1. Introduction

The interaction of a strong laser field with atoms and molecules resulted in a
variety of phenomena ^{[1}^{,}^{2]}. Because of that, particular attention has been dedicated
to this “problem”, both theoretical and experimental ^{[3}^{,}^{4]}. The theoretical approaches are based on the numerical
solution of the time-dependent Schrodinger equation (TDSE) ^{[5}^{,}^{6]}, the strong-field approximation (SFA) ^{[7]}, and the semiclassical model for
the strong-field ionization ^{[8]}.

In this paper, we consider the case when the conditions for the semiclassical
approximation are satisfied (low frequency and field intensity in the range of ^{[9]}. It is also well known by Keldysh
parameter, ^{[10]} tunnel and multiphoton
ionization in strong laser field co-exist as two channels of ionization. Keldysh’s
theory is improved by Perelomov, Popov, and Terentev (PPT) ^{[11]}, and later extended by Ammosov, Delone, and
Krainov (ADK) ^{[12]}. The ADK theory
is one of the most used ones.

Here we will deal with elliptical polarization of the laser pulses. Compared with the case of linear (the most often used), the electron kinematics in elliptically polarized laser field are quite different. With elliptical polarization, an emitted electron is pulled away transversely because of the additional polarization direction and its trajectory becomes elliptical, reducing the probability of recolliding with its parent ion.

The quasistatic tunneling theory in an elliptically polarized laser field for a small
Keldysh parameter has been very successful in explaining experimental data ^{[18]}. However, as the Keldysh
parameter increases to the intermediate range, it was shown that the ADK theory
quantitatively deviates from the experimental results ^{[19]}. The reason for this deviation lies in the fact
that above mentioned theory is based on the independent particle (single active
electron). So, in order to avoid this problem, it is necessary to extend the
quasistatic tunneling theory with the presence of electron interaction in the system
^{[20}^{,}^{21]}.

Zon ^{[22]} introduced the idea
ofinelastic “tunneling”, whereby the parent ion can be left in an excited state
following the ionization of one electron. Release of the electron through the
process of photoionization may leave the residual positive ion either in the ground
state or in an excited state of higher energy in which at least one electron is
promoted to some empty orbital. Excitation is entirely caused due to
electron-electron interaction and probes the electron correlation in the ground and
final state.

In this paper, we introduced the excitation as well as electron-electron correlation,
and as a result we obtained the formula for transition rate and energy distribution
for the simultaneous core ionization and core-excitation of a helium atom (^{[23}^{,}^{24]}. We observed a non-relativistic domain in which the
influence of the magnetic field can be neglected ^{[23]}. That is reason why the transition rate and the
energy distribution of the ejected photoelectrons are determined by the electric
component of the laser field.

2. Theorical Concept

One of the most used theory for description of the ionization process of atoms in a
laser field, the ADK theory, is based on the tunneling of an electron through the
suppressed potential barrier of the combined atomic field and the external electric
field. For a monochromatic, elliptically polarized laser field, the atomic tunneling
ionization rate can be calculated using the following formula ^{[12]}:

where ^{[25]},

But, this theory neglected many aspects of the mentioned process, such as correlation
^{[23]}. But, it is fact that an
atom with more than one electron is a complex system of mutually interacting
electrons moving in the field of the nucleus. Because of that, we reported
theoretical calculations concerning electrons correlation. Additionally, according
to ^{[26]}, parallel with ionization
there is an excitation process. So, based on that, we modified the aforementioned
formula by treating the ionization rate as a cumulative contribution of simultaneous
processes, ionization and excitation, as a sequence of events.

We calculated a helium (and helium like) atoms within ^{[27]}. We assumed that the electron velocity is small
compared to the speed of light and applied a nonrelativistic calculation.

At the end, based on obtained formula, we formulated the expression for the energy
distribution. As we said, we considered the general case of a monochromatic wave,
with elliptical polarization,

We started with the adiabatic Landau-Dykhne approximation ^{[28]} of the saddle-point method for estimating the
time integrals in the quantum theory of transitions in an external AC field. In
order for this approximation be valid, it is necessary that the photon energy of the
was small compared to the ionization potential ^{[9}^{,}^{29]} (with exponential accuracy):

*i.e.* the transition rate ^{[30]}, where last term denotes the
electron’s energy in the core field.

We applied the described formalism on a two-electron transition in a helium (and
helium like) atoms after the absorption of a single photon. Also, we analyzed the
photoelectron distribution from

We firstly considered the excitation process. Simultaneous excitation ionization is
only possible due to electron-electron correlations ^{[31]}. To introduce it into transition’s formula, we
included the correlation effect into the initial energy, which now can be written
as:

where the second term describes the correlation effect ^{[32]}. Here, we omit the Stark shift of the initial
binding state. Applied laser field causes a shift of the atom’s energy levels and
this displacement of the energy level is determined by expression ^{[33]}. Also,
we included the ponderomotive potential which correlates to the oscillating movement
of charged particles in the final expression for the initial energy:

Now, we incorporated excitation of the second electron by modifying the final energy, ^{[34]}, and the Coulomb interaction, ^{[35]}, where the terms ^{[36]}. The lower sign describes
the state of lower energy, thus making the configuration

