Abstract

Taking the TEM1,0-mode Hermite–Gaussian (H–G) beam as a numerical calculation example, and based on the method of the perturbation series expansion, the higher-order field corrections of H–G beams are derived and used to study the electron acceleration by a tightly focused H–G beam in vacuum. For the case of the off-axis injection the field corrections to the terms of order f3 (f=1kw0, k and w0 being the wavenumber and waist width, respectively) are considered, and for the case of the on-axis injection the contributions of the terms of higher orders are negligible. By a suitable optimization of injection parameters the energy gain in the giga-electron-volt regime can be achieved.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 26, 219-224 (1985).
    [CrossRef]
  2. M. D. Perry, D. Pennington, B. C. Stuart, G. Tiethohl, J. A. Britten, C. Brown, and S. Herman, “Petawatt laser pulses,” Opt. Lett. 24, 160-162 (1999).
    [CrossRef]
  3. G. Malka, E. Lefebvre, and J. L. Miquel, “Experimental observation of electron accelerated in vacuum to relativistic energies by a high-intensity laser,” Phys. Rev. Lett. 78, 3314-3317 (1997).
    [CrossRef]
  4. G. Malka and J. L. Miquel, “Experimental confirmation of ponderomotive-force electrons produced by an ultrarelativistic pulse on a solid target,” Phys. Rev. Lett. 77, 75-78 (1996).
    [CrossRef] [PubMed]
  5. M. O. Scully and M. S. Zubairy, “Simple laser accelerator: optics and particle dynamics,” Phys. Rev. A 44, 2656-2663 (1991).
    [CrossRef] [PubMed]
  6. E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46, 6640-6653 (1991).
    [CrossRef]
  7. E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443-5453 (1995).
    [CrossRef]
  8. L. J. Zhu, Y. K. Ho, J. X. Wang, and Q. Kong, “Violent acceleration of electrons by an ultra-intense pulsed laser beam,” Phys. Lett. A 248, 319-324 (1998).
    [CrossRef]
  9. J. X. Wang, Y. K. Ho, Q. Kong, L. J. Zhu, L. Feng, S. Scheid, and H. Hora, “Electron capture and violent acceleration by an extra-intense laser beam,” Phys. Rev. E 58, 6575-6577 (1998).
    [CrossRef]
  10. J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473-7478 (1999).
    [CrossRef]
  11. Y. I. Salamin, F. H. M. Faisal, and C. H. Keitel, “Exact analysis of ultrahigh laser-induced acceleration of electrons by cyclotron autoresonance,” Phys. Rev. A 62, 053809 (2000).
    [CrossRef]
  12. Q. Kong, Y. K. Ho, J. X. Wang, P. X. Wang, L. Feng, and Z. S. Yuan, “Conditions for electron capture by an ultraintense stationary laser beam,” Phys. Rev. E 61, 1981-1984 (2000).
    [CrossRef]
  13. J. L. Hirshfield and C. Wang, “Laser-driven electron cyclotron autoresonance accelerator with production of an optically chopped electron beam,” Phys. Rev. E 61, 7252-7255 (2000).
    [CrossRef]
  14. Y. I. Salamin and F. H. M. Faisal, “Ultrahigh electron acceleration and Compton emission spectra in a superintense laser pulse and a uniform axial magnetic field,” Phys. Rev. A 61, 043801 (2000).
    [CrossRef]
  15. J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, “Mechanism of electron violent acceleration by extra-intense lasers in vacuum,” Phys. Lett. A 280, 121-128 (2001).
    [CrossRef]
  16. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  17. 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, 101301 (2002).
    [CrossRef]
  18. Y. I. Salamin, “Relativistic electron dynamics in intense crossed laser beams: acceleration and Compton harmonics,” Phys. Rev. E 67, 016501 (2003).
    [CrossRef]
  19. S. Liu, H. Guo, H. Tang, and M. Liu, “Direct acceleration of electron using single Hermite-Gaussian beam and Bessel beam in vacuum,” Phys. Lett. A 324, 104-113 (2004).
    [CrossRef]
  20. Y. I. Salamin, “Electron acceleration in a tightly-focused vacuum laser beat wave,” Phys. Lett. A 335, 289-294 (2005).
    [CrossRef]
  21. Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys. 8, doi:10.1088/1367-2630/8/8/133 (2006).
    [CrossRef]
  22. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  23. G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693-1695 (1983).
    [CrossRef]
  24. H. Laabs, “Propagation of Hermite-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1-4 (1998).
    [CrossRef]
  25. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Herlmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404-412 (2002).
    [CrossRef]
  26. K. Duan, B. Wang, and B. Lü, “Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 22, 1976-1980 (2005).
    [CrossRef]

