#### 4.1. Theory

We assume the laser field to be classical, monochromatic, spatially homogeneous over atomic dimensions, and linearly polarized. Working in the Coulomb gauge, we have for the electric field $\mathcal{E}\left(t\right)={\mathcal{E}}_{0}sin(\omega t+\phi )$ the corresponding vector potential $\mathit{A}\left(t\right)={\mathit{A}}_{0}cos(\omega t+\phi )$ with ${\mathit{A}}_{0}=c\phantom{\rule{4pt}{0ex}}{\mathcal{E}}_{0}/\omega $. Remembering that in the Ehrhardt asymmetric coplanar geometry, a fast electron of momentum ${\mathit{k}}_{i}$ is incident on the helium target, and a fast scattered electron of momentum ${\mathit{k}}_{a}$ is detected in coincidence with a slow ejected electron of momentum ${\mathit{k}}_{b}$ , the three momenta ${\mathit{k}}_{i}$, ${\mathit{k}}_{a}$, and ${\mathit{k}}_{b}$ being in the same plane. In addition, the scattering angle ${\theta}_{a}$ of the fast electron is fixed and small, while the angle ${\theta}_{b}$ of the slow electron is varied.

The central quantity to be evaluated is therefore the direct first Born

S-matrix element [

29]

with

In this equation

${\mathit{r}}_{0}$ denotes the coordinate of the incident (and scattered) electron,

${\mathit{r}}_{1}$ is the coordinate of the target electron, and

${r}_{01}=\mid {\mathit{r}}_{0}-{\mathit{r}}_{1}\mid $. The wave functions

${\chi}_{{k}_{i}}$ and

${\chi}_{{k}_{a}}$ are Volkov wave functions describing, respectively, the motion of the incident and scattered electrons in the presence of the laser field. They are given by Equation (

8).

The wave functions

${\varphi}_{0}({\mathit{r}}_{1},t)$ and

${\varphi}_{{k}_{b}}({\mathit{r}}_{1},t)$ appearing in Equation (

22) are “dressed” states of the hydrogen atom embedded in the laser field. The first one corresponds to the initial bound state, and the second one to the final continuum state consisting of an ejected electron of momentum

**k**${}_{b}$ moving under the combined influence of the proton and the laser field. For electric field strengths

${\mathcal{E}}_{0}$ small with respect to the atomic unit of field strength (

${\mathcal{E}}_{0}\ll \frac{e}{{a}_{0}^{2}}\simeq 5\times {10}^{11}$ V·m

^{−1}), which is the case considered in this work, we can use the first-order time-dependant perturbation theory to obtain explicit expressions of the dressed atomic bound states. The dressed ground-state wave function of the hydrogen atom is given by Equation (

12), while the dressed continuum wavefunction we have used is given by Equation (

21).

Substituting the expressions (

8), (

12) and (

21) into the first Born S-matrix element (

22) and performing the time integration by using the Fourier expansion of the factors

${e}^{-i{\mathit{k}}_{b}.{\mathit{\alpha}}_{0}sin\left(\omega t\right)}$, we obtain

here

${f}_{ion}^{{B}_{1},\ell}$ is the first Born amplitude for the laser-assisted (e, 2e) scattering amplitude with the transfer of

ℓ photons. This quantity is given by

with

and

In these equations, ${J}_{\ell}$ is a Bessel function of order ℓ, ${J}_{\ell}^{\prime}$ is its first derivative, and we have introduced the quantity $\lambda =(\Delta -{\mathit{k}}_{b}).{\mathit{\alpha}}_{0}$, where $\Delta ={\mathit{k}}_{i}-{\mathit{k}}_{a}$ is the momentum transfer of the collision.

