Abstract

A solution to the Klein Gordon equation for a laser-driven electron is constructed from a superposition of Volkov states. The time-and space-dependent three-dimensional superposition integral can be evaluated analytically for an initial Gaussian momentum distribution when the expression for relativistic energy is expanded in a Taylor series over the scaled initial momenta. The solution preserves many complicated wave-packet dynamics in a strong field, including so-called wave-packet shearing and the formation of multiple peaks when the wave packet spreads to the scale of the driving-field wavelength. The range of applicability of the solution applies to much of the parameter space accessible by current intense ultra-short laser systems.

© 2007 Optical Society of America

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References

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  1. G. R. Mocken and C. H. Keitel, "Quantum Signatures in Laser-Driven Relativistic Multiple Scattering," Phys. Rev. Lett. 91, 173202 (2003).
    [CrossRef] [PubMed]
  2. G. R. Mocken and C. H. Keitel, "Quantum Dynamics of Relativistic Electrons," J. Comput. Phys. 199,558-588 (2004).
    [CrossRef]
  3. M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
    [CrossRef]
  4. P. Panek, J. Z. Kaminski, and F. Ehlotzky, "Laser-Induced Compton Scattering at Relativistically High Radiation Powers," Phys. Rev. A 65,022712 (2002).
    [CrossRef]
  5. D. M. Volkov, Z. Phys. 94, 250 (1935).
    [CrossRef]
  6. J. S. Román, L. Roso, and H. R. Reiss, "Evolution of a Relativistic Wavepacket Describing a Free Electron in a Very Intense Laser Field," J. Phys. B: At. Mol. Opt. Phys. 33,1869-1880 (2000).
    [CrossRef]
  7. J. S. Román, L. Plaja, and L. Roso, "Relativistic quantum dynamics of a localized Dirac Electron Driven by an Intense-Laser-Field Pulse," Phys. Rev. A 64,063402 (2001).
    [CrossRef]
  8. M. Verschl and C. H. Keitel, "Analytical Approach to Wave-Packet Dynamics of Laser-Driven Particles beyond the Dipole Approximation," Laser Phys. 15,529-535 (2005).
  9. Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
    [CrossRef]
  10. J. H. Eberly, "Interaction of Very Intense Light with Free Electrons," Progress in Optics 7, 361-414 (1969).
    [CrossRef]
  11. W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd ed., Springer, Berlin (2000).
  12. J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

2006 (1)

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
[CrossRef]

2005 (1)

M. Verschl and C. H. Keitel, "Analytical Approach to Wave-Packet Dynamics of Laser-Driven Particles beyond the Dipole Approximation," Laser Phys. 15,529-535 (2005).

2004 (1)

G. R. Mocken and C. H. Keitel, "Quantum Dynamics of Relativistic Electrons," J. Comput. Phys. 199,558-588 (2004).
[CrossRef]

2003 (1)

G. R. Mocken and C. H. Keitel, "Quantum Signatures in Laser-Driven Relativistic Multiple Scattering," Phys. Rev. Lett. 91, 173202 (2003).
[CrossRef] [PubMed]

2002 (2)

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

P. Panek, J. Z. Kaminski, and F. Ehlotzky, "Laser-Induced Compton Scattering at Relativistically High Radiation Powers," Phys. Rev. A 65,022712 (2002).
[CrossRef]

2001 (1)

J. S. Román, L. Plaja, and L. Roso, "Relativistic quantum dynamics of a localized Dirac Electron Driven by an Intense-Laser-Field Pulse," Phys. Rev. A 64,063402 (2001).
[CrossRef]

2000 (1)

J. S. Román, L. Roso, and H. R. Reiss, "Evolution of a Relativistic Wavepacket Describing a Free Electron in a Very Intense Laser Field," J. Phys. B: At. Mol. Opt. Phys. 33,1869-1880 (2000).
[CrossRef]

1969 (1)

J. H. Eberly, "Interaction of Very Intense Light with Free Electrons," Progress in Optics 7, 361-414 (1969).
[CrossRef]

1935 (1)

D. M. Volkov, Z. Phys. 94, 250 (1935).
[CrossRef]

Eberly, J. H.

J. H. Eberly, "Interaction of Very Intense Light with Free Electrons," Progress in Optics 7, 361-414 (1969).
[CrossRef]

Ehlotzky, F.

P. Panek, J. Z. Kaminski, and F. Ehlotzky, "Laser-Induced Compton Scattering at Relativistically High Radiation Powers," Phys. Rev. A 65,022712 (2002).
[CrossRef]

Hatsagortsyan, K. Z.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
[CrossRef]

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

Hu, S. X.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
[CrossRef]

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

Kaminski, J. Z.

P. Panek, J. Z. Kaminski, and F. Ehlotzky, "Laser-Induced Compton Scattering at Relativistically High Radiation Powers," Phys. Rev. A 65,022712 (2002).
[CrossRef]

Keitel, C. H.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
[CrossRef]

M. Verschl and C. H. Keitel, "Analytical Approach to Wave-Packet Dynamics of Laser-Driven Particles beyond the Dipole Approximation," Laser Phys. 15,529-535 (2005).

