Bootstrapping to the Molecular Frame with Time-domain Photoionization Interferometry

TL;DR

This paper introduces a time-domain photoionization interferometry method using impulsively aligned molecules to determine ionization matrix elements and reconstruct molecular frame interferograms, enabling comprehensive photoionization analysis.

Contribution

The work presents a novel time-domain approach to perform complete photoionization experiments and reconstruct molecular frame data without prerequisites for any molecule.

Findings

Able to determine ionization matrix element magnitudes and phases

Reconstructs molecular frame interferograms from lab frame data

Applicable to all molecules, energies, and channels

Abstract

Photoionization of molecular species is, essentially, a multi-path interferometer with both experimentally controllable and intrinsic molecular characteristics. In this work, XUV photoionization of impulsively aligned molecular targets ($N_2$) is used to provide a time-domain route to "complete" photoionization experiments, in which the rotational wavepacket controls the geometric part of the photoionization interferometer. The data obtained is sufficient to determine the magnitudes and phases of the ionization matrix elements for all observed channels, and to reconstruct molecular frame interferograms from lab frame measurements. In principle this methodology provides a time-domain route to complete photoionization experiments, and the molecular frame, which is generally applicable to any molecule (no prerequisites), for all energies and ionization channels.

Figures

Equations

$$S(\theta ,\varphi ,t)=\sum _{L,M}{\beta }_{L,M}(t)Y_{L,M}(\theta ,\varphi )$$

$${\beta }_{L,M}(t)=\sum _{K,Q}\left(\sum _{\alpha ,{\alpha }^{\prime }}{\gamma }_{K,Q}^{\alpha ,{\alpha }^{\prime }}{\mathbf{D}}_{\alpha }^{\ast }{\mathbf{D}}_{{\alpha }^{\prime }}\right)A_{K,-Q}(t)$$

$${\beta }_{L,M}(t)=\sum _{K,Q}{C}_{K,Q}^{L,M}{A}_{K,-Q}(t)$$

Full text

Photoionization is an interferometric process, in which the final observable results from a coherent sum over multiple quantum paths to a set of final continuum photoelectron states $|k,l,m\rangle$ Cohen and Fano (1966); Dill (1976). Interferences between these components are manifest in the observable energy spectra and photoelectron angular distributions (PADs) Cooper and Zare (1969); Dill (1976); Reid (2003), the latter of which can be considered as a particularly high information content observable, extremely sensitive to the phases of the partial waves $|l,m\rangle$ Reid (2003); Ramakrishna and Seideman (2012); consequently, PADs have been investigated in a large range of control and metrology scenarios Reid (2012); Hockett et al. (2015a) . In the context of phase-sensitive metrology, the goal is to obtain the full set of complex photoionization matrix elements, hence characterise the photoelectron wavefunction, by analysis of sets of PAD measurements - this is a “complete” photoionization experiment Klar and Kleinpoppen (1982); Becker (1998).

In the molecular case, the number of final $|l,m\rangle$ states is typically large, and obtaining a sufficient dataset for a complete experiment remains a challenge. In the energy domain, a number of different schemes have been demonstrated in both the laboratory (LF) and molecular frames (MF) Reid et al. (1992); Shigemasa et al. (1995); Geßner et al. (2002); Lucchese et al. (2002); Hockett et al. (2009); Tang et al. (2010); Hockett et al. (2014). The common theme to all these measurements is some form of control over the experimental contributions to the photoionization interferometer (e.g. rotational state, polarization geometry), to which the intrinsic molecular contributions remain invariant. The difficulties in such cases are the ability to obtain a sufficiently large dataset, and molecular specificity in the methodologies, i.e. prerequisites such as low density of states Hockett et al. (2009), resonances Tang et al. (2010) or dissociative channels Shigemasa et al. (1995); Geßner et al. (2002); Lucchese et al. (2002).