The case of the absorption

$(\ell =1)$ or emission

$(\ell =-1)$ of one laser photon by the electron-atom system is of particular interest. Indeed, in addition to the expressions (

26a)–(

26c) which are valid to all orders in the laser-projectile interaction and to first order in the laser-atom and electron-atom interactions, one can then also obtain the lowest-order perturbative version of these amplitudes by using a second-order treatment in the combined electron-atom and laser-atom system (electron + atom) interactions. The corresponding amplitudes are given for the case

$\ell =1$ by

and

where

${G}_{c}$ denotes the Coulomb Green’s function. For the case

$\ell =-1$ (emission of one photon) the corresponding formulas are obtained by changing

$\omega $ into

$-\omega $. We remark that these limiting forms can be recovered directly from the corresponding amplitudes of Equation (

26a)–(

26c) by retaining only terms up to first order (in the field strength) in the power series expansions of the Bessel functions.

The first Born triple differential cross section corresponding to the (e, 2e) reaction accompanied by the transfer of

ℓ photons is then given by

We remark that the amplitude

${f}_{1}$ corresponds to a first Born treatment in which the target dressing effects are neglected. In this approximation, the first Born TDCS reduces to

with

is the field-free first Born ionization amplitude.

A similar analysis of the laser-assisted electron-impact ionization can be made for the second-order term of the Born series. Thus, the second Born S-matrix element accompanied by the transfer of

ℓ photons can be given by

where

${G}_{0}^{(+)}$ is the causal propagator defined by

It should be noted that this term is second order in the electron-atom interaction potential ${V}_{d}$, and contains atomic wave functions corrected to first-order correction in ${\mathcal{E}}_{0}$ for the target dressed states. If one retains a global first-order correction in ${\mathcal{E}}_{0}$ for the target states, one finds that ${S}_{ion}^{{B}_{2}}$ is the sum of two terms which are respectively of zeroth and first-order in ${\mathcal{E}}_{0}$. We shall neglect the second-order contribution to the S-matrix element for laser-assisted collisions calculated in first order in ${\mathcal{E}}_{0}$, and concentrate our discussion on the computation of the dominant term ${S}_{ion}^{{B}_{2},0}$, which describes the collision of a Volkov electron with the undressed atom.

Making explicit the time dependence of the matrix elements and using the integral representation of

$\Theta $
one obtains

with

${\Delta}_{i}={\mathit{k}}_{i}-\mathit{q}$,

${\Delta}_{f}=\mathit{q}-{\mathit{k}}_{a}$, and

${\tilde{V}}_{d}(\Delta ,\mathit{r})={e}^{i\Delta \mathit{r}}-1$. The expression (

31) is obtained from the quantity Equation (

34) by shifting the pole of the integrand, respectively, below and above the real

$\omega $-axis by a small positive quantity

$\epsilon \to {0}^{+}$. The sum over

n runs over the complete set of hydrogenic states, and one has to integrate over the virtual projectile states

$\mid {\chi}_{q}({\mathbf{r}}_{0},t)>$ with wave vectors

$\mathit{q}$. After using the generating function expansion

for each of the sinusoidal exponentials, the

${t}^{\prime}$ integral reduces to a

$\delta $ function. This makes it trivial to perform the

$\xi $ integration for

$\xi ={E}_{{k}_{i}}+{E}_{0}-{E}_{{k}_{b}}+{\ell}^{\u2033}\omega $ with considering the exchange of

$\ell ={\ell}^{\prime}+{\ell}^{\u2033}$ photons, the corresponding second Born S-matrix component then reads

Using the soft-photon approximation [

84] which assumes that the quantity

${\ell}^{\prime}\omega $ entering the denominator of the second-order matrix element in Equation (

36) is much smaller than any energy difference

${\ell}^{\prime}\omega \ll \phantom{\rule{4pt}{0ex}}\mid {E}_{{k}_{i}}+{E}_{0}-{E}_{q}-{E}_{{k}_{b}}+\ell \omega -{E}_{n}\mid $, and performing the infinite summation over

${\ell}^{\prime}$ by using the addition theorem for Bessel function

Thus, the lowest-order component

${S}_{ion}^{{B}_{2},0}$ evaluated at the shifted momenta

${\Delta}_{i}$ and

${\Delta}_{f}$ can be expressed in terms of a second-Born amplitude as

where

with

is the field-free second-Born ionization amplitude evaluated at the shifted momenta

${\Delta}_{i}$ and

${\Delta}_{f}$.

is the Coulomb Green’s function with argument

$\Omega ={E}_{{k}_{i}}-{E}_{{k}_{b}}-{E}_{q}+{E}_{0}+\ell \omega $, where

${E}_{0}=-0.5\phantom{\rule{4pt}{0ex}}$ a.u. is the ground state energy of atomic hydrogen, and

${E}_{q}$ is the virtual projectile energy. We have the definition

$\lambda ={\mathit{\alpha}}_{0}.(\Delta -{\mathit{k}}_{b})$ with

$\Delta ={\Delta}_{i}+{\Delta}_{f}$.

