## Acceleration of electrons by a tightly focused intense laser beam

Optics Express, Vol. 17, Issue 14, pp. 11850-11859 (2009)

http://dx.doi.org/10.1364/OE.17.011850

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### Abstract

The recent proposal to use Weinger transformation field (WTF) [Opt. Express 17, 4959-4969 (2009)] for describing tightly focused laser beams is investigated here in detail. In order to validate the accuracy of WTF, we derive the numerical field (NF) from the plane wave spectrum method. WTF is compared with NF and Lax series field (LSF). Results show that LSF is accurate close to the beam axis and divergent far from the beam axis, and WTF is always accurate. Moreover, electron dynamics in a tightly focused intense laser beam are simulated by LSF, WTF and NF, respectively. The results obtained by WTF are shown to be accurate.

© 2009 Optical Society of America

## 1. Introduction

1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. **55(6)**, 447–449 (1985).
[CrossRef]

2. 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]

4. Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. ST Accel. Beams **5(10)**, 101301 (2002).
[CrossRef]

*w*

_{0}is much larger than the laser wavelength

*λ*. If a laser beam is focused down to the order of the laser wavelength, a Gaussian beam description becomes insufficient. In 1975, Lax

*et al*. [5

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

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

7. 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]

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

9. 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 tightly focused laser beams

### 2.1 Lax series approach and Weniger transformation

*x*direction and propagates along the

*z*axis. The electromagnetic field can be described in form of the vector potential

**A**=

*x̂A*

_{0}

*ψ*(

**r**)exp(

*iη*), where

**A**

_{0}is a constant amplitude, and

*η*=

*ωt-k*. The vector potential satisfies the following wave equation:

_{z}*x*=

*ξw*

_{0},

*y*=

*υw*

_{0},

*z*=ζ

*z*, where

_{r}*w*

_{0}and

*z*=

_{r}*kw*

^{2}

_{0}/2 are the beam radius and Rayleigh length, respectively. Equation (2) can be rewritten as

^{2}

_{⊥}=

*∂*

^{2}/

*∂ξ*

^{2}+

*∂*

^{2}/

*∂υ*

^{2}and the diffraction angle

*ε*=

*w*

_{o}/

*z*. Since

_{r}*ε*

^{2}is small, one can expand

*ψ*as a sum of even power of

*ε*[5

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

*ψ*and the purpose of gaining physical insight, we follow the work of Davis

_{2n}*et al*[10

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

*z*axis from the origin. Such a wave has an exponential factor, which can be expanded as

*z*≫

*z*the condition

_{r}*f*→

*iz*holds, this line of reasoning suggests that

_{r}/z*C*

_{2n}is determined by inserting this function into Eq. (6).

*ψ*

_{2n}can be easily derived by simple recurrence relations [11

11. 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]

*D*

_{0}=1, and

*a*

_{2n}(

*ρ,f*) is given by the Eq. (8). These recurrence relations can be used to obtain accurate results with arbitrary order of

*ε*. Such as,

*ϕ*=

*i*∇·

**A**/

*k*by using the Lorentz gauge. Then, the components of LSF can be obtained by substituting

**A**and

*ϕ*into Maxwell’s equations

**E**=-

*ik*

**A**-∇

*ϕ*and

**B**=∇×

**A**, accurate up to

*ε*

^{2m+1}, as [7

7. 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]

12. Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B **86(2)**, 319–326 (2007).
[CrossRef]

*m*≥0,

*E*=

*E*

_{o}exp[

*i*(

*ωt*-

*kz*+

*ϕ*

_{0}),

*E*

_{0}=

*kA*

_{0}and

*ϕ*

_{0}is the constant phase. For example, the components of electromagnetic field accurate up to

*ε*

^{7}can be expressed as

13. 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]

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

7. 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]

*s*=∑

_{m}*=0*

^{m}_{n}*a*(

_{n}*m*≥0), can convert them into the following sequence:

*b*)

*denotes the Pochhammer symbol.*

_{j}*z*component of Eq. (16), accurate to

*ε*

^{2m+1}, can be rewritten by using Weniger transformation:

*s*=∑

^{ez}_{z}

^{n}_{i=0}

*ε*

^{2i+1}

*E*(

^{z}_{i}*f,ρ*). Equation (27) gives the

*z*component of electric field of WTF, and other components can be obtained by the same process.

### 2.2 Plane wave spectrum method

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

9. 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]

**i, j**, and

**k**to denote unit vectors of in the positive

*x, y*and

*z*directions. Equations (28) and (29) are the exact solutions of Maxwell’s equations when the boundary value

*ψ*(

*x, y*, 0) of one of the components of the Hertz vector is given in the

*z*=0 plane. The Gaussian function exp(-

*ρ*

^{2}) is usually chosen as the boundary value. However, for a non-paraxial laser beam, the terms of high order corrections should be included due to the Lax series theory [5

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

1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. **55(6)**, 447–449 (1985).
[CrossRef]

*J*is the Bessel function,

_{n}*n*order, first kind.

