A plano–convex thick-lens velocity map imaging apparatus for direct, high resolution 3D momentum measurements of photoelectrons with ion time-of-flight coincidence

Velocity map imaging (VMI)¹ is a widely used and extremely versatile technique for measuring a large range of ion or electron energies with a high resolution.^2–5 The VMI technique utilizes a series of cylindrically symmetric electrodes to create an electrostatic field that serves as a lens to map charged particles of a given transverse momentum (or energy) to a specific position on the surface of a detector. Over time, a variety of lensing geometries have been developed to increase the relative energy range and resolution to >1 keV^2,4 and <2%,^3,4 respectively. One such development pushing previous limits is the thick lens VMI, first introduced for ions by Lin et al.⁶ The thick-lens design was subsequently modified and applied to electrons by Kling et al.,⁴ where it was shown to extend the VMI detectable energy range while maintaining a high resolution by using 11 VMI electrodes rather than the standard three-electrode system. Additionally, new applications of mesh electrodes have become common.^2,7–9 These electrodes allow for equipotential surfaces that span the electrostatic lens and give a fine control of the field while being transparent enough to limit particle loss.

Although these changes have enhanced the VMI capabilities, the net result of the VMI largely remained a 2D projection of the 3D momentum distribution. Despite this traditional limit of the VMI, there has been constant and continued interest in attaining the full 3D momentum distribution for use in studying various phenomena such as ionization^10,11 and photodissociation.^12,13 Other 3D methods such as the cold-target recoil-ion-momentum spectroscopy (COLTRIMS)^14–16 have proven effective to this end by using delay-line detectors, but this can be difficult to implement on existing VMI systems.

3D VMI momentum measurements have been achieved through either reconstruction techniques or, more recently, direct measurement. Of the two methods, the direct measurement is generally considered the ideal in most circumstances as reconstruction often comes with considerable drawbacks. The most common reconstruction technique utilizes the inverse Abel transform,^17,18 which requires a cylindrical symmetry and is, thus, unsuitable for experiments using elliptically polarized light. Alternatively, tomographic imaging methods have been implemented,¹⁹ which recover 3D information by rotating the momentum distribution (via the exciting laser field polarization) and reconstructing from multiple 2D projections. This has the obvious drawback of requiring multiple datasets to be taken to recover a single 3D distribution, which is not practical in some situations, e.g., pump–probe experiments. Furthermore, both reconstruction methods are liable to introduce noise and potential artifacts into the reconstructed distribution.

The direct measurement of the 3D distribution can be performed by using the particle’s time-of-flight (TOF), which encodes its momentum along the axis of the electrostatic lens. DC slicing²⁰ is a technique by which the VMI detector is gated on a range of TOFs by a fast voltage switch as to image only a “slice” of the 3D momentum distribution. Scanning the gate, then, allows for the full 3D distribution to be measured. Unfortunately, not only does this method require multiple datasets to be taken to produce a single distribution, but it also effectively discards the majority of data by measuring only the small fraction of particles that fall within a given slice. It is also worth noting that this technique has been used in combination with a thick-lens VMI for ions by Lin et al.,⁶ where the imaging camera was gated rather than the detector itself.

The most straightforward, though technically challenging, method for measuring the 3D distribution is by coincident measurement of each particle’s position and TOF. In this method, the momentum components p_x, p_y, and p_z are measured by x, y, and t, respectively, where x and y are the position of the particle on the surface of a detector and t is its TOF. This type of measurement has attracted significant interest in recent years despite the high temporal resolution (generally <1 ns) needed to measure TOF, particularly with respect to electrons. Lee et al.²¹ performed such 3D momentum measurements for electrons by correlating images with TOF information from an MCP pickoff measured by a high-speed digitizer with a resulting temporal resolution of 30 ps. This group went on to introduce an event-driven camera, TPX3CAM (TPX3),^22,23 to their setup, which natively records both time-over-threshold (TOT) and time-of-arrival (TOA) for every pixel activation event.²⁴ They utilized TOT and TOA to synchronize electron TOF measurements (from the digitizer) and estimate pixel brightness, respectively. Although TPX3 is equipped with two time-to-digital converters (TDCs), which can be used to record the electron TOF, they would be limited to a resolution of 260 ps. This constitutes almost an order of magnitude reduction in resolution as compared to that achieved by the digitizer approach, so they do not pursue this. Separately, Cheng et al.²⁵ accomplished such measurements using a comparable MCP pickoff coupled to the TDC of TPX3. Although Cheng et al. sacrificed temporal resolution, it is important to note that this scheme is quite powerful as it is easily implementable on most standard VMIs due to the inherent synchronization of spatial and temporal information for each electron. Furthermore, they showed 260 ps to be sufficient for resolving the structure in the momentum distribution along the TOF axis and performed 3D momentum measurements for electrons and ions in coincidence using a single detector with a fast-switching voltage supply, building upon their previous 2D work.²⁶ 

