## Simulations of vacuum laser acceleration: Hidden errors from particle’s initial positions

Optics Express, Vol. 18, Issue 13, pp. 14144-14151 (2010)

http://dx.doi.org/10.1364/OE.18.014144

Acrobat PDF (956 KB)

### Abstract

Simulation of vacuum laser acceleration, because of its scheme’s simplicity, attracts many people involved in. However, how to put the particle in the initial positions in the field has not been considered seriously in some such schemes. An inattentive choice of electron’s initial conditions may lead to misleading results. Here we show that arbitrarily placing the particle within the laser field leads to an overestimation of its energy gain, and offer suggestions for selecting appropriate initial conditions.

© 2010 OSA

## 1. Introduction

1. K. Shimoda, “Proposal for an electron accelerator using an optical maser,” Appl. Opt. **1**(1), 33–36 (1962). [CrossRef]

2. T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. **43**(4), 267–270 (1979). [CrossRef]

3. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. **56**(3), 219–221 (1985). [CrossRef]

4. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. **78**(2), 309–371 (2006). [CrossRef]

5. Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. **427**(2-3), 41–155 (2006). [CrossRef]

6. V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. **4**(6), 447–453 (2008). [CrossRef]

7. G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. **86**(23), 5274–5277 (2001). [CrossRef] [PubMed]

8. G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. **78**(17), 3314–3317 (1997). [CrossRef]

*Rayleigh*length. The particles are usually given initial positions close to the focus, but may have zero or nonzero velocities. Some works begin simulating the particles at an early time, before the arrival of the pulse, while others begin with the pulse contained the particles or even use a continuous laser beam. In the latter cases, the particles are in a strong laser field region from the very start, so would have required a large injection energy to reach that point. Even in the first case, the initial distance between the particle and the laser pulse may also be too short for the field to be negligible. It is important for VLA simulations to take the injection energy into account when predicting the energy gain of the particle.

9. S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. **66**(16), 2072–2075 (1991). [CrossRef] [PubMed]

## 2. Simulation method

*ω*is the laser’s angular frequency. The electron’s charge and rest mass are

*e*and

*ω*, length in units of 1/

*k*(

*k*is the laser wave number), momentum in units of

10. P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. **78**(15), 2253–2255 (2001). [CrossRef]

11. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. **88**(9), 095005 (2002). [CrossRef] [PubMed]

*x*-direction and propagating along the

*z*-axis is described as follows [12]:where

*Rayleigh*length, and

*ωτ*) or higher have been neglected [13

13. J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. **85**(17), 3705–3707 (2004). [CrossRef]

13. J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. **85**(17), 3705–3707 (2004). [CrossRef]

14. Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. **8**(8), 133 (2006). [CrossRef]

## 3. Some typical examples

*t*is shown in Fig. 2(a) (solid line). Its outgoing energy is

*z*-coordinate is shown in Fig. 2(c). The dotted line is a time-reversed simulation, while the dot-dashed line is for a laser propagating along the negative

*z*-axis.

*z*-direction: at rest (

*Rayleigh*lengths behind the laser focus (Nos. 10, 11, 12); and e) the electron is at the origin and the center of the laser pulse is five pulse durations behind (Nos. 13, 14, 15). For each case we run the simulation forwards and backwards to determine the outgoing energy

15. T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. **16**(23), 1054–1056 (1966). [CrossRef]

7. G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. **86**(23), 5274–5277 (2001). [CrossRef] [PubMed]

*i. e.*

5. Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. **427**(2-3), 41–155 (2006). [CrossRef]

16. B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(3), 3719–3732 (1998). [CrossRef]

17. P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids **16**(2), 321–328 (1973). [CrossRef]

18. E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **48**(4), 3003–3021 (1993). [CrossRef] [PubMed]

8. G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. **78**(17), 3314–3317 (1997). [CrossRef]

16. B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(3), 3719–3732 (1998). [CrossRef]

## 4. Identifying proper initial positions

*ε*.

7. G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. **86**(23), 5274–5277 (2001). [CrossRef] [PubMed]

11. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. **88**(9), 095005 (2002). [CrossRef] [PubMed]

*x-y*plane (

*z*= 0) at time

*t*= 0 in a state of free motion, uninfluenced by the laser field. The electron’s initial velocity is

*ε*is the small threshold mentioned above. From Eq. (3), we find that in order to satisfy

*τ*,

19. C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B **74**, s83–s88 (2002). [CrossRef]

*ε*should be chosen differently under the same initial conditions.

