Angle-dependent ionization rate of DCl

The theoretical treatment of the orientation-dependent ionization rates of DCl includes both HOMOs and goes beyond the one-center approximation. Therefore the ansatz of Ref. [58] is useful to calculate strong-field ionization probabilities in DCl [26]. The molec- ular orbitals of DCl were computed in the adiabatic approximation using a static electric field with the quantum chemistry package Molpro [92] on the CASSCF(6,9)/6-311G++ (2DF,2PD) level of theory and with electric field strength of 0.06 a.u. (corresponding to

Figure 4.9: Calculated orientation-dependent ionization probability for the HOMOs of DCl at intensity of 1.3×1014 W cm−2, obtained for ionization at the field maximum of a cosine-type pulse (ϕ= 0). Adapted from [26].

Figure 4.10: (a) Absolute value of the asymmetry amplitude |A(W, θ)| of D+ fragments. The image is left-right symmetrized along the direction of the laser polarization (0◦). (b) Energy-integrated (over 6–16 eV) asymmetry amplitude |A(W, θ)| derived from (a) (blue dots) as compared to the calculated symmetric ionization probability from the HOMOs of DCl (red line). From [26].

4.2 Waveform control of orientation-dependent ionization of DCl in few-cycle

laser fields 57

the peak intensity in the experiment). The calculated asymmetric angle-dependent ioniza- tion probabilities for an ideal cosine-type pulse (ϕ= 0, ionization only at one maximum) are plotted in Fig. 4.9. Note that when the electric field points upwards (the polarization of the laser is vertical), then the DCl molecules with the Cl-atom pointing upwards are preferentially ionized. Characteristic features of DCl molecule are distinct maxima of the ionization probability around ±45◦ and ±135◦ (see Fig. 4.9) and minima at 0◦, ±90◦ and 180◦. In order to shed light on the origin of the observed asymmetry in DCl the angular distributions of D+ and Cl+ fragments were analyzed. Under the axial recoil approxima- tion and neglected dynamic alignment of the molecules in the few-cycle laser pulse, the measured D+ ion angular distribution from an unaligned sample of DCl follows the ioniza- tion probability, i.e. the probability to remove an electron from DCl. The angle-dependent asymmetries obtained via Eq. 3.10 as functions of kinetic energy W, fragment emission angle θ and phase ϕCEP were fitted to a ϕCEP dependent cosine function (Eq. 3.13). The

obtained result for D+ ions is shown in Fig. 4.10a. The absolute value of the asymmetry amplitude|A(W, θ)|is maximal in the energy range between 9 and 14 eV and in the angular range between 30◦ and 60◦. It becomes smaller towards emission angles of 0◦ and ±90◦, in other words along and perpendicular to the laser polarization direction. In order to show the angular dependence more clearly, the absolute value of the asymmetry amplitude |A(W, θ)| has been integrated from 6 to 16 eV. The resulting energy-integrated asymmetry amplitude |A(θ)| is shown in Fig. 4.10b (blue dots).

The energy-integrated asymmetry amplitude |A(θ)| is not dependent on the CEP. Therefore, the orientation-dependent ionization probability for a random CEP was ob- tained. For comparison to theory, averaging of the ionization probability over the two opposite field directions (ϕ = 0◦ and ϕ = 90◦) was done. For clarity, the direct com- parison of experimental |A(θ)| and calculated ionization probability from the HOMOs of DCl is shown in Fig. 4.10b over the range of 360◦. The butterfly shape of the asymmetry amplitude (blue dots) peaks at 45◦and resembles the orientation-dependent symmetric ion- ization probability (red solid line). The momentum distributions of D+fragments are more pronounced at θ about ±60◦. These fragment angular distributions differ from the calcu- lated ionization probability from the HOMOs of DCl (which peaks at ±45◦, Fig. 4.10b). This indicates that the angular distribution of preferentially ionized DCl+ ions is further modified in processes following ionization, i.e. RCE and dissociation.

The good agreement between the angular dependence of the asymmetry amplitude and the ionization probability from the HOMOs of DCl suggests that the ionization is responsible for the observed asymmetry in the emission of charged fragments. The observed

π-phase shift in the CEP-dependence of the asymmetry of D+ and Cl+ fragments (shown in Fig. 4.7) further supports the proposed mechanism.

4.2.4

Conclusions and outlook

Few-cycle laser pulses preferentially ionize DCl oriented at certain angles relative to the direction of the electric field, so that the resulting DCl+ ions are partially oriented. A

π-phase shift of the electric field results in a π-phase shift of the DCl+ orientation and fragment asymmetries. Thus, the orientation of DCl+ is directly imprinted in the asymme- try of the fragment ion emission. The dissociation step has only been modeled for longer wavelengths [122]. Modeling the dissociation process for shorter wavelengths is still in progress.

Similar control can be achieved in other heteronuclear molecules with potentially higher degrees of orientation. Taking advantage of this control scheme (Fig. 4.11), near-single- cycle light fields may be used to produce samples of oriented molecular ions under field-free conditions for studies on their dynamics in the molecular frame.

Figure 4.11: Schematic view of CEP-stabilized few-cycle laser pulse (ϕCEP = 0) permit-

ting the control over the orientation of DCl+ in the ionization of randomly oriented DCl molecules. Figure courtesy: Bernd Ullmann, MPQ.

5

Two-color laser-induced field-

free molecular orientation

Aligned molecules have attracted widespread interest for applications such as ultrafast dynamic imaging [123], high harmonic orbital tomography [124, 125], laser-induced electron diffraction [126, 127], gas electron diffraction [128, 129], and photoelectron imaging using a HHG source [130]. Molecules have been successfully aligned in one and even three dimensions using strong, linearly-polarized laser fields [131]. For polar molecules, however, orientation is necessary to overcome the effect of averaging in two opposite directions.

Whereas the orientation of polar molecules is possible in strong DC fields [132] or by the combination of a laser field and a DC field [133, 134, 135] the presence of a strong DC field might influence the outcome of experiments on oriented molecular samples. The development of field-free orientation methods is crucial to avoid this limitation.

Laser-induced field-free orientation of polar molecules is much less studied than align- ment. This chapter describes the first experimental observation of the nonadiabatic field- free orientation of the heteronuclear diatomic molecule carbon monoxide using a two-color femtosecond laser field. To aid the theoretical interpretation of the results, two alternative orientation mechanisms within the studied intensity range are proposed and discussed.

5.0.1

Orientation vs. alignment of molecules

A standard way to measure the molecular orientation is given by the expectation value of hcosθi whereθ is the angle between the molecular axis and a reference axis. An ensemble of molecules is considered to be oriented when its angular distribution is asymmetric with respect to the reflection plane perpendicular to the reference axis, i.e. hcosθi 6= 0. Ori- entation is always observed together with alignment, however, the opposite is not always the case. Alignment is measured as hcos2θi. Orientation differs from alignment where angular distribution is considered to be symmetric but not isotropic, i.e. hcos2θi 6= 1₃ (see Fig. 5.1). Perfectly peaked along the reference axis alignment would be described by value of hcos2θi= 1 and angular distribution perpendicular to reference axis by hcos2θi= 0.

Figure 5.1: Schematic illustration of the difference between orientation and alignment for a sample of diatomic polar molecules. The reference axis is vertical.