Picosecond pulse-shaping for strong three-dimensional field-free alignment of generic asymmetric-top molecules
Terry Mullins, Evangelos Thomas Karamatskos, Joss Wiese, Jolijn Onvlee, Arnaud Rouzée, Andrey Yachmenev, Sebastian Trippel, Jochen Küpper
PPicosecond pulse-shaping for strong three-dimensional field-free alignmentof generic asymmetric-top molecules
Terry Mullins, Evangelos T. Karamatskos,
1, 2
Joss Wiese,
1, 3, 4
Jolijn Onvlee,
1, 4
Arnaud Rouzée, Andrey Yachmenev,
1, 4
Sebastian Trippel,
1, 4 and Jochen Küpper
1, 2, 3, 4, ∗ Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany Department of Physics, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Department of Chemistry, Universität Hamburg, Martin-Luther-King-Platz 6, 20146 Hamburg, Germany Center for Ultrafast Imaging, Universität of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Max Born Institute, Max-Born-Straße 2a, 12489 Berlin, Germany (Dated: 2020-09-18)We demonstrate three-dimensional (3D) field-free alignment of the prototypical non-rotation-symmetric molecule indole using elliptically polarized, shaped, off-resonant laser pulses. A truncatedlaser pulse is produced using a combination of extreme linear chirping and controlled phase andamplitude shaping using a spatial-light-modulator (SLM) based pulse shaper of a broadband laserpulse. The angular confinement is detected through velocity-map imaging of H + and C fragmentsresulting from strong-field ionization and Coulomb explosion of the aligned molecules by intensefemtosecond laser pulses. The achieved three-dimensional alignment is characterized by comparingthe result of ion-velocity-map measurements for different alignment directions and for different timesduring and after the alignment laser pulse to accurate computational results. The achieved strongthree-dimensional field-free alignment of (cid:68) cos δ (cid:69) = 0 . demonstrates the feasibility of both, strongthree-dimensional alignment of generic complex molecules and its quantitative characterization. Laser-induced alignment of gas-phase molecules hasproven to be an efficient way to access the molecularframe [1, 2]. It was extensively used in high-harmonic-generation-spectroscopy [3, 4], strong-field-ionization [5–7], x-ray-diffraction [8, 9] and electron-diffraction [10–14]experiments, enabling the imaging of molecular structureand dynamics directly in the molecular frame. Further-more, it was crucial for retrieving the shapes of molecularorbitals [15–17].Such advanced imaging technologies are especially im-portant for complex molecules, i. e., asymmetric topswithout any rotational symmetry, which is the case foralmost all molecules on earth. Thus it is of utmost im-portance to develop laser alignment into a practical toolfor such molecules. This would, for instance, maximizethe information content of atomic-resolution imaging ex-periments [18, 19], as already suggested for the coherentdiffractive x-ray imaging of biological macromoleculesmore than fifteen years ago [20]. In order to minimize per-turbations by external fields this should be achieved in alaser-field-free environment. The associated problems aretwofold: The rotational dynamics of these non-rotation-symmetric molecules are very complicated and incommen-surate [21, 22]. Moreover, the standard approaches tocharacterize the 3D degree of alignment, using ion imag-ing of atomic fragments, mostly halogen atoms, recoilingalong a well-defined molecular axis, do not work.One-dimensional alignment of linear and (near) sym-metric top molecules has been demonstrated extensivelyand really pushed to the limits [1, 2, 16, 23–26]. Further-more, also the three-dimensional (3D) control of rotation-symmetric molecules, typically during long laser pulses,was demonstrated by multiple groups, making use of highly polarizable halogen atoms for large polarizabilityeffects as well as their symmetric fragmentation dynam-ics for characterization [16, 27–32]. This was extendedto the field-free 3D alignment of molecules in heliumnanodroplets [33] and to the alignment of one slightlynon-rotation-symmetric molecule 6-chloropyridazine-3-carbonitrile using long laser pulses [21, 34].Here, we demonstrate and characterize the strong laser-field-free three-dimensional alignment of the prototypical(bio)molecule indole (C H N, Figure 1 a), a good repre-sentative of the general