*P*(

*κ*), which is derived from the boundary field

*ψ*(

*x, y,*0). Therefore, the accurate boundary field should be taken into account.

*z*component of NF for different beam waist sizes in Fig. 1. For the beam waist

*w*

_{0}=5

*λ*, the propagating field derived by the boundary field of

*ε*

^{0}model is same as that obtained by high order model, as shown in Fig. 1(a). On the contrary, in the case of

*w*

_{0}=

*λ*, the propagating field based on the

*ε*

^{0}model deviates from that of the high order model, as shown in Fig. 1(b). Therefore, the terms of

*ε*

^{2}should be included in the boundary field. In Fig. 1(c), we can see that the boundary field of

*ε*

^{2}model is also inaccurate for

*w*

_{0}=0.5

*λ*. The terms of

*ε*

^{4}should be included into boundary field. Thus, we should consider the high order correction for a tightly focused laser beam.

## 3. Results and discussion

### 3.1 Comparison of LSF, WTF and NF

*z*and

*x*components for a tightly focused beam with waist

*w*

_{0}=

*λ*. The boundary field of NF is accurate to

*ε*

^{2}. Results of Figs. 2(a) and 2(c) show that LSF is accurate close to the beam axis, and divergent far from the beam axis. The divergence of Lax series becomes more serious as increasing of the order of

*ε*. We employ Weniger transformation to eliminate the divergence of LSF and obtain WTF, which are given in Figs. 2(b) and 2(d). It can be seen that the divergence of LSF has been eliminated. Moreover, WTF is accurate more and more as increasing of the order of

*ε*.

### 3.2 Simulation of electron dynamics in a tightly focused laser beam

2. 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]

4. Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. ST Accel. Beams **5(10)**, 101301 (2002).
[CrossRef]

**p**=

*γmc*

**β**, the energy

*χ*=

*γmc*

^{2}, the Lorentz factor

*γ*=(1-

*β*

^{2})-

^{1/2}, and

**β**is the velocity scaled by

*c*, the speed of light in vacuum. The peak field intensity I

_{0}, will be given in terms of the dimensionless parameter

*q*=

*eE*

_{0}/

*mcω*, where I

_{0}

*λ*

^{2}≈1.375×10

^{18}

*q*

^{2}(W/cm

^{2})(

*µ*m)

^{2}. The boundaries of the beam are described by the curves

*x*=±

*w*(

*z*), where

*w*(

*z*)=

*w*

_{0}[1+(

*z/z*)

_{r}^{2}]

^{1/2}. An electron will be transmitted if its trajectory crosses the line

*x*=

*w*(

*z*), and will be reflected if its trajectory crosses the line

*x*=-

*w*(

*z*) twice or never. Otherwise, it will be captured by the beam.

**17(7)**, 4959–4969 (2009).
[CrossRef]

*ε*

^{7}and

*ε*

^{39}, respectively, due to that as the correction order of

*ε*increases, the divergence of LSF is more serious and WTF is more accurate. It is clearly shown that for all cases electron dynamics of WTF are consistent with those of NF, but, electron dynamics of LSF deviate from those of NF. Hence, electron dynamics of WTF are accurate.

## 4. Conclusion

## Acknowledgements

## References and links

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

2. | 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. |

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

4. | Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. ST Accel. Beams |

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

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

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

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

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

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

11. | 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. |

12. | Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B |

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

14. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 8, 2009

Revised Manuscript: June 15, 2009

Manuscript Accepted: June 16, 2009

Published: June 29, 2009

**Citation**

Jian-Xing Li, Wei-Ping Zang, Ya-Dong Li, and Jian-Guo Tian, "Acceleration of electrons by a tightly focused intense laser beam," Opt. Express **17**, 11850-11859 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11850

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### References

- D. Strickland, and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 55(6), 447-449 (1985). [CrossRef]
- 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, and C. H. Keitel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88(9), 095005 (2002). [CrossRef]
- Y. I. Salamin, G. R. Mocken, and C. H. Keitel, "Electron scattering and acceleration by a tightly focused laser beam," Phys. Rev. ST Accel. Beams 5(10), 101301 (2002). [CrossRef]
- M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11(4), 1365-1370 (1975). [CrossRef]
- R. Borghi, and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28(10), 774-776 (2003). [CrossRef]
- 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]
- 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]
- L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19(3), 1177-1179 (1979). [CrossRef]
- 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]
- Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86(2), 319-326 (2007). [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]
- B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. A 253(1274), 358-379 (1959). [CrossRef]

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