Laser-assisted inelastic scattering of electrons by helium atoms

The differential cross section for electron-impact excitation of helium atoms in the presence of a linearly polarized laser field is calculated. The interaction of the laser field with both the projectile electron and the target atom is treated in a fully nonperturbative way, while the electron-atom interaction is treated within the first Born approximation. We are interested in studying two cases where the photon energy of the laser field is chosen to be far from resonance and when it matches with $2{}^{1}S⟶2{}^{1}P$ and $3{}^{1}S⟶3{}^{1}P$ transition frequencies. The agreement between perturbative and nonperturbative results is good, except close to the resonance where the perturbative cross sections diverge while the nonperturbative ones predict no maximum of the cross sections. Another interesting effect is the presence of an avoided crossing of the Floquet pseudoenergies at resonance.

Figure 1

Selected scattering geometry for electron-helium excitation in the presence of linearly polarized laser field.

Figure 2

First Born differential cross sections corresponding to the laser-assisted excitation of $2{}^{1}S$ state of helium atom as a function of the scattering angle $\theta$ for the exchange of $\ell =0,\pm 1$ photons and for various laser intensities $I=1\times {10}^9,1\times {10}^{10},1\times {10}^{11}$, and $1\times {10}^{12}{\mathrm{W}/\mathrm{cm}}^2$. The incident electron energy is $E_{k_i}=300\mathrm{eV}$, and the laser photon energy is $\omega =1.17\text{eV}$. The laser polarization vector is taken to be parallel to the momentum transfer $\mathit{K}$. Black solid lines, results with the absorption of one photon; green solid lines, results with no photon exchange; blue dotted lines, results with the emission of one photon; black dashed lines, field-free results.

Figure 3

All the parameters are the same as Fig. 2, but for the excitation of $2{}^{1}P$ state of helium.

Figure 4

First Born differential cross sections as a function of the laser photon energy $\omega$. The incident electron energy is $E_{k_i}=300\text{eV}$ and the the scattering angle is ${20}^{\circ }$. The laser polarization vector is taken to be parallel to the momentum transfer $\mathit{K}$. (a) Our nonperturbative results using the scattering amplitude given by Eq. (22) for a laser intensity $I_0=1\times {10}^9\text{W}{\text{cm}}^{-2}$: black solid line, results corresponding to the excitation of the Floquet state ${\varphi }_{2{}^{1}S}$ with $\ell =-1$; green solid line, results corresponding to the excitation of the Floquet state ${\varphi }_{2{}^{1}P}$ with $\ell =0$. Perturbative results using the scattering amplitude given by Eq. (24): black dashed line, results corresponding to the excitation of the $2{}^{1}S$ state with $\ell =-1$; green dashed line, results corresponding to the excitation of the $2^{P}S$ state with $\ell =0$. (b) Similar to (a), but for the excitation of the Floquet states ${\varphi }_{3{}^{1}S}$ and ${\varphi }_{3{}^{1}P}$ compared to the excitation of the nonperturbed states $3{}^{1}S$ and $3{}^{1}P$ for a laser intensity $I_0=1\times {10}^8\text{W}{\text{cm}}^{-2}$.

Figure 5

Comparison of the Floquet pseudoenergies ${\epsilon }_{n{}^{1}S}+\hslash \omega$ and ${\epsilon }_{n{}^{1}P}$ with the unperturbed energies $E_{n{}^{1}S}+\hslash \omega$ and $E_{n{}^{1}P}$. Black solid lines, floquet pseudoenergies ${\epsilon }_{n{}^{1}S}+\hslash \omega$; green solid lines, floquet pseudoenergies ${\epsilon }_{n{}^{1}P}$; black dashed lines, unperturbed energies $E_{n{}^{1}S}+\hslash \omega$; green dashed lines, unperturbed energies $E_{n{}^{1}P}$. (a) For $n=2$ and a laser field intensity $I_0=1\times {10}^9\text{W}{\text{cm}}^{-2}$. (b) For $n=3$ and a laser field intensity $I_0=1\times {10}^8\text{W}{\text{cm}}^{-2}$.

Figure 6

All the parameters are the same as Fig. 4, but with the absorption of one $(\ell =1)$ and two photons $(\ell =2)$.