The electron-atom amplitude with the transfer of

ℓ photons may be written in the second Born approximation as

We note that the integral in Equation (

40) over the virtual projectile states

${\chi}_{q}({\mathbf{r}}_{0},t)$ with wave vector

$\mathit{q}$ is prohibitively difficult, which is actually zero at some values of incident electron energies. We shall overcome this difficult by using the exact upper boundary of the integral Equation (

40) over the virtual projectile energies (see

Appendix B), which is obtained by the requirement

The first and second Born amplitudes corresponding to the first and second-order contributions to the S-matrix element, for the laser-assisted electron-impact ionization, have been computed exactly without further approximation with the help of a Sturmian approach described in

Appendix A.

The contribution of laser-assisted (e, 2e) collisions to the S-matrix of exchange scattering leads to some conceptual difficulties but would not significantly alter the results of the present discussion. We have considered in the present work a first Born exchange amplitude

${g}_{ion}^{\ell}$ with the transfer of

ℓ photons [

79,

85]

where

Finally, the second Born triple differential cross section corresponding to the ionization process, with the transfer of

ℓ photons, is given by

#### 4.2. Results and Discussion

Let us turn to a discussion of the results obtained in the case of (e, 2e) collisions in several geometrical configurations. In the present investigation, our results are interpreted by estimating the first and second Born triple differential cross sections, where the scattering angle is kept fixed at ${5}^{\circ}$. We present and analyze our findings for the TDCS of the laser-assisted (e, 2e) reaction in the coplanar asymmetric geometry. Without loss of generality, we assume the origin of the coordinate system to be the target nucleus and the z-axis to be along the incident momentum. The x-axis is in the plane defined by the incident momentum and the polarization vector of the external field. The scattering angle of the scattered electron and the emission angle of the ejected electron are denoted respectively by ${\theta}_{a}$ and ${\theta}_{b}$. The former is measured in anticlockwise direction, and the latter clockwise.

In

Figure 3, we give the triple differential cross sections corresponding to the ionization of the atomic hydrogen from the ground state by electron-impact, in the presence of a laser field, as a function of the ejected electron angle

${\theta}_{b}$. The incident electron energy is

${E}_{{k}_{i}}=40$ eV, the ejected electron energy is

${E}_{{k}_{b}}=5$ eV, the scattering angle is

${\theta}_{a}={5}^{\circ}$, and the laser photon energy is

$\omega =1.17$ eV which corresponds to first harmonic of Nd-YAG laser. We are working in a geometry in which the polarization vector

$\widehat{\epsilon}$ of the field (which is along

${\mathcal{E}}_{0}$ for the case of linear polarization considered here) is parallel to the incident momentum

**k**${}_{i}$. We present the results of our complete computation of the TDCS in the second Born approximation (SBA), and compared them with the first Born approximation (FBA) ones, and with those obtained when ignoring the dressing of the target. One remarks that the values obtained from the simplified treatment ignoring the dressing of the target correctly reproduce the shape of the angular distribution although the magnitudes of the cross sections are underestimated. With no net exchange of photons, the results show a distinct two peaks structure. The binary peak is attributed to the electron-electron interaction, while the recoil peak is governed by the attraction between the electron and the nucleus. We also remark that when the laser field is applied, but with no photon exchange, the results are governed by binary collisions. The comparison between FBA and SBA results indicates that the binary peak is strongly enhanced in SBA while the recoil peak is suppressed slightly. With the net exchange of one photon

$(\ell =\pm 1)$, the recoil collision becomes important and the dressing of the atomic target can also significantly affect the TDCS corresponding to the ionizing process. This is shown in

Figure 3, where both the shape and the magnitude of our FBA and SBA cross sections exhibit important departures with respect to the results obtained by ignoring dressing effects for a laser frequency

$\omega =1.17$ eV and an electric field strength

${\mathcal{E}}_{0}={10}^{7}$ V/cm.

