1993). [CrossRef]

20. K. Lee, Y. H. Cha, M. S. Shin, B. H. Kim, and D. Kim, “Relativistic nonlinear Thomson scattering as attosecond x-ray source”, in print in Phys. Rev. E **67**2003. [CrossRef]

^{17}W/cm

^{2}and 10

^{18}W/cm

^{2}, respectively. For a better comparison of the shape of the angular distributions between different harmonics, the intensities are normalized to their maxima. The peaks of the linear scattering, that is, the first order harmonic line lie on the plane perpendicular to the laser field (θ = 0°, 180°), as in usual dipole radiations. As the laser intensity increases, it radiates more in the direction of the laser propagation due to the relativistic effect. For the nonlinear scattering(higher harmonics), the primary peak directions of all the harmonics noticeable in the calculated spectra are plotted, up to the 8th order for 10

^{17}W/cm

^{2}and the 20th order for 10

^{18}W/cm

^{2}. The primary peaks are off the y-z plane (θ = 0°, 180°). As the harmonic order increases, they move from the backward to the forward direction of the laser propagation, approaching the direction of the electron’s velocity denoted by the horizontal line in Fig. 3(c). In the forward direction, there are two small peaks whereas in the backward direction, there exists the different number of peaks for different harmonic orders. The number of peaks, excluding two small forward peaks, is equal to the harmonic order number. The secondary peaks get weaker relative to the primary peaks as the harmonic order increases. The contour plots of the angular distributions (θ, ϕ) for the laser intensity of 10

^{18}W/cm

^{2}are plotted in Fig. 4. For the contour plot of each harmonic (Fig. 4(b)–(g)), the logarithmic scale is used to display detailed structures clearly. In the backward radiation, it can be noticed that there are line-shaped nodes perpendicular to the direction of the laser electric field and the number of the nodes is the harmonic order number minus one.

^{17}W/cm

^{2}is just 2 % of the unshifted frequency. This is why the small peaks that appear in the case of 10

^{18}W/cm

^{2}do not appear in the case of 10

^{17}W/cm

^{2}(Fig. 1(a)).

**48**, 3003 (1993). [CrossRef]

13. E. S. Sarachik and G. T. Schappert, “Classical Theory of the Scattering of Intense Laser Radiation by Free Electrons,” Phys. Rev. D **1**, 2738 (1970). [CrossRef]

19. R. E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,”, Phys. Rev. A **60**, 3233 (1999). [CrossRef]

^{17}W/cm

^{2}and 10

^{18}W/cm

^{2}amounts to 0.022 and 0.19, respectively. This drift motion of an electron along the laser propagation makes the electron feel a longer oscillation period (

*T*) than that of the laser field oscillation (

_{e}*T*).

_{L}*r*⃗(

*t*) is the position of the electron. ω

_{L}and

*k*⃗

_{L}=

*k*

_{L}

*z*̂ are the angular frequency and the wave vector of the driving laser, respectively. With

*t*′ =

*t*+

*T*

_{e}and

*z*′ =

*z*+ Δ

*z*(see Fig. 5), the oscillation period seen by the electron is given by

*c*is the speed of light. This is just a mathematical statement that the laser pulse propagates more than the electron by its wavelength during one cycle of the electron’s motion. Obtaining Δ

*z*form the solution of the electron’s motion under a laser pulse [4

**48**, 3003 (1993). [CrossRef]

*T*

_{e}and

*T*

_{L}is obtained from Eq. (4).

*T*

_{D}, detected by a detector located at a far distance is the difference between the arrival times of the radiations departed from A and B in Fig. 5:

*T*is the same as

_{D}*T*and at θ = 90°, the detector sees the same oscillation period as that of the electron oscillation. Then the insertion of Eq. (5) into Eq. (8) using

_{L}*E*=

*h*/

*T*reproduces Eq. (2). In Fig. 6 (a), the simulation and analytic results for the

*T*and

_{D}*E*at a laser intensity of 10

_{s}^{18}W/cm

^{2}are compared. In this figure, the simulation results for

*T*and the

_{D}*E*(solid squares and circles, respectively) are obtained from radiation power and spectrum, respectively. The oscillation of the electron plotted in Fig. 6(b) shows that the period of the electron’s oscillation is the same as

_{s}*T*at θ = 90°.