In the time-domain, rotational wavepackets can be utilized to control the geometric part of the interferometer. In this case, a high degree of spatio-temporal control of the axis distribution (alignment) of the ionizing molecular ensemble in the LF can be obtained via preparation of a broad rotational wavepacket. Although this idea is conceptually obvious, the theory is complex; it has been elucidated by multiple authors (e.g. refs. Underwood and Reid (2000); Seideman (2002); Suzuki (2006); Suzuki and Suzuki (2007); Ramakrishna and Seideman (2012); Hockett (2015)), but - to date - there have been no experimental demonstrations beyond the limiting case of a narrow wavepacket prepared via resonant excitation Suzuki (2006); Tang et al. (2010); Suzuki et al. (2012) (c.f. rotational coherence spectroscopy Felker and Zewail (1987); Felker (1992)), and an exploration in the related case of high-harmonic spectroscopy from an impulsively aligned sample Lock et al. (2012). While experimental methods for preparing rotational wavepackets, and measuring PADs, are relatively well established Eppink and Parker (1997); Suzuki (2006); Reid (2012); Rouzée et al. (2012), the analysis of such experiments remains challenging, since both the rotational wavepacket and the photoionization dynamics must now be fully characterised. However, the benefits are significant - the time-domain methodology is, in principle, completely general with no molecule-specific prerequisites; additionally the use of high harmonics for ionization provides channel and energy multiplexing in each time-domain measurement, resulting in an extremely high information content metrology ichi Suzuki and Seideman (2005); Ramakrishna and Seideman (2012); Rouzée et al. (2012). Furthermore, if the determination of the photoionization matrix elements is of sufficient fidelity, one can reconstruct the MF interferograms without the necessity for direct MF measurements. In general, such measurements are desirable for detailed understanding of molecular processes Lucchese and Stolow (2012).

Here we present a general time-domain approach to complete photoionization experiments in molecules. Experimentally, a double-pulse impulsive alignment scheme is used to create a broad rotational wavepacket in N2 Bisgaard et al. (2004); Cryan et al. (2009); Ren et al. (2013). High harmonics of a 267 nm driving field are used to ionize the aligned ensemble, and velocity-map imaging (VMI) provides energy, angle and time-resolved photoelectron interferograms. A bootstrapping methodology is employed to analyse the data, in which (1) the prepared rotational wavepacket is characterised without knowledge of the intrinsic molecular photoionization dynamics Makhija et al. (2016) and (2) these results feed into a protocol for the determination of the channel-specific ionization matrix elements. As a test of the results so obtained, the MF interferograms (as a function of polarization geometry) are reconstructed and compared with ab initio calculations. The key results are presented herein; an extended presentation and discussion, and numerical data, can be found in the Supplementary Material (SM) Hockett (2016).

Experiment: laser pulses were generated by a Ti:sapphire amplifier (100 Hz, 11 mJ, 800 nm, 50 fs). Approximately 3.5 mJ was used to generate the third harmonic (267 nm, $\sim$0.05 mJ, $\sim$100 fs) via double and sum mixing stages in $\beta$-BBO. The output was filtered by dichroic mirrors and then sent into a vacuum chamber where high harmonics were generated from a pulsed gas jet of argon. Harmonics were then sent through a 200 nm-thick aluminum foil to filter out the driving laser light; the final spectrum was dominated by harmonics 5 (H5, $h\nu$=23.3 eV) and 7 (H7, $h\nu$=32.6 eV). The remainder of the initial 800 nm beam (7.5 mJ) was sent to a Michelson interferometer; the two replicas of the pulse generated formed alignment pulses. The alignment and filtered harmonic beams were recombined (on a holey mirror) and focussed into the interaction region of a VMI spectrometer Eppink and Parker (1997). Spatial and temporal superposition of the alignment pulses and the harmonic pulses was achieved by maximizing the AC Stark shifts in the photoelectron spectrum of argon. The estimated laser parameters in the interaction region for each alignment pulse are 0.5 mJ, 100 fs at focus, resulting in a peak intensity $I$ = 20 TW/cm2. All beams were linearly polarized, and the data reported herein was obtained for a parallel polarization geometry (Fig. 1(a)). The delay between the two pump pulses was experimentally optimized (for maximal alignment) at 3.76 ps, near the rising edge of the half revival induced by the first pulse Ren et al. (2013).