class of molecules without anyrotational symmetry and without any good leaving-groupfragments for standard characterization. We use a combi-nation of a shaped, truncated, elliptically-polarized laserpulse with a short kick pulse before truncation to in-duce strong three-dimensional alignment. The degree ofalignment is characterized through strong-field multipleionization and subsequent velocity-map imaging (VMI) ofH + , C + , C , and CH x N + (x=0,1,2) fragments, combinedwith computational results to disentangle the temporaland angular dependence of the alignment. Our approachshows that the molecular frame even of molecules withoutrotational symmetry can be accessed.The experimental setup was described elsewhere [35].Briefly, molecules were cooled in a supersonic expansionfrom a pulsed Even-Lavie valve [36], operated at a temper-ature of ◦ C and at a repetition rate of 100 Hz. Around1.4 mbar of indole was seeded in 95 bar of He, which was ex-panded into vacuum. The lowest-energy rotational stateswere selected using an electrostatic deflector [37, 38]. In-side a VMI spectrometer, the strongly deflected moleculeswere aligned using a shaped 250 ps long laser pulse witha peak intensity of ∼ . · W/cm . These pulses a r X i v : . [ phy s i c s . a t m - c l u s ] S e p were produced by a commercial laser system (CoherentLegend Elite Duo) with a kHz repetition rate and aspectrum similar to a rounded saw tooth. The pulse wasstrongly negatively chirped to a duration of ∼ ps usinga grating based compressor [35] before further shaping.The alignment laser pulses were elliptically polarized witha intensity ratio between major and minor axes.The strongly chirped mJ pulses were sent through azero dispersion f pulse shaper [39] with a spatial lightmodulator (SLM, Jenoptik S640d) situated at the Fourierplane in order to generate a truncated pulse with a fast fall-off time. The most relevant part of the shaped temporalintensity profile, around the cutoff, is shown in Figure 2 a.The pulse consisted of a slow rise beginning at − ps(not shown), followed by some amplitude modulation, ashort kick, and, finally, a fast truncation. The SLM wasspecifically used for spectral phase modulation of the long-wavelength side of the laser spectrum, i. e., componentslonger than nm. In addition, wavelengths longer than ∼ nm were blocked by a razor blade, situated in frontof the SLM. This was necessary due to Nyquist-samplinglimits encountered. We note that the combination ofphase shaping and spectral truncation with the razorblade improved the temporal fall-off time by a factor of . down to . ps, i. e., to within the noise level of themeasurement, which was below 1 % of the signal peak,compared to simply cutting the spectrum [32].The post-pulse observed at ∼ ps is unwanted andprobably originates from imperfect phase compensationfrom the SLM or space-time coupling in the pulse shapingsetup. However, the post pulse is irrelevant to the degreeof alignment within the first ps after the temporaltruncation, which corresponds to the important temporalregion investigated in this experiment.A second, time-delayed, laser pulse was used to multiplyionize indole, resulting in Coulomb explosion. Thesepulses were circularly polarized in order to avoid anysecondary dynamics induced by electron rescattering [40]and any additional alignment induced by the probe laserfield itself.Velocity-mapped fragments were detected on a mi-crochannel plate (MCP) detector equipped with a phos-phor screen. The voltage on the MCP was switchedbetween 2050 V (MCP “on”) and 1150 V (MCP “off”)using a fast switch (Behlke HTS 31-03-GSM) with nsrise- and fall-times to select the different ion fragmentsbased on their time-of-flight (TOF). A camera (Optro-nis CL600) recorded single-shot images of the phosphorscreen at Hz. Images without pulses from the molec-ular beam were subtracted from those with the molecularbeam to account for any signal from background moleculesin the interaction region. After selection of a suitable two-dimensional (2D) radial range, the degree of alignment (cid:10) cos θ (cid:11) was computed.The in-plane principal axes of inertia ( a, b ) and polar-izability ( z I , x I ) are shown in the ball-and-stick represen- a bc
123 74 65 d Mass / Charge (u / e) S i g n a l ( a r b . u . ) C +2 C + C x H + y H + C H N +2 C H N + HNCH + H + t = 0 C +2 t = 3 .