For the correlation of two electrons, the Coulomb repulsion and exchange integrals
have the following form: ^{[36]}. Additionally,
the Coulomb interaction is described as ^{[37]}, where

In Eq. 6, ^{[28]}:

We used some simple transformations and Maclaurin expansion in order to express the
turning point,

here,

In the interest of calculating the action,

Following

We would like to note that Eq. 10
strongly depends, among other, on the momentum ^{[37]}, where ^{[38]}. The momentum is conserved
along the classical path, ^{[29]} when a system’s total energy
is independent of the parabolic coordinate

Finally, in order to obtain the expression for the ionization rate we incorporated
Eq. (8) and Eq. (10) into already mentioned formula

For the sake of optimizing Eq. (11) we
introduced the effective Keldysh parameter

During our calculation, we supposed that the term

Next, we were interested to examine how mentioned effects influence the energy
distribution spectra. We started from the expression for the energy distribution
spectra ^{[40]}: ^{[30]}. Combined with the well-known expressions for
longitudinal energy of the ejected electron ^{[40]}, the energy distribution of
the ejected photoelectrons for standard ADK formula can be written as:

while our theoretical result based on Eq. (12), takes the form:

Eq. (13) and Eq. (14) describes the exponential
dependence of the energy distribution on the amplitude of the laser field,

3. Results and Discussion

In this section we investigated the ratio between the transition rate and the energy
distribution spectra of the ejected photoelectrons, obtained based on our analytical
formula for the ionization rate and the energy distribution (Eq. (12) and Eq. (14), respectively) and the
standard formula (Eq. (1) and Eq. (13)), for single ionized helium
atom, He, Z=1. The calculations were made for the linearly, circularly and
elliptically polarized laser pulses obtained by Ti:sapphire laser which provides
pulses of a wavelength

We started from the comparative review of the energy distribution spectra obtained
based on the standard ADK formula, Eq.
(13), (left plot) and our formula, Eq. (14), (right plot), for limiting case of the laser field
polarization,

From Fig. 1, it can be seen that both
theoretical curves are qualitatively similar. They continuously increase, reach
prominent peak and then decrease, but on the different energy range. The theoretical
ADK curve reaches a peak at

One can observe the shift to the lower intensity of the curve obtained based on our
formula, which is in accordance with^{[41]} where this movement to lower field intensity was
distinguished. Also, its energy range is significantly narrower. This is in
accordance with ^{[42]}. The ADK
curve lies above our curve by a few orders of magnitude. Significant deviation of
the ADK curve in comparison to experimental results was observed in ^{[43]}, where it was concluded that
ADK theory often overestimates the ionization rate ^{[23}^{,}^{43]}. This is in accordance to our results for the same
range of intensities. Also, our curve follows the trend of the experimental data and
has a similar shape to ^{[44]} .

Next, we repeated procedure for the case of a circularly polarized laser field,

Unlike the previous, Fig. 2 shows significantly
different behavior of the observed theoretical curves. For both curves is common
that they decline after reaching their maximum values. The difference between these
curves lies in the fact that the ADK curve decreases slowly, compared to ours which
approaches to the energy axis on about ^{[42}^{,}^{45]}. It is also important to note that the curves for the
case of circularly polarized laser field are a few magnitudes higher that in the
case of linear ^{[24]}.

Next, we examined how the ellipticity influences the transition rate and the energy
distribution range of the ejected photoelectrons. Figure 3 displays theoretical curves obtained based on our formula for
the transition rate, Eq. (12), (left
plot) and the energy distribution, Eq.
(14), (right plot), for ellipticities in the range of

As we said, on the left graph, we considered transition rates curves, in the given
ellipticity range. For the higher values of ellipticity ^{[46]}. From the first curve on the left with the
ellipticity ^{[47]}. In the Fig. 3
(right plot), we presented how the change in ellipticity affects observed energy
distribution spectra. It is obvious that the shape of the curves is maintained with
the change of ellipticity. For ellipticities until approximately ^{[45}^{,}^{46]}. Described curve’s behavior is in accordance
with experimental investigation by Chen *et al*. ^{[44]} and Dietrich *et
al*. ^{[48]}. Based on
all aforementioned, our results are closer to experimental data than those by the
standard ADK. Conclusion is that additional processes (which we included in our
formula), lead to better agreement between theoretical and experimental results.
That is why the behavior of our curves is consistent with ^{[23}^{,}^{44}^{,}^{48]}.

In Fig. 4, we wanted to show how the energy
distribution depends from two parameters. First, on left plot, we displayed the 3D
graph which demonstrates the transition rate obtained from our analytical formula,
Eq. (14), as a function of the
energy,

From Fig. 4 (left plot), it can be seen that 3D
curve raises faster for the change of the parabolic coordinate, ^{[46]}.

From all aforementioned, we can conclude that the photoelectron energy distribution spectra is very sensitive to the parabolic coordinate and ellipticity.

4. Conclusion

In conclusion, by applying a semiclassical model, we observed the tunneling ionization process in an elliptically polarized laser field. We presented results for the transition rate and energy distribution spectra with the contribution of additional processes, such as excitation and electron-electron correlation. The obtained results substantially deviate from the predictions of the ADK tunneling theory. We attributed the difference in results to the electrons correlation and excitation. Related to the influence of laser field polarization on the energy distribution spectra, we showed that it plays an important role.