G. R. Mocken and C. H. Keitel, "Quantum Dynamics of Relativistic Electrons," J. Comput. Phys. 199,558-588 (2004).
[CrossRef]

G. R. Mocken and C. H. Keitel, "Quantum Signatures in Laser-Driven Relativistic Multiple Scattering," Phys. Rev. Lett. 91, 173202 (2003).
[CrossRef] [PubMed]

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

Mocken, G. R.

G. R. Mocken and C. H. Keitel, "Quantum Dynamics of Relativistic Electrons," J. Comput. Phys. 199,558-588 (2004).
[CrossRef]

G. R. Mocken and C. H. Keitel, "Quantum Signatures in Laser-Driven Relativistic Multiple Scattering," Phys. Rev. Lett. 91, 173202 (2003).
[CrossRef] [PubMed]

Panek, P.

P. Panek, J. Z. Kaminski, and F. Ehlotzky, "Laser-Induced Compton Scattering at Relativistically High Radiation Powers," Phys. Rev. A 65,022712 (2002).
[CrossRef]

Plaja, L.

J. S. Román, L. Plaja, and L. Roso, "Relativistic quantum dynamics of a localized Dirac Electron Driven by an Intense-Laser-Field Pulse," Phys. Rev. A 64,063402 (2001).
[CrossRef]

Reiss, H. R.

J. S. Román, L. Roso, and H. R. Reiss, "Evolution of a Relativistic Wavepacket Describing a Free Electron in a Very Intense Laser Field," J. Phys. B: At. Mol. Opt. Phys. 33,1869-1880 (2000).
[CrossRef]

Román, J. S.

J. S. Román, L. Plaja, and L. Roso, "Relativistic quantum dynamics of a localized Dirac Electron Driven by an Intense-Laser-Field Pulse," Phys. Rev. A 64,063402 (2001).
[CrossRef]

J. S. Román, L. Roso, and H. R. Reiss, "Evolution of a Relativistic Wavepacket Describing a Free Electron in a Very Intense Laser Field," J. Phys. B: At. Mol. Opt. Phys. 33,1869-1880 (2000).
[CrossRef]

Roso, L.

J. S. Román, L. Plaja, and L. Roso, "Relativistic quantum dynamics of a localized Dirac Electron Driven by an Intense-Laser-Field Pulse," Phys. Rev. A 64,063402 (2001).
[CrossRef]

J. S. Román, L. Roso, and H. R. Reiss, "Evolution of a Relativistic Wavepacket Describing a Free Electron in a Very Intense Laser Field," J. Phys. B: At. Mol. Opt. Phys. 33,1869-1880 (2000).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
[CrossRef]

Urbach, D. J.

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

Verschl, M.

M. Verschl and C. H. Keitel, "Analytical Approach to Wave-Packet Dynamics of Laser-Driven Particles beyond the Dipole Approximation," Laser Phys. 15,529-535 (2005).

Volkov, D. M.

D. M. Volkov, Z. Phys. 94, 250 (1935).
[CrossRef]

Walser, M. W.

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

J. Comput. Phys. (1)

G. R. Mocken and C. H. Keitel, "Quantum Dynamics of Relativistic Electrons," J. Comput. Phys. 199,558-588 (2004).
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys. (1)

J. S. Román, L. Roso, and H. R. Reiss, "Evolution of a Relativistic Wavepacket Describing a Free Electron in a Very Intense Laser Field," J. Phys. B: At. Mol. Opt. Phys. 33,1869-1880 (2000).
[CrossRef]

Laser Phys. (1)

M. Verschl and C. H. Keitel, "Analytical Approach to Wave-Packet Dynamics of Laser-Driven Particles beyond the Dipole Approximation," Laser Phys. 15,529-535 (2005).

Phys. Rep. (1)

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, "Relativsitic High-Power Laser-Matter Interactions," Phys. Rep. 427, 41-155 (2006).
[CrossRef]

Phys. Rev. A (3)

M. W. Walser, D. J. Urbach, K. Z. Hatsagortsyan, S. X. Hu, and C. H. Keitel, "Spin and radiation in intense laser fields," Phys. Rev. A 65, 043410 (2002).
[CrossRef]

P. Panek, J. Z. Kaminski, and F. Ehlotzky, "Laser-Induced Compton Scattering at Relativistically High Radiation Powers," Phys. Rev. A 65,022712 (2002).
[CrossRef]

J. S. Román, L. Plaja, and L. Roso, "Relativistic quantum dynamics of a localized Dirac Electron Driven by an Intense-Laser-Field Pulse," Phys. Rev. A 64,063402 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

G. R. Mocken and C. H. Keitel, "Quantum Signatures in Laser-Driven Relativistic Multiple Scattering," Phys. Rev. Lett. 91, 173202 (2003).
[CrossRef] [PubMed]

Progress in Optics (1)

J. H. Eberly, "Interaction of Very Intense Light with Free Electrons," Progress in Optics 7, 361-414 (1969).
[CrossRef]

Z. Phys. (1)

D. M. Volkov, Z. Phys. 94, 250 (1935).
[CrossRef]

Other (2)

W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd ed., Springer, Berlin (2000).