In

Figure 4,

Figure 5 and

Figure 6, we give the triple differential cross sections as a function of the ejected angle

${\theta}_{b}$ and for different incident energies

${E}_{i}=30,\phantom{\rule{4pt}{0ex}}50$ and 100 eV with and without exchange of photons (

$\ell =0,1,-1$). One remarks that, with the increase of

$\mid \ell \mid $, the recoil collision becomes prominent. As is already noted before by several authors [

29,

32,

76], the dressing effects are seen to be important with the exchange of one photon (

$\ell =\pm 1)$. This is due to the presence in the atomic term in first Born approximation of

$s-p$ transition amplitudes which behave like

${\Delta}^{-1}$ for small transfer momentum

$\Delta $. We notice that, in

Figure 4, when the incident electron energy decreases, the recoil peaks in the first and second Born approximations are depressed, and the difference between the values of the second Born and first Born TDCS becomes larger. In fact, the second order correction becomes significantly important in the vicinity of the maximum of the binary peaks and almost invisible elsewhere. Furthermore, the absolute magnitude of the first and second Born triple differential cross sections increases with the incident energy.

In

Figure 5 and

Figure 6, the dependance of the cross sections on the ejected angle is shown for

$\ell =\pm 1$. For the absorption of one photon, the angular distribution is strongly modified as the binary and recoil peaks are now split into big and smaller lobes with different magnitudes. The overall magnitude of the cross section increases steadily with the incident electron energy. Another interesting point is the fact that the binary peaks remains unchanged for the first and second Born approximation, while the recoil peaks marks a small change in the order of their cross sections. This indicates that, for the absorption of one photon (

$\ell =1$), and at relatively low incident energies, the electron-atom interaction in the second Born approximation does not play a dominant role in the physics of the process. For the emission of one photon (

$\ell =-1$) the laser-assisted triple differential cross sections corresponding to the (e, 2e) reaction in atomic hydrogen show different behavior for the cross sections to the case of absorption. In fact, the magnitude of the cross sections is significantly smaller, meaning that the system absorbs net energy from the radiation background. Furthermore, a major suppression of the magnitude of the recoil peaks in the SBA with respect to that predicted by the first Born approximation is observed. Note that our second-order Born results are in excellent agreement with those obtained in the first-order Born approximation at high impact energies. We also observe the occurrence of a shift in the position of the peaks. In fact, when decreasing the incident electron energy, the maximum of the binary and recoil peaks moves to

${\theta}_{b}=0$ and

${\theta}_{b}=180-{\theta}_{b}$ respectively. This is due to the momentum transfer

$\Delta $ who changes with the incoming electron energy (according to the energy conservation equation) leading to the observed shift of the pics to the right or left to the momentum transfer.

The last set of figures address the role of the laser polarization orientation. In

Figure 7,

Figure 8,

Figure 9 and

Figure 10, we vary the polarization orientation and everything else being kept fixed. It can be observed that the geometry in which

$\widehat{\epsilon}$ is kept parallel to the momentum transfer of the projectile

$\Delta $ maximizes the cross section (see

Figure 7 and

Figure 9). This can be understood as resulting from the fact that this geometry maximizes the argument of the Bessel functions entering the expression of the transition amplitudes. In a more physical perspective, it corresponds to the fact that the coupling between the projectile and the field is maximum for laser-assisted TDCS which are located at angles for which

**k**${}_{b}$ is parallel to

$\Delta $, as the coupling between the ejected electron and the field is also maximized. We also note that, in this case, the symmetry of the field-free TDCS with respect to

$\Delta $ is restored. On the other hand, in the presence of the laser, the symmetry with respect to

$\Delta $ is broken and one recovers the splitting of the recoil and binary peaks. The magnitudes of the cross sections, though close from each other, being notably smaller than in the preceding case. The similarity with the angular distributions in

Figure 5 and

Figure 7 and in

Figure 6 and

Figure 9 comes form the fact that, for the kinematics of the collision chosen here, the incident momentum

**k**${}_{i}$ is almost perpendicular to

$\Delta $ for every value

**k**${}_{b}$. We note also that, at the laser intensity considered here, dressing effects are seen once again to produce a dramatic enhancement of the cross sections, and that the approximate treatment neglecting the dressing of the target is not adequate for the geometries considered here.