_{D}## 3. Summary

## Acknowledgment

## References and links

1. | M. D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers,” Science |

2. | G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-Intensity Lasers: Physics of the Extreme on a Tabletop,” Phys. Today |

3. | P. Sprangle, A. Ting, E. Esarey, and A. Fisher, “Tunable, short pulse hard x-rays from a compact laser synchrotron source,” J. Appl. Phys. |

4. | E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson Scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E |

5. | F. V. Hartemann, “High-intensity scattering processes of relativistic electrons in vacuum,” Phys. Plasmas |

6. | Y. Ueshima, Y. Kishimoto, A. Sasaki, and T. Tajima, “Laser Larmor X-ray radiation from low-Z matter,” Laser and Particle Beams |

7. | A. E. Kaplan and P. L. Shkolnikov, “Lasetron: A Proposed Source of Powerful Nuclear-Time-Scale Electromagnetic Bursts,” Phys. Rev. Lett. |

8. | Dong-Eon Kim, Csaba Toth, and Christopher P. J. Barty, “Population inversion between atomic inner-shell vacancy states created by electron-impact ionization and Coster-Kronig decay,” Phys. Rev. A Rap. Comm. |

9. | D. Kim, S. H. Son, J. H. Kim, C. Toth, and C. P. J. Barty, “Gain characteristics of inner-shell photoionization-pumped L23M1 transition in Ca,” Phys. Rev. A |

10. | J. Sheffield, |

11. | Vachaspati, “Harmonics in the Scattering of Light by Free Electrons,” Phys. Rev |

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

13. | E. S. Sarachik and G. T. Schappert, “Classical Theory of the Scattering of Intense Laser Radiation by Free Electrons,” Phys. Rev. D |

14. | E. Esarey and P. Sprangle, “Generation of stimulated backscattered harmonic radiation from intense-laser interactions with beams and plasmas,” Phys. Rev. A |

15. | E. Esarey, A. Ting, P. Sprangle, D. Umstadter, and X. Liu, “Nonlinear analysis of relativistic harmonic generation by intense lasers in plasmas,” IEEE Trans. Plasma Sci. |

16. | Wei Yu, M. Y. Yu, J. X. Ma, and Z. Xu, “Strong frequency up-conversion by nonlinear Thomson scattering from relativistic electrons,” Phys. Plasmas |

17. | S.-Y. Chen, A. Maksimchuk, and D. Umstadter, “Experimental observation of relativistic nonlinear Thomson scattering,” Nature (London) |

18. | S.-Y. Chen, A. Maksimchuk, E. Esarey, and D. Umstadter, “Observation of Phase-Matched Relativistic Harmonic Generation,” Phys. Rev. Lett. |

19. | R. E. Wagner, Q. Su, and R. Grobe, “High-order harmonic generation in relativistic ionization of magnetically dressed atoms,”, Phys. Rev. A |

20. | K. Lee, Y. H. Cha, M. S. Shin, B. H. Kim, and D. Kim, “Relativistic nonlinear Thomson scattering as attosecond x-ray source”, in print in Phys. Rev. E |

21. | J. Zhou, J. Peatross, M. M. Murnane, and H. C. Kapteyn, “Enhanced High-Harmonic Generation Using 25 fs Laser Pulses”, Phys. Rev. Lett. |

**OCIS Codes**

(190.5890) Nonlinear optics : Scattering, stimulated

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(350.4990) Other areas of optics : Particles

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 17, 2002

Revised Manuscript: February 13, 2003

Published: February 24, 2003

**Citation**

Kitae Lee, Y. Cha, M. Shin, B. Kim, and D. Kim, "Temporal and spatial characterization of harmonics structures of relativistic nonlinear Thomson scattering," Opt. Express **11**, 309-316 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-4-309

