Bicircular-laser-field-assisted electron-ion radiative recombination

Electron-ion radiative recombination assisted by a bicircular laser field that consists of two circularly polarized fields counterrotating in the $xy$ plane and having the frequencies $r\omega$ and $s\omega$, which are integer multiples of the fundamental frequency $\omega$, is considered using the $S\text{-matrix}$ theory. The energy and polarization of soft x rays generated in this process are analyzed as functions of the incident electron energy and incident electron angle with respect to the $x$ axis. Numerical results for the process of direct recombination of electrons with ${\text{He}}^{+}$ ionic targets are presented. Abrupt cutoffs of the plateau structures in the emitted x-ray energy spectra are explained by classical analysis. Simpler or more complex oscillatory structures in the spectrum may appear as a result of the interference of a different number of classical orbits. Symmetry analysis and the numerical results show that the x-ray power spectrum and ellipticity are invariant with respect to a rotation of the incident electron momentum by the angle $2\pi /(r+s)$. We have visualized this by presenting the logarithm of the differential power spectrum and polarization of the emitted x rays in false colors as functions of the incident electron angle and the x-ray energy. We have also shown that the change of the relative phase of the bicircular field is equivalent to the change of the incident electron angle. By controlling this relative phase it is possible to control the polarization of the emitted soft x rays.

Figure 1

Normalized electric-field vector $\mathbf{E}\left(t\right)$ (solid black lines) and vector potential $\mathbf{A}\left(t\right)$ (dashed red lines) of the $r\omega -s\omega$ bicircular laser field given by Eq. (7) for equal laser field component intensities and the relative phases ${\varphi }_r={\varphi }_s=0$, plotted for $0\le t\le T, T=2\pi /\omega$. The electric-field vector starts from the point $\mathbf{E}\left(0\right)=(0,0)$ and develops in the clockwise direction for $t>0$, while the vector potential develops in the counterclockwise sense. The various panels depict the field for different combinations of the values of $r$ and $s$ as indicated in the upper right corners.

Figure 2

Logarithm of the differential power spectrum for the laser-assisted radiative recombination of electrons with ${\text{He}}^{+}$ ions in the presence of the bicircular laser field (6) with the fundamental wavelength of 800 nm, $I_r=I_s={10}^{15}\mathrm{W}/{\mathrm{cm}}^2$, and ${\varphi }_r={\varphi }_s=0$ ($r=1, s=2$), as a function of the emitted x-ray energy ${\omega }_{\mathbf{K}}$. The incident electron energies for the curves from bottom to top are $E_{\mathbf{p}}=10$, 20, 50, 100, and 200 eV, respectively. The incident electron angle is $\theta =0^{\circ }$. The ordinate of the bottom curve ($E_{\mathbf{p}}=10$ eV) is in atomic units, while all other curves are shifted up by three orders of magnitude, consecutively.

Figure 3

Classical analysis of the direct recombination solutions for the parameters of Fig. 2. The emitted x-ray energy ${\omega }_{\mathbf{K}}$ is presented as a function of the recombination time $t$, expressed is units of $T$. The incident electron energy is $E_{\mathbf{p}}=50$ eV.

Figure 4

Classical analysis of the direct recombination solutions for the parameters of Fig. 2. The emitted x-ray energy ${\omega }_{\mathbf{K}}$ is presented as a function of the recombination time $t$, expressed is units of $T$. The incident electron energy for each curve is indicated in the top right corner.

Figure 5

Same as in Fig. 2 but for $\theta ={180}^{\circ }.$

Figure 6

Same as in Fig. 3 but for $\theta ={180}^{\circ }$.

Figure 7

Same as in Fig. 4 but for $\theta ={180}^{\circ }$.

Figure 8

Logarithm of the differential power spectrum in false color as a function of the incident electron angle and the x-ray energy for $E_{\mathbf{p}}=20$ eV and the same laser and atomic parameters as in Fig. 2.

Figure 9

Same as in Fig. 8 but for $E_{\mathbf{p}}=200$ eV.

Figure 10

Logarithm of the differential power spectrum in false color as a function of the incident electron angle and the x-ray energy for fixed incident electron energy $E_{\mathbf{p}}=50$ eV and different combinations $(r,s)$: $(r,s)=(1,3)$ (top left panel), $(r,s)=(1,4)$ (top right panel), $(r,s)=(1,5)$ (bottom left panel), and $(r,s)=(2,3)$ (bottom right panel). The other laser and atomic parameters are the same as in Fig. 2.

Figure 11

Ellipticity ${\varepsilon }_{\mathbf{K}}$ in false color as a function of the incident electron angle and the x-ray energy for the same laser and atomic parameters as in Fig. 2 and for $E_{\mathbf{p}}=200$ eV and $(r,s)=(1,2)$ (top) and $E_{\mathbf{p}}=50$ eV and $(r,s)=(1,4)$ (bottom).

Figure 12

Polarization ${\varepsilon }_{\mathbf{K}}$ as a function of the incident electron angle $\theta$ for $E_{\mathbf{p}}=50$ eV, $(r,s)=(1,2)$, and the x-ray energy ${\omega }_{\mathbf{K}}=109$ eV. The other laser and atomic parameters are the same as in Fig. 2.

Figure 13

Bicircular $\omega$–$2\omega$ field for fixed phase ${\varphi }_s=0^{\circ }$ and for two different values of the phase ${\varphi }_r, {\varphi }_r=0^{\circ }$ (black solid line) and ${\varphi }_r={60}^{\circ }$ (red dashed line). The change of phase ${\varphi }_r$, for fixed value of ${\varphi }_s$, corresponds to the rotation field around the $z$ axis by the angle $\alpha$.