2006

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys. 8, doi:10.1088/1367-2630/8/8/133 (2006).
[CrossRef]

2005

2004

S. Liu, H. Guo, H. Tang, and M. Liu, “Direct acceleration of electron using single Hermite-Gaussian beam and Bessel beam in vacuum,” Phys. Lett. A 324, 104-113 (2004).
[CrossRef]

2003

Y. I. Salamin, “Relativistic electron dynamics in intense crossed laser beams: acceleration and Compton harmonics,” Phys. Rev. E 67, 016501 (2003).
[CrossRef]

2002

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

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, 101301 (2002).
[CrossRef]

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Herlmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404-412 (2002).
[CrossRef]

2001

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, “Mechanism of electron violent acceleration by extra-intense lasers in vacuum,” Phys. Lett. A 280, 121-128 (2001).
[CrossRef]

2000

Y. I. Salamin, F. H. M. Faisal, and C. H. Keitel, “Exact analysis of ultrahigh laser-induced acceleration of electrons by cyclotron autoresonance,” Phys. Rev. A 62, 053809 (2000).
[CrossRef]

Q. Kong, Y. K. Ho, J. X. Wang, P. X. Wang, L. Feng, and Z. S. Yuan, “Conditions for electron capture by an ultraintense stationary laser beam,” Phys. Rev. E 61, 1981-1984 (2000).
[CrossRef]

J. L. Hirshfield and C. Wang, “Laser-driven electron cyclotron autoresonance accelerator with production of an optically chopped electron beam,” Phys. Rev. E 61, 7252-7255 (2000).
[CrossRef]

Y. I. Salamin and F. H. M. Faisal, “Ultrahigh electron acceleration and Compton emission spectra in a superintense laser pulse and a uniform axial magnetic field,” Phys. Rev. A 61, 043801 (2000).
[CrossRef]

1999

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473-7478 (1999).
[CrossRef]

M. D. Perry, D. Pennington, B. C. Stuart, G. Tiethohl, J. A. Britten, C. Brown, and S. Herman, “Petawatt laser pulses,” Opt. Lett. 24, 160-162 (1999).
[CrossRef]

1998

H. Laabs, “Propagation of Hermite-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1-4 (1998).
[CrossRef]

L. J. Zhu, Y. K. Ho, J. X. Wang, and Q. Kong, “Violent acceleration of electrons by an ultra-intense pulsed laser beam,” Phys. Lett. A 248, 319-324 (1998).
[CrossRef]

J. X. Wang, Y. K. Ho, Q. Kong, L. J. Zhu, L. Feng, S. Scheid, and H. Hora, “Electron capture and violent acceleration by an extra-intense laser beam,” Phys. Rev. E 58, 6575-6577 (1998).
[CrossRef]

1997

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

1996

G. Malka and J. L. Miquel, “Experimental confirmation of ponderomotive-force electrons produced by an ultrarelativistic pulse on a solid target,” Phys. Rev. Lett. 77, 75-78 (1996).
[CrossRef] [PubMed]

1995

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443-5453 (1995).
[CrossRef]

1991

M. O. Scully and M. S. Zubairy, “Simple laser accelerator: optics and particle dynamics,” Phys. Rev. A 44, 2656-2663 (1991).
[CrossRef] [PubMed]

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46, 6640-6653 (1991).
[CrossRef]

1985

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 26, 219-224 (1985).
[CrossRef]

1983

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

1975

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

J. Opt. Soc. Am. A

New J. Phys.