Although the traditional (2D) VMI can measure electrons with a transverse kinetic energy on the order of 1 keV, doing so requires increasing the gradient of the electrostatic potential, which reduces the TOF range of the particles; in turn, this necessitates unrealizable temporal resolutions to achieve 3D measurement. This effectively means that measurements of this kind are limited to low energy while still requiring a temporal resolution on the order of 100 ps. For reference, Cheng et al. ran their VMI at voltages that limit it to the collection of electrons with a kinetic energy less than ∼5 eV, but the electron TOF spread is still only ∼8 ns.

Here, we expand on the work of both Cheng et al.²⁵ and Kling et al.⁴ by using a TPX3 camera in conjunction with a coincident plano–convex thick-lens (PCTL) VMI and an ion TOF spectrometer for the measurement of 3D electron momentum distributions. By simply introducing a grounded mesh electrode at the far end of a thick-lens VMI, the electrostatic field forms a plano–convex lens that extends the detectable electron cutoff energy range while maintaining a high spatial energy resolution. Additionally, this VMI design increases the electron TOF range by reducing the potential gradient. We achieve this through a small potential difference (13 V) between repeller and V1 (see Table I) and through a large opening in the lens electrodes (52 mm), which further reduces the curvature of the potential and, consequently, its gradient. This extended electron TOF range serves to lessen the temporal resolution required in the measurement to achieve a similar energy resolution. The net result is the collection of electrons with transverse energies up to ∼7 eV while extending the electron TOF spread to ∼30 ns. Furthermore, by combining signals for the laser timing and ion TOF to be recorded by a single TDC channel, we effectively expand the capacity of the TPX3 for the simultaneous, inherently synchronized, measurement of both electron and ion TOF. Extending the TPX3 capability to multiple coincidence measurements is relevant to the broader audience as TPX3 is currently utilized in a variety of applications, including proton beam radiography and spectral x-ray imaging.²⁷ Here, the coincident ion TOF measurement allows for a significant noise reduction and is shown to be effective in extracting the unique 3D electron momentum distributions for multiple ion species measured simultaneously.

These techniques for high resolution 3D momentum information with ion coincidence have the potential to lend themselves to more advanced measurements, such as 3D holography²⁸ and photoionization with intricate exciting fields (e.g., elliptical¹⁰ and two-color²⁹), and other systems for which electron momentum distributions are not cylindrically symmetric. Subsequently, these measurements can be applied to a variety of previously observed phenomena in new systems where a traditional VMI study is insufficient, for instance, resonant multiphoton ionization³⁰ by elliptically polarized light or ionization by pump–probe pulses with different polarization vectors.¹⁰ 

The general setup of the coincident PCTL-VMI and ion TOF spectrometer is depicted in Fig. 1. Note that the axes shown in Fig. 1(c) are used throughout this manuscript; particularly, $ẑ$ is the axis of the electrostatic lens and $x̂$ is the axis of laser propagation.

