Deconvoluting nonaxial recoil in Coulomb explosion measurements of molecular axis alignment

We report a quantitative study of the effect of nonaxial recoil during Coulomb explosion of laser-aligned molecules and introduce a method to remove the blurring caused by nonaxial recoil in the fragment-ion angular distributions. Simulations show that nonaxial recoil affects correlations between the emission directions of fragment ions differently from the effect caused by imperfect molecular alignment. The method, based on analysis of the correlation between the emission directions of the fragment ions from Coulomb explosion, is used to deconvolute the effect of nonaxial recoil from experimental fragment angular distributions. The deconvolution method is then applied to a number of experimental data sets to correct the degree of alignment for nonaxial recoil, to select optimal Coulomb explosion channels for probing molecular alignment, and to estimate the highest degree of alignment that can be observed from selected Coulomb explosion channels.

Figure 1

Sketch of the geometry used in the simulations of the recoil direction of the fragment ions during Coulomb explosion: (a) the detector plane and the definition of ${\theta }_{\text{2D}}$, (b) (one axis of) the molecular coordinate system with respect to the laboratory-fixed coordinate system, and (c) the fragment recoil velocity vector (${\tilde{\mathrm{v}}}_{\mathrm{f}}$) with respect to the molecular coordinate system. Note that in this work ${\theta }_{\text{2D}}$ is used to describe the direction of both $y_{\mathrm{mol}}$ and ${\tilde{\mathrm{v}}}_{\mathrm{f}}$ in the detector plane.

Figure 2

Examples of ion images (column 1) and angular covariance maps (column 2) with different combinations of the degrees of alignment and axial recoil. In each row the ion image and the angular covariance map are produced from the same data set. The combination of $\left({\mathrm{\Theta }}_{\text{mol}},{\mathrm{\Theta }}_{\text{frag}}\right)$ used are $\left(4^{\circ },4^{\circ }\right)$ [row (a)], $\left(4^{\circ },{50}^{\circ }\right)$ [row (b)], $\left({50}^{\circ },4^{\circ }\right)$ [row (c)], and $\left({50}^{\circ },{50}^{\circ }\right)$ [row (d)]. Row (b) shows the definition of ${\theta }_{\text{2D}}$ and the $x$ and $y$ axes used to determine the width and length shown in (d2).

Figure 3

(a) Width and (b) length of the island in the angular covariance maps represented by the FWHM for different combinations of ${\mathrm{\Theta }}_{\text{mol}}$ and ${\mathrm{\Theta }}_{\text{frag}}$.

Figure 4

Color map showing the length, $l_{\text{mol}}$, of the distribution of the molecular axes for different combinations of ${\mathrm{\Theta }}_{\text{mol}}$ and ${\mathrm{\Theta }}_{\text{frag}}$. It is seen that $l_{\text{mol}}$ is almost independent of ${\mathrm{\Theta }}_{\text{frag}}$. The green color represents data where the deconvolution of the shape of the covariance map produces unphysical results. See text for detail.

Figure 5

Color maps comparing the results of the estimated degrees of alignment to the actual degree of alignment for different combinations of ${\mathrm{\Theta }}_{\text{mol}}$ and ${\mathrm{\Theta }}_{\text{frag}}$: (a) ${\langle {cos}^2{\theta }_2\mathrm{D}\rangle }_{\text{mol}}$ for the molecular axis distribution, (b) ${\langle {cos}^2{\theta }_2\mathrm{D}\rangle }_{\text{frag}}$ determined from the projection of the fragment-ion velocities on the 2D detector, (c) ${\langle {cos}^2{\theta }_2\mathrm{D}\rangle }_{\text{mol}}$ for the molecular axis estimated from the deconvolution of the covariance map island, and (d) the error in the estimate represented by the estimated ${\langle {cos}^2{\theta }_2\mathrm{D}\rangle }_{\text{mol}}$ divided by the actual ${\langle {cos}^2{\theta }_2\mathrm{D}\rangle }_{\text{mol}}$ [(c) divided by (a)]. The upper color bar shows the color scale for (a), (b), and (c). The lower color bar shows the scale for (d). The green color represents data where the deconvolution of the covariance map island produces unphysical results. See discussion of Fig. 4 for details.

Figure 6

Examples of ion images (column 1) and angular covariance maps (column 2) of ${\mathrm{I}}^{+}$ ions from different experiments performed on DIB. (a) Unaligned DIB, $I_{\text{probe}}=238\mathrm{TW}{\mathrm{cm}}^{-2}$. (b) Unaligned DIB, $I_{\text{probe}}=589\mathrm{TW}{\mathrm{cm}}^{-2}$. (c) DIB nonadiabatically aligned by four 200-fs, 16 TW ${\mathrm{cm}}^{-2}$, 800-nm alignment laser pulses, $I_{\text{probe}}=358\mathrm{TW}{\mathrm{cm}}^{-2}$. (d) DIB adiabatically aligned by a 10-ns, 0.66 TW ${\mathrm{cm}}^{-2}$, 1064-nm laser pulses, $I_{\text{probe}}=326\mathrm{TW}{\mathrm{cm}}^{-2}$.

Figure 7

Result of the deconvolution to obtain the degree of alignment of the molecular axis of the data displayed in Fig. 6(d2). Solid red circles show the projections of the covariance island (a) onto the $y$ axis, termed ${\mathrm{proj}}_y$, and (b) onto the $x$ axis, termed ${\mathrm{proj}}_x$. (c) The result of deconvolving ${\mathrm{proj}}_y$ from ${\mathrm{proj}}_x$. The blue line in (b) is the convolution of the data in (a) and (c).

Figure 8

(a) Ion images and (b) angular covariance maps of ${\mathrm{S}}^{+}$ ions from ${\mathrm{CS}}_2$ molecules undergoing Coulomb explosion after irradiation by 800-nm, 35-fs, 421 TW ${\mathrm{cm}}^{-2}$ laser pulses.