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys. 8, doi:10.1088/1367-2630/8/8/133 (2006).
[CrossRef]

Opt. Commun.

H. Laabs, “Propagation of Hermite-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 147, 1-4 (1998).
[CrossRef]

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 26, 219-224 (1985).
[CrossRef]

Opt. Lett.

Phys. Lett. A

S. Liu, H. Guo, H. Tang, and M. Liu, “Direct acceleration of electron using single Hermite-Gaussian beam and Bessel beam in vacuum,” Phys. Lett. A 324, 104-113 (2004).
[CrossRef]

Y. I. Salamin, “Electron acceleration in a tightly-focused vacuum laser beat wave,” Phys. Lett. A 335, 289-294 (2005).
[CrossRef]

J. X. Wang, W. Scheid, M. Hoelss, and Y. K. Ho, “Mechanism of electron violent acceleration by extra-intense lasers in vacuum,” Phys. Lett. A 280, 121-128 (2001).
[CrossRef]

L. J. Zhu, Y. K. Ho, J. X. Wang, and Q. Kong, “Violent acceleration of electrons by an ultra-intense pulsed laser beam,” Phys. Lett. A 248, 319-324 (1998).
[CrossRef]

Phys. Rev. A

Y. I. Salamin, F. H. M. Faisal, and C. H. Keitel, “Exact analysis of ultrahigh laser-induced acceleration of electrons by cyclotron autoresonance,” Phys. Rev. A 62, 053809 (2000).
[CrossRef]

M. O. Scully and M. S. Zubairy, “Simple laser accelerator: optics and particle dynamics,” Phys. Rev. A 44, 2656-2663 (1991).
[CrossRef] [PubMed]

E. J. Bochove, G. T. Moore, and M. O. Scully, “Acceleration of particles by an asymmetric Hermite-Gaussian laser beam,” Phys. Rev. A 46, 6640-6653 (1991).
[CrossRef]

Y. I. Salamin and F. H. M. Faisal, “Ultrahigh electron acceleration and Compton emission spectra in a superintense laser pulse and a uniform axial magnetic field,” Phys. Rev. A 61, 043801 (2000).
[CrossRef]

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

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

Phys. Rev. E

Y. I. Salamin, “Relativistic electron dynamics in intense crossed laser beams: acceleration and Compton harmonics,” Phys. Rev. E 67, 016501 (2003).
[CrossRef]

E. Esarey, P. Sprangle, and J. Krall, “Laser acceleration of electrons in vacuum,” Phys. Rev. E 52, 5443-5453 (1995).
[CrossRef]

Q. Kong, Y. K. Ho, J. X. Wang, P. X. Wang, L. Feng, and Z. S. Yuan, “Conditions for electron capture by an ultraintense stationary laser beam,” Phys. Rev. E 61, 1981-1984 (2000).
[CrossRef]

J. L. Hirshfield and C. Wang, “Laser-driven electron cyclotron autoresonance accelerator with production of an optically chopped electron beam,” Phys. Rev. E 61, 7252-7255 (2000).
[CrossRef]

J. X. Wang, Y. K. Ho, Q. Kong, L. J. Zhu, L. Feng, S. Scheid, and H. Hora, “Electron capture and violent acceleration by an extra-intense laser beam,” Phys. Rev. E 58, 6575-6577 (1998).
[CrossRef]

J. X. Wang, Y. K. Ho, L. Feng, Q. Kong, P. X. Wang, Z. S. Yuan, and W. Scheid, “High-intensity laser-induced electron acceleration in vacuum,” Phys. Rev. E 60, 7473-7478 (1999).
[CrossRef]

Phys. Rev. Lett.

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

G. Malka and J. L. Miquel, “Experimental confirmation of ponderomotive-force electrons produced by an ultrarelativistic pulse on a solid target,” Phys. Rev. Lett. 77, 75-78 (1996).
[CrossRef] [PubMed]

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

Phys. Rev. ST Accel. Beams

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, 101301 (2002).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Laser-electron interaction geometry.

