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

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    [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]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2006 (1)

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 (2)

2004 (1)

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 (1)

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

2002 (3)

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]

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]

2001 (1)

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 (4)

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 (2)

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 (3)

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]

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

1997 (1)

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 (1)

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 (1)

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

1991 (2)

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 (1)

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

1983 (1)

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

J. Opt. Soc. Am. A (2)

New J. Phys. (1)

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. (2)

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. (1)

Phys. Lett. A (4)

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, 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]

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]

Phys. Rev. A (6)

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]

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]

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 (6)

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

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

Phys. Rev. Lett. (3)

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

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]

Phys. Rev. ST Accel. Beams (1)

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]

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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 (44)

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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
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 ) ] ,

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