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. VolkovZ. 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]

Bjorken, J.D.

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

Drell, S.D.

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

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]

Greiner, W.

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

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. VolkovZ. 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]

Other (3)

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).

D. M. VolkovZ. Phys.94, 250 (1935).
[Crossref]

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.

Download Full Size | PPT Slide | PDF

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]

Download Full Size | PPT Slide | PDF

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]

Download Full Size | PPT Slide | PDF

Equations (22)

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− ħ 2 ∂ 2 Ψ ∂ t 2 = m 2 c 4 Ψ + c 2 ( − iħ ∇ + 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 ⇀ ( ℓ ) + e 2 A 2 ( ℓ ) 2 ] dℓ } .
E 2 = p 2 c 2 + m 2 c 4 .
〈 p ⇀ 1 ∣ p ⇀ 2 〉 = iħ 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 ℓ , 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 ∫ − ∞ ∞ dζ e 1 2 ( 1 + ip o ′ η t − ) ζ 2 + ip o ′ η zζ 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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