The Bessel function occurring in different amplitudes causes the oscillating structure in the results for a fixed laser field strength. When the magnitude of the total momentum transfer $\Delta $ (the recoil ion momentum) is large, the Bessel functions arguments varies in a wide range when the electron angle is scanned, leading thus to the observed oscillations in TDCS. The curves for ℓ and $-\ell $ present similar features since ${J}_{-\ell}\left(\lambda \right)={(-1)}^{\ell}{J}_{\ell}\left(\lambda \right)$. Nevertheless, the magnitudes of the cross sections for ℓ and $-\ell $ are different. The origin of this difference lies in the terms in different amplitudes other than ${J}_{\ell}\left(\lambda \right)$. For a different direction of the electric field, i.e., different direction of ${\mathit{\alpha}}_{0}$, the scalar product $\Delta .{\mathit{\alpha}}_{0}$ may become sizeable, and hence the difference seen between $\widehat{\epsilon}//\Delta $ and $\widehat{\epsilon}\perp \Delta $. It is generally believed that the recoil peak involves significant interaction with the residual ion.

It is seen in the set of

Figure 7,

Figure 8,

Figure 9 and

Figure 10 that the target dressing has a relatively large effect on the cross sections. When the laser polarization is parallel to the momentum transfer (see

Figure 7 and

Figure 9), the difference between the FBA and SBA triple differential cross sections persists in the low energies of the incident electron region in the vicinity of the recoil peaks. Note that, for the case of the absorption of one photon (

$\ell =1)$ the second-order correction is small. When the polarization vector of the field is perpendicular to the momentum transfer (see

Figure 8 and

Figure 10), a significant difference is noticed between the triple differential cross sections predicted by the second and first-order Born approximations at energies

$\le 50\phantom{\rule{4pt}{0ex}}$eV especially for (

$\ell =-1)$. In fact, the margins between the SBA and FBA results are large at the maxima of the binary and recoil peaks. With impact energy decreasing, the binary collision is enhanced and the recoil collision is suppressed. To sum up, for a particular choice of the scattering geometry (

$\widehat{\epsilon}\perp \Delta $), the absolute magnitudes of the binary and recoil peaks are given extremely well by our method, while the first Born approximation is seen to be unsatisfactory. These results clearly demonstrate the importance of second-order effects in understanding the dynamics of the ionization processes at low energies. Note that the effect of the second term of the Born series decreases with the increase of the incident energy.

In

Figure 11, we give the triple differential cross sections corresponding to the laser-assisted electron-impact ionization of a hydrogen target as a function of the incident electron energies, where the angle of the ejected electron is kept fixed. The polarization vector of the field

$\widehat{\epsilon}$ (which is along

${\mathcal{E}}_{0}$ for the case of linear polarization considered here) is set parallel to the impact momentum

**k**${}_{i}$. The complete results obtained by using the ionization amplitude Equations (

25) and (

42) for the first and second Born approximations are compared with those obtained by neglecting the dressing of the target which coincide with the electronic amplitude

${f}_{1}$ and with field-free results. It is interesting to note that the results are notably sensitive to the second Born approximation at low incident electron energies. As the incident energy increases, the ratio of the second Born triple differential cross section to the first one becomes smaller, and at high incoming energies such proportionality factor is completely absent, where the second Born approximation does not offer a significant improvement over the first Born treatment.

