Scaling law for energy-momentum spectra of atomic photoelectrons

Huiliang Ye; Yan Wu; Jingtao Zhang; D.-S. Guo

doi:10.1364/OE.19.020849

Optics Express
Vol. 19,
Issue 21,
pp. 20849-20856
(2011)
•https://doi.org/10.1364/OE.19.020849

Scaling law for energy-momentum spectra of atomic photoelectrons

Huiliang Ye, Yan Wu, Jingtao Zhang, and D.-S. Guo

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Huiliang Ye, Yan Wu, Jingtao Zhang, and D.-S. Guo, "Scaling law for energy-momentum spectra of atomic photoelectrons," Opt. Express 19, 20849-20856 (2011)

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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.

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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

\sqrt{K}

times in the scaling transformation.

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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.

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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.

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\frac{dW}{d Ω_{p_{f}}} = \frac{{(2 m_{e}^{3} ω^{5})}^{1 / 2}}{{(2 π)}^{2}} \sum_{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 |),

\begin{array}{l} T_{fi}^{r} \propto - i π \sum_{ℰ_{P, n} = ℰ_{f}} \sum_{ℰ_{P^{'}, n^{'}} = ℰ_{P, n}} 〈 ϕ_{f}, n_{f} | Ψ_{P, n} 〉 \\ \times 〈 Ψ_{P, n} | U | Ψ_{P^{'}, n^{'}} 〉 〈 Ψ_{P^{'}, n^{'}} | V | Φ_{i, n_{i}} 〉, \end{array}

\begin{array}{l} T_{fi}^{r} & = & \frac{i m_{e}}{4 π^{5 / 2}} \sum_{j_{i}} 𝒳_{- j_{f}} (ζ_{f}, η) (u_{p} - j_{i}) | P | Φ_{i} (| P |) \\ \times & \int d Ω_{P} 𝒳_{q - j_{i} + j_{f}} (ζ - ζ_{f}) 𝒳_{- j_{i}} {(ζ, η)}^{*} U (P_{f} - P - q k) \end{array}

\begin{array}{l} X_{n} (z) & \equiv & J_{n} (| z |) e^{in arg (z)}, \\ 𝒳_{j} (z, z^{'}) & \equiv & \sum_{m = - \infty}^{\infty} X_{j - 2 m} (z) X_{m} (z^{'}), \end{array}

ζ_{f} = ζ_{0} P_{f} \cdot ɛ, ζ = ζ_{0} P \cdot ɛ, η = u_{p} ɛ \cdot ɛ / 2,

U (r) = \frac{- Z e^{2}}{4 π r} exp (- λ r),

U (P_{f} - P) = \frac{- Z e^{2}}{{| P_{f} - P |}^{2} + λ^{2}} .

Φ_{i} | P_{f} | = \frac{8 {(π β^{5})}^{1 / 2}}{{(β^{2} + {| P_{f} |}^{2})}^{2}},

W_{γ} (ℰ) = \frac{γ^{2^{n}}}{{(H_{0} - ℰ)}^{2^{n}} + γ^{2^{n}}},

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