## Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method

Optics Express, Vol. 17, Issue 7, pp. 4959-4969 (2009)

http://dx.doi.org/10.1364/OE.17.004959

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### Abstract

For an electron accelerated by a tightly focused Gaussian laser beam, its dynamics are usually simulated through the field obtained by Lax approach [Phys. Rev. A 11, 1365 (1975)]. However, as Lax series field (LSF) is not always convergent, the obtained results are usually inaccurate and even illogical. Here we report that the divergence of LSF can be eliminated by using Weniger transformation, and the electron dynamics simulated by this new field are logical and accurate.

© 2009 Optical Society of America

## 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**, 160–162 (1999). [CrossRef]

^{21}W/cm

^{2}. Laser-based accelerators [2

2. E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma-based accelerator concepts,” IEEE Trans. Plasma Sci. **24**, 252–288 (1996). [CrossRef]

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

4. 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**, 7–15 (2002). [CrossRef]

5. A. Sokolow, J. M. M. Pfannes, R. L. Doney, M. Nakagawa, J. H. Agui, and S. Sen, “Absorption of short duration pulses by small, scalable, tapered granular chains,” Appl. Phys. Lett. **87**, 254104 (2005). [CrossRef]

*w*

_{0}is much larger than the laser wavelength

*λ*. However, if a laser beam is focused down to the order of the laser wavelength, a Gaussian beam description becomes inaccurate. Even for a laser beam that is linearly polarized outside the focal region, tight focusing results in non-Gaussian field components in all three dimensions [6

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

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

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

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

## 2. Laser field descriptions: LSF and WTF

*x*direction and propagates along the

*z*axis, and its electromagnetic field can be described in form of the vector potential

**A**=

*x*̂

**A**

_{0}

*ψ*(

**r**) exp(

*iη*), where

*A*

_{0}is a constant amplitude, and

*η*=

*ωt*-

*kz*. The vector potential satisfies the following wave equation:

*x*=

*ξw*

_{0},

*y*=

*υw*

_{0},

*z*=

*ζz*, where

_{r}*w*

_{0}is the beam radius, and

*z*= k

_{r}*w*

_{0}

^{2}/2, Eq. (2) can be rewritten as

_{⊥}

^{2}=

*∂*

^{2}/

*∂ξ*

^{2}+

*∂*

^{2}/

*∂υ*

^{2}and

*ε*=

*w*

_{0}/

*z*is the diffraction angle. Since

_{r}*ε*

^{2}is small, one can expand [8

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

*ψ*as a sum of even power of

*ε*,

*ψ*

_{2n}and the purpose of gaining physical insight, we follow the work of Davis

*et al*[8

8. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**, 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*/

_{r}*z*holds, this line of reasoning suggests that

*C*

_{2n}is determined by inserting this function into Eq. (6), and this process is complicated. However,

*ψ*

_{2n}can be easily derived by simple recurrence relations [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**, 1692–1694 (2007). [CrossRef] [PubMed]

*C*

_{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*. After getting vector potential, one can obtain the scalar potential

*ϕ*=

*i*∇ ·

**A**/

*k*by using the Lorentz gauge. And then, the electromagnetic field components can be obtained with

**E**= -

*ik*

**A**- ∇

*ϕ*and

**B**= ∇ ×

**A**, as [10

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

*E*

_{0}=

*kA*

_{0}and

*ε*

_{0}is the initial phase.

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

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

*s*= Σ

_{n}^{n}

_{j=0}

*a*(

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

*b*)

_{m}denotes the Pochhammer symbol.

*x*component of Eq. (11), accurate to

*ε*

^{2n}, can be rewritten by using Weniger transformation:

*s*= Σ

_{j}^{j}

_{i=0}

*ε*

^{2i}

*E*(

^{x}_{i}*f*,

*ρ*,

*ξ*).

14. A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Optics.Comm. **136**, 114–124 (1997). [CrossRef]

*x*and

*z*components of electric field in a line of

*x*= 0, perpendicular to the beam axis at

*z*= 19

*z*. Electric field

_{r}*x*and

*z*components of LSF with different order corrections are shown in Figs. 1(a) and 1(c). The exact numerical field is obtained by the plane wave spectrum method, whose boundary value is defined by

*ψ*

_{2n}is the 2

*n*-order correction to the

*ψ*

_{0}, derived by Eq. (10). For a waist radius of the order of wavelength, it is enough to set

*m*= 4 to get a high accuracy. Results show that LSF is accurate inside beam boundary and divergent outside beam boundary. Moreover, the divergence of LSF becomes more serious as the order of

*ε*increases. Consequently, higher-order correction terms of Lax series do not always bring better results. To eliminate the divergence of LSF, we employ Weniger transformation to handle LSF, and the results are given in Figs. 1(b) and 1(d) respectively. It can be seen that the divergence of LSF has been eliminated. Moreover, WTF is more accurate when the higher order terms of

*ε*are included. In the following sections we will use WTF to discuss electron acceleration and compare the results with those calculated by LSF.

