Alignment dependence of photoelectron angular distributions in the few-photon ionization of molecules by ultraviolet pulses

We probe the time-dependent ionization dynamics of impulsively excited rotational wave packets of ${\mathrm{N}}_2, {\mathrm{CO}}_2$, and ${\mathrm{C}}_2{\mathrm{H}}_4$ using broadband ultraviolet pulses centered at 262 nm. Photoelectron momentum distributions recorded by velocity-map imaging show a strong dependence on alignment, on multiphoton order, and on the electronic and vibrational states of the cation. We show that substantial information about the molecular-frame photoelectron angular distribution can be obtained from the high-order laboratory-frame asymmetry parameters without any prior knowledge of the photoionization process. We also compare few-photon ionization with one-photon ionization and strong-field ionization.

Figure 1

All angles are defined in the LF, where the $Z$ axis is the laser polarization axis and $X$ is the laser propagation direction. The molecular axis is described by the polar angle $\theta$ and the azimuthal angle $\varphi$. The asymptotic photoelectron momentum is described by the polar angle ${\theta }_k$, the azimuthal angle ${\varphi }_k$, and the magnitude $k$ of the momentum. The angle $\varphi$ is not physically relevant, and the relevant angle ${\varphi }_k-\varphi$ is lost because of the axial symmetry of the measurement, so the molecular axis is shown here at an arbitrarily chosen angle $\varphi =0^{\circ }$. We would like to determine the AR-LFPAD, $d\sigma /d\theta d{\theta }_{k}dk$, which is the photoionization DCS that depends on both the photoelectron energy and the orientation of the molecule in the laser field. This AR-LFPAD is closely related to the MFPAD, as discussed in the text.

Figure 2

VMI images of the photoelectron momentum from ${\mathrm{N}}_2$ at the (a) isotropic, (b) aligned, and (c) antialigned distributions. The laser polarization is along the $p_z$ direction. The linear color scale expresses the yield of the electrons in arbitrary units; the same scale is used in all images. Using the pBasex algorithm [36], the VMI image of the isotropic distribution is converted to energy and angular distributions as shown in (d), with ionic states and photon numbers $n$ identified using spectroscopic data [40]. (e) A delay-momentum map of the pBasex coefficients. These coefficients describe the modulation of the time-resolved LFPAD as a function of the electron asymptotic momentum $k$ and the pump-probe delay $t$ through Eq. (7). Signals for $k>0.53$ a.u. were multiplied by 10 for clarity. Even these faint channels show clear modulations with time. We implemented the Nyquist theorem by dropping the low-$k$, high-$L$ basis functions; that is, data near the center of the image (low $k$) do not have enough resolution to resolve complex angular structure (high $L$). (f) The AR-PES [see Eq. (4)]. Each vertical line is the angle-dependent ionization rate at a specific photoelectron momentum $k$ describing the ionization yield at different orientations of the molecular axis $\theta$ in the laser field.

Figure 3

Results for ionization of ${\mathrm{N}}_2$ into the $X{}^2{\mathrm{\Sigma }}_g(n=4)$ ionic state ($k$ is integrated from 0.44 to 0.53 a.u.): (a) The delay-dependent pBasex coefficients $C_L\left(t\right)$ (dashed lines with squares) and their corresponding fits (solid lines) as described in Eq. (9). From these fits, we can retrieve (b) the angle-dependent ionization rate and (c) the YCAR-LFPAD $S(\theta ,k,{\theta }_k)$ as in Eq. (3). The angle-dependent ionization rate looks similar to the previous result from SFI [46]. We compare the vertical slice of $S(\theta ,k,{\theta }_k)$ at $\theta =0^{\circ }$ with the LFPAD at the alignment peak in (d), and the slice of $S(\theta ,k,{\theta }_k)$ at $\theta ={90}^{\circ }$ with the LFPAD at the antialignment peak in (e). The angle in the polar plots in (d) and (e) is ${\theta }_k$. Their similarities indicate these measurements in the LF are a good representation of the MF. The laser polarization axis is indicated by the red arrow.

Figure 4

Results for ionization of ${\mathrm{N}}_2$ into three different vibrational levels of the first excited state $A{}^2{\mathrm{\Pi }}_u(n=4)$ of the cation. Left: The angle-dependent ionization rate. Right: The YCAR-LFPAD $S(\theta ,k,{\theta }_k)$. All three 2D plots use the same color scale shown at the top. Different vibrational levels are $\nu =1–3$ from top to bottom. The momentum $k$ is integrated in the ranges of 0.267–0.316 a.u. for $\nu =1$, 0.316–0.337 a.u. for $\nu =2$, and 0.344–0.387 a.u. for $\nu =3$.

Figure 5

Three-photon ionization of ${\mathrm{CO}}_2$ into the ground state, ${\tilde{X}}^2{\mathrm{\Pi }}_g$, of the ion. Top: VMI electron images at the (a) alignment and (b) antialignment peaks. (c) The delay-dependent coefficients $C_L(k,t)$ and their corresponding fits. The range of $k$ averaged is from 0.12 to 0.19 a.u. (d) The angle-dependent ionization rate. (e) The YCAR-LFPAD. Bottom: Slices of the YCAR-LFPAD (f) at $\theta =0^{\circ }$ and ${90}^{\circ }$ and (g) at ${45}^{\circ }$. The laser polarization axis is indicated by the red arrow.

Figure 6

Results on three-photon ionization of ${\mathrm{C}}_2{\mathrm{H}}_4$ into the ground state, ${\tilde{X}}^2{B}_{3u}$, and the first excited state, ${\tilde{A}}^2{B}_{3g}$, of the ion. Top: (a) An example of a VMI electron image and (b) the calibrated energy with ionic states identified by using spectroscopic data [52]. (c) The delay-dependent electron yields of the two channels and their corresponding fits. The first excited state, ${\tilde{A}}^2{B}_{3g}$, was shifted up by 0.5 for clarity. Bottom: The angle-dependent ionization rates $R(\theta ,\chi )$ of (d) the ground state and (e) the excited state. The momentum $k$ is integrated in the ranges of 0.22–0.34 a.u. for the ${\tilde{X}}^2{B}_{3u}$ state and 0.46–0.54 a.u. for ${\tilde{A}}^2{B}_{3g}$.

Figure 7

(a) Electron image with [probe, gas]. (b) Electron image with [probe, background] subtracted from [probe, gas]. (c) Delay-dependent electron yield with and without probe-alone normalization. The red curve has been normalized to its mean and shifted up by 0.25. See text for more details.