33 ps HNCH + t = 7 .
33 ps XX t = 13 .
34 ps a bx I z I C a C a C C + H + C H N C H N + XX CH x N + (x = 0 , , H + x (x = 1 , , CH x N + (x = 0 , , C α = 0 ◦ α = 90 ◦ FIG. 1. (a) The structure of indole molecule with its principalaxes of inertia and polarizability, labeled by a, b, c and x I , y I , z I ( α y I < α x I < α z I ), respectively. (b) TOF mass spectrum ofindole. (c) 2D momentum distributions for H + , C andCH x N + (x=0,1,2) fragments at peak alignment at t = 3 . ps,with the major axis of the alignment laser polarization parallel(first row, α = 0 ◦ ) or perpendicular (second row, α = 90 ◦ )to the detector plane. (d) Time-snapshots of 2D momentumdistributions of H + fragments for the case of parallel laserpolarization. tation of indole in Figure 1 a. Both axis frames lie inthe plane of the molecule with an angle of . ◦ betweenthem, whereas the c and y I axes are perpendicular to thatplane. Upon Coulomb explosion, several fragmentationchannels were detected. The resulting time-of-flight massspectrum is depicted in Figure 1 b.Despite the low symmetry of indole, several frag-ments showed anisotropic momentum distributions.The ion-momentum distributions of H + , C andCH x N + (x=0,1,2) for a delay time of t = 3 . ps, cor-responding to the highest observed degree of alignment,are shown in Figure 1 c for two orientations of the align-ment laser, i. e., with the main polarization axis beingparallel, α = 0 ◦ , or perpendicular, α = 90 ◦ , to the planeof the detector. t = 0 corresponds to the peak intensityof the alignment laser field. Furthermore, ion-momentumdistributions of H + for time delays of t = 0 , 3.3, 7.3,and 13.3 ps are shown in Figure 1 d. The strongest field-free alignment was observed near t = 3 . ps. At laterdelay times, the dephasing of the rotational wavepacketleads to a decrease of the molecular alignment, as seenin the momentum distributions recorded at time delaysof t = 7 . ps and t = 13 . ps. Momentum distributionsof other fragments displaying alignment are shown in theSupplementary Information.Indole does not contain unique markers, like halogenatoms, which would allow us to easily experimentally ac-cess the degree of alignment. Therefore, all ions with agiven mass to charge ratio, produced through multiple ion-ization with subsequent Coulomb explosion, potentiallycontributed to the measured 2D momentum distributions.There are seven sites in the indole molecule from whichthe H + fragments originate and eight sites for the C fragments, see Figure 1 a. Each molecular site will resultin different momentum and recoil axis of the ionic frag-ment, and the total measured distribution is the sum ofall of them.The delay-dependent measured 2D degree of alignmentis shown for a variety of fragments in Figure 2. Assum-ing axial recoil, H + fragments would have measurablemomentum components only within the ab plane of in-dole [41]. Hence, the H + fragments are a priori a goodmeasure of the planar alignment of indole in the labora-tory frame. The slow rise of the alignment pulse confinedthe plane of the indole molecules in a quasi-adiabatic fash-ion [24, 35] to a measured maximum degree of alignment of (cid:10) cos θ (cid:11) expH + = 0 . . Following the kick at the end of thealignment pulse, the degree of alignment increased slightlyto (cid:10) cos θ (cid:11) expH + = 0 . before monotonically decreasingover ∼ ps to (cid:10) cos θ (cid:11) expH + = 0 . . The permanentalignment of (cid:10) cos θ (cid:11) expH + = 0 . was slightly higher thanthe value (cid:10) cos θ (cid:11) expH + = 0 . observed without alignmentlaser; the latter is due to the geometric alignment from anisotropic distribution. At a delay of . ps the intensityof the alignment pulse decreased to 1 % of its maximum,and the “field-free” region began. At this delay the de-gree of alignment was (cid:10) cos θ (cid:11) expH + = 0 . , which waseven larger than the alignment measured just before thekick, confirming that the