The importance of molecular axis alignment and symmetry-breaking in photoelectron elliptical dichroism

The first experiment of this kind was conducted by Böwering et al. in 2001,⁴ measuring the forward–backward PECD asymmetry in bromocamphor directly using two fixed-angle photoelectron spectrometers positioned opposite each other. The measured dichroism was around 3%, an effect much larger than the typical values obtained in conventional CD spectroscopy. Soon after this, in 2003, the first photoelectron imaging^5,6 PECD experiment was performed on the related camphor system by Powis and co-workers.⁷ This provided complete energy- and angle-resolved information in a single measurement using a single detector, with a similar observed asymmetry of ∼3%. Both measurements, though, required vacuum ultraviolet (VUV) radiation with single-photon energies exceeding 9 eV to ionize their respective molecular targets. Due to this requirement, early PECD experiments were confined to synchrotron facilities.

Combining Eqs. (3) and (5) shows that $G=2b1{+1}$⁠, and thus, the two quantitative definitions of PECD are equivalent.⁹ Early single-photon experiments also demonstrated that when elliptical ionizing polarizations were used (resulting in photoelectron elliptical dichroism or PEELD), the asymmetry effect, neatly described by the G-value, scales linearly with the third Stokes parameter, S₃.^10,11 As expanded upon later, this term describes the degree of elliptical polarization, with $S3=1$ denoting circular, S₃ = 0 being linear, and intermediate values describing elliptical behavior.

The first MPI-PECD measurements were performed independently by Lehmann et al.⁹ and Lux et al.¹³ on the camphor and fenchone systems, reporting G-values of around 8%. This initial demonstration made gas-phase chiral imaging available to a far broader range of experimentalists due to the wide availability of table-top, turn-key femtosecond laser systems that can efficiently drive MPI processes.^14–16 The PECD technique is now well on the way to revolutionizing the study of chiral molecules, allowing for real-time measurement of enantiomeric excess to incredibly high precision, even in multi-component mixtures.^15,17–19 Such precision measurements are incredibly important, particularly in the pharmaceuticals sector, where synthesis of enantiopure compounds is a crucial concern.

Going further, the effects of elliptical polarization under MPI conditions can also be considered. The combination of resonance-enhanced multiphoton ionization (REMPI) with PEELD has been studied previously and is starting to reveal the importance of intermediate state alignment effects on the resulting PADs.^{15,16,20–24} During the resonant excitation step, each ellipticity accesses different subsets of the initially randomized molecular axis distribution, generating different degrees of alignment. This effect may distort a plot of G vs $S3$ away from being linear and, in contrast to the single-photon case, can lead to G being maximized at a value of $S3$ other than 1.^15,20,22,23 The use of elliptical polarizations therefore provides a route to effectively enhance the utility of PECD measurements by directly exploiting these resonance effects. Experimentalists in search of novel chiral analysis methods based on MPI and table-top laser sources may therefore turn to PEELD-based approaches to obtain the highest possible enantiomeric sensitivity, rather than simply relying on PECD measurements.