Fig. 2
Fig. 2

(a) Normalized transverse electric field component E ξ E ξ max , and (b) longitudinal electric-field components E ς E ς max along the electron trajectories. The calculation parameters are seen in the text.

Fig. 3
Fig. 3

Energy gain Δ W versus the waist width w 0 .

Fig. 4
Fig. 4

Energy gain Δ W versus the initial phase ψ 0 .

Fig. 5
Fig. 5

Energy gain Δ W versus the injection energy γ 0 .

Fig. 6
Fig. 6

Energy gain Δ W as a function of the injection angle θ.

Fig. 7
Fig. 7

Maximum energy gain Δ W max and optimal injection energy γ opt as a function of the injection angle θ.

Fig. 8
Fig. 8

Maximum energy gain Δ W max and optimal injection angle θ opt as a function of the injection energy γ 0 .

Fig. 9
Fig. 9

Energy gain Δ W versus the waist width w 0 .

Fig. 10
Fig. 10

Energy gain Δ W versus the injection energy γ 0 .

Fig. 11
Fig. 11

Maximum energy gain Δ W max and optimal injection energy γ opt as a function of the waist width w 0 .

Equations (45)

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

E = E T + E z e ̂ z = ( F T + F z e ̂ z ) exp [ i ( k z ω t ψ 0 ) ] ,
F ξ = F ξ ( 0 ) + f 2 F ξ ( 2 ) + f 4 F ξ ( 4 ) + ,
F ς = f F ς ( 1 ) + f 3 F ς ( 3 ) + f 5 F ς ( 5 ) + ,
ξ = x w 0 , η = y w 0 , ς = z l , l = w 0 f ,
F T F ξ , F z F ς
( 2 ξ 2 + 2 η 2 2 i ς ) F ξ ( 0 ) = 0 ,
( 2 ξ 2 + 2 η 2 2 i ς ) F ξ ( 2 s + 2 ) = 2 ς 2 F ξ ( 2 s ) ,
F ς ( 2 s + 1 ) = q = 0 s ( i ) s + 1 q s q ς s q ξ F ξ ( 2 q ) ,
F ξ , m , n ( 0 ) = TEM m , n ( p ) = HGM m ( p ) ( ξ , ς ) HGM m ( p ) ( η , ς ) ,
HGM m ( p ) ( ξ , ς ) = A ( 2 π ) 1 4 σ 2 m m ! w 0 ( σ σ * ) m 2 H m ( 2 σ σ * ξ ) exp ( σ ξ 2 ) ,
F ξ , m , n ( 2 s ) = p = 1 s c p ( 2 s ) ( 2 i ς ) 1 2 2 s + 2 p q = 0 s + p ( s + p q ) 2 s + 2 p ξ 2 s + 2 p 2 q η 2 q TEM m , n ( p ) ,
