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Scaling law for energy-momentum spectra of atomic photoelectrons

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Abstract

A scaling law which was used to classify photoelectron angular distributions (PADs) is now extended to photoelectron kinetic energy spectra. Both a theoretical proof and an independent verification are presented. Considering PADs are of photoelectron momentum spectra, this extension really extends the scaling law to the entire energy-momentum spectra. The scaling law for photoelectron energy-momentum spectra applies to both directly ionized and rescattered photoelectrons. Re-scaling experimental input parameters without loosing the physical essence with this scaling law may ease the experimental conditions and reduce the material and the energy consumptions in the experiments.

© 2011 Optical Society of America

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Figures (3)

Fig. 1
Fig. 1 Polar plots of the PADs of the 23rd ATI peaks. (a) serves as a reference which is for hydrogen atoms irradiated by laser field of intensity 1.04 × 1014 W/cm2 and of wavelength 800 nm. (b), (c) and (d) are from a model atoms with binding energy 27.20 eV, irradiated by laser field of wavelength 400 nm and of intensity 4.16, 8.32, and 16.64 × 1014 W/cm2, respectively. The similarity between the PADs in (a) and (c) verifies the scaling law. We set λ = 0.1 and change it as K times in the scaling transformation.
Fig. 2
Fig. 2 (Color online) The calculated energy spectra for different scaling ratios. The original one is for the hydrogen atoms irradiated by laser field of wavelength 800 nm and intensity 1.04 × 1014 W/cm2. The rest two spectra are these calculated according to the scaling law with different scaling ratios. The value of λ is chosen as 0.1.
Fig. 3
Fig. 3 The kinetic energy spectra of photoelectrons calculated by the TDSE method: (a) is for argon atoms in laser field of wavelength 800nm and intensity 6.80 × 1013 W/cm2; (b)–(d) are calculated spectra for a model atom with binding energy 31.52 eV irradiated by laser field of wavelength 400 nm and intensity 2.72 × 1014 W/cm2 (b); 5.44 × 1014 W/cm2 (c); 1.09 × 1015 W/cm2 (d). For convenience of comparison, the abscissa is plotted in the unit of ω.

Equations (10)

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dW d Ω p f = ( 2 m e 3 ω 5 ) 1 / 2 ( 2 π ) 2 q ( q ɛ b ) 1 / 2 | T fi d + T fi r | 2 ,
T fi d = ( u p j i ) 𝒳 j i ( ζ f , η ) * 𝒳 j f ( ζ f , η ) Φ i ( | P f q k | ) ,
T fi r i π P , n = f P , n = P , n ϕ f , n f | Ψ P , n × Ψ P , n | U | Ψ P , n Ψ P , n | V | Φ i , n i ,
T fi r = i m e 4 π 5 / 2 j i 𝒳 j f ( ζ f , η ) ( u p j i ) | P | Φ i ( | P | ) × d Ω P 𝒳 q j i + j f ( ζ ζ f ) 𝒳 j i ( ζ , η ) * U ( P f P q k )
X n ( z ) J n ( | z | ) e in arg ( z ) , 𝒳 j ( z , z ) m = X j 2 m ( z ) X m ( z ) ,
ζ f = ζ 0 P f ɛ , ζ = ζ 0 P ɛ , η = u p ɛ ɛ / 2 ,
U ( r ) = Z e 2 4 π r exp ( λ r ) ,
U ( P f P ) = Z e 2 | P f P | 2 + λ 2 .
Φ i | P f | = 8 ( π β 5 ) 1 / 2 ( β 2 + | P f | 2 ) 2 ,
W γ ( ) = γ 2 n ( H 0 ) 2 n + γ 2 n ,
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