According to the domain of validity of the treatment used for taking into account the laser-atom interaction, the Nd-YAG laser frequency is taken to be

$\hslash \omega =0.043\phantom{\rule{4pt}{0ex}}$a.u. For no net transfer of photons

$(\ell =0)$, our results of TDCS are nearly of the same order of magnitude as field-free TDCS, while for

$\mid \ell \mid =1$ the field-free results are much larger (not presented here in our figures). This results from the fact that the laser itself does not contribute to the ionizing process. In fact, the laser redistributes the ejected electrons in new channels associated to indices

$\ell \ne 0$ in the energy conservation relation Equation (

2), which are accessible in the dressed continuum of the atomic target. One observes significant departures of the results obtained by the simplified treatment neglecting dressing effects with respect to those obtained by the first and second born approximations. This difference in magnitude is traced to the role played by the explicit introduction of the atomic dressing states. This directly reflects the role of the dressing of the projectile target system by the external laser field. This is one of the interesting typical signatures of the dressing of the electron-target system in the TDCS which clearly shows the effects of the internal structure of the atomic target. Such a distorted atom also acts on the projectile by a long-range dipole potential (

$\sim 1/{r}^{2}$), which requires a non-perturbative treatment of laser-atom interactions. The long-range dipole potential affects mainly the distant collisions, which contribute when the energies of the primary electron are weak. Another interesting point is the fact that the overall magnitude of the cross sections corresponding to the difference between SBA and FBA decreases with the increase of the incident electron energies. Moreover, our second and first Born TDCS present an absolute maximum corresponding to the zero of the momentum transfer (i.e.,

$\phantom{\rule{4pt}{0ex}}1/{\Delta}^{2}$) contained in various TDCS expressions. The same remark appears in the results obtained by neglecting the dressing of the target and laser-off.

Figure 12 and

Figure 13 shows triple differential cross sections versus the field amplitude. Apparently, the (e, 2e) reaction process can be controlled by the field strength. Two special geometries of laser polarization are considered:

${\widehat{\epsilon}}_{0}//{\mathit{k}}_{i}$ (the laser polarization vector parallel to the incident momentum) and

${\widehat{\epsilon}}_{0}\perp {\mathit{k}}_{i}$ (the laser polarization vector perpendicular to the incident momentum). For no net exchange of photons, the margins between the results of the first and second Born approximations reaches maxima at several values of field strengths in the case of

${\widehat{\epsilon}}_{0}//{\mathit{k}}_{i}$, while for

${\widehat{\epsilon}}_{0}\perp {\mathit{k}}_{i}$ the margins occurs at weak field strength and at

${\mathcal{E}}_{0}=7\times {10}^{7}\phantom{\rule{4pt}{0ex}}$V/cm. Note that, for both geometries, dressing effects become very important with the increase of the field strength, this is because the stronger is the laser the more the atomic states are distorted. The second order correction is seen to be significant in the vicinity of the maxima of the peaks and decreases with the increase of the laser field amplitude. We also observe a small influence of the laser field at low field strength with the net exchange of photons. This is due to the chosen geometry that coincides approximatively with the region of the binary peak. This situation change in the recoil peak region when even a weak field strength leads to sizeable changes in the cross section.

In

Figure 12 and

Figure 13, the corresponding dispersion curves in terms of

${\mathcal{E}}_{0}$ are characterized by the occurrence of sharp maxima separated by deep minima. The number of lobes increases with the laser intensity. This behavior can be traced back to the fact that the argument of the Bessel functions

${J}_{\ell}\left(\lambda \right)$, entering the expressions of the amplitudes Equations (

25) and (

42), grows with

${\mathcal{E}}_{0}$. By comparing

Figure 12 and

Figure 13, one observes that changing the polarization orientation significantly affects the triple differential cross sections. Indeed, the amplitudes

${f}_{ion}^{{B}_{2},\ell ,0}$ and

${f}_{ion}^{{B}_{1},\ell}$ depend on the laser polarization direction in a quite intricate way since

${\widehat{\epsilon}}_{0}$ enters their expressions though the second-order matrix elements and also via the scalar products

${\mathit{\alpha}}_{0}.\Delta $ and

${\mathit{\alpha}}_{0}.{\mathit{k}}_{b}$. Dressing effects i.e., the contributions of the amplitudes

${f}_{2}$ and

${f}_{3}$ significantly affect the TDCS corresponding to the ionizing process. This observation is applied well to the SBA where the contribution of the correction term is seen to be important and improves triple differential cross sections calculations.

