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High-order above-threshold ionization with few-cycle pulse: a meter of the absolute phase

D. B. Milošević, G. G. Paulus, and W. Becker

Open Access Open Access

Abstract

Angle-resolved energy spectra of high-order above-threshold ionization are calculated in the direction of the laser polarization for a linearly polarized four-cycle laser pulse (two cycles FWHM) as a function of the carrier-envelope relative phase (absolute phase). The spectra exhibit a characteristic left-right (backward-forward) asymmetry, which should allow one to determine the value of the absolute phase in a given experiment by comparison with the theoretical spectra. A classical analysis of the spectra calculated is presented. High-energy electron emission is found to occur in one or two ultrashort (≲0.7 fs) bursts. In the latter case, the spectra display a peak structure whose analysis reveals a time-domain image of electron emission.

©2003 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. Angle-resolved ATI energy spectrum generated by the 4-cycle pulse (11) with intensity I=E02=6×1013 Wcm-2 and λ=800 nm in krypton (IP =14 eV) for an absolute phase ϕ=0°, for the two opposite detector positions θ=0° and θ=180° along the laser-field polarization. For comparison, the envelope of the ATI spectrum for the same atomic and laser parameters but for an infinitely long pulse with flat envelope (amplitude E 0) is also shown (green dashed line with circles). For this case, the high-energy cutoff is at 10UP =35.9 eV. The high-energy part of the 4-cycle spectrum for θ=0° virtually agrees with the envelope of the infinitely-long-pulse spectrum. The explanation will be found in the classical model considered below.

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Fig. 2.
Fig. 2. The same as Fig. 1, but for nine different absolute phases between ϕ=0° and ϕ=180°. The uppermost left panel with ϕ=0° shows the same spectra as Fig. 1. The spectra for θ=0° are designated by black lines, those for θ=180° by thin red (gray) lines.

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Fig. 3.
Fig. 3. Solutions of the classical rescattering model formulated in Eqs. (16)–(18) for the parameters of Fig. 1, absolute phase ϕ=0°, and θ=0° (upper panel) and θ=180° (lower panel). The various black curves specify the energy E p at the detector as a function of the ionization time t. The dashed magenta curve represents the electric field given by Eq. (11) with the scale given on the right-hand ordinate. The solid blue curves that are identical in both panels, being symmetric with respect to t=2 o.c. and having two maxima at E p =7.1 eV refer to direct ionization. They are trivial solutions of Eq. (16) for t=t′.

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Fig. 4.
Fig. 4. Rescattering times t′ as a function of the ionization time t for the solutions of the classical model, given by Eqs. (16)–(18). The parameters are those of Fig. 1 and the absolute phase is ϕ=0°. Rescattering times that give rise to E p >10 eV, E p >20 eV, and E p >30 eV, are marked in red (medium gray), blue (dark gray), and cyan (light gray), respectively.

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Equations (18)

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M p = − i lim t → ∞ ∫ 0 t d t ′ 〈 ψ p ( t ) ∣ U ( t , t ′ ) r · E ( t ′ ) ∣ ψ 0 ( t ′ ) 〉 .
H ( t ) = − ∇ 2 2 + r · E ( t ) + V ( r ) ,
U ( t , t ′ ) = U F ( t , t ′ ) − i ∫ t ′ t d t ″ U F ( t , t ″ ) V U ( t ″ , t ′ ) ,
∣ χ p ( t ) 〉 = ∣ p + A ( t ) 〉 exp [ − i S p ( t ) ]
S p ( t ) = 1 2 ∫ t d t ′ [ p + A ( t ′ ) ] 2 .
U F ( t , t ′ ) = ∫ d 3 k ∣ χ k ( t ) 〉 〈 χ k ( t ′ ) ∣ .
p ~ = p − A ( T p )
M p SFA = M p ( 0 ) + M p ( 1 ) ,
M p ( 0 ) = − i ∫ 0 T p d t 〈 χ p ~ ( t ) ∣ r · E ( t ) ∣ ψ 0 ( t ) 〉 ,
M p ( 1 ) = − ∫ 0 T p d t ∫ t ∞ d t ′ 〈 χ p ~ ( t ′ ) ∣ V U F ( t ′ , t ) r · E ( t ) ∣ ψ 0 ( t ) 〉 .
E ( t ) = e ̂ E 0 sin 2 ω p t 2 cos ( ω t + ϕ ) = e ̂ ∑ i = 0 , 1 , 2 ℰ i cos ( ω i t + ϕ )
A ( t ) = − ∫ t d t ′ E ( t ′ ) = − e ̂ ∑ i = 0,1,2 ℰ i ω i sin ( ω i t + ϕ ) .
w ( θ , ϕ ) = ∣ M p ∣ 2 d 3 p d Ω p ̂ d E p = p ∣ M p ∣ 2 ,
〈 k ∣ r · E ( t ) ∣ ψ 0 〉 = − i 2 7 2 ( 2 I p ) 5 4 k · E ( t ) π ( k 2 + 2 I p ) 3 .
v ( t 1 ) = A ( t 1 ) − A ( t ) .
r ( t ′ ) = α ( t ′ ) − α ( t ) − A ( t ) ( t ′ − t ) = 0 ,
E p = v 2 ( T p ) 2 ,
v ( T p ) = A ( T p ) − A ( t ′ ) + ∣ A ( t ′ ) − A ( t ) ∣ v ̂ ( t ′ + ) ≡ p ,
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