Figure 1 presents three example photoelectron images from molecular nitrogen (99.999% purity, 4.8 bar backing pressure, 250 $\mu m$ nozzle) and corresponding photoelectron spectra. The complete dataset consists of 150 temporal steps in 67 fs increments from $t=$-0.5 ps to +9.5 ps, where $t$=0 is defined as the peak intensity of the second alignment pulse. On average, about 30000 laser shots were accumulated at each delay. Additional signals that monitored gas density in the VMI, harmonic source brightness and background contributions (ATI from the alignment pulses, scattered light, residual background gas signals) were taken together with the data, providing calibration and background measurements.

Theory: The observed angular distributions are most generally expressed as an expansion in spherical harmonics:

[TABLE]

where $\beta_{L,M}(t)$ are the expansion coefficients. For cylindrically symmetric cases, as for all LF quantities discussed herein, $M=0$, hence the angular dependence is reduced to a function of $\theta$, and the angle $\phi$ is redundant. The LF symmetry also restricts terms to even-$L$ only. The experimental data analysis consisted of inverting each background-subtracted VMI image using the pBaseX algorithm Garcia et al. (2004), providing expansion coefficients for each photoelectron image as a function of radius, subsequently averaged over each photoelectron band (Fig. 1(d)) to provide $\beta_{L,M}(t)$ for each ionization channel, hereafter denoted as the $X$, $A$ and $B$ channels. Selected time-domain results are shown in Figs. 2, 3 for H5 $X$ channel (full presentation in SM Hockett (2016)).

The $\beta_{L,M}(t)$ can further be expanded in terms of the contributing physical factors:

[TABLE]

In this form, all of the angular momentum coupling terms are denoted by $\gamma$, and can be defined analytically. $A_{K,-Q}(t)$ are the time-dependent axis distribution moments (ADMs) arising from the rotational wavepacket, and define the spatial alignment of the molecular ensemble: the full axis distribution in the LF can be described as $P(\theta,\phi,t)=\sum_{K,Q}A_{K,-Q}(t)Y_{K,Q}(\theta,\phi)$ (cf. Eqn. 1). The ADMs couple geometrically to the ionization dynamics - the intricate details of this coupling dictates the response of different partial waves to the ADMs, and different revivals in the rotational wavepacket are sensitive to different aspects of the ionization dynamics Hockett (2015). The $\boldsymbol{D}_{\alpha}$ are the symmetrized ionization matrix elements (complex) to be determined. All other required quantum numbers are denoted $\alpha$, and the coherent summation is obtained by summing over all possible pairs of each quantum number. The full form of Eqn. 2, which explicitly shows all summations (all interfering paths) which contribute to each observable $\beta_{L,M}(t)$, is given in the SM Hockett (2016). This treatment follows the formalism of Underwood & Reid Underwood and Reid (2000); Reid and Underwood (2000); Stolow and Underwood and is applicable to single-photon ionization in the dipole limit.

Evidently, the above equation can be recast as

[TABLE]

where the coefficients $C_{K,Q}^{L,M}$ contain all the terms in the brackets in Eq. 2. Since this equation is linear in the ADMs, the measured $\beta_{L,M}(t)$ can be used to determine the ionization dynamics in this phenomenological form - such solutions of Eq. 3 constitute the first part of the bootstrapping procedure Makhija et al. (2016).

