## Vacuum electron acceleration driven by a tightly focused radially polarized Gaussian beam |

Optics Express, Vol. 19, Issue 10, pp. 9303-9308 (2011)

http://dx.doi.org/10.1364/OE.19.009303

Acrobat PDF (1002 KB)

### Abstract

Electron acceleration in vacuum driven by a tightly focused radially polarized Gaussian beam has been studied in detail. Weniger transformation method is used to eliminate the divergence of the radially polarized electromagnetic field derived from the Lax series approach. And, electron dynamics in an intense radially polarized Gaussian beam is analyzed by using the Weniger transformation field. The roles of the initial phase of the electromagnetic field and the injection angle, position and energy of electron in energy gain of electron have been studied in detail.

© 2011 OSA

## 1. Introduction

1. M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. **24**(3), 160–162 (1999). [CrossRef]

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

6. J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. **35**(19), 3258–3260 (2010). [CrossRef] [PubMed]

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

7. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**(23), 233901 (2003). [CrossRef] [PubMed]

4. Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A **73**(4), 043402 (2006). [CrossRef]

8. S. X. Hu and A. F. Starace, “Laser acceleration of electrons to giga-electron-volt energies using highly charged ions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(6), 066502 (2006). [CrossRef] [PubMed]

9. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A **11**(4), 1365–1370 (1975). [CrossRef]

10. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**(3), 1177–1179 (1979). [CrossRef]

11. Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. **31**(17), 2619–2621 (2006). [CrossRef] [PubMed]

12. H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. **32**(12), 1692–1694 (2007). [CrossRef] [PubMed]

*ε*in the description of the tightly focused radially polarized Gaussian field. However, Lax series appears to be divergent for nonparaxial beams, and the higher order corrections of Lax series cannot always produce better results [12

12. H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. **32**(12), 1692–1694 (2007). [CrossRef] [PubMed]

15. J. X. Li, W. P. Zang, Y.-D. Li, and J. G. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express **17**(14), 11850–11859 (2009). [CrossRef] [PubMed]

13. R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. **28**(10), 774–776 (2003). [CrossRef] [PubMed]

15. J. X. Li, W. P. Zang, Y.-D. Li, and J. G. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express **17**(14), 11850–11859 (2009). [CrossRef] [PubMed]

16. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. **136**(1-2), 114–124 (1997). [CrossRef]

17. P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. **152**(1-3), 108–118 (1998). [CrossRef]

## 2. Electromagnetic field of a radially polarized Gaussian beam beyond the paraxial approximation

*r*,

*θ*,

*z*), where the

*z*-axis is aligned with the laser beam propagation direction, the vector potential for a radially polarized Gaussian beam is of the formwhere

*A*

_{0}is a constant amplitude. Using Lax series method [9

9. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A **11**(4), 1365–1370 (1975). [CrossRef]

10. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**(3), 1177–1179 (1979). [CrossRef]

*ψ*(

*r*,

*z*) in Eq. (1) [11

11. Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. **31**(17), 2619–2621 (2006). [CrossRef] [PubMed]

12. H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. **32**(12), 1692–1694 (2007). [CrossRef] [PubMed]

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

*ε*

^{2m + 2}can be represented as [12

**32**(12), 1692–1694 (2007). [CrossRef] [PubMed]

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

*kA*

_{0}is field amplitude,

*φ*

_{0}is an initial phase,

*f*=

*i*/(

*i*+

*ζ*),

*ζ*=

*z*/

*z*

_{r},

*z*

_{r}is the Rayleigh length,

*ρ*=

*r*/

*w*

_{0}, and

*w*

_{0}is the beam waist radius. Other components

*E*,

_{θ}*B*, and

_{r}*B*vanish identically.

_{z}*E*

_{2}

*,*

_{n}*E*

_{2n}_{+1}, and

*B*

_{2n}_{+1}, as well as

*ψ*

_{2}

_{n}_{,}are available in Refs. [12

**32**(12), 1692–1694 (2007). [CrossRef] [PubMed]

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

13. R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. **28**(10), 774–776 (2003). [CrossRef] [PubMed]

15. J. X. Li, W. P. Zang, Y.-D. Li, and J. G. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express **17**(14), 11850–11859 (2009). [CrossRef] [PubMed]

19. E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. **10**(5-6), 189–371 (1989). [CrossRef]

21. E. J. Weniger, “Summation of divergent power series by means of factorial series,” Appl. Numer. Math. **60**(12), 1429–1441 (2010). [CrossRef]

*w*(

*z*) =

*w*

_{0}(1 +

*ζ*

^{2})

^{1/2}is the beam radius for an arbitrary

*z*.

16. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. **136**(1-2), 114–124 (1997). [CrossRef]

17. P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. **152**(1-3), 108–118 (1998). [CrossRef]

16. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. **136**(1-2), 114–124 (1997). [CrossRef]

17. P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. **152**(1-3), 108–118 (1998). [CrossRef]

**A**in Eq. (1) can be represented by its angular spectrum representationwhere

*k*

_{x}^{2}

**+**

*k*

_{y}^{2}

**+**

*k*

_{z}^{2}

**=**

*k*

^{2}, andTo simplify Eq. (8), polar coordinates are introducedNote that the coordinates

*x*and

*y*are completely symmetrical, and so

*P*(

*k*,

_{x}*k*) always degenerates into the form

_{y}*P*(

*κ*). Consequently, Eq. (8) can be represented aswhere

*J*

_{0}is the zeroth Bessel function of the first kind. Substituting Eq. (11) into Eqs. (3), the angular spectrum representations of the fields are where

*J*

_{1}is the first Bessel function of the first kind. When exploring the integrals in Eqs. (12)–(14), the integration range can be decomposed into two parts:where

**136**(1-2), 114–124 (1997). [CrossRef]

**152**(1-3), 108–118 (1998). [CrossRef]

*n*= 10 for instance.

*E*and

_{r}*B*both vanish completely in the longitudinal axis and peaks at

_{θ}11. Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. **31**(17), 2619–2621 (2006). [CrossRef] [PubMed]

**8**(8), 133 (2006). [CrossRef]

## 3. Electron dynamics in a tightly focused radially polarized Gaussian laser beam

*d*

**p**/

*dt*= -

*e*(

**E**+

**β**×

**B**), and

*dΞ*/

*dt*= -

*ec*

**β·E**, where the momentum

**p**= γ

*mc*

**β**, the energy

*Ξ*= γ

*mc*

^{2}, the Lorentz factor γ = (1-

**β**

^{2})

^{-1/2}, and

**β**is the velocity scaled by

*c*,

*e*is the charge of an electron. It is convenient to introduce a normalized factor

*q*=

*eE*

_{0}

*/mωc*indicating the intensity of fields. We define the lines

*x*= ±

*w*(

*z*) as the beam boundaries where the field intensity falls to

*e*

^{−2}of its maximum value on the beam axis.

*E*and

_{r}*B*both vanish completely, it only interacts with

_{θ}*E*and may be accelerated to high energy along the longitudinal axis [4

_{z}4. Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A **73**(4), 043402 (2006). [CrossRef]

*θ*

_{0}= 25° relative to the longitudinal axis. As the electron moves outside the longitudinal axis, transverse electric field also makes significant contribution for its energy gain. Due to the intensity diffraction of Gaussian beam, the electron gains its energy mainly in the focal area. As the electron moves far away from the axis, where the field is very weak, it will gain little or no net energy, as simulated by WTF. The divergence of LSF causes the serious errors of electron dynamics simulation. Even though the injected angle of electron is very small, the transverse electric field still deflects the electron outside the beam axis, where LSF is divergent. Therefore, WTF should be used to simulate electron dynamics in a tightly focused radially polarized Gaussian beam instead of LSF.

*θ*

_{0}, position, and energy

*γ*

_{0}of electron in energy gains of electron are studied in Fig. 4 . In Fig. 4(a),

*E*can afford maximum accelerating force at

_{z}*E*initially affords negative or few accelerating force, and the electron will experience accelerating and decelerating phase alternately. So, the electron gains few or no net energy. For electron acceleration by a radially polarized Gaussian beam,

_{z}*E*affords mainly accelerating force for electron energy gain, and

_{z}*E*forces electron to deflect outside the intense field region. Therefore, an electron injected with smaller angle can be accelerated to higher energy, as shown in Fig. 4(b). For the same reason, an electron injected at a closer distance from the axis can gain higher energy, as shown in Fig. 4(c). The phase velocity of field around the longitudinal axis is very fast, an electron with finite initial energy cannot be synchronously accelerated to much high energy, as shown in Fig. 4(d). Furthermore, the results simulated by LSF have obvious differences from those of WTF. Therefore, electron acceleration by an intense radially polarized Gaussian beam should be studied by using WTF.

