Strong-field approximation for ionization of a diatomic molecule by a strong laser field. II. The role of electron rescattering off the molecular centers

The strong-field approximation for ionization of diatomic molecules by a strong laser field [D. B. Milošević, Phys. Rev. A 74, 063404 (2006)] is generalized to include rescattering of the ionized electron wave packet off the molecular centers (the electron’s parent ion or the second atom). There are four rescattering contributions to the ionization rate, which are responsible for the high-energy plateau in the electron spectra and which interfere in a complicated manner. The spectra are even more complicated due to the different symmetry properties of the atomic orbitals of which a particular molecular orbital consists. Nevertheless, a comparatively simple condition emerges for the destructive interference of all these contributions, which yields a curve in the $(E_{{\mathbf{p}}_f},\theta )$ plane. Here $\theta$ is the electron emission angle and $E_{{\mathbf{p}}_f}$ is the electron kinetic energy. The resulting suppression of the rescattering plateau can be strong and affect a large area of the $(E_{{\mathbf{p}}_f},\theta )$ plane, depending on the orientation of the molecule. We illustrate this using the examples of the $3{\sigma }_g$ molecular orbital of ${\mathrm{N}}_2$ and the $1{\pi }_g$ molecular orbital of ${\mathrm{O}}_2$ for various orientations of these molecules with respect to the laser polarization axis. For ${\mathrm{N}}_2$, for perpendicular orientation and the equilibrium internuclear distance $R_0$, we find that the minima of the ionization rate form the curve $E_{{\mathbf{p}}_f}{\cos }^2\theta ={\pi }^2∕\left(2R_0^2\right)$ in the $(E_{{\mathbf{p}}_f},\theta )$ plane. For ${\mathrm{O}}_2$ the rescattering plateau is absent for perpendicular orientation.

Figure 1

(Color online) Schematic diagram of the six $T$-matrix contributions to the strong-field ionization of a molecule having two centers $A$ and $B$. The electron with the final momentum ${\mathbf{p}}_f$ is emitted in the direction $\theta$ with respect to the relative nuclear coordinate ${\mathbf{R}}_0$. The laser field polarization vector and, in consequence, the intermediate rescattering electron momentum ${\mathbf{k}}_{\mathrm{st}}$ are in the direction ${\theta }_L$. In the upper left panel the two direct ionization contributions ($T^{+}$ for the ionization at the center $A$ and $T^{-}$ for the center $B$) and the corresponding electron path difference $\Delta$ are presented. In the upper right panel the contributions $T^{++}$ and $T^{--}$, where the electron rescatters at the same center at which it was born, are depicted. The two lower panels show the contributions of the processes where ionization and rescattering happen at different centers. The path difference of electrons in the lower left and lower right panels, before the rescattering, is $2{\Delta }_L$.

Figure 2

(Color online) Logarithm of the differential ionization rate of ${\mathrm{N}}_2$ as a function of the electron kinetic energy $E_{{\mathbf{p}}_f}$ in units of the ponderomotive energy $U_p$, for emission in the direction $\theta =0°$, for a linearly polarized laser field having the intensity $3\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and the wavelength $800\mathrm{nm}$. The angle between the molecular axis and the laser polarization axis is ${\theta }_L=0°$. The results are obtained using the improved dressed modified MSFA in length gauge (blue dot-dashed line denoted as “total”). Various partial contributions to the total rate are also exhibited, especially the contribution of only the $s$ (green dashed line) and the $p$ (red dotted line) components of the ${\mathrm{N}}_2$ ground state wave function in the partial $T$-matrix element “$T^{++--}$.” The remaining notation is explained in Eqs. (24, 25). The curves “$T^{+-}$” and “$T^{-+}$,” which correspond to the $T^{+-}$ (cyan two-dots-dashed curve) and $T^{-+}$ (magenta two-dash dotted curve) contributions to the $T$ matrix, are shifted up by two orders of magnitude for better visual clarity.

Figure 3

(Color online) Same as in Fig. 2 but for the laser intensity $4\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 4

(Color online) Same as in Fig. 2 but ${\mathrm{O}}_2$, $\theta ={\theta }_L=45°$, and for $p$ and $d$ states.

Figure 5

(Color online) Same as in Fig. 3 but for ${\mathrm{O}}_2$, $\theta ={\theta }_L=45°$, and for $p$ and $d$ states.

Figure 6

(Color online) Logarithm of the differential ionization rate of ${\mathrm{N}}_2$, coded in false colors in the $(E_{{\mathbf{p}}_f},\theta )$ plane, for four different orientations of the molecular axis with respect to the polarization vector of the laser field, from the top to the bottom panel: ${\theta }_L=0°$, 30°, 60°, and 90°. Intensity of the linearly polarized laser field is $1.68\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and the wavelength is $800\mathrm{nm}$. The results are obtained using the undressed length-gauge modified MSFA.

Figure 7

(Color online) Logarithm of the differential ionization rates of ${\mathrm{N}}_2$ as functions of the electron kinetic energy $E_{{\mathbf{p}}_f}$ in units of the ponderomotive energy $U_p$, for emission in the direction $\theta =60°$, for ${\theta }_L=60°$. The laser parameters are as in Fig. 6. The spectra are obtained using the dressed length-gauge modified MSFA and only the atomic $s$ orbitals, which are dominant in the high-energy part of the spectrum of ${\mathrm{N}}_2$. The total rate is denoted by total (blue dot-dashed line). The partial contributions $T^{++--}$ (black solid line), $T^{+-}$ (cyan two-dots-dashed curve), and $T^{-+}$ (magenta two-dash dotted curve) are also presented.

Figure 8

(Color online) Logarithm of the differential ionization rates of ${\mathrm{N}}_2$ as functions of $E_{{\mathbf{p}}_f}∕U_p$, for emission in the direction $\theta =40°$ (red dashed curve) and $\theta =50°$ (black solid curve). The results are obtained using the dressed length-gauge modified MSFA. The laser parameters are as in Fig. 6 and ${\theta }_L=90°$.

Figure 9

(Color online) Logarithm of the differential ionization rate of ${\mathrm{O}}_2$, coded in false colors in the $(E_{{\mathbf{p}}_f},\theta )$ plane, obtained using the undressed length-gauge modified MSFA, for three different orientations of the molecular axis with respect to the polarization vector of the laser field ${\theta }_L=0.01°$ (upper), 45° (middle), and 90° (bottom panel). The presentation and the laser parameters are as in Fig. 6.

Figure 10

(Color online) Logarithm of the differential ionization rates of ${\mathrm{O}}_2$ as functions of $E_{{\mathbf{p}}_f}∕U_p$, for emission in the direction $\theta =45°$ (black dotted curve), 60° (blue solid curve), 64° (red dashed curve), and 70° (green dot-dashed curve). The results are obtained using the dressed length-gauge modified MSFA. The laser parameters are as in Fig. 6 and ${\theta }_L=45°$.