planar alignment in the field-freeregion was even better than for an adiabatic alignmentpulse [42, 43]. All other fragments showed similar distri-butions to the H + fragment, with the measured maximumdegree of alignment being largest for the C fragment.The differences in the measured alignment between theH + , C + , C and CH x N + (x=0,1,2) fragments can be at-tributed to non-axial recoil or to the geometry of Coulombexplosion fragmentation, i. e., the velocity vectors of thefragments in the molecular frame.To determine the 3D alignment of indole, an additionalobservable is required that characterizes the in-plane align-ment, i. e., the alignment of the most polarizable axis ofindole z I with respect to the main polarization axis ofthe alignment field. This information can be accessed bymeasuring the angular distribution of the ionic fragmentswithin the indole plane. By rotating the polarization el- – – Delay (ps) . . . . . . I n t e n s i t y ( W / c m ) . . . . . . . . (cid:31) c o s θ D (cid:30) e x p X + C + C H + CH x N + Delay (ps) . . . (cid:31) c o s θ D (cid:30) H + H + exp . H + sim . Delay (ps) . . . (cid:31) c o s δ (cid:30) -
10 0 10 200 . . . abc (cid:31) cos θ Xx I (cid:30) = 0 . (cid:31) cos θ Y y I (cid:30) = 0 . (cid:31) cos θ Zz I (cid:30) = 0 . FIG. 2. (a) Temporal evolution of the alignment of indole.The solid lines show the measured 2D degree of alignment (cid:68) cos θ (cid:69) expX + for different fragments X + and the dashed linesindicate values of the 2D degree of alignment obtained withoutalignment laser. Statistical error bars, representing the stan-dard error, are shown for selected delays. The grey area showsthe intensity profile of the alignment laser pulse. (b) Thealignment revival structure of H + fragments for longer times isshown in red. The dotted green line shows the fitted simulationfor the H + fragment. (c) Simulated 3D degree of alignment,characterized through the single-scalar metric (cid:68) cos δ (cid:69) , seetext/SI for details. Note the peak after truncation reaching (cid:68) cos δ (cid:69) = 0 . at t = 3 . ps. lipse of the alignment laser around the laser propagationaxis at a fixed delay time of . ps, the laboratory axesto which the a and b axes of indole align, were commen-surately rotated. In the laboratory frame, the transversemomenta of ionic fragments recoiling within the plane ofindole will depend on the ellipse-rotation angle α , between α = 0 for parallel and α = 90 ◦ for perpendicular orien-tation, see Figure 1 c. By counting only those fragmentsimpinging at the center of the detector, within a smallradius of 20 pixel, the distribution of fragments within theplane can be determined [44]. Note that the size of the –
30 0 30 60 90 120 α (degree) . . . . . . R e l a t i v e i o n r a t e C H + C simulationH + simulation FIG. 3. Measured masked-VMI ion count rates of H + andC fragments as the major axis of the alignment laser’spolarization ellipse is rotated; see text for details. Circles(red and orange) indicate measured values, the dashed lines(green and black) are simulations based on fitted atomic-ioncontributions, see text for details. VMI images, as shown in Figure 1 c and Figure 1 d, was × pixels. A full tomographic 3D distribution for α = 0 – ◦ (∆ α = 2 ◦ ) was obtained from the measure-ments of H + momentum distributions, see SupplementaryInformation. As no fragments were observed at low 3Dmomenta, these signals at the center of the VMI canbe attributed to in-plane fragments recoiling along thedetector normal.Angular scans of this “masked VMI” are shown in Fig-ure 3 for H + and C . For both fragments, we observeda clear angle-dependent structure on top of a significantisotropic background. C ions show two smaller peaksat α ≈ ± ◦ and a much stronger peak at α ≈ ◦ .The H + signal shows a peak at α ≈ ◦ , similar to C ,and a smaller peak at α ≈ ◦ . Note that at ◦ thealignment laser’s major polarization axis is pointing alongthe detector