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

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  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, V. Yanovsky, "Petawatt laser pulses," Opt. Lett. 24, 160-162 (1999).
    [CrossRef]
  2. 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]
  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, 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]
  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]
  9. 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]
  10. Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86, 319-326 (2007).
    [CrossRef]
  11. R. Borghi and M. Santarsiero, "Summing Lax series for nonparaxial beam propagation," Opt. Lett. 28, 774-776 (2003).
    [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, 1692-1694 (2007).
    [CrossRef] [PubMed]
  13. 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]
  14. Q3. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Optics.Comm. 136, 114-124 (1997).
    [CrossRef]

2007 (2)

2006 (1)

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]

2005 (1)

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]

2003 (1)

2002 (2)

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]

1999 (1)

1997 (1)

Q3. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Optics.Comm. 136, 114-124 (1997).
[CrossRef]

1996 (1)

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]

1989 (2)

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]

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]

1979 (1)

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1975 (1)

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

Agui, J. H.

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]

Alexander, D. R.

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]

Barton, J. P.

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]

Borghi, R.

Britten, J. A.

Brown, C.

Cao, N.

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]

Chan, C. T.

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Doicu, A.

Q3. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Optics.Comm. 136, 114-124 (1997).
[CrossRef]

Doney, R. L.

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]

Esarey, E.

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]

Golick, B.

Herman, S.

Ho, Y. K.

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]

Hu, S. X.

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]

Ito, H.

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]

Kartz, M.

Kertel, C. H.

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

Kong, Q.

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]

Krall, J.

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]

Lax, M.

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

Lin, Z. F.

Liu, S. Y.

Louisell, W. H.

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

Luo, H.

McKnight, W. B.

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

Miller, J.

Nakagawa, M.

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]

Nishida, Y.

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]

Pennington, D.

Perry, M. D.

Pfannes, J. M. M.

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]

Powell, H. T.

Salamin, Y. I.

Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86, 319-326 (2007).
[CrossRef]

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

Santarsiero, M.

Sen, S.

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]

Sokolow, A.

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]

Sprangle, P.

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]

Starace, A. F.

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]

Stuart, B. C.

Tietbohl, G.

Ting, A.

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]

Vergino, M.

Wang, P. X.

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]

Weniger, E. J.

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]

Wriedt, T.

Q3. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Optics.Comm. 136, 114-124 (1997).
[CrossRef]

Yanovsky, V.

Yuan, X. Q.

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]

Yugami, N.

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]

Appl. Phys. B (1)

Y. I. Salamin, "Fields of a Gaussian beam beyond the paraxial approximation," Appl. Phys. B 86, 319-326 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

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]

Comm. (1)

Q3. A. Doicu and T. Wriedt, "Plane wave spectrum of electromagnetic beams," Optics.Comm. 136, 114-124 (1997).
[CrossRef]

Comput. Phys. Rep. (1)

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]

IEEE Trans. Plasma Sci. (1)

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]

J. Appl. Phys. (1)

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]

Opt. Commun (1)

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]

Opt. Lett. (3)

Phys. Rev. A (2)

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]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (1)

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

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Figures (10)


               Fig. 1.
Fig. 1.

(Color online) The x and z components of electric field as a function of transverse coordinate y in the line of x = 0 , at longitudinal coordinate z = 19zr for a Gaussian beam with a spot size w 0 = λ , and initial phase ϕ 0 = 0 . The Lax series approximation orders are ε 4(red), ε 12 (green) and ε 38 (blue). The black curves represent the exact solutions with m = 4. The x component of electric field represented by (a) LSF and (b) WTF; the z component of electric field represented by (c) LSF and (d) WTF.

Fig. 2.
Fig. 2.

(Color online) (a) The trajectories. (b) The energy gains of the electron dynamics in a laser beam. The insets in (a) and (b) show magnifications of local portions of the same data. The red curves and blue curves represent the electron dynamics obtained by LSF and WTF respectively. The two black curves represent the beam boundaries. Parameters used in the simulations are injected angle θ = 10°,q = 10,w 0= 5 μm , λ = 1μm , (z0, y, x0) = (-5mm, 0, -5tanθmm), initial phase ϕ 0=0 , the initial injection energy γ 0 = 16.03, Energy Gain = m 0 c 2(γ-; 0) and the full interaction time ωt = 1.96 × 106.

Fig. 3.
Fig. 3.

(Color online) (a) The y component of magnetic field, (b) the x component and (c) the z component of electric field, and, (d) the x component and (e) the z component of the force sensed by the reflected electron along its trajectories.