b ξ b HGM m ( p ) ( ξ , ς ) = b ! o = 0 floor ( b 2 ) ( 2 σ ) b o 2 o o ! α = 0 b 2 o ( 1 ) α ( q α 2 o ) !
( m α ) 1 α ! ( ξ ) b α 2 o HGM m α ( p ) ( ξ , ς ) ,
F ξ , m , n = s = 1 f 2 s F ξ , m , n ( 2 s ) .
E ¯ ξ = [ F ξ ( 0 ) + f 2 F ξ ( 2 ) + f 4 F ξ ( 4 ) + ] exp [ i ( 2 k z r ς ω t ψ 0 ) ] ,
F ξ ( 0 ) = TEM 1 , 0 ( p ) ,
F ξ ( 2 ) ( ξ , η , ζ ) = ( 2 i ς ) σ 2 [ ( 2 4 σ ρ 2 + σ 2 ρ 4 ) TEM 1 , 0 ( P ) + 4 σ ξ ( 2 σ ρ 2 ) TEM 0 , 0 ( p ) ] ,
F ξ ( 4 ) ( ξ , η , ζ ) = 2 σ 3 ς { [ 12 i 2 i σ ρ 2 ( 6 + σ ρ 2 ) ( 3 + σ ρ 2 ) + σ ς ( 24 + σ ρ 2 ( 4 + σ ρ 2 ) ( 24 12 σ ρ 2 + σ 2 ρ 4 ) ) ] TEM 1 , 0 ( p ) + 4 σ ξ [ 3 i ( 6 6 σ ρ 2 + σ 2 ρ 4 ) + 2 σ ς ( 24 + σ ρ 2 ( 6 + σ ρ 2 ) 2 ) ] TEM 0 , 0 ( p ) } ,
ρ = r w 0 , r = x 2 + y 2 ,
z r = k w 0 2 2
( w 0 , the waist width of the corresponding Gaussian beam ) ,
E ¯ ς = [ f F ς ( 1 ) + f 3 F ς ( 3 ) + f 5 F ς ( 5 ) + ] exp [ i ( 2 k z r ς ω t ψ 0 ) ] ,
F ς ( 1 ) = i ξ F ξ ( p ) = 2 i σ [ TEM 0 , 0 ( p ) ξ TEM 1 , 0 ( p ) ] ,
F ς ( 3 ) = i ξ F ξ ( 2 ) ς ξ F ξ ( 0 ) = 2 σ { [ 2 i σ ( 2 + σ ξ 2 σ * ξ 2 + σ ρ 2 ) + 2 σ 2 ς ( 6 12 σ ξ 2 6 σ ρ 2 + 4 σ 2 ξ 2 ρ 2 + σ 4 ρ 4 ) ] TEM 0 , 0 ( p ) + [ i ξ ( σ * + 5 σ 2 σ 2 ρ 2 ) 2 σ 2 ξ ς ( 6 6 σ ρ 2 + σ 2 ρ 4 ) ] TEM 1 , 0 ( p ) } ,
F ς ( 5 ) = i ξ F ξ ( 4 ) ς ξ F ξ ( 2 ) + i 2 ς 2 ξ F ξ ( 0 ) = 4 i σ 2 { 18 σ + ( 5 σ σ * ) 2 ξ 2 + 2 σ 2 [ 9 2 ( 2 σ σ * ) ξ 2 ] ρ 2 3 σ 3 ρ 4 + 2 i σ 2 ς [ 48 6 ( σ * 23 σ ) ξ 2 + 72 σ ρ 2 + 6 σ ( σ * 15 σ ) ξ 2 ρ 2 24 σ 2 ρ 4 + σ 2 ( 11 σ σ * ) ξ 2 ρ 4 + 2 σ 3 ρ 6 ] + ς 2 σ 3 [ 120 240 ( 2 ξ 2 + ρ 2 ) + 120 σ 2 ( 4 ξ 2 + ρ 2 ) ρ 2 20 σ 3 ( 6 ξ 2 + ρ 2 ) ρ 4 + σ 4 ( 8 ξ 2 + ρ 2 ) ρ 6 ] } TEM 0 , 0 ( p ) + 2 i σ ξ { 10 σ σ * + 47 σ 2 σ * 2 4 σ 2 ( σ * + 10 σ ) ρ 2 + 6 σ 4 ρ 4 2 i ς σ 2 [ [ 6 ( σ * + 17 σ ) + 6 σ ( σ * + 25 σ ) ρ 2 σ 2 ( 49 σ + σ * ) ρ 4 + 4 σ 4 ρ 6 ] ] 2 σ 4 ς 2 ( 120 240 σ ρ 2 + 120 σ 2 ρ 4 20 σ 3 ρ 6 + σ 4 ρ 8 ) } TEM 1 , 0 ( p ) ,
