Rotational wave-packet imaging of molecules

We propose and illustrate numerically the possibility of imaging rotational wave packets in angular space and time by using different pump-probe spectroscopic techniques. A general theoretical framework to perform such rotational mapping is derived and three specific spectroscopies, namely, birefringence, high harmonic generation, and angle-resolved photoelectron spectroscopy, are numerically explored. All three approaches are shown to provide direct mapping of the rotational coherences of molecules but they are not equivalent; comparison of their results yields interesting insights into their relative merits. Finally, we illustrate the role played by the symmetry of the molecular orbitals in determining the quality of the images generated by high harmonic and photoelectron signals. The potential of rotational imaging as a route to both intramolecular coupling mechanisms and the interaction of molecules with different environments is discussed.

Figure 1

The rotational probability density for (a) N${}_2$ and (b) O${}_2$ vs time and the polar Euler angle $\theta$ compared with the averaged alignment measure $\langle {\cos }^2\theta \rangle$ (left). The Gaussian pump pulse of 25 fs pulse width and a peak intensity of 75 TW cm${}^{-2}$ for N${}_2$ and 25 TW cm${}^{-2}$ for O${}_2$ and a rotational temperature of 30 K is used in the calculations.

Figure 2

Rotational mapping using RIPS for (a) N${}_2$, (b) O${}_2$. The pulse and system parameters are as in Fig. 1.

Figure 3

Cuts through the N${}_2$ rotational probability distributions (RPD) of Fig. 1a (black) compared with cuts through the RIPS signal of Fig. 2a (red dashed) and the HHG signal of Fig. 5a (solid blue/gray) at polar angles (a) $\theta =0$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={60}^{\circ }$, and (d) $\theta ={90}^{\circ }$. The rotational shift discussed in the text is included in the HHG cuts to simplify the comparison.

Figure 4

As in Fig. 3 for the case of O${}_2$ molecules.

Figure 5

Rotational mapping using HHG for (a) N${}_2$ and (b) O${}_2$. The pump pulse and system parameters are as in Fig. 1, the probe pulse used to calculate the matrix elements is long (akin to continuous wave) with an intensity of 200 TW, and the 23rd harmonic is detected at an angle of ${45}^{\circ }$ with respect to the probe polarization vector.

Figure 6

Rotational mapping using PES for (a) N${}_2$ and (b) O${}_2$. The pump pulse and system parameters are as in Fig. 1, and the photoelectron signal is detected at an angle of $0^{\circ }$ with respect to the probe polarization vector.

Figure 7

Cuts through the N${}_2$ rotational probability distributions of Fig. 1a (black) compared with cuts through the PES signal of Fig. 6a (red dashed) and the HHG signal (solid blue/gray) at polar angles (a) $\theta =0$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={60}^{\circ }$, and (d) $\theta ={90}^{\circ }$. The HHG emission direction is along the probe polarization vector.

Figure 8

Cuts through the O${}_2$ rotational probability distributions of Fig. 1b (black) compared with cuts through the PES signal of Fig. 6b (red dashed) and the RIPS signal (solid blue/gray) at polar angles (a) $\theta =0$, (b) $\theta ={30}^{\circ }$, (c) $\theta ={60}^{\circ }$, and (d) $\theta ={90}^{\circ }$. The RIPS signal shown as red dashed curves in Fig. 4 is included to simplify comparison.