## 3. Electron acceleration and numerical simulations

**p**=

*γmc*

**β**, the energy

*χ*=

*γmc*

^{2}, the Lorentz factor

*γ*= (1-

*β*

^{2})

^{-1/2}, and

**β**is the velocity normalized by the speed of light

*c*in vacuum. The peak field intensity I

_{0}can be expressed in terms of I

_{0}

*λ*

^{2}≈ 1.375×10

^{18}

*q*

^{2}(W/cm

^{2})(

*μ*m)

^{2}, where

*q*=

*eE*

_{0}/

*mcω*. The boundary of the beam is described by the curves in the

*xz*plane defined by

*x*= ±

*w*(

*z*),

*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 it will be reflected if its trajectory crosses the line

*x*= -

*w*(

*z*) twice or never crosses it. Otherwise, it will be captured by the beam. Strictly speaking, the field intensity on the curves

*x*= ±

*w*(

*z*) falls down to 1/

*e*

^{2}of its maximum value on axis. Thus a transmitted or reflected electron will be weakly affected by the laser field beyond the beam boundary, and its energy should keep constant or vary slightly when it moves far away from the focus.

## 3.2 Capture

## 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. | E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Overview of plasma-based accelerator concepts,” IEEE Trans. Plasma Sci. |

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

4. | 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 |

5. | A. Sokolow, J. M. M. Pfannes, R. L. Doney, M. Nakagawa, J. H. Agui, and S. Sen, “Absorption of short duration pulses by small, scalable, tapered granular chains,” Appl. Phys. Lett. |

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

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

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

9. | J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for fundamental Gaussian beam,” J. Appl. Phys. |

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

11. | R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” 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. | E. J. Weniger, “Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, ” Comput. Phys. Rep. |

14. | A. Doicu and T. Wriedt, “Plane wave spectrum of electromagnetic beams,” Optics.Comm. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: December 9, 2008

Revised Manuscript: February 10, 2009

Manuscript Accepted: February 11, 2009

Published: March 16, 2009

**Citation**

Jianxing Li, Weiping Zang, and Jianguo Tian, "Simulation of Gaussian laser beams and electron dynamics by Weniger transformation method," Opt. Express **17**, 4959-4969 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4959

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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, V. Yanovsky, "Petawatt laser pulses," Opt. Lett. 24, 160-162 (1999). [CrossRef]
- Q1. E. Esarey, P. Sprangle, J. Krall, and A. Ting, "Overview of plasma-based accelerator concepts," IEEE Trans. Plasma Sci. 24, 252-288 (1996). [CrossRef]
- Y. I. Salamin and C. H. Kertel, "Electron acceleration by a tightly focused laser beam," Phys. Rev. Lett. 88, 095005 (2002). [CrossRef] [PubMed]
- N. Cao, Y. K. Ho, Q. Kong, P. X. Wang, X. Q. Yuan, Y. Nishida, N. Yugami, H. Ito, "Accurate description of Gaussian laser beams and electron dynamics," Opt. Commun 204, 7-15 (2002). [CrossRef]
- A. Sokolow, J. M. M. Pfannes, R. L. Doney, M. Nakagawa, J. H. Agui, and S. Sen, "Absorption of short duration pulses by small, scalable, tapered granular chains," Appl. Phys. Lett. 87, 254104 (2005). [CrossRef]
- S. X. Hu and A. F. Starace, "Laser acceleration of electrons to giga-electron-volt energies using highly charged ions," Phys. Rev. E 73, 066502 (2006). [CrossRef]
- M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975). [CrossRef]
- L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979). [CrossRef]
- J. P. Barton and D. R. Alexander, "Fifth-order corrected electromagnetic field components for fundamental Gaussian beam," J. Appl. Phys. 66, 2800-2802 (1989). [CrossRef]
- Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86, 319-326 (2007). [CrossRef]
- R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28, 774-776 (2003). [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, 1692-1694 (2007). [CrossRef] [PubMed]
- Q2. E. J. Weniger, "Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, " Comput. Phys. Rep. 10, 189-371 (1989). [CrossRef]
- Q3. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Optics.Comm. 136, 114-124 (1997). [CrossRef]

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