In

Figure 14,

Figure 15 and

Figure 16, we give the triple differential cross sections corresponding to the ionization of the atomic hydrogen from the ground state by electron-impact, in the presence of a laser field, as a function of the ejected electron angle

${\theta}_{b}$. The incident electron energy is

${E}_{{k}_{i}}=30\phantom{\rule{4pt}{0ex}}$eV, the ejected electron energy is

${E}_{{k}_{b}}=5\phantom{\rule{4pt}{0ex}}$eV, and the scattering angle is

${\theta}_{a}={5}^{\circ}$. We are working in a geometry in which the polarization vector

${\widehat{\epsilon}}_{0}$ of the field is parallel to the incident momentum

${\mathit{k}}_{i}$. We present the results of our complete computation of the TDCS in the second Born approximation, and compare them with the first Born approximation ones, and with those obtained by neglecting the dressing effects by the laser field. One remarks that dressing effects have a relatively large effect on the cross sections with the exchange of one photon (

$\ell =\pm 1)$. The frequency regime have also controlling effects on the collision process. In

Figure 4, we observe that the binary peak remains unchanged for the first and second term of the Born series, while the recoil peaks is suppressed. Furthermore, the binary peak is dominant and the magnitude of the cross sections is considerably smaller when (

$\ell =-1)$ compared to the case of the absorption of a photon (

$\ell =1)$.

Let us now consider the case of higher frequency lasers. The preceding discussion remains qualitatively valid as long as the condition

$\omega <{E}_{{k}_{b}}$ is satisfied. This is well illustrated by our results presented in

Figure 4 and

Figure 5 which display the angular distribution for the frequencies

$\omega =1.17\phantom{\rule{4pt}{0ex}}$eV and

$\omega =2.34\phantom{\rule{4pt}{0ex}}$eV. By comparing these results, one observes that the shapes of the angular distributions are mostly the same, the main difference lying in the overall magnitude of the TDCS. Indeed, if everything else being kept fixed, the frequency is increased by a factor of 2, the electric field coupling parameter

${\alpha}_{0}$ is four time smaller, which correspondingly affects the magnitude of the cross sections. The high frequency regime where

$\omega >{E}_{{k}_{b}}$ is satisfied leads to strong modifications of the angular distribution of the ejected electron. This is confirmed by the results given in

Figure 16 for

$\omega =6.42\phantom{\rule{4pt}{0ex}}$eV and

${E}_{{k}_{b}}=5\phantom{\rule{4pt}{0ex}}$eV. One observes a typical splitting of the binary and recoil peaks. For the absorption of one photon

$\ell =1$, the margins between SBA and FBA results are negligible, but for the case of emission

$\ell =-1$, the modification is large. In fact, the margins between the SBA and FBA results are large except in the vicinity of the binary peak where the two curves coincide. As seen in

Figure 16, at a given field strength, the overall magnitude of the TDCS for

$\ell =-1$ is smaller by three orders of magnitude than in the case of

$\ell =1$.

The results displayed in this paper show that the photon absorption processes dominate the photon emission ones, meaning that the system absorbs net energy from the laser field background. The curves for

ℓ and

$-\ell $ present similar features since

${J}_{-\ell}\left(\lambda \right)={(-1)}^{\ell}{J}_{\ell}\left(\lambda \right)$. Nevertheless, the magnitudes of the cross sections for

ℓ and

$-\ell $ are different. The origin of this difference lies in terms in different expressions of (e, 2e) scattering amplitudes other than

${J}_{\ell}\left(\lambda \right)$. The cross sections in

Figure 12 and

Figure 13 for both geometries are different. In both cases, the oscillation structure in the results is determined by the Bessel function entering the expressions of the ionization amplitudes. When the argument

$\lambda $ and the order

ℓ are approximately equal, the value of the function

${J}_{\ell}\left(\lambda \right)$ diminishes rapidly. Physically, in the extreme case where