normal. A direct extraction of the in-planedegree of alignment from the experimentally obtainedin-plane angular distribution was not possible due to alarge isotropic background. Furthermore, the degree ofmolecular alignment retrieved from the angular momen-tum distributions of H + and C can be misrepresentedmainly due to two reasons – the many-body Coulombbreak-up of the multiply charged indole cation, with ionicfragments violating the axial-recoil approximation, aswell as the indistinguishability of fragments emitted atdifferent molecular sites.In order to extract the actual 3D degree of alignment,we performed comprehensive variational simulations ofthe rotational dynamics of indole in the presence of thealignment field. We employed the general variationalapproach RichMol [45, 46] to compute time-dependentrotational probability density distributions for differentdelay times. In order to incroporate the experimentalconditions and to achieve better agreement, we took intoaccount the non-thermal distribution of rotational states in the deflected part of the molecular beam and laserfocal-volume averaging.The total probability density distributions of C andH + were modeled as the weighted sums of contributionsfrom the individual atoms. Based on chemical similar-ity, see Figure 1 a, we used equal weights for pairs ofatoms H & H , H & H , H & H , C a & C a , C &C , and C & C . As the recoil axes we choose vectorsconnecting carbon atoms to the center of mass of themolecule for C and vectors along molecular C–H andN–H bonds for H + . To reproduce the experimental datathe simple axial recoil approximation yielded excellentagreement for C , whereas for H + we had to accountfor non-axial recoil by convoluting the calculated prob-ability density distributions of hydrogen atoms with aGaussian function of a solid angle representing angulardisplacement from the recoil vector. The weights and theFWHM parameter of this Gaussian function were deter-mined in a least-squares fitting procedure to the measuredalignment revival trace and the angle-dependent maskedVMI data. The obtained parameters are specified in theSupplementary Information. The results of the fit showvery good agreement with the experimental alignmentrevivals in Figure 2 b and excellent agreement with theintegrated in-plane angle-dependent projections throughthe 3D momentum distribution in Figure 3.This excellent agreement confirms the correct repre-sentation of the experiment by our quantum simulations.The obtained planar alignment in terms of squared di-rection cosines [47] at t = 3 . ps is (cid:10) cos θ Y y I (cid:11) = 0 . and (cid:10) cos θ Xx I (cid:11) = 0 . . These values are higher thanthe measured values, which is due to non-axial recoilof the H + fragments and different recoil axes contribut-ing to the measured ion-momentum distributions. Thecomputed in-plane degree of alignment at t = 3 . ps is (cid:10) cos θ Zz I (cid:11) = 0 . , which we also assign as the experimen-tal value due to the excellent match of the angular distri-butions in Figure 3. Simulated time-dependent alignmentrevivals can be found in the Supplementary Information.A single scalar metric describing the overall degree of 3Dalignment cos δ = (1+cos θ Zz I +cos θ Y y I +cos θ Xx I ) ,is shown in Figure 2 c. A maximum degree of field-freealignment of (cid:10) cos δ (cid:11) = 0 . was obtained, which is com-parable to or even larger than one can achieve for complexasymmetric top molecules using adiabatic alignment tech-niques [34, 48] and clearly sufficient for molecular-framecoherent diffractive imaging [18, 19].In summary, we demonstrated strong laser-field-free3D alignment of the prototypical complex non-rotation-symmetric molecule indole induced by shaped truncatedquasi-adiabatic laser pulses. Both, the amplitude and thephase of a strongly-chirped broad-band alignment laserpulse were tailored using an SLM, which allowed us toproduce very short truncation