B ¯ ξ = 1 c { f 2 [ 4 i σ 2 η ( TEM 0 , 0 ( p ) ξ TEM 1 , 0 ( p ) ) ] + f 4 [ 8 σ 3 η i ( 3 + σ ξ 2 σ * ξ 2 + σ ρ 2 ) + σ ς ( 12 16 σ ξ 2 8 σ ρ 2 + 4 σ 2 ξ 2 ρ 2 + σ 2 ρ 4 ) TEM 0 , 0 ( p ) + 4 σ 2 ξ η i σ * + i σ ( 7 + 2 σ ρ 2 ) + 2 σ 2 ς ( 12 8 σ ρ 2 + σ 2 ρ 4 ) TEM 1 , 0 ( p ) ] + } exp [ i ( 2 k z r ς ω t ψ 0 ) ] ,
B ¯ η = 1 c { TEM 1 , 0 ( p ) + f 2 [ 2 σ ξ 4 i σ 2 ς ( 2 σ 2 ρ 2 ) + ( 3 σ + σ * ) TEM 0 , 0 ( p ) + 2 i σ 2 ς ( 2 4 σ ρ 2 + σ 2 ρ 4 ) ( σ + σ * ) + 2 σ 2 ( 2 ξ 2 + ρ 2 ) TEM 1 , 0 ( P ) ] + f 4 [ 4 σ 2 ξ σ * + 9 σ + 2 σ σ * ξ 2 2 σ 2 ( ξ 2 + ρ 2 ) + 4 σ 3 ς 2 ( 24 + 36 σ ρ 2 12 σ 2 ρ 4 + σ 3 ρ 6 ) + i σ ς ( 2 σ * + 62 σ 32 σ 2 ξ 2 4 σ ( σ * + 10 σ 2 σ 2 ξ 2 ) ρ 2 + σ 2 ( 5 σ + σ * ) ρ 4 ) TEM 0 , 0 ( p ) 2 σ σ * 3 σ + 2 σ ( 7 σ + σ * ) ξ 2 2 σ 2 ( 1 + 2 σ ξ 2 ) ρ 2 + σ 3 ρ 4 + i σ ς ( 2 σ * 10 σ + 48 σ 2 ξ 2 4 ( σ σ * 2 σ 2 + 8 σ 3 ξ 2 ) ρ 2 + ( σ * σ 2 σ 3 + 4 σ 4 ξ 2 ) ρ 4 ) + σ 3 ς 2 ( 24 96 σ ρ 2 + 72 σ 2 ρ 4 16 σ 3 ρ 6 + σ 4 ρ 8 ) TEM 1 , 0 ( p ) ] + } exp [ i ( 2 k z r ς ω t ψ 0 ) ] ,
B ¯ ς = 1 c { f [ 2 i σ η TEM 1 , 0 ( p ) ] + f 3 [ 4 σ 3 η ς 4 σ ξ ( 3 + σ ρ 2 ) TEM 0 , 0 ( p ) + ( 6 + 6 σ ρ 2 σ 2 ρ 4 ) TEM 1 , 0 ( p ) ] + f 5 ( 4 σ 4 η ς ) [ 4 σ ξ 3 ( 12 8 σ ρ 2 + σ 2 ρ 4 ) + 2 σ ς ( 60 + 60 σ ρ 2 15 σ 2 ρ 4 + σ 3 ρ 6 ) TEM 0 , 0 ( p ) + 2 ( 24 + 36 σ ρ 2 12 σ 2 ρ 4 + σ 3 ρ 6 ) + σ ς ( 120 240 σ ρ 2 + 120 σ 2 ρ 4 20 σ 3 ρ 6 + σ 4 ρ 8 ) TEM 1 , 0 ( p ) ] + } exp [ i ( 2 k z r ς ω t ψ 0 ) ] ,