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### References

- M. D. Perry and G. Mourou, "Terawatt to Petawatt Subpicosecond Lasers,�?? Science 264, 917 (1994) [CrossRef] [PubMed]
- G.A. Mourou, C. P. J. Barty, and M.D. Perry, �??Ultrahigh-Intensity Lasers: Physics of the Extreme on a Tabletop,�?? Phys. Today 51, 22 (1998). [CrossRef]
- P. Sprangle, A. Ting, E. Esarey, and A. Fisher, "Tunable, short pulse hard x-rays from a compact laser synchrotron source,�?? J. Appl. Phys. 72, 5032 (1992). [CrossRef]
- E. Esarey, S. K. Ride, and P. Sprangle, �??Nonlinear Thomson Scattering of intense laser pulses from beams and plasmas,�?? Phys. Rev. E 48, 3003 (1993). [CrossRef]
- F. V. Hartemann, �??High-intensity scattering processes of relativistic electrons in vacuum,�?? Phys. Plasmas 5, 2037 (1998). [CrossRef]
- Y. Ueshima, Y. Kishimoto, A. Sasaki, and T. Tajima, �??Laser Larmor X-ray radiation from low-Z matter,�?? Laser and Particle Beams 17, 45 (1999). [CrossRef]
- A. E. Kaplan and P. L. Shkolnikov, �??Lasetron: A Proposed Source of Powerful Nuclear-Time-Scale Electromagnetic Bursts,�?? Phys. Rev. Lett. 88, 074801 (2002) [CrossRef] [PubMed]
- Dong-Eon Kim, Csaba Toth, and Christopher P. J. Barty, �??Population inversion between atomic inner-shell vacancy states created by electron-impact ionization and Coster-Kronig decay,�?? Phys. Rev. A Rap. Comm. 59, R4129 (1999)
- D. Kim, S. H. Son, J. H. Kim, C. Toth, and C. P. J. Barty, �??Gain characteristics of inner-shell photoionization-pumped L23M1 transition in Ca,�?? Phys. Rev. A 63, 023806 (2001). [CrossRef]
- J. Sheffield, Plasma Scattering of Electromagnetic Radiation, (Academic Press, New York, 1975).
- Vachaspati, �??Harmonics in the Scattering of Light by Free Electrons,�?? Phys. Rev. 128, 664 (1962). [CrossRef]
- L. S. Brown and T. W. B. Kibble, �??Interaction of Intense Laser Beams with Electrons,�?? Phys. Rev. 133, A705 (1964). [CrossRef]
- E. S. Sarachik and G. T. Schappert, �??Classical Theory of the Scattering of Intense Laser Radiation by Free Electrons,�?? Phys. Rev. D 1, 2738 (1970). [CrossRef]
- E. Esarey and P. Sprangle, �??Generation of stimulated backscattered harmonic radiation from intense-laser interactions with beams and plasmas,�?? Phys. Rev. A 45, 5872 (1992). [CrossRef] [PubMed]
- E. Esarey, A. Ting, P. Sprangle, D. Umstadter, and X. Liu, �??Nonlinear analysis of relativistic harmonic generation by intense lasers in plasmas,�?? IEEE Trans. Plasma Sci. 21, 95 (1993). [CrossRef]
- Wei Yu, M. Y. Yu, J. X. Ma, Z. Xu, �??Strong frequency up-conversion by nonlinear Thomson scattering from relativistic electrons,�?? Phys. Plasmas 5, 406 (1998). [CrossRef]
- S.-Y. Chen, A. Maksimchuk, and D. Umstadter, �??Experimental observation of relativistic nonlinear Thomson scattering,�?? Nature (London) 396, 653 (1998). [CrossRef]
- S.-Y. Chen, A. Maksimchuk, E. Esarey, and D. Umstadter, �??Observation of Phase-Matched Relativistic Harmonic Generation,�?? Phys. Rev. Lett. 84, 5528 (2000). [CrossRef] [PubMed]
- R. E. Wagner, Q. Su, and R. Grobe, �??High-order harmonic generation in relativistic ionization of magnetically dressed atoms,�?? Phys. Rev. A 60, 3233 (1999). [CrossRef]
- K. Lee, Y. H. Cha, M. S. Shin, B. H. Kim, and D. Kim, �??Relativistic nonlinear Thomson scattering as attosecond x-ray source,�?? in print in Phys. Rev. E 67 2003. [CrossRef]
- J. Zhou, J. Peatross, M. M. Murnane, and H. C. Kapteyn, �??Enhanced High-Harmonic Generation Using 25 fs Laser Pulses,�?? Phys. Rev. Lett. 76, 752 (1996). [CrossRef] [PubMed]

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