times, unachievable withamplitude truncation alone. The combination of quasi-adiabatic alignment with a kick pulse directly before thesudden truncation produced a higher degree of alignmentunder field-free conditions than in the field. The alreadyachieved strong degree of alignment is limited by theinitially populated states in the molecular beam [49] andcould be further improved through even colder molecularbeams [37, 38].We have developed and tested a new approach to char-acterize molecular alignment in 3D by performing sep-arate measurements of planar alignment and in-planetomography. Planar alignment was measured as the time-dependent alignment trace of the H + fragments. In-planealignment was extracted from the angular dependence ofH + and C fragment distributions at the center of the de-tector, obtained by rotating the laser polarization ellipseand thus the molecule in the plane perpendicular to thedetector. Robust variational simulations of the alignmentdynamics of indole, considering weighted contributions offragments emitted non-axially at different molecular sites,reproduced the experiment with high accuracy.This demonstration of strong field-free alignment foran asymmetric top rotor without rotational symmetriesand without any good ionic fragments for the characteri-zation of the alignment paves the way for strong field-freealignment of any arbitrary molecule. This opens up newprospects for probing native (bio)molecules in the molecu-lar frame [19, 20, 50] without chemically attaching markeratoms that influence the function and properties of themolecule.We thank Stefanie Kerbstadt for helpful discussions.This work has been supported by the Deutsche Forschungs-gemeinschaft (DFG) through the priority program “Quan-tum Dynamics in Tailored Intense Fields” (QUTIF,SPP1840, KU 1527/3, AR 4577/4, YA 610/1) and bythe Clusters of Excellence “Center for Ultrafast Imaging”(CUI, EXC 1074, ID 194651731) and “Advanced Imag-ing of Matter” (AIM, EXC 2056, ID 390715994), andby the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013)through the Consolidator Grant COMOTION (ERC-Küpper-614507). J.O. gratefully acknowledges a fellow-ship of the Alexander von Humboldt Foundation. Appendix
Fragments of indole showing alignment
In Figure 1, ion-momentum distributions for the H + ,C and CH x N + (x=0,1,2) fragments were shown at adelay time of t = 3 . ps, corresponding to the time atwhich the highest degree of field-free alignment was ob-served. Further ion-momentum distributions for the sameconditions are shown in Figure 4, i. e., for the fragmentsC + , C +2 , C H + x , and C H + x , where x depicts fragmentswith different number of hydrogens whose masses couldnot be resolved by the high-voltage gating of the detector. a b c d FIG. 4. VMI images of the fragments (a) C + , (b) C +2 , (c)C H + x , and (d) C H + x . The top row shows images obtained forthe major alignment-laser-polarization axis vertical and paral-lel to the detector surface whereas the bottom row shows corre-sponding images with the minor alignment-laser-polarizationaxis vertical and parallel to the detector surface; cf. Figure 1. Images in the top row show the ion-momentum distribu-tions with the major polarization axis of the alignmentlaser being parallel and the minor polarization axis beingperpendicular to the detector plane ( α = 0 ◦ in main pa-per), whereas in the bottom row the major polarizationaxis is perpendicular and the minor polarization axis isparallel to the detector plane ( α = 90 ◦ in main paper). Tomographic reconstruction of H +
3D momentumdistribution