For the full potential of PEELD measurements to be realized, it is crucial to better understand the role played by the elliptical ionizing field on the measured PAD. In particular, the way in which G and, more fundamentally, the underlying odd-order angular anisotropy parameters evolve with $S3$ is of considerable interest as this contains encoded information about the alignment introduced by the resonant photoexcitation step. One complication here, however, is that unlike in the case of purely circular (or purely linear) ionizing laser polarizations, the 3D-PADs produced under elliptical ionization will no longer possess an axis of cylindrical symmetry. This loss of symmetry also manifests in optical schemes that combine linear and circular laser polarizations or exploit non-parallel linear polarization geometries, such as in time-resolved PECD (TR-PECD),^25–29 photoexcitation circular dichroism (PEXCD),³⁰ and bichromatic field experiments.^31–33 For such experiments, complete 3D angular information can no longer be extracted from a single 2D projection image using standard data-processing techniques (based around the inverse-Abel transform) that are commonplace within the photoelectron/ion imaging community.^34–39 Alternatively, a 3D imaging apparatus may be used, where the arrival time of each photoelectron at the detector is recorded in addition to the position.^40–43 This provides a direct route to obtaining the 3D-PAD but requires more advanced detectors and data-acquisition procedures. The technique is also typically limited to only a few electron hits per laser shot to retain accuracy in correlating time-of-flight information with pixel location. To continue working with significantly higher count rates, tomographic methods may instead be employed, requiring multiple different 2D projections of the 3D-PAD. Until recently, the number of such projections required for robust quantitative analysis was in the region of 50 or more, presenting a significantly more complex and time-consuming experimental challenge.^{15,20,44–47} The novel advancement of image reconstruction techniques within our group has, however, dramatically reduced the heavy data acquisition burden associated with these forms of measurement. Specifically, Hankel–Transform Reconstruction (HTR) offers an efficient strategy for photoelectron tomography, requiring far fewer projections than other algorithms in use today.^48,49 Furthermore, the Arbitrary Image Reinflation (AIR) neural network has been developed with the goal of obtaining accurate 3D reconstructions from just a single projection image when cylindrical symmetry is absent.⁵⁰ These innovative tools provide exciting new possibilities for more in-depth investigations into photoelectron dichroism. As an instructive first step, we present PEELD measurements undertaken on the (1R,4R)-(+) and (1S,4S)-(−) enantiomers of camphor [hereafter referred to simply as (R)- and (S)-camphor, respectively] using 2 + 1 REMPI at 400 nm, employing both the HTR and AIR approaches in the reconstruction of our experimental data. Our findings reveal quantitively, for the first time, significant contributions to the 3D-PAD and PEELD from higher-order non-cylindrically symmetric spherical harmonics. Critically, in similar experiments conducted previously, such terms have typically been assumed to be negligible—in part, because they cannot be quantified when image reconstruction procedures based around the inverse Abel transform are employed.^{21,25–28,30,51}

Laser pulses were focused into a velocity map imaging (VMI)⁶ photoelectron spectrometer using an N-BK7 lens (f = 25 cm, resulting in a peak intensity of around 9 × 10¹³ W cm^-2). This is a reconfigured variant of an instrument used previously for laser-based thermal desorption studies of non-volatile molecules^53–55—including recent PECD measurements on phenylalanine.⁵² The main spectrometer body has now been split into two differentially pumped sections: a source chamber, where a gas-phase sample delivery system is housed, and the interaction chamber, where a sample molecular beam is ionized, and photoelectrons detected (see Fig. 1). Both chambers were independently maintained at a base pressure of ∼1.0 × 10⁻⁷ mbar using turbomolecular pumps.

Inside the source chamber, pre-loaded samples of (R)- or (S)-camphor were retained within a cylindrically bored phosphor–bronze gas-flow cartridge with a 20 µm exit pinhole (see Fig. 1 callout). A heating collar attached around the cartridge maintained the sample temperature at 30 °C using a thermocouple feedback loop. Passing helium over the gently heated camphor sample and through the pinhole generated a continuous flow molecular beam. The center of this expansion then entered the interaction chamber via a skimmer (0.63 mm in diameter) before passing through a set of annular VMI electrostatic lenses. Here, the molecular sample beam and the focused laser pulses cross in a perpendicular geometry between the first two electrodes (see Fig. 1 callout). Camphor molecules are ionized via a 2 + 1 REMPI process, where the resonant excited state is predominantly of 3s Rydberg character.⁹ The expanding photoelectron distribution then accelerates towards a 40 mm diameter dual micro-channel plate/phosphor screen detector, positioned at the end of a short flight tube. A 480 × 640-pixel CCD camera then captures the image projected onto the detector. Typical acquisition rates were around 25–35 photoelectrons per laser shot across all values of S₃ used in the present measurement. The use of mu-metal shielding prevents external magnetic fields influencing the photoelectron trajectories.

A LabVIEW program developed in-house was used to fully automate the polarization control and acquisition of image data. In total, 20 distinct elliptical polarization geometries (ten different $S3$ values, with both left- and right-handedness) were repeatedly sampled by rotating the half waveplate in 2.5° increments over a 45° range. This produces a smooth variation from purely circularly polarized light through pure vertically polarized light and back again.

At each ellipticity, the major axis alignment of the polarization ellipse was rotated through a series of angles α (defined as the angle between the major polarization axis and the detection plane) so that multiple different projections of the 3D-PAD could be recorded. To achieve this, the angles θ_HWP and θ_QWP (see Fig. 1) were advanced by α/2 and α, respectively.

