Theoretical study of photoelectron angular distributions in single-photon ionization of aligned ${\mathrm{N}}_2$ and ${\mathrm{CO}}_2$

We calculate photoelectron angular distributions (PADs) resulting from single-photon (43 eV) ionization of molecules that have been transiently aligned with a short laser pulse. The total ionization cross sections of ${\mathrm{N}}_2$ and ${\mathrm{CO}}_2$ vs the time delay between the aligning laser pulse and the soft x-ray photon are calculated and compared to experimental results reported by I. Thomann et al. [J. Phys. Chem. A 112, 9382 (2008)]. We present the PADs from these aligned molecules in the laboratory frame which can be compared directly with future experiments from aligned ${\mathrm{N}}_2$ and ${\mathrm{CO}}_2$. The alignment dependence of single-photon ionization, multiphoton ionization, and high-order harmonic generation are also analyzed.

Figure 1

(a) Integrated photoionization cross sections and (b) doubly differential cross sections for electrons ejected in the polarization direction for ${\mathrm{N}}_2$ aligned at an angle $\theta$ from the polarization axis, by a single photon at 43 eV. (c) Alignment dependence of the multiphoton ionization rate of ${\mathrm{N}}_2$ by a laser with intensity of $2\times {10}^{14}$ W/cm${}^2$. (d) Alignment dependence of the 27th harmonic (43 eV) of ${\mathrm{N}}_2$ by a laser with intensity of $2\times {10}^{14}$ W/cm${}^2$, wavelength of 800 nm, and duration (FWHM) of 30 fs. HOMO (solid lines), HOMO-1 (dotted lines), and HOMO-2 (dot dashed lines).

Figure 2

(a) Calculated degree of alignment $\langle \cos {}^2\theta \rangle$ for ${\mathrm{N}}_2$ vs time delay near first half-revival. (b) Single-photon ionization yield from transiently aligned ${\mathrm{N}}_2$ by 43-eV photons vs time delay: theory (solid line) and experiment (solid squares) [9]. (c) Angular dependence of the ionization rate in single-photon (43-eV) ionization (solid line), and by multiphoton ionization by an IR laser with intensity of $2\times {10}^{14}$ W/cm${}^2$ (dashed line).

Figure 3

(a) Calculated degree of alignment $\langle \cos {}^2\theta \rangle$ for ${\mathrm{CO}}_2$ vs time delay near first half-revival. (b) Single-photon ionization yield from transiently aligned ${\mathrm{CO}}_2$ by 43-eV photons vs time delay: theory (solid line) and experiment (solid squares) [9]. (c) Integrated photoionization cross section for ionization leading to the $X$ (solid line), $A$ (dotted line), and $B$ (dot-dashed line) ionic states of CO${}_2^{+}$, with alignment angle $\theta$, by single-photon (43-eV) ionization of ${\mathrm{CO}}_2$. (d) Angular dependence of the ionization rate for single-photon (43-eV) ionization (solid line), and for multiphoton ionization by an IR laser with intensity of $1.1\times {10}^{14}$ W/cm${}^2$ (dashed line).

Figure 4

Photoionization cross sections in the laboratory frame for single-photon (43-eV) ionization of fixed-in-space ${\mathrm{N}}_2$ vs emission angle ${\theta }_{k^{\prime }}$ at alignment angles indicated and for ionization leading to ${\mathrm{N}}_2^{\hspace{0.5em}+}$ in the $X$, $A$, and $B$ states, shown in panels (a)–(c), respectively. In panels (d)–(g) the same distributions are shown for the $X$, $A$, and $B$ channels at each fixed-in-space molecular alignment angle. See text.

Figure 5

PADs in the laboratory frame for single-photon (43-eV) ionization of ${\mathrm{N}}_2$ as a function of emission angle ${\theta }_{k^{\prime }}$ and pump-probe time delay. (a)–(c): Molecules are maximally aligned ($\tau =4.00$ ps), antialigned ($\tau =4.55$ ps), and isotropically distributed, for ionization leading to ${\mathrm{N}}_2^{\hspace{0.5em}+}$ ions in $X$, $A$, and $B$ states, respectively. (d) and (e): The same distributions are compared for maximally aligned and antialigned molecules. (f)–(h): PADs vs time delay for the $X$, $A$, and $B$ channels, respectively.

Figure 6

Same as Fig. 5 except that a strong aligning pump laser is assumed. See text.

Figure 7

Fixed-in-space photoionization angular distributions (vs ${\theta }_{k^{\prime }}$) in the laboratory frame, for photon energies of 20, 30, or 46 eV, and the four alignment angles shown. First row: $X$ channel. Second row: $A$ channel. Third row: $B$ channel. The last row shows that the PAD becomes featureless if the molecules are isotropically distributed.

Figure 8

Photoionization cross sections in the laboratory frame for single-photon (43-eV) ionization of fixed-in-space ${\mathrm{CO}}_2$ vs emission angle ${\theta }_{k^{\prime }}$ at alignment angles indicated and for ionization leading to CO${}_2^{+}$, in panels (a)–(c), in the $X$, $A$, and $B$ states, respectively. In panels (d)–(g) the same distributions are shown for the $X$, $A$, and $B$ channels at each fixed-in-space molecular alignment angle. See text.

Figure 9

PADs in the laboratory frame for single-photon (43-eV) ionization of ${\mathrm{CO}}_2$ as a function of emission angle ${\theta }_{k^{\prime }}$ and pump-probe time delay. (a)–(c): Molecules are maximally aligned ($\tau =20.82$ ps), antialigned ($\tau =22.14$ ps), and isotropically distributed, for ionization leading to CO${}_2^{\hspace{0.5em}+}$ ions in the $X$, $A$, and $B$ states, respectively. (d) and (e): The same distributions are compared for maximally aligned and antialigned molecules. (f)–(h): PADs vs time delay for the $X$, $A$, and $B$ channels, respectively.