$\Delta -{\mathit{k}}_{b}$ is very small, e.g., when the nucleus is a spectator during the collision,

$\lambda $ is as well very small and the laser field plays a minor role. This is because the energy absorbed by the electrons from the radiation field needs to be converted into a linear momentum via a re-scattering from the massive residual ion. Small

$\mid \Delta -{\mathit{k}}_{b}\mid $ means that the re-scattering did not take place and hence the weak influence of the laser field on the outcome of the collision process. For large

$\mid \Delta -{\mathit{k}}_{b}\mid $, the scattering processes take place near the nucleus and hence the probability for the electrons to experience a violative transition is generally much higher, except for

$\lambda =0$ (the electric field is ⊥ to

$\Delta -{\mathit{k}}_{b})$.

In

Figure 17 and

Figure 18, we discuss the influence of the laser polarization orientation on the angular distribution of the ejected electron. We compare the TDCSs computed within the second Born approximation for two laser polarization directions where the polarization vector of the field is taken to be either parallel to the incident momentum

$\widehat{\epsilon}//{\mathit{k}}_{i}$ or parallel to the momentum transfer

$\widehat{\epsilon}//\Delta $. As shown in

Figure 17 corresponding to

$\ell =0$, the laser assisted TDCS for

$\widehat{\epsilon}//{\mathit{k}}_{i}$ deviates form that for

$\widehat{\epsilon}//\Delta $ so long as the incident electron energy is larger than 80 eV. At

${E}_{{k}_{i}}=40\phantom{\rule{4pt}{0ex}}$eV, the angular distribution of the ejected electron is the same for both laser polarization orientations. This can be understood as resulting from the fact that, at low incident energy, these orientations give substantially the same argument of Bessel functions entering the expression of the transition amplitudes Equations (

25) and (

42). The absolute magnitude of the TDCSs for the ionization of hydrogen by electron impact in the presence of a laser field increases with the increase of the incident electron energy for both laser polarization orientations

$\widehat{\epsilon}//{\mathit{k}}_{i}$ and

$\widehat{\epsilon}//\Delta $. Furthermore, the angular distribution is symmetrical with respect to the maxima of the binary and recoil peaks. With decreasing

${E}_{{k}_{i}}$, the maximum corresponding to the binary peak moves towards

${\theta}_{b}\simeq 0$, while the recoil peak disappears. The finding results have not been provided by a first Born treatment of laser assisted (e, 2e) collisions.

We present in

Figure 18 our results of ionization cross sections in the case

$\ell =1$, corresponding to the net absorption of one laser photon. The angular distribution for

$\widehat{\epsilon}//{\mathit{k}}_{i}$ differs markedly from that of

$\widehat{\epsilon}//\Delta $ when the incident energy is

$\ge 100\phantom{\rule{4pt}{0ex}}$eV i.e., when the first Born approximation is sufficient to describe the projectile-target interaction and the higher order terms could be negligible compared to the first Born term. Indeed, the Born series will converge if the incident particle is sufficiently high so that it can not interact many times with the target and for it the interaction potential is weak enough. At high incoming energies, the choice of the orientation

$\widehat{\epsilon}//\Delta $ maximizes the cross sections, this results from the fact that this geometry maximizes the argument of the Bessel functions entering the expressions of the ionization amplitudes, as the coupling between the projectile and the laser field is maximum in this laser polarization direction. In

Figure 18, the angular distribution is strongly modified as the binary and recoil peaks are split into smaller lobes with different magnitudes. The splitting of the peaks or, more precisely, the occurrence of increasing numbers of zeros in the TDCS can be traced to the behavior of the Bessel functions which appear in the expressions (

25) and (

42). Moreover, the magnitude of the cross section for net exchange of one photon decreases with the incident energy. The binary peak remains intense relative to the recoil peak for both laser orientations which reflects the fact that the electron-electron interaction is important and remains dominant regardless of the incident energy. The second Born approximation gives same magnitude of the cross sections and similar behavior of the angular distribution in the low energy regime for net exchange of one photon for both laser polarization orientations. The very small gap between the curves may be removed by taking into account second order exchange effects and higher order terms of the Born series.