The electrostatic lens is formed by 11 stainless-steel electrodes, five on the electron side (V1–V5) and six on the ion side (repeller and iV1–iV5). The position and potential of each electrode are detailed in Table I. Both the electron and ion side electrodes are connected in a chain by 100 MΩ resistors, but the two sides are isolated from one another. In doing so, only four voltages need to be applied to the PCTL-VMI lens, one at either end of each electrode chain (V1, V5, repeller, and iV5), and the equal resistance resistors ensure a uniform voltage step between every two adjacent electrodes in each chain. The electron side geometry is based on a previous design.⁴ 

In addition to the electrodes, there is a grounded, copper mesh disk on the far end of the electron side. The mesh is 12 mm from the V5 electrode and 71 mm in diameter; it is commercially available from Precision Eforming with 3.15 lines per mm and an 80% transmission efficiency. The grid is crucial to the formation of the plano–convex electrostatic lens on the electron side as it serves to be the “plano” of the lens by providing a uniform voltage spanning the transverse profile of the lens. This is in contrast to the typical annular electrodes that result in a convex field. The plano–convex field is clearly visible in Fig. 1(c), a SIMION schematic of the VMI cross section in which the red lines indicate equipotential surfaces. Each of the five electrodes on the electron side is annular, and moving away from the interaction region, with each electrode, the field lines become flatter up to the surface of the grid. The repeller is equipped with the same mesh except 23 mm in diameter; such a mesh on the repeller is not uncommon for a double sided VMI (or VMI-TOF). Note that when a mesh is referred to throughout the rest of this manuscript, it refers to the electron-side mesh, not the repeller mesh.

The electron detector is a stack of two 80 mm diameter microchannel plates (MCPs) in the chevron configuration equipped with a P47 phosphor anode. The front face of the electron detector is grounded, so given that the mesh is also grounded, there is a 45 mm field free drift region between the exit of the plano–convex lens and the electron detector. The ion detector is a stack of two 25 mm MCPs (also in the chevron configuration) equipped with a stainless-steel anode. Note that the front face of the ion-side MCP stack is kept at −3210 V. This face cannot be grounded as to maintain the directionality of the electrostatic field, and this specific voltage is necessitated by in-vacuum voltage dividers that are in place for high energy applications of the VMI.

The lens electrodes, mesh, and detectors are encased in a mu-metal cylinder that shields the charged particles from any external magnetic field. The entire assembly is mounted in a high vacuum chamber with a baseline pressure of 2.5 × 10⁻⁸ Torr.

External to the vacuum, the electron MCP and phosphor stack is imaged by a TPX3 camera that is equipped with two time-to-digital converters (TDC 1 and TDC 2), both with a temporal resolution of 260 ps. Additionally, the TPX3 also natively records the time at which each pixel is activated (time-of-arrival, TOA) with a resolution of 1.6 ns. In our detection scheme, TDC 1 is used to measure both the TOF for ions and a laser trigger, while TDC 2 is used for electron TOF. In this regard, the TPX3 camera represents a convenient platform for multiple laser-event synchronization. Pixel TOA is used to distinguish laser correlated electron events from noise by removing pixel events outside of a 1 µs window for each laser pulse. Furthermore, because a pixel TOA is recorded for every pixel activation, a clustering algorithm is implemented to identify multiple pixel events as belonging to the same electron by evaluating their proximity in space and time. Clustering serves to effectively increase the spatial resolution of the camera.³¹ Here, we make use of an open-source algorithm, DBSCAN.³² Because we are grouping pixel events into clusters, we will associate each cluster with a cluster TOA that is the mean TOA, weighted by TOT, for the pixel events comprising that cluster. In practice, TOA is recorded by the TPX3 as an absolute time; however, throughout the rest of this manuscript, we will refer to TOA as it is processed and referenced to a specific laser pulse. In other words, TOA is now the time difference between a laser timing reference signal and the recorded (absolute) TOA. This effectively makes the cluster TOA a measure of the electron TOF, but the 1.6 ns resolution is insufficient to resolve the structure in p_z, so its use is limited to noise reduction. In addition, note that the TPX3 can handle pixel rates up to 80 MHz; for reference, the standard delay line detector (utilized in COLTRIMS 3D momentum measurements) is limited to rates on the order of 1 MHz; see the commercially available products from Surface Concept and RoentDek for reference.

A diagram of the electronic schemes for electron and ion detection is shown in Fig. 1(a). Both the electron and ion detector anodes are supplied with a voltage through simple decoupling circuits that allow for single particles to be detected electronically. Signal pulses from the decoupling circuits are amplified and subsequently processed by constant fraction discriminators (CFDs, Ortec models 9327 and 583) with a temporal jitter <20 and <75 ps for the electrons and ions, respectively. Because the TPX3 TDCs call for a positive transistor–transistor logic (TTL) pulse, the output of each CFD triggers a delay generator (SRS model DG535 for ions and DG645 for electrons) to produce such a pulse. We use two delay generators to avoid dead-time issues with the ion and electron signals arriving too closely in time. The delay generators have a nominal jitter <60 and <30 ps for electrons and ions, respectively. Altogether, the temporal resolution of the electronic configuration for both electrons and ions is limited by the resolution of the TDCs themselves at 260 ps.