E ξ = E { [ C 11 H 1 ( 2 w 0 ξ w ) ] + f 2 ( 2 ς ) [ 2 ( 2 S 31 4 S 41 ρ 2 + S 51 ρ 4 ) H 1 ( 2 w 0 ξ w ) 8 ξ ( 2 S 40 S 50 ρ 2 ) ] + f 4 ( 2 ς ) [ 2 ( 6 S 41 + 18 S 51 ρ 2 9 S 61 ρ 4 + S 71 ρ 6 ) ς ( 24 C 51 96 C 61 ρ 2 + 72 C 71 ρ 4 16 C 81 ρ 6 + C 91 ρ 8 ) H 1 ( 2 w 0 ξ w ) + 4 2 ξ 3 ( 6 S 50 6 S 60 ρ 2 + S 70 ρ 4 ) + 2 ς ( 24 C 60 + 36 C 70 ρ 2 12 C 80 ρ 4 + C 90 ρ 6 ) ] + } ,
E ς = E { 2 f [ 2 S 20 ξ S 21 H 1 ( 2 w 0 ξ w ) ] + 2 f 3 [ 4 S 20 ξ 2 w 0 2 w 2 + 2 S 30 S 40 ( ξ 2 + ρ 2 ) + 4 ς 6 C 40 6 C 50 ( 2 ξ 2 + ρ 2 ) + C 60 ( 4 ξ 2 + ρ 2 ) ρ 2 2 ξ ( w 0 w S 20 + 5 S 31 2 S 41 ρ 2 ) + 2 ς ( 6 C 41 6 C 51 ρ 2 + C 61 ρ 4 ) H 1 ( 2 w 0 ξ w ) ] + f 5 E ς ( 5 ) + } ,
E ς ( 5 ) = 4 2 { 18 S 40 ( 25 S 50 10 S 30 w 0 2 w 2 + S 10 w 0 4 w 4 ) ξ 2 [ 18 S 50 4 ( 2 S 60 S 40 w 0 2 w 2 ) ξ 2 ] ρ 2 + 3 S 60 ρ 4 2 ς [ 48 C 50 6 ( C 40 w 0 2 w 2 23 C 60 ) ξ 2 + 72 C 60 ρ 2 + 6 ( C 50 w 0 2 w 2 15 C 70 ) ξ 2 ρ 2 24 C 70 ρ 4 + ( 11 C 80 C 60 w 0 2 w 2 ) ξ 2 ρ 4 + 2 C 80 ρ 6 ] ς 2 [ 120 S 60 240 S 70 ( 2 ξ 2 + ρ 2 ) + 120 S 80 ( 4 ξ 2 + ρ 2 ) ρ 2 20 S 90 ( 6 ξ 2 + ρ 2 ) ρ 4 + S 100 ( 8 ξ 2 + ρ 2 ) ρ 6 ] } + 2 ξ { 10 S 30 w 0 w 47 S 41 + S 10 w 0 3 w 3 + 4 ( S 40 w 0 w + 10 S 51 ) ρ 2 6 S 61 ρ 4 + 2 ς [ 6 ( C 40 w 0 w + 17 S 51 ) + 6 ( C 50 w 0 w + 25 C 61 ) ρ 2 ( 49 C 71 + C 60 w 0 w ) ρ 4 + 4 C 81 ρ 6 ] + 2 ς 2 ( 120 S 61 240 S 71 ρ 2 + 120 S 81 ρ 4 20 S 91 ρ 6 + S 101 ρ 8 ) } H 1 ( 2 w 0 ξ w ) ,
B ξ = E c ( 2 2 η ) { f 2 [ 2 C 30 + 2 ξ C 31 H 1 ( 2 w 0 ξ w ) ] + f 4 [ 4 C 30 ξ 2 w 0 2 w 2 3 C 40 + C 50 ( ξ 2 + ρ 2 ) + ς ( 12 S 50 8 S 60 ( 2 ξ 2 + ρ 2 ) + S 70 ( 4 ξ 2 + ρ 2 ) ρ 2 ) + 2 ξ ( C 30 w 0 w + 7 C 41 2 C 51 ρ 2 ) 2 ς ( 12 S 51 8 S 61 ρ 2 + S 71 ρ 4 ) H 1 ( 2 w 0 ξ w ) ] + } ,
B η = E c { C 11 H 1 ( 2 w 0 ξ w ) + f 2 [ 2 2 ξ 4 ς ( 2 S 40 + S 50 ρ 2 ) + ( 3 C 30 + C 10 w 0 2 w 2 ) 2 ς ( 2 S 31 4 S 41 ρ 2 + S 51 ρ 4 ) + ( C 21 + C 10 w 0 w ) + 2 C 31 ( 2 ξ 2 ρ 2 ) H 1 ( 2 w 0 ξ w ) ] + f 4 B η ( 4 ) + } ,