Analysis: The ADMs can be computed by solving the Time Dependent Schrödinger Equation (TDSE) for a linear rigid rotor rotor in a linearly polarized non-resonant pulse. Although the $C_{K,Q}^{L,M}$ are channel dependent, the axis distribution is universal; hence fits to different $L$ and/or ionization channels can be performed as a rigourous cross-check on the results. In this case, computation of the ADMs was carried out for rotational temperatures $T_{rot}$ = 1 K to 30 K in 1 K steps and laser intensities $I$ = 10 TW/cm2 to 30 TW/cm2 in 2 TW/cm2 steps for each pulse. The delay between pulses and their durations were kept fixed to the experimentally determined values. The calculated ADMs were stored and a linear regression to solve Eq. 3 was carried out for each parameter set on the measured $\beta_{0,0}(t)$, $\beta_{2,0}(t)$ and $\beta_{4,0}(t)$ parameters for the $X$-channel. Each regression was performed independently, and $I$ = 20 TW/cm2 for each pulse and $T_{rot}$ = 15 K provided the best fit in all three cases, in good agreement with the experimentally estimated values. The fits are shown in Fig. 2, along with a comparison with $\beta_{6,0}(t)$ obtained using the same parameter set ($I$ and $T_{rot}$). Given the fidelity of the fits, the ADMs ($K_{max}=6$) obtained for these parameters are assumed to accurately describe the experimentally prepared axis distributions, and the resultant axis distribution $P(\theta,t)$ is shown in Fig. 2.

Although the retrieved coefficients $C_{K,Q}^{L,M}$ contain information on the ionization dynamics Makhija et al. (2016), it is preferable to use the resulting ADMs within the full ionization framework of Eq. 2 to directly retrieve the intrinsic molecular ionization dynamics defined by $\boldsymbol{D}_{\alpha}$. This constitutes the second stage of the bootstrapping procedure. The multiple interfering pathways set the requirement for a high information-content dataset, and the equations are highly non-linear multivariate quadratic equations. Careful sampling of the solution hyperspace is required in order to ensure unique results: this step was therefore broken down into sub-steps to allow (a) statistical sampling and (b) further bootstrapping. In (a) repeated coarse fits with minimal information content (selected experimental measurements with distinct ADMs) and randomised seed parameters were employed Hockett et al. (2015b), while (b) took the best fit result(s) as seeds for fits with higher information content and/or more stringent convergence criteria, based on computational time and desired precision.

Fig. 3 illustrates this method and example results. The $X$-channel is particularly sensitive to the axis distribution, with significant changes in both the yields and angular distributions. In this case, two unique fit results (sets of matrix elements $\boldsymbol{D}_{\alpha}$) were obtained via statistical sampling at a coarse level (11 temporal points over the revival features), and fine-tuning via the bootstrapping methodology outlined above (finally incorporating 89 temporal points), led to a single best-fit solution (6 complex-valued matrix elements, results as shown in Fig. 3(a)). For the $A$ and $B$-channels the data becomes increasingly noisy as the yields decrease, and also indicates much less dramatic dependence on the axis distribution. In these cases, adequate fits were obtained at the coarse level (as indicated in fig. 3(b)), but further bootstrapping was not explored in either case. For the $A$-channel, three best fit parameter sets were obtained (7 matrix elements), differing only in the perpendicular continuum waves. For the $B$-channel data , a single best fit parameter set was obtained (5 matrix elements), although four additional parameter sets were within 1% of the best fit (defined by the minimal residual, $\chi^{2}$ Bevington and Robinson (1992)), and differed in the parallel continuum. In these cases, additional data and/or cross-checks on the determined matrix elements are therefore desirable to confirm their validity. The full set of fit results and matrix elements determined (including the associated uncertainties) are given in the SM.