In an event-driven experiment, it is necessary to label each event; we achieve this by recording each laser pulse non-ambiguously using the TDC 1 channel of the TPX3. For this purpose, the ion TTL signal is first combined with a laser trigger TTL through a DC pass splitter/combiner (Mini-Circuits, model ZX10R-2-183-S+). This combined ion and laser trigger signal is critical as the current TPX3 model has only two TDCs, but to attain the 3D momentum distribution, we require both ion and electron signals, as well as the laser trigger to serve as a reference for the electron and ion TOF. The laser trigger TTL pulse arrives well before the ion signal (∼260 µs), meaning that there is no concern of overlap, and has a significantly longer pulse width relative to the ion TTL (1 µs vs 50 ns) so that the two are easily distinguishable. The time sequence of the three signals as recorded by the two TDCs is shown in Fig. 1(b).

To optimize the operating voltages and determine the energy range and resolution capabilities of the PCTL-VMI geometry shown in Fig. 1(c), electron trajectory simulations were performed using SIMION³³ (version 8.1). A key feature of the PCTL-VMI apparatus presented here is its capability to directly measure the 3D momentum distribution of electrons, which requires the range and resolution of the PCTL-VMI to be assessed in two parts. First, the momentum of the electron transverse to the axis of the electrostatic lens (xy-plane) is related to the position of the electron on the MCP/phosphor detector; this is the standard operation mode for a 2D VMI. The corresponding (spatial) energy resolution, ΔE/E, is proportional to the range of radii (distance from the center of the detector, R) for electrons of a given energy. Specifically, because a VMI exhibits a linear relationship between the electron momentum and R, we can approximate ΔE/E ≃ 2ΔR/R as is standard.⁴ Here, we take ΔR to be the full width at half maximum (FWHM) of the electron radial distribution.

Additionally, the component of momentum along the axis of the detector, p_z, is measured by the electron TOF. Assessing this (temporal) ΔE/E is less straightforward as there is no linear relationship between the TOF and momentum. Furthermore, for electrons emitted with a momentum ±p_z, there is a lack of symmetry up vs down in the electrostatic field. Thus, to evaluate the temporal energy resolution, we require an additional simulation to serve as a reference and reliably relate a given electron TOF to its initial momentum. This reference simulation is performed with electrons originating from a single, “ideal,” point with energies from 0 to 7 eV in steps of 0.001 eV and an initial velocity in both $±ẑ$⁠. The reference simulation is presented later in the text as a comparison to experimental results and is shown in Fig. 4(d). Using this simulation to correlate electron TOF to momentum ΔE/E is now evaluated in a similar manner described before: ΔE is the energy range corresponding to the FWHM of the TOF distribution for a given kinetic energy.

The simulated spatial and temporal energy resolutions of the PCTL-VMI are presented in Figs. 2(a) and 2(b), respectively. The electron source volume was taken to be a cylinder of 1 mm radius and 2 mm length with its axis along $x̂$⁠. The electrons have an initial momentum corresponding to kinetic energies from 0 to 7.5 eV in steps of 0.25 eV; this range is particularly relevant as it corresponds to low (up to fifth) order above-threshold ionization (ATI) from an 800 nm light. The initial electron momenta are along $ŷ$ to determine the spatial energy resolution [Fig. 2(a)] or $±ẑ$ for the temporal energy resolution [Fig. 2(b)]. Note that in Fig. 2(b), the positive and negative kinetic energies correspond to the initial momenta in the respective $±ẑ$ direction.