*z*-axis with various kinematic parameters. The solid circles in this plot correspond to the cases listed in Table 1. In Fig. 3 we see that the absolute value of

*ε*could be larger in this case. The dotted and dot-dashed curves in Fig. 3 show that changing the waist size

## 5. Sideways injection examples

10. P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. **78**(15), 2253–2255 (2001). [CrossRef]

11. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. **88**(9), 095005 (2002). [CrossRef] [PubMed]

*x-z*plane sideways injection at

*z*-axis) into the focus. The initial velocity is

*Rayleigh*length from the focus. Figure 4(a) also reveals that the best acceleration case may not be the maximum impact case. For some phases, the net energy gain is negative (shown with an arrow in Fig. 4(a)). In the simulations of Fig. 4(a),

*Rayleigh*lengths away from the focus. A brief calculation gives

## 6. Summary

## Acknowledgments

## References and links

1. | K. Shimoda, “Proposal for an electron accelerator using an optical maser,” Appl. Opt. |

2. | T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. |

3. | D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. |

4. | G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. |

5. | Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. |

6. | V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. |

7. | G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. |

8. | G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. |

9. | S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. |

10. | P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. |

11. | Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. |

12. | E. Siegman Lasers, (University Science Books, Mill Valley, California, 1986). |

13. | J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. |

14. | Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. |

15. | T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. |

16. | B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

17. | P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids |

18. | E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

19. | C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B |

**OCIS Codes**

(320.7120) Ultrafast optics : Ultrafast phenomena

(350.4990) Other areas of optics : Particles

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: May 5, 2010

Revised Manuscript: June 6, 2010

Manuscript Accepted: June 8, 2010

Published: June 16, 2010

**Citation**

P. X. Wang, S. Kawata, and Y. K. Ho, "Simulations of vacuum laser acceleration: Hidden errors from particle’s initial positions," Opt. Express **18**, 14144-14151 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14144

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### References

- K. Shimoda, “Proposal for an electron accelerator using an optical maser,” Appl. Opt. 1(1), 33–36 (1962). [CrossRef]
- T. Tajima and J. M. Dawson, “Laser electron-accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979). [CrossRef]
- D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985). [CrossRef]
- G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006). [CrossRef]
- Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser–matter interactions,” Phys. Rep. 427(2-3), 41–155 (2006). [CrossRef]
- V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles and applications of compact laser–plasma accelerators,” Nat. Phys. 4(6), 447–453 (2008). [CrossRef]
- G. V. Stupakov and M. S. Zolotorev, “Ponderomotive laser acceleration and focusing in vacuum for generation of attosecond electron bunches,” Phys. Rev. Lett. 86(23), 5274–5277 (2001). [CrossRef] [PubMed]
- G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electrons accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78(17), 3314–3317 (1997). [CrossRef]
- S. Kawata, T. Maruyama, H. Watanabe, and I. Takahashi , “Inverse-bremsstrahlung electron acceleration,” Phys. Rev. Lett. 66(16), 2072–2075 (1991). [CrossRef] [PubMed]
- P. X. Wang, Y. K. Ho, X. Q. Yuan, Q. Kong, N. Cao, A. M. Sessler, E. Esarey, and Y. Nishida, “Vacuum electron acceleration by an intense laser,” Appl. Phys. Lett. 78(15), 2253–2255 (2001). [CrossRef]
- Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002). [CrossRef] [PubMed]
- E. Siegman Lasers, (University Science Books, Mill Valley, California, 1986).
- J. F. Hua, Y. K. Ho, Y. Z. Lin, Z. Chen, Y. J. Xie, S. Y. Zhang, Z. Yan, and J. J. Xu, “High-order corrected fields of ultrashort, tightly focused laser pulses,” Appl. Phys. Lett. 85(17), 3705–3707 (2004). [CrossRef]
- Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006). [CrossRef]
- T. W. Kibble, “Refraction of electron beams by intense electromagnetic waves,” Phys. Rev. Lett. 16(23), 1054–1056 (1966). [CrossRef]
- B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3719–3732 (1998). [CrossRef]
- P. K. Kaw and R. M. Kulsrud, “Relativistic acceleration of charged particles by superintense laser beams,” Phys. Fluids 16(2), 321–328 (1973). [CrossRef]
- E. Esarey, S. K. Ride, and P. Sprangle, “Nonlinear Thomson scattering of intense laser pulses from beams and plasmas,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 48(4), 3003–3021 (1993). [CrossRef] [PubMed]
- C. Varin and M. Piché, “Acceleration of ultra-relativistic electrons using high-intensity laser beams,” Appl. Phys. B 74, s83–s88 (2002). [CrossRef]

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