These distributions should be symmetric and antisymmetric, respectively, with respect to reflection in the xy-plane (i.e., the plane perpendicular to the laser propagation axis). As such, the distributions may be symmetrized (the forward and backward hemispheres of the 3D-PAD are averaged together) and anti-symmetrized (the 3D-PEELD is averaged with a copy which has been reflected in the xy-plane and had its sign reversed). Some selected example 3D-PADs/PEELDs of (R)-camphor are rendered in Fig. 3.

Qualitatively, the shape of the 3D-PAD can be seen to change subtly with S₃. Although the angular profile in the yz-plane remains essentially invariant, the out-of-plane contributions for the 3D-PAD are different in each case. A far more pronounced difference can be seen in the 3D-PEELD distributions, with the overall magnitude of the PEELD effect appearing to scale approximately with S₃. It is more challenging, though, to see any φ-dependence by eye. Further quantitative information can, however, be extracted from the Fig. 3 distributions by decomposing them into their individual spherical harmonic contributions. This is done by Fourier transforming the 3D-PAD/PEELD distributions about the angle that lies in the xy-plane. This filters the different spherical harmonics by their cos mφ terms. The l components of the spherical harmonics can then be calculated by expanding the mth Fourier transform profiles in terms of the associated Legendre polynomials $Plm$⁠. All l and m values from 0 to 6 were considered in the fitting procedure, but only even m values were found to be non-zero—indicating the further symmetry restriction considered earlier would have been appropriate. The analysis of the accompanying B_lm coefficients is discussed in Sec. III.

Figures 4 and 5 plot the various B_lm parameters extracted from the (R)-camphor reconstructions in Fig. 3 as a function of photoelectron kinetic energy and the S₃ Stokes parameter. Even B_lm terms contributing to the 3D-PAD are shown in Fig. 4, while the odd B_lm values are presented in Fig. 5. These parameters are scaled such that the maximum value of B₀₀ for all photoelectron energies and pulse ellipticities is 1. Note that these parameters (and all those discussed throughout the remainder of this publication) are spherical harmonic coefficients and so differ by a normalization factor from the Legendre polynomial coefficients that are commonly used in cases where cylindrical symmetry is present [denoted simply as b_l—see Eqs. (3), (4), (6) and (7)]. To make meaningful comparisons between similar anisotropy parameter values reported elsewhere in the literature, this difference in normalization factor must be accounted for. To further understand these data, we begin by adopting the same analysis methodology established by Comby et al.¹⁵ and decompose the elliptical laser field into its separate linear and circular contributions. The circular portion is simply proportional to $S3$⁠, and the remaining linear contribution is then given by $1−S32$⁠. For the α = 0° projection, $1−S32=S1$ and so $S1$ will be used to describe the linear polarization fraction of the elliptical field. The relative values of S₁ and S₃ not only determine the overall magnitude of the PEELD effect but also adjust the selection of different molecular alignments that contribute to the full 3D-PAD. For small values of $S3$⁠, molecules with transition dipole tensors directed along the linear polarization axis (the y-axis in this case—see Figs. 2 and 3) are preferentially excited. As $S3$ increases, the likelihood of the final ionizing photon being circularly polarized correspondingly rises, and so the magnitude of the PEELD signal increases proportional to $S3$⁠. In this regime, the overall 2 + 1 REMPI process can be viewed as a single-photon PECD measurement from a pre-aligned sample of excited camphor molecules. However, because the alignment is generated about the y-axis and the final ionizing photon acts on molecules preferentially lying in the xy-plane (see Figs. 2 and 3), the interaction no longer possesses an axis of cylindrical symmetry about the z-axis. In the other extreme case, when $S3≈1$⁠, most of the laser field is circularly polarized. This will now preferentially excite and ionize molecules oriented in the xy-plane of the circular polarization, restoring the cylindrical symmetry about the z-axis. The resulting distribution is then just the conventional MPI-PECD of camphor that has been studied extensively. Recasting the B₁₀, B₃₀, and B₅₀ spherical harmonic parameters obtained using pure circular polarization in terms of Legendre polynomial coefficients yields values of b₁ = 0.025 ± 0.002, b₃ = −0.049 ± 0.002, and b₅ = 0.020 ± 0.001, which are in excellent agreement with those reported elsewhere.^9,24,59,60