Finally, from the matrix elements obtained, the molecular frame (MF) photoelectron interferograms may be calculated, and compared with ab initio results. In this case, the interferograms are no longer restricted by the LF symmetry, hence this provides a sensitive test of the retrieved matrix elements. Fig. 4 shows these results, and comparison with ab initio results obtained using ePolyScat Gianturco et al. (1994); Natalense and Lucchese (1999). Results are shown for three different polarization geometries in the MF. The parallel and perpendicular cases cleanly separate sets of matrix elements $\boldsymbol{D}_{\alpha}$ by symmetry, while the diagonal result mixes these components and provides a particularly stringent test of the results Lucchese et al. (2002). Compared to the LF, the MF results show a wealth of structure. The experimentally reconstructed and computed results for the $X$-channel are in good agreement, while for the $A$ channel the agreement is variable, consistent with larger uncertainties in the retrieved matrix elements Hockett (2016). Photoionization of $N_{2}$ has attracted much interest due to the $\sigma$ continuum shape resonance Lucchese et al. (1986). Significantly, the MF results (Fig. 4) for the $X$ and $B$ $\sigma$ continua are similar to direct MF measurements in the same photoelectron energy region presented in refs. Shigemasa et al. (1995); Yagishita et al. (2005) for 2 and 9 eV photoelectrons; while those studies investigated dissociative core ionization, the scattering and shape-resonance behaviour of the $\sigma$ continuum accessed is analogous.

From the ab initio perspective, photoionization calculations present a formidable challenge and there are a dearth of MF measurements for comparison and validation, due to the experimental difficulty of such measurements; thus the MF results herein can alternatively be viewed as an excellent test for theory Lucchese et al. (1986); Semenov et al. (2000); Lin and Lucchese (2002); Lucchese et al. (2002); Jin et al. (2010). From this perspective, consistency between the ab initio and experimental reconstructions can be taken as a good indicator that both methodologies are robust.

In this work, time-domain measurements, utilizing molecular alignment techniques, have been demonstrated as a means to “complete” photoionization studies. The time-domain data provided sufficiently high information content to reliably extract the axis distributions and matrix elements for three different ionization channels. A bootstrapping fitting methodology provided a flexible and robust analysis route, and MF reconstructions provided an additional stringent test of the physical parameters so determined, and an illustration of their predictive power. The main advantage of this methodology is that it is completely general in principle (although may be restricted by symmetry in certain cases Hockett (2016)), and applicable to any molecular ionization problem, provided that the preparation and propagation of the rotational wavepacket remains decoupled from other molecular dynamics. All relative magnitudes and phases of the ionization matrix elements can be determined and MF interferograms can be obtained without experimentally challenging MF measurements, which require molecular orientation or fragmentation. The disadvantage is that this fitting methodology is not a black-box procedure, and requires detailed analysis (see SM). While this methodology has been demonstrated herein for the simplest case of a homonuclear linear molecule, the theory and analysis protocol allows for arbitrary asymmetric molecules, although such cases will clearly be more challenging Makhija and Hockett (2017). Some improvements may be possible here, for example the use of other fitting algorithms (genetic algorithms and homotopy methods Sommese and Wampler II (2005)), which may allow for a more robust approach less sensitive to local minima in the solution hyperspace, and the use of GPUs to massively parallelise the computations.

Experimentally, the use of an extended harmonic spectrum would allow for the observation and analysis of many more photoelectron bands Rouzée et al. (2012), thus providing a route to obtaining even higher degrees of multiplexing in the measurements, and MF reconstruction for a range of channels and energies from a single time-domain experimental dataset. In this light, the generality of the method suggests such measurements, combined with high-repetition rate laser sources, provide a route to highly multiplexed “complete” experiments and MF reconstruction for a range of molecules, and a tractable method for high-resolution dynamical MF imaging Hockett (2014), which is otherwise typically restricted by experimental prerequisites and data-acquisition time-scales Hockett et al. (2013).

Acknowledgements: We thank the US Army Research Office, for funding under contract W911NF-14-1-0383. AS thanks the NSERC Discovery program for financial support.

Supplementary Material Hockett (2016) available at DOI: 10.6084/m9.figshare.4480349.