Relativistic photoelectron spectra in the ionization of atoms by elliptically polarized light

Recently an increasing interest in the investigation of relativistic ionization phenomena has been observed [1–10]. Relativistic effects will appear if the electron velocity in the initial bound state or in the final state is comparable with the speed of light. The initial state should be considered relativistic in the case of inner shells of heavy atoms [3,4]. In a recent paper [10] the photoionization of an atom from a shell with relativistic velocities has been considered for the case of elliptically polarized laser light. In the present paper we will study the effect of a relativistic final-state of an electron on the ionization of an atom by elliptically polarized light. The initial state will be considered as nonrelativistic. The final-state electron will have an energy in the laser field measured by the ponderomotive energy. If the ponderomotive energy approaches the electron rest energy, then a relativistic treatment of the ionization process is required. For an infrared laser the necessary intensities are of the order of 10W cm−2. Ionization phenomena influenced by relativistic final state effects have been studied for the cases of linearly and circularly polarized laser radiation both in the tunnel [6,9] and above threshold regimes [2,7]. The ionization rate for relativistic electrons has been found to be very small for the case of linear polarization [6,7]. On the contrary a circularly polarized intense laser field produces mainly relativistic electrons [2,9]. In the papers of Reiss and of Crawford and Reiss [1,2,7] a covariant version of the so-called strong field approximation [11] has been given for the cases of linear and circular polarization. Within this approximation one calculates the transition amplitude between the initial state taken as the solution for the Dirac equation for the hydrogen atom and the final state described by the relativistic Volkov solution. Coulomb corrections are neglected in the final Volkov state. Analytical results for the ionization rate have been given in Refs. [1,2,7] These results apply to above barrier cases as well as to tunneling cases. However, the corresponding expressions are complicated and numerical calculations are needed to present the final results. The present paper is aimed to investigate the relativistic electron energy spectra in the ionization of atoms by intense elliptically polarized laser light. In contrast to the more sophisticated strong field approximation we would like to obtain simple analytical expressions from which the dependence of the ionization process on the parameters, such as binding energy of the atom, field strength, frequency and ellipticity of the laser radiation may be understood without the need of numerical calculations. Therefore we restrict the considerations to the case of tunnel ionization. Our results will be applicable only for laser field strengths smaller than the inner atomic field F Fa. In order to observe relativistic effects, the inequality = F/ωc > 0.1 should be fulfilled. (The atomic system of units is used throughout the paper, m = e = h̄ = 1.) Both inequalities yield a limitation for the laser frequency ω from above. For the ionization of multi-charged ions an infrared laser satisfies this condition. The non-relativistic sub-barrier ionization with elliptically polarized light was studied in [12]. In the tunnel limit the simple expression