Each plot of ΔE/E for the PCTL-VMI shown in Fig. 2 is accompanied by the simulation results for the same VMI geometry except with a conventional (convex) thick-lens VMI (labeled CTL-VMI). This conversion from plano–convex to convex is achieved simply by removing the mesh and reoptimizing the electrode voltages while keeping the repeller voltage fixed at −100 V. Note that for both configurations, the electrode voltages are optimized with the restriction that they are connected by a chain of equal resistance resistors; that is, there is a uniform voltage drop between adjacent electrodes. Thus, for a fixed repeller voltage, the optimization of the electron-side lens is a two-parameter problem consisting of the voltages applied at the ends of the electrode stack (electrodes V1 and V5). With optimized electrode voltages, it is shown in Fig. 2(a) that both the PCTL-VMI and CTL-VMI configurations have a spatial energy resolution better than or comparable to the resolution of most modern VMI designs over their respective energy ranges.^2–4,9

Although electrons were simulated with energies up to 7.5 eV, Fig. 2(a) shows that, for initial velocities in $ŷ$ and the repeller at −100 V, the PCTL-VMI collects electrons up to a cutoff of 6.75 eV as compared to just 4.25 eV for the CTL-VMI. Furthermore, while the CTL-VMI has about twice the resolution of the PCTL-VMI for electron energies between 0.5 and 2 eV, the PCTL-VMI configuration retains a high energy resolution (ΔE/E < 1%) over a range more than twice that of its convex counterpart. The energy resolution curve of the CTL-VMI is tunable by the voltage on V1 such that the ΔE/E minimum can be shifted between 1.5 and 3.5 eV, but the cutoff energy cannot be extended without significantly compromising the resolution. In fact, tuning the voltage for a better resolution at the higher end of this energy range further limits the cutoff energy by about 0.75 eV (∼18%). The PCTL-VMI configuration exhibits a similar dependence on the V1 electrode, but because it already has a high energy resolution over much of its collection range, optimizing the ΔE/E curve for specific energies comes at the cost of resolution over the rest of the range. In general, the PCTL-VMI is shown more valuable for a reliable spatial energy resolution over a large range of electron energies, whereas the CTL-VMI may be preferred in instances where an extremely high resolution is required over a narrow range of energies. With respect to 3D momentum measurements, the measurement of p_x and p_y is already extremely limited by the low voltages required to maintain a resolvable TOF spread. In this case, the PCTL-VMI design offers a means to extend the detectable energy range without increasing voltages, thus retaining a broad TOF range.

Figure 2(b) shows the temporal energy resolution for the two configurations. Here, the PCTL and CTL are almost indistinguishable as the electron TOF is most dependent on the potential difference between the repeller and V1 rather than the lensing itself. The temporal energy resolution is shown to be less than its spatial resolution, but it compares well to the resolution of some current, conventional VMI designs (e.g., this design by Garcia et al.³⁴). There are two other points to be made regarding Fig. 2(b): first, although the simulated electrons only have velocity along the axis of the lens $(ẑ)$⁠, it is shown that the CTL-VMI cannot collect electrons with velocities corresponding to more than ∼5.8 eV in −$ẑ$ [E < −5.8 eV in Fig. 2(b)]. This is because, given an enough velocity along −$ẑ$⁠, an electron can escape the region between the repeller and V1 into the (largely) field free ion side of the apparatus and will not have a trajectory that ends at the electron detector. To this end, the PCTL does have a slight advantage over the CTL because its optimized V1 voltage is slightly less negative than in the CTL configuration. On the other hand, although trajectories for electrons with an initial velocity in +$ẑ$ will always reach the electron detector and are shown to have a reasonable energy resolution, this does not mean that, in practice, we can resolve energies in this regime. The issue is that although ΔE/E appears reasonable, the differences in the electron TOF become so small for similar energies that the TPX3 TDC does not have the temporal resolution to distinguish them. We have found (experimentally) for our setup that we are able to resolve energetic structures for electrons up to ∼3.5 eV with velocities along +$ẑ$ and ∼7.5 eV for velocities along −$ẑ$ [see Figs. 4(b) and 4(d)].