The intermediate regime, where $S3≈S1$⁠, is of most interest. Here, the laser field is approximately equally linearly and circularly polarized, and the combination of the alignment and PECD effects is maximized. This can lead to measured G-value asymmetry factors that are larger for elliptical polarizations than for the pure circular case.^20,22,23,61 The effect of intermediate state alignment effects has been studied previously using the PEELD technique^15,20,21 and also by using two-color linear-circular pump–probe schemes to photoexcite and ionize chiral molecules.⁶² Both methods show that the measured PECD/PEELD effect is highly sensitive to any degree of alignment introduced in the resonant excitation step. Some of the first PEELD measurements made use of both tomographic and coincidence detection methods to reconstruct 3D-PADs and 3D-PEELDs without making any symmetry assumptions, but no quantitative description of the 3D-PEELD in terms of spherical harmonics was provided.^15,20 In subsequent works, attempts have been made to give a quantitative description of 3D-PEELD in terms of angular anisotropy parameters.²¹ For this analysis, however, an initial assumption that the 3D-PEELD will be cylindrically symmetric about the laser propagation direction (here, the z-axis) has always been made. The authors of these works acknowledge that this is an assumption, and the calculated anisotropy parameters do not necessarily represent the true 3D-PAD/PEELD but nevertheless use them to build a quasi-quantitative description of the ionization process. This, though, limits the description of the 3D-PAD/PEELD to only the m = 0 cylindrically symmetric B_l0 anisotropy parameters, and any φ-dependent (i.e. m ≠ 0) terms are necessarily assumed to be zero. As clearly seen in Figs. 4 and 5, this is not always the case. Specifically, Fig. 5 reveals significant contributions to the 3D-PEELD from the Y₃₂(θ, φ) and Y₅₂(θ, φ) spherical harmonics that are of similar magnitude to those of the m = 0 terms (and so are far from negligible).

The nature of the anisotropic excitation may be extracted by analyzing the even B_lm contributions to the 3D-PAD. For a quantitative measure of these parameters, the contour plots in Fig. 4 are averaged over the region of 0.4–0.6 eV (i.e. the peak in the photoelectron spectrum corresponding to ionization of the resonant-excited 3s Rydberg state⁹) and normalized by the isotropic B₀₀ intensity over the same range. The l = 2 and 4 plots for m = 0 are shown in the first two panels of Fig. 6. The l = 6 contribution is effectively zero within error bars for all S₃ and is omitted from this figure. Interestingly, the change in excitation anisotropy has no appreciable influence on the cylindrically symmetric component of the 3D-PAD as these B_l0 are essentially constant over all S₃ (with averaged values of B₂₀ = −0.44 ± 0.02 and B₄₀ = 0.05 ± 0.01 shown as an overlaid dotted orange line). More interesting information is, however, obtained from the m = 2 contributions to the 3D-PAD. The right-hand side panel of Fig. 6 shows the B₂₂ contribution as a function of S₃. There is a clear and strong dependence here, which can be related directly to the different excitation anisotropy introduced for different S₃. Initially, B₂₂ is zero when $S3=1$⁠, which is expected since this corresponds to a purely circular ionizing laser field. The 3D-PAD should therefore be cylindrically symmetric about the z-axis. Furthermore, B₂₂ is clearly maximized when S₃ = 0. At this point, $S1=1$⁠, and the laser is perfectly linearly polarized. Under this polarization geometry, the excitation anisotropy is cylindrically symmetric about the y-axis and aligned perpendicular to the cylindrical symmetry axis for the $S3=1$ case. Thus, the B₂₂ parameter appears to be a direct reflection of the anisotropy introduced in the two-photon resonant absorption step in the overall MPI scheme.