In Figure 5 isosurfaces of the 3D momentum distribu-tion of H + obtained from a tomographic reconstructionof the individual 2D projections measured at a delay timeof . ps are shown. The individual 2D projections wereutilized to extract the in-plane distributions in Figure 3.In Figure 5 a the plane spanned by the a and b axes,in Figure 5 b the plane spanned by the a and c axes, andin Figure 5 c the plane spanned by the b and c axes areshown.Due to the presence of the 2 kV/cm dc field in the inter-action point of the VMI spectrometer, minor orientationeffects were observed. For the CH x N + (x=0,1,2) fragment,a degree of orientation (cid:104) cos θ (cid:105) ranging from − . to . was measured when the laser polarization was ro-tated by ◦ . The 2D ion-momentum distributions of H + were thus symmetrized prior to the 3D reconstruction,such that the 3D momentum distributions in Figure 5represent the average over the four degenerate orienta-tions of indole, which are related via rotations of ◦ about the a and b axes. a b c a b c FIG. 5. Reconstructed isosurfaces from H + ion-tomographyshowing three different orientations of the molecule with a a-baxes in plane b c-a axes in plane c c-b axes in plane. Theorientation of indole is indicated by the ball-and-stick modelof the indole molecule. Computations
The field-free rotational motion of indole was mod-elled using the rigid-rotor Hamiltonian with rotationalconstants A = 3877 . MHz, B = 1636 . MHz, and C = 1150 . MHz [51, 52]. The electric polarizabilitytensor for the equilibrium molecular geometry was com-puted ab initio at the CCSD/aug-cc-pVTZ [53, 54] levelof theory in the frozen core approximation. Electronicstructure calculations employed the quantum chemistrypackage Dalton [55].Time-dependent quantum dynamics simulations usedthe general purpose code for quantum mechanical mod-elling of molecule-field interactions RichMol [45]. In thesimulations, the time-dependent wavefunction was builtfrom a superposition of field-free eigenstates and the time-dependent coefficients were obtained by numerically solv-ing the time-dependent Schrödinger equation. The latterwas solved using the iterative approximation based onKrylov subspace methods, as implemented in the Expokitcomputational library [56]. The elliptically polarizedalignment laser field was modelled by the function E ( t ) = E ( t ) (cid:110) e x cos( ωt ) / √ , e z sin( ωt ) (cid:111) , (1)with the electric field amplitude E ( t ) computed fromthe measured experimental peak intensity. The carrierfrequency was fixed to ω = c/ (2 π nm ) . The time-dependent wavefunction was expressed in the basis offield-free rotational eigenstates of indole with all rota-tional states with J ≤ included. The time step usedfor wavefunction propagation was fixed to 10 fs. Conver-gence with respect to the size of the rotational basis setand the time step were carefully verified. Since alignmentdepends nonlinearly on the laser intensity, which is notconstant within the focal volume of the laser, integrationof all simulated observables over the interaction volume is required. This has been approximated by repeating thecalculations for five individual laser amplitudes, obtainedby scaling the originally measured peak intensity I ( t ) withfactors 0.2, 0.4, 0.6, 0.8, and 1.0. The focal volume aver-aging was carried out using the measured Gaussian beamprofiles with widths (FWHM) of σ align = 56 . µm and σ probe = 28 . µm. Furthermore, an incoherent averageover the initial rotational state distribution, determinedfrom measured experimental deflection profiles, was car-ried out.In order to characterize the degree of alignment inthe experiment, we computed the rotational probability-density distributions for all hydrogen and carbon atomsin the molecule. The evaluation of the rotational-densityfunctions required the calculation of the Wigner rota-tion matrices, which was carried out using a Fourier-series based algorithm [57]. For different time delays,two-dimensional projections of the rotational probabilitydensity onto the Y Z laboratory plane were computed foreach atom individually, assuming axial recoil of the hydro-gen ions along the C-H and N-H bond vectors, whereasfor C the vectors connecting the center of mass witheach carbon atom were chosen as recoil axes. The 2Dprojections yielded simulated