J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964).

Supplementary Material (5)

» Media 1: MOV (59 KB)     
» Media 2: MOV (271 KB)     
» Media 3: MOV (813 KB)     
» Media 4: MOV (82 KB)     
» Media 5: MOV (176 KB)     

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Figures (3)

Fig. 1.
Fig. 1.

(64 KB, 276 KB, 820 KB) Movies of an electron wave packet in a linearly polarized plane-wave field during (a) 0-10 cycles, [Media 1] (b) 50-60 cycles, [Media 2] and (c) 100-110 cycles. [Media 3] The intensity is 2.1×1018 W/cm2 at 800 nm wavelength; the initial momentum spread is po = 0.005mc.

Fig. 2.
Fig. 2.

(88 KB) Movie of an electron wave packet in a linearly polarized plane-wave field during 0-6 cycles. The intensity is 5.3×1019 W/cm2 at 800 nm wavelength; the initial momentum spread is po = 2×10-6 mc. [Media 4]

Fig. 3.
Fig. 3.

(180 KB) Movie of an electron wave packet in a circularly polarized plane-wave field during 50-60 cycles. The intensity is 4.2×1018 W/cm2 at 800 nm wavelength; the initial momentum spread is po = 0.005mc. [Media 5]

Equations (22)

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ħ 2 2 Ψ t 2 = m 2 c 4 Ψ + c 2 ( + e A ) 2 Ψ ,
Ψ p v r t = mc 2 ( 2 πħ ) 3 E exp { i ( p r Et ) ħ i ħ ( p z E c ) z ct [ e p A ( ) + e 2 A 2 ( ) 2 ] dℓ } .
E 2 = p 2 c 2 + m 2 c 4 .
p 1 p 2 = 2 mc 2 All space d 3 r [ Ψ p 1 V * r t Ψ p 2 V r t t Ψ p 2 V r t Ψ p 1 V * r t t ] = δ ( p 2 p 1 ) .
Ψ r t = All momenta d 3 pa ( p ) Ψ p V r t .
a ( p ) = 1 ( p o π ) 3 2 exp { p 2 2 p o 2 } ,
E mc 2 [ 1 + p x 2 + p y 2 + p z 2 2 m 2 c 2 + O ( p 4 m 4 c 4 ) ] .
1 E cp z 1 mc 2 [ 1 + p z mc p x 2 + p y 2 p z 2 2 m 2 c 2 p x 2 p z + p y 2 p z m 3 c 3 + O ( p 4 m 4 c 4 ) ] .
p o 4 k c c t 1 ; p o 3 f x , p o 3 f y 1 ; p o 4 f 1 ,
p o ' p o mc , k c mc ħ , f x e ħ z ct A x dℓ , f y e ħ z ct A y d , and f e 2 2 ħmc z ct A 2 dℓ .
Ψ r t = ( k c p o π ) 3 2 e i η t 2 η e 1 2 ( 1 + ip o η t ) ζ 2 + ip o η e p o 2 η x 2 + η y 2 + 2 p o ( f x η x + f y η y ) ζ + p o 2 ( f x 2 + f y 2 ) ζ 2 2 α [ 1 + i 2 p o 3 f α ζ ] α [ 1 + i 2 p o 3 f α ζ ] ,
1 1 + i 2 p o 3 f α ζ 1 i 2 p o 3 f α ζ 4 p o 6 f 2 α 2 ζ 2 +
Ψ r t = ( k c p o π ) 3 2 1 α β exp { i η t p o 2 η x 2 + η y 2 2 α p o 2 η 2 2 β }
η η z + i p ' o 2 f x η x + f y η y α + C η and β 1 + i p ' o 2 η t + p ' o 4 f x 2 + f y 2 α + C β .
C η p ' o 4 f η x 2 + η y 2 α 2 and C β 4 p ' o 8 f 2 α 3 ( η x 2 + η y 2 ) i 4 p ' o 6 f α 2 ( f x η x + f y η y )
ρ i ħ 2 mc 2 [ Ψ * r t Ψ r t t Ψ r t Ψ * r t t ] .
Ψ p , s V D r t = M u p , s Ψ p V KG ( r , t ) ,
M 1 = e 2 ( E c p z ) [ 0 A x iA y 0 A x i A y A x iA y 0 A x + i A y 0 0 A x i A y 0 A x i A y A x + i A y 0 A x i A y 0 ] .
u p , s = E + mc 2 2 mc 2 [ 1 0 p z c E + mc 2 p x c + ip y c E + mc 2 ]
Ψ D r t [ 1 e A x + i A y 2 mc 0 e A x + i A y 2 mc ] Ψ KG r t .
ρ D [ 1 + e 2 ( A x 2 + A y 2 ) 2 m 2 c 2 ] Ψ KG r t 2 .
x e A x mc , y e A y mc , and z 1 e 2 ( A x 2 + A y 2 ) 2 m 2 c 2 .

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