In many areas of physical chemistry, two-photon absorption processes are modelled with a two-step sequential mechanism. The absorption of the first photon excites a virtual intermediate state before the second photon goes on to further excite these molecules into their final state. Since this model depends on two independent absorption events, the overall efficiency is expected to increase with the square of the driving photon intensity.⁶³ Therefore, in the context of this experiment, the symmetry breaking B₂₂ term would naïvely be expected to vary with $S12$—the square of the linear portion of the laser field. A fit to the B₂₂ distribution proportional to $S1n$ reveals, however, that the best fit value is n = 1.02 ± 0.03 (overlaid in Fig. 6 as a dotted orange line). Unintuitively, this suggests that the anisotropy created in the excitation step is directly proportional to $S1$⁠, rather than its square. This is what may be expected from a single-photon ionization measurement, where no resonant excitations are involved. The precise nature of the alignment introduced in the two-photon absorption step may be determined theoretically, but this is not trivial^64,65 and is further complicated by the broad band excitation induced by a femtosecond laser pulse. Investigating this specific phenomenon will form the basis of future research within our group and is not discussed further here.

With this excitation anisotropy model, we can now examine the odd B_lm contributions to the 3D-PEELD. As with the even parameter analysis, the contour plots in Fig. 5 can be averaged over the peak in the photoelectron spectrum to yield a plot of B_lm vs S₃ (see Fig. 7). For the m = 0 terms, a clear monotonic dependence of B_l0 with S₃ can be seen (with a straight line fit overlaid). This simple linear dependence is that expected in the case of a single-photon PEELD measurement.^15,20,21,66 Single-photon ionization, though, only yields information relating to a B₁₀ parameter. In our multiphoton data, the same behavior is also observed when l = 3 and 5. Thus, any non-linear and/or non-monotonic features previously observed in MPI-PEELD experiments (either in the G-value or b_l parameters) cannot be attributed to these m = 0 anisotropy terms. This result stands in contrast with previous PEELD measurements where other, more complex, variations in the m = 0 parameters have been reported.^21,22 Analysis of this earlier data was, however, undertaken with the assumption of cylindrical symmetry in the 3D-PEELD, something which is not assumed in our present study. Here, the m = 2 terms (only accessible without initially assuming cylindrical symmetry) instead exhibit the more complex non-linear S₃ dependence, as seen in the bottom row of Fig. 7.

As with the 3D-PAD, the symmetry breaking terms in the 3D-PEELD can be interpreted by considering the excitation anisotropy introduced by the elliptical pulse. For the limiting case of $S3=1$⁠, these terms must be equal to zero because for this polarization, all the excitation anisotropy will be cylindrically symmetric about the z-axis. As demonstrated using the 3D-PAD, any symmetry breaking terms will vary proportionally with $S1$⁠, but the B_lm terms in the 3D-PEELD must also depend on S₃. Of course, when S₃ = 0, there will be no PECD or PEELD effect, and the odd B_lm terms must vanish. Based on these requirements for the competing alignment and chiral effects, symmetry breaking terms in the 3D-PEELD are predicted to vary proportional to $S1×S3$⁠. This assumption is used to fit to the B₃₂ and B₅₂ data points in Fig. 7 (orange dotted line), and the model describes the data incredibly well, correctly predicting the maxima and minima locations of the data at $S1=S3=1/2$⁠. This is the polarization that simultaneously maximizes the combination of competing alignment effects caused by the linear part of the elliptical laser field and the chiral effects caused by the circular field. It is also apparent from Fig. 7 that these contributions are not at all negligible, with the m = 2 terms being of a similar magnitude to the m = 0 terms. This is completely at odds with assumptions made previously in similar experiments (such as in TR-PECD measurements^25–29) where these terms have been assumed to make only a negligible contribution to the resulting PADs. A closer inspection of the full 3D distributions produced in such studies without making any symmetry assumptions will allow for a more sensitive and accurate measure of the photoionization dynamics of chiral molecules.^{25–27,30,51,67} This will be important for fully realizing future practical applications of MPI-PECD experiments in chiral analysis.

The importance of the m ≠ 0 terms to the overall PEELD distribution can be further highlighted by reconstructing the angular distributions returned from the fitted B_lm values in Fig. 7. The PEELD has no significant radial dependence (see Fig. 5), and so this dimension is omitted from the reconstructions for clarity. The θ- and φ-dependence of the PEELD with $S3$ is shown in Fig. 8. Again, the break in cylindrical symmetry is clear, even for $S3$ values as large as 0.94, and becomes far more pronounced as $S3$ decreases. It is therefore anticipated that the m ≠ 0 angular parameters will be a crucial component to the photoelectron anisotropy produced whenever elliptical or, alternatively, independent circular and linear laser fields are used to ionize chiral molecules.