momentum distributionsfor all hydrogen atoms, from which the (cid:10) cos θ (cid:11) wasextracted just the same as from the measured momentumdistributions. Furthermore, by rotating the simulatedrotational probability-density around the laboratory Y -axis in steps of ◦ and carrying out the 2D projectionfor each rotation angle, the tomography measurementswere mimicked. The signal within a radius of 20 pixelswas integrated for each angle and for every hydrogen andcarbon atom, where both, the experimental and simulatedVMI images had a total image size of × pixels.Finally, for hydrogens, the weighted simulated align-ment traces and the angle-dependent integrated probabil-ity density from the center of the detector were simulta-neously fitted to the experiment employing least-squaresfitting. The resulting time-dependent alignment-revivalfit and the angle-dependent integrated probability-densityfit are shown in Figure 2 b and Figure 3, respectively. Forcarbon ions only the integrated probability density wasfitted to the experimental data, shown in Figure 3; thecorresponding experimental time-dependent alignmenttrace was too noisy.The fitting procedures yielded weights representing thecontributions of all individual hydrogen and carbon atomsto the measured ion-momentum distributions. Due tothe structure of indole, hydrogen atoms that have al-most similar angles with respect to the z I axis of indole,e. g., H & H , see Figure 1 a, result in revivals andangle-dependent probability distributions that are indis-tinguishable from each other in the experiment. Thetotal probabilities for these equivalent hydrogens werethus summed and calculated to be P ( H +H ) ≈ . , P ( H ) ≈ . , P ( H +H ) ≈ . , P ( H +H ) ≈ . Time (ps) - -
10 0 10 200 . . . (cid:31) c o s δ (cid:30)
150 300 450 600 7500 . . . . (cid:31) c o s θ Z z I (cid:30) . . . . (cid:31) c o s θ Y y I (cid:30) . . . . (cid:31) c o s θ X x I (cid:30) abcd FIG. 6. Simulated degree of 3D alignment character-ized through the expectation values (a) (cid:68) cos θ Zz I (cid:69) , (b) (cid:68) cos θ Y y I (cid:69) , (c) (cid:68) cos θ Xx I (cid:69) and (d) (cid:68) cos δ (cid:69) with the carte-sian principal axes of the polarizability tensor frame z I , y I , x I ,the cartesian axes of the laboratory-fixed frame X, Y, Z and cos δ = (1+cos θ Zz I +cos θ Y y I +cos θ Xx I ) , a single scalarmetric to characterize the 3D alignment [47]. with the squared difference between the experimental andthe simulated data χ min ≈ . .For carbon the same procedure yielded P ( C ) ≈ . , P ( C ) ≈ . , P ( C a +C a ) ≈ . , P ( C +C ) ≈ . and P ( C +C ) ≈ . with χ min ≈ . .The expectation values of the alignment cosines werecomputed for the three main polarizability axes in theprinciple-axis polarizability frame with respect to thelaboratory-fixed frame by employing Monte-Carlo integra-tion, converged to better than − using ∼ samplingpoints. The simulated degree of alignment for the mainpolarizability axes of the molecule α z I > α x I > α y I withrespect to the laboratory axes XY Z is shown in Figure 6.The highest achieved 3D degree of alignment was thuscharacterized to be (cid:10) cos θ Zz I (cid:11) = 0 . , (cid:10) cos θ Y y I (cid:11) =0 . , (cid:10) cos θ Xx I (cid:11) = 0 . and (cid:10) cos δ (cid:11) = 0 . , where cos δ = (1 + cos θ Zz I + cos θ Y y I + cos θ Xx I ) [47].Calculations of the theoretically expected alignment pulse shape, taking into account SLM pixellation, SLMpixel gaps, the laser beam diameter, and the spectralspread at the Fourier plane, resulted in an expected laserintensity at times between t = 4 ps and t = 10 ps to be afactor of lower than at the peak of the alignment pulseat t = 0 ps.Finally, we note that further rigid rotor calculationsfor indole (not shown) indicate that a truncation time of ≤ ps is required to obtain essentially identical dynamicsto having an instantaneous truncation. 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