Strong-field ionization of homonuclear diatomic molecules by a bicircular laser field: Rotational and reflection symmetries

We investigate above-threshold ionization (ATI) of homonuclear diatomic molecules by the so-called bicircular field using the improved molecular strong-field approximation. Bicircular field is a two-color laser field having coplanar circularly polarized counter-rotating components of frequencies $r\omega$ and $s\omega$, with $r$ and $s$ integers. Our analysis includes the high-energy part of the corresponding spectra, i.e., high-order ATI (HATI). The obtained molecular (H)ATI spectra are more complicated than the corresponding atomic spectra. We have identified four symmetries which are satisfied in (H)ATI of homonuclear diatomic molecules. Two of these symmetries are general rotational symmetries valid both for direct and rescattered HATI electrons. The remaining two symmetries are reflection symmetries valid only for the direct ATI electrons. Analytical proof of these symmetries is also given. These symmetries are illustrated using numerical examples of HATI spectra of the ${\mathrm{N}}_2$ molecule for various molecular orientations.

Figure 1

Schematic presentation of the coordinate systems used in the paper. Molecule AB is along the $z$ axis in the $zx$ coordinate system, while the laser field is defined in the $z_L{x}_L$ system. The ionized electron is emitted with the momentum $\mathbf{p}$ whose direction is determined by the angle $\theta \left({\theta }_e\right)$ with respect to the $z\left(z_L\right)$ axis.

Figure 2

Upper panel: Tips of the electric field $\mathbf{E}\left(t\right)$ and the vector potential $\mathbf{A}\left(t\right)$ are presented, with first third of a cycle in red, second third in green, and the last third in blue. Position of the molecule rotated by $-{120}^{\circ }$ about the $y$ axis is also shown. Lower panel: Position of the new laser field coordinate system (denoted with the primes) rotated by ${120}^{\circ }$ about the $y$ axis. The electron emission angles ${\theta }_e$ and ${\theta }_e^{\prime }$ for two laser field coordinate systems are shown.

Figure 3

Electron HATI spectra obtained using the improved MSFA (both the direct and the rescattered electrons are included) for the ${\mathrm{N}}_2$ molecule, laser intensity given by $E_1^2=E_2^2=1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and the fundamental wavelength of 800 nm. The electron momentum plane is defined with respect to the laser field coordinate system: $p_z=pcos{\theta }_e$, $p_x=psin{\theta }_e$. Upper panel: ${\theta }_L=0^{\circ }$. Lower panel: ${\theta }_L={120}^{\circ }$. The false color scale covers five orders of magnitude.

Figure 4

Same as in Fig. 3, but for ${\theta }_L={30}^{\circ }$ (upper panel) and ${\theta }_L={90}^{\circ }$ (lower panel).

Figure 5

Reflection of the molecule with respect to the $yz$ plane.

Figure 6

Direct electron spectra for the same parameters as in Fig. 3, but for ${\theta }_L=0^{\circ }$ (top panel), ${\theta }_L={120}^{\circ }$ (middle panel), and ${\theta }_L={60}^{\circ }$ (bottom panel).

Figure 7

Direct electron spectra for the same parameters as in Fig. 3, but for ${\theta }_L={90}^{\circ }$ (upper panel) and ${\theta }_L={30}^{\circ }$ (lower panel).

Figure 8

Upper panel: Diatomic molecule AB with the denoted axes. Lower panel: The same molecule rotated by ${180}^{\circ }$ about the $y$ axis that passes through the center of the molecule.