A focus of recent research within our group has been streamlining and simplifying the analysis of VMI images recorded from source distributions that do not exhibit cylindrical symmetry. As has been highlighted in this article and elsewhere,⁴⁹ the HTR approach provides a mathematically robust way of dealing with this reconstruction problem, although it does require multiple VMI projections. This was a primary motivation behind the development of the Arbitrary Image Reinflation (AIR) neural network,⁵⁰ which provides the possibility to process distributions lacking any cylindrical symmetry while still only requiring a single input projection image. The present experiment provides an excellent test system for comparing the AIR machine learning approach with the numerical HTR.

The original version of AIR presented in our introductory publication was designed to process image data comprised of even spherical harmonic terms only. From a symmetry-breaking perspective, this is relevant to 3D-PADs produced, for example, using perpendicular linear polarization geometries in REMPI schemes.⁶⁸ In such instances, only one quadrant of any given VMI image is required to “reinflate” the entire 3D distribution. This breaks down when odd spherical harmonic terms are considered (as required to describe the MP-PEELD effect), and at least one half of an image now needs to be considered. To reconstruct the MP-PEELD data presented here, a modified version of the AIR neural network was therefore developed.

Figure 9 shows the modified network structure for this new version of AIR. The input to this structure is 40 × 80 pixels (one half of an 80 × 80-pixel projection image). This (reduced) image size was chosen as a compromise between having sufficient resolution for robust data analysis and manageable memory requirements for the ANN training data. The contraction path of the modified autoencoder network consists of two convolutional layers (with 64 and 128 filters, respectively), followed by a rectilinear function and a 2 × 2 max-pooling window. The expansion path is built from two deconvolutional operations (each followed by a rectilinear function), making use of skip connections to concatenate each layer with the corresponding layer in the contraction path. The output of the decoder is then put through a final convolutional filter to produce an output that is 40 × 80 × 40 voxels. This represents two octants of the volume of a full 3D distribution. Rotated copies of these octants can be assembled to yield the full 3D AIR prediction for the PAD. Following the same training procedures established in our original publication,⁵⁰ 10 000 distributions were simulated using Eq. (10) with randomized B_lm expansion coefficients. Each of these angular distributions was applied to a single Gaussian spherical shell with a randomized radius and width to produce a series of numerically generated 3D-PADs. In the original AIR publication,⁵⁰ we highlight that being able to constrain the reinflation procedure to just a single ring feature allows for anisotropy parameters to be recovered with a higher level of accuracy. We also demonstrate that it is possible to use AIR in situations where multiple different rings (corresponding to ionization from different states) are present in the image data. In future publications, we hope to apply AIR in these more challenging scenarios by using different excitation wavelengths and/or other chiral molecules in MPI-PEELD experiments. The projection of each single ring 3D-PAD was obtained by summing along the projection direction of the 3D grid. Network training was then undertaken using the simulated 3D distributions (9000 for the training and 1000 for validation testing) for 3000 epochs using root mean square error loss and the Adam optimizer.⁶⁹ This took around 15 h on an NVIDIA RTX 6000. With training complete, the network can process a single 80 × 80-pixel input image in around 35 ms. Full 3D distributions are extracted from the α = 0° projections (Fig. 2) using the new AIR network. This projection corresponds to the single image that would normally be recorded under typical VMI experimental conditions. The predicted 3D distributions are then decomposed into their spherical harmonic contributions and averaged using the same strategy as for the HTR approach.

As seen in the upper panels of Fig. 10, plots of B_lm vs S₃ produced for (R)-camphor using AIR are in excellent agreement with those obtained with HTR (Fig. 7). We stress here that no other restrictions (i.e., any constraints relating to the S₃ or S₁ dependencies discussed earlier) are imposed upon the training data or the output of AIR—only the general angular distribution formula given in Eq. (10). To compliment these results, the MP-PEELD of the (S)-camphor enantiomer was also measured over the same range of ellipticities, but in this case, only the α = 0° projections were recorded (see Fig. 2). Using AIR, averaged plots of odd order B_lm vs S₃ were then produced (see lower panels of Fig. 10). For each B_lm parameter, the same overall trends are seen for both enantiomers, but the plots are of opposite sign—as expected given the discussion in the Introduction. This demonstration serves to highlight the potential that machine learning techniques have for streamlining future imaging experiments. In the initial adoption phase, a blended approach (as used here), where subsets of the data are processed with alternative reconstruction methods such as HTR tomography, will help build overall confidence in the AIR approach. Furthermore, for scenarios where no multiple image projections are available, any experimental data previously analyzed with the inverse-Abel transform (and thereby assuming cylindrical symmetry) may now instead be processed with AIR. This, in principle, will allow for retrospective data analysis without needing to repeat complicated and time-consuming measurements.

