Effect of the corrected ionization potential and spatial distribution on the angular and energy distribution in tunnel i
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TOMS, MOLECULES, OPTICS
Effect of the Corrected Ionization Potential and Spatial Distribution on the Angular and Energy Distribution in Tunnel Ionization1 V. M. Petrović and T. B. Miladinović* Department of Physics, Faculty of Science, Kragujevac University 34 000, Kragujevac, Serbia * e-mail: [email protected] Received November 9, 2015
Abstract—Within the framework of the Ammosov–Delone–Krainov theory, we consider the angular and energy distribution of outgoing electrons due to ionization by a circularly polarized electromagnetic field. A correction of the ground ionization potential by the ponderomotive and Stark shift is incorporated in both distributions. Spatial dependence is analyzed. DOI: 10.1134/S1063776116050101
1. INTRODUCTION Tunnel ionization of atoms is an important mechanism of laser ionization. There are several theoretical approaches to the calculation of the tunnel ionization rate. The oldest is Keldysh’s approach [1] based on the assumption known as the strong-field approximation [2]. In [1], Keldysh introduced the well-known parameter γ = E i /2U p (Ei is the unperturbed ionization energy and Up is the ponderomotive potential) that determines whether the photoionization process lies in the tunneling or multiphoton region. For γ ≪ 1, the tunnel ionization is a dominant ionization process. Soon after this work, Perelomov, Popov, and Terent’ev developed the PPT model [3]. Finally, Ammosov, Delone, and Krainov extended the PPT theory and derived the ADK theory, which we use here [4]. Within the framework of the ADK theory, the ionization rate for a circularly polarized laser field is [5]
potential caused by laser irradiation taken into account. 2. THEORETICAL FRAMEWORK In particular cases of circular polarizations of the ionizing field, the angular distribution of photoelectrons emerging from tunnel ionization is described by [5]
⎛ F 2E ⎞ W (θ) = W (0) exp ⎜ − cir 2 i θ 2 ⎟ , ω ⎝ ⎠
where θ is the azimuthal angle and W(0) = W cirADK is the tunneling rate. The nonzero initial electron influences the tunneling rate, and we therefore use the modified expression [6]
Wcir, p = ADK
ADK
W cir
=
2 ⎛ ⎞ Fcir Dcir exp ⎜ − 3Z ⎟, 8π Z ⎝ 3n* Fcir E i ⎠
n*
Fcir = 19 I 9 , 5.1 × 10
I is the laser field intensity, and n* = Z/ 2E i is the effective principal quantum number (Z is the ion charge). The purpose of this paper is to improve the angular and energy distributions of ejected photoelectrons with the corrections of the unperturbed ionization 1The article is published in the original.
2 ⎛ Fcir Dcir p 2γ 3 ⎞ exp ⎜ − 3Z − ⎟. 8πZ ⎝ 3n* Fcir E i 3ω ⎠
Just after leaving the barrier, the momentum of the ejected photoelectron is p. Because the ionization probabilities in static and alternating electric fields are different only by a preexponential factor [3], it is convenient to use the parabolic coordinate to express the initial momentum outside the barrier as
where 3 ⎞ ⎛ Dcir ≡ ⎜ 4Z e4 ⎟ , ⎝ Fcir n* ⎠
(1)
⎛ ⎞ 1 p = 1 ⎜ Fcir η − 1 − ⎟, 2⎝ η Fcir η − 1⎠ where η is the parabolic coordinate, η > 1
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