B η ( 4 ) = 4 2 ξ { [ C 20 + 2 C 30 ξ 2 ς ( 2 S 30 4 S 40 ρ 2 + S 50 ρ 4 ) ] w 0 2 w 2 + 9 C 40 2 C 50 ( ξ 2 + ρ 2 ) ς [ 62 S 50 S 60 ( 8 ξ 2 + 10 ρ 2 ) + S 70 ( 8 ξ 2 + 5 ρ 2 ) ρ 2 ] + 4 ς 2 ( 24 C 60 + 36 C 70 ρ 2 12 C 80 ρ 4 + C 90 ρ 6 ) } 2 { [ C 20 + 2 C 30 ξ 2 ς ( 2 S 30 4 S 40 ρ 2 + S 50 ρ 4 ) ] w 0 w 3 C 31 + 2 C 41 ( 7 ξ 2 ρ 2 ) + C 51 ( 4 ξ 2 + ρ 2 ) ρ 2 ς [ 10 S 41 + 8 S 51 ( 6 ξ 2 + ρ 2 ) S 61 ( 32 ξ 2 + ρ 2 ) ρ 2 + S 71 ξ 2 ρ 4 ] + ς 2 ( 24 C 51 96 C 61 ρ 2 + 72 C 71 ρ 4 16 C 81 ρ 6 + C 91 ρ 8 ) } H 1 ( 2 w 0 ξ w ) ,
B ς = E c { f [ 2 η S 21 H 1 ( 2 w 0 ξ w ) ] + f 3 ( 4 η ς ) [ ( 6 C 41 + 6 C 51 ρ 2 C 61 ρ 4 ) H 1 ( 2 w 0 ξ w ) 4 2 ξ ( 3 C 50 + C 60 ρ 2 ) ] + f 5 ( 4 η ς ) [ 4 2 ξ 3 ( 12 C 60 C 70 ρ 2 + C 80 ρ 4 ) + 2 ς ( 60 S 70 + 60 S 80 ρ 2 15 S 90 ρ 4 + S 100 ρ 6 ) + 2 ( 24 C 51 + 36 C 61 ρ 2 12 C 71 ρ 4 + C 81 ρ 6 ) + ς ( 120 S 61 240 S 71 ρ 2 + 120 S 81 ρ 4 20 S 91 ρ 6 + S 101 ρ 8 ) H 1 ( 2 w 0 ξ w ) ] + } ,
E = E 0 exp ( w 0 2 ρ 2 w 2 ) , E 0 = A ( w 0 π 1 2 ) , w = w 0 1 + 4 ς 2 ,
C α m = ( w 0 w ) α cos [ ψ + ( α + l ) ψ p ] , S α m = ( w 0 w ) α sin [ ψ + ( α + l ) ψ p ] ( α , l = 0 , 1 , 2 )
ψ = ψ 0 2 k z r ς + ω t k w 0 2 ρ 2 ( 2 R ) , R = 2 z r ς + z r 2 ς , ψ p = arctan ( 2 ς ) .
E ξ ( 0 ) = E [ C 11 H 1 ( 2 w 0 ξ w ) ] , E η ( 0 ) = 0 , E ς ( 1 ) = 2 E f [ 2 S 20 ξ S 21 H 1 ( 2 w 0 ξ w ) ] ,
B ξ ( 0 ) = 0 , B η ( 0 ) = E C 11 H 1 ( 2 w 0 ξ w ) , B ς ( 1 ) = 2 E f η S 21 H 1 ( 2 w 0 ξ w ) .
d P d t = e [ E + c β × B ] ,
d W d t = e c β E ,
Δ W = W γ 0 m c 2 = ( γ γ 0 ) m c 2 ,
E ξ = E η = 0 , B ξ = B η = B ς = 0 ,
E ς = 2 2 E 0 w 0 2 w 2 [ f sin ψ 2 + 4 f 3 w 0 w ( sin ψ 3 + 3 ς w 0 w cos ψ 4 ) + 12 f 5 w 0 2 w 2 ( 3 sin ψ 4 + 16 ς w 0 w cos ψ 5 20 ς 2 w 0 2 w 2 sin ψ 6 ) ] ,

Metrics