We have presented a full analysis of the 3D-PEELD of (R)- and (S)-camphor following a 2 + 1 REMPI measurement at 400 nm. Whereas similar studies on other molecules have made simplifying postulates about the symmetry of the distributions, the analysis presented here makes no such supposition. This unambiguously reveals the importance of higher-order symmetry breaking spherical harmonic terms in 3D-PAD/PEELD experimental data and overturns previously made symmetry assumptions in other works. This key finding is important for developing a full understanding of PEELD and related effects and, furthermore, is crucial if PEELD is to ultimately find practical applications within chiral analysis.

A full experimental demonstration of two novel image reconstruction approaches, HTR and AIR, as detailed in earlier publications,^49,50 was presented. The results produced using the machine learning based AIR methodology agree near perfectly with those obtained via the mathematically rigorous HTR technique. This highlights the potential for the use of neural networks for greatly simplified data acquisition and analysis in imaging experiments where cylindrical symmetry is absent. During the initial adoption phase of machine learning approaches, however, it is still crucial to have alternative analytical reconstruction approaches available for benchmarking and confirmation purposes. In this sense, the HTR technique provides a significant advantage over other tomographic reconstruction algorithms for VMI applications as it drastically lowers the burden of data acquisition for these experiments. It is therefore anticipated that a blended approach to machine learning will provide an excellent balance between experimental rigor and simplicity moving forward. For example, the AIR method could be used as the default reconstruction technique throughout an experiment, with HTR only being used to validate a subset of the data, as we demonstrate here in the case of (R)- and (S)-camphor. This form of approach will help in building overall confidence in machine learning techniques and aid in the rapid adoption of AIR within the chemical imaging community.

In addition to appearing in photoionization measurements using elliptically polarized light, it is expected that m ≠ 0 symmetry breaking terms will also manifest whenever circular and linear polarizations are mixed in pump–probe schemes. These anisotropy parameters therefore provide a new form of observable that is sensitive to the underlying molecular axis distribution during a photochemical reaction with a chiral molecule. This also has clear implications for the study of aligned or partially aligned chiral systems, where a more detailed picture of the ionization dynamics is accessible in the molecular frame. In more typical photoionization experiments (i.e., those not involving PECD or PEELD effects), laser-induced ground state alignment of a gas-phase molecular ensemble has been used to significantly enhance the angular structure of the resulting PADs.^70–72 It is anticipated that similar enhancement will be possible in the PECD of an aligned chiral molecule. With the use of the advanced HTR and/or AIR imaging techniques, both the acquisition and analysis of data from such future experiments can be greatly streamlined simplified.

This work was supported by Leverhulme Trust Research Project Grant No. RPG-2012-735, Carnegie Trust Research Incentive Grant No. 70264, EPSRC Platform Grant No. EP/P001459, and EPSRC Quantum Imaging Grant No. EP/T00097X/1. Heriot-Watt University is also acknowledged for providing C.S., A.R., and L.I. with PhD funding. Finally, we thank Yann Mairesse (Université de Bordeaux) for helpful discussions.

The authors have no conflicts to disclose.

Chris Sparling: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (equal); Writing – review & editing (equal). Alice Ruget: Formal analysis (supporting); Methodology (supporting). Lewis Ireland: Formal analysis (supporting); Investigation (supporting). Nikoleta Kotsina: Methodology (supporting). Omair Ghafur: Methodology (supporting). Jonathan Leach: Methodology (supporting); Supervision (supporting). Dave Townsend: Conceptualization (supporting); Funding acquisition (lead); Project administration (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

Data supporting the findings of this study and the MATLAB and Python codes related to the HTR and AIR reconstruction techniques are available from the corresponding author upon reasonable request.