_{r}## 4. Conclusion

## Acknowledgment

## References and links

1. | M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. |

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

3. | N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. |

4. | Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A |

5. | J. X. Li, W. P. Zang, and J. G. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. |

6. | J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. |

7. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

8. | S. X. Hu and A. F. Starace, “Laser acceleration of electrons to giga-electron-volt energies using highly charged ions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

9. | M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A |

10. | L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A |

11. | Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. |

12. | H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. |

13. | R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. |

14. | J. X. Li, W. P. Zang, and J. G. Tian, “Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method,” Opt. Express |

15. | J. X. Li, W. P. Zang, Y.-D. Li, and J. G. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express |

16. | A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. |

17. | P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. |

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

19. | E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. |

20. | U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, “Resummation of QED perturbation series by sequence transformations and the prediction of perturbative coefficients,” Phys. Rev. Lett. |

21. | E. J. Weniger, “Summation of divergent power series by means of factorial series,” Appl. Numer. Math. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 7, 2011

Revised Manuscript: April 13, 2011

Manuscript Accepted: April 18, 2011

Published: April 27, 2011

**Citation**

Lin Dai, Jian-Xing Li, Wei-Ping Zang, and Jian-Guo Tian, "Vacuum electron acceleration driven by a tightly focused radially polarized Gaussian beam," Opt. Express **19**, 9303-9308 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9303

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### References

- M. D. Perry, D. Pennington, B. C. Stuart, G. Tietbohl, J. A. Britten, C. Brown, S. Herman, B. Golick, M. Kartz, J. Miller, H. T. Powell, M. Vergino, and V. Yanovsky, “Petawatt laser pulses,” Opt. Lett. 24(3), 160–162 (1999). [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]
- N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, and H. Ito, “Accurate description of Gaussian laser beams and electron dynamics,” Opt. Commun. 204(1-6), 7–15 (2002). [CrossRef]
- Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006). [CrossRef]
- J. X. Li, W. P. Zang, and J. G. Tian, “Electron acceleration in vacuum induced by a tightly focused chirped laser pulse,” Appl. Phys. Lett. 96(3), 031103 (2010). [CrossRef]
- J. X. Li, W. P. Zang, and J. G. Tian, “Analysis of electron capture acceleration channel in an Airy beam,” Opt. Lett. 35(19), 3258–3260 (2010). [CrossRef] [PubMed]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
- S. X. Hu and A. F. Starace, “Laser acceleration of electrons to giga-electron-volt energies using highly charged ions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6), 066502 (2006). [CrossRef] [PubMed]
- M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11(4), 1365–1370 (1975). [CrossRef]
- L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19(3), 1177–1179 (1979). [CrossRef]
- Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31(17), 2619–2621 (2006). [CrossRef] [PubMed]
- H. Luo, S. Y. Liu, Z. F. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 32(12), 1692–1694 (2007). [CrossRef] [PubMed]
- R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28(10), 774–776 (2003). [CrossRef] [PubMed]
- J. X. Li, W. P. Zang, and J. G. Tian, “Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method,” Opt. Express 17(7), 4959–4969 (2009). [CrossRef] [PubMed]
- J. X. Li, W. P. Zang, Y.-D. Li, and J. G. Tian, “Acceleration of electrons by a tightly focused intense laser beam,” Opt. Express 17(14), 11850–11859 (2009). [CrossRef] [PubMed]
- A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Opt. Commun. 136(1-2), 114–124 (1997). [CrossRef]
- P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152(1-3), 108–118 (1998). [CrossRef]
- Y. I. Salamin, “Accurate fields of radially polarized Gaussian laser beam,” N. J. Phys. 8(8), 133 (2006). [CrossRef]
- E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series,” Comput. Phys. Rep. 10(5-6), 189–371 (1989). [CrossRef]
- U. D. Jentschura, J. Becher, E. J. Weniger, and G. Soff, “Resummation of QED perturbation series by sequence transformations and the prediction of perturbative coefficients,” Phys. Rev. Lett. 85(12), 2446–2449 (2000). [CrossRef] [PubMed]
- E. J. Weniger, “Summation of divergent power series by means of factorial series,” Appl. Numer. Math. 60(12), 1429–1441 (2010). [CrossRef]

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