Fig. 4.
Fig. 4.

(Color online) (a) The electron energy gains. (b) The trajectories. The dimensionless parameter q = 100 , the full interaction time ωt = 0.93 × 106 , the initial injection energy λ 0 = 200 , and the other parameters are the same as those of Fig. 2.

Fig. 5.
Fig. 5.

(Color online) (a) The y component of magnetic field, (b) the x component and (c) the z component of electric field sensed by the transmitted electron along its trajectories.

Fig. 6.
Fig. 6.

(Color online) Case of capture with an injection point outside the boundary. (a) The electron energy gains. (b) The trajectories. The full interaction time εt = 3 × 106, and the initial injection energy λ 0 = 16 , q = 100 and the other parameters are the same as those of Fig. 2.

Fig. 7.
Fig. 7.

(Color online) [(a), (b)] The y component of magnetic field near the injection point, [(c), (d)] the z component, and [(e), (f)] the x component of electric field sensed by the captured electron along its trajectories. Field components sensed by the electron in the region [(a), (c), (e)] near the injection point, and [(b), (d), (f)] beyond the focus.

Fig. 8.
Fig. 8.

(Color online) (a) The z component, and (b) the x component of velocity of the electron along its trajectories.

Fig. 9.
Fig. 9.

(Color online) Case of capture with the injection point inside the beam boundary. (a) The electron energy gains. (b) The trajectories. The initial injection energy λ 0 =16.7 , q = 100 , the z component of initial coordinate z 0 = -0.0005 cm and the other parameters are the same as those of Fig. 2.

Fig. 10.
Fig. 10.

(Color online) (a) The x component and (b) the z component of electric field sensed by the electron along its trajectories.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

2 A 1 c 2 2 A t 2 = 0 .
2 ψ 2 ik ψ z = 0 .
2 ψ 4 i ψ ζ + ε 2 2 ψ ζ 2 = 0 ,
ψ = n = 0 ε 2 n ψ 2 n ,
2 ψ 0 4 i ψ 0 ζ = 0 ,
2 ψ 2 n + 2 4 i ψ 2 n + 2 ζ + 2 ψ 2 n ζ 2 = 0 . n 0
ψ 0 = f e f ρ 2 , f = i / ( ζ + i ) , ρ 2 = ξ 2 + υ 2 .
exp [ ik ( z 2 + r 2 ) 1 / 2 ] = exp [ ikz i ( z r / z ) ρ 2 ] n = 0 ε 2 n a 2 n ( ρ , z r / z ) .
ψ 2 n = ( C 2 n ( f ) + a 2 n ( ρ , f ) ) ψ 0 .
ψ 2 n = C 2 n ψ 0 , C 2 n = a 2 n ( ρ , f ) + ( n + 1 ) f C 2 n 2 / 4 ,
E x = i E 0 ψ 0 exp ( i ( ωt kz + ϕ 0 ) ) n = 0 ε 2 n E n x ( f , ρ , ξ ) ,
E y = i E 0 ψ 0 exp ( i ( ωt kz + ϕ 0 ) ) ξυ n = 0 ε 2 n E n y ( f , ρ ) ,
E z = i E 0 ψ 0 exp ( i ( ωt kz + ϕ 0 ) ) ξ n = 0 ε 2 n + 1 E n z ( f , ρ ) ,
B x = 0 ,
B y = i E 0 ψ 0 exp ( i ( ωt kz + ϕ 0 ) ) n = 0 ε 2 n B n y ( f , ρ ) ,
B z = E 0 ψ 0 exp ( i ( ωt kz + ϕ 0 ) ) n = 0 ε 2 n + 1 B n z ( f , ρ ) ,
δ n = j = 0 n ( n j ) ( 1 + j ) n 1 S j a j + 1 j = 0 n ( n j ) ( 1 + j ) n 1 1 a j + 1 ,
b m = b ( b + 1 ) ( b + 2 ) ( b + m 1 ) , ( n j ) = n ! ( n j ) ! j ! .
E x = E E 0 ψ 0 exp ( i ( ωt kz + ϕ 0 ) ) j = 0 n ( n j ) ( 1 + j ) n 1 S j ε 2 ( j + 1 ) E j + 1 x j = 0 n ( n j ) ( 1 + j ) n 1 1 ε 2 ( j + 1 ) E j + 1 x ,
ψ ( x , y , 0 ) = n = 0 m ε 2 n ψ 2 n ( x , y , 0 ) ,
d p dt = e ( E + β × B ) , dt = ec β · E ,

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