To demonstrate the 3D measurement capabilities of the PCTL-VMI, above-threshold ionization was observed in xenon (Xe). For these experiments, the VMI chamber was filled with Xe gas through a leak valve to a pressure of 2.2 × 10⁻⁷ Torr. The voltages applied to the electrostatic lens were measured to be within ±0.2% of the target voltages shown in Table I. Ionization is achieved with a titanium:sapphire system producing 30 fs pulses at a repetition rate of 1 kHz. Pulses were spectrally filtered to a 10 nm width centered at 800 nm using an interferometric bandpass filter, yielding a temporal FWHM (in intensity) of 80 fs with a pulse energy of 12.6 μJ. A wire grid polarizer and λ/2 plate (both Thorlabs products) are used to clean and orient the laser polarization before being focused into the VMI chamber by an f = +300 mm focusing lens. The intensity in the interaction region is estimated to be on the order of 10¹³ W/cm². The net result of these conditions is that ∼15% of laser pulses have a Xe ionization event.

In Fig. 3, we compare the VMI images of Xe⁺ before and after processing. Figures 3(a) and 3(b) present the ion TOF spectrum and clustered VMI image for linearly polarized light oriented along $ŷ$⁠. Also included in Fig. 3(a) is a histogram of the cluster TOA. Figure 3(b) is clustered, but otherwise, there is no data processing, meaning that it includes every pixel activation during data collection. Figure 3(c) is a processed image where cluster events are gated by two means. First, we remove clusters with a TOA outside a 1 µs window [centered about the cluster TOA peak in Fig. 3(a)] as they are not correlated with electrons emitted by a laser pulse. In doing so, ∼40% of clusters are removed, which include any faulty or dead pixels. Second, we gate on the ion species of interest. Specifically, we only take clusters that occur in coincidence with an ion measured within a particular TOF range. Here, we gate on a 1.75 µs window in the ion TOF [vertical black lines in Fig. 3(a)] corresponding to Xe⁺. Note that the ion TOF spectrum does not resolve the various Xe isotopes; this lack of resolution is a consequence of the low running voltages applied to the apparatus. The resolution here is sufficient for the purposes of broad coincidence gating but could be improved upon with a redesign (e.g., extending the ion drift region).

Much in the same way that the energy resolution for the PCTL-VMI was simulated spatially and temporally, we now require both spatial and temporal energy calibrations to access the 3D momentum distribution; this is shown in Fig. 4. The spatial calibration is done in a standard manner: ATI peaks for a VMI image (with laser polarization in the xy-plane) are fit under the condition that each is separated by a photon energy and that clusters at R = 0 correspond to having a zero kinetic energy. The ATI peaks and corresponding calibration fit are shown to be in good agreement compared to SIMION simulations [Figs. 4(a) and 4(c)]. The temporal energy calibration requires a separate dataset to be taken with the laser polarization oriented along $ẑ$ so that the ATI peaks are now clearly discernible in the electron TOF spectrum [shown in Fig. 4(b)]. Notice that in this ATI spectrum, there is an additional peak corresponding to a zero kinetic energy along $ẑ$ present in the electron TOF spectrum. This feature is also shown in Fig. 3(c) as the projection of the image on the polarization axis, which is now effectively being measured as a TOF distribution. By calibrating the spatial ATI, first, we have the absolute energy for each remaining (E ≠ 0) peak in the electron TOF spectrum. The ATI energies are plotted as a function of TOF and fit to a fifth order polynomial to serve as a conversion from TOF to energy. Figure 4(d) shows the measured ATI TOFs and conversion fit to be in good agreement with SIMION simulations. It is important to note that the temporal energy calibration is very sensitive and requires a very good fit for conversion.

It is important to note that with the PCTL-VMI design, the electron TOF spectrum is far broader than with previously reported 3D VMIs. For example, Cheng et al. observed electrons of ±2.12 eV over a range of ∼5 ns,²⁵ as compared to ∼10 ns for the PCTL-VMI as shown in Fig. 4. This means that, for the same timing resolution, the PCTL-VMI design presented here will have a greater overall TOF energy resolution. Furthermore, even in the low voltage configuration required for a broad electron TOF spectrum, the PCTL-VMI also retains a higher spatial photoelectron cutoff energy than that shown by Cheng et al.

Figure 5(a) presents the calibrated 3D histogram of the electron momentum distribution, for laser polarization along $ŷ$⁠, in coincidence with Xe⁺. This is the same dataset presented in Figs. 3, 4(a), and 4(c). Figure 5 also presents slices of the 3D histogram for p_z = 0 (b) and p_x = 0 (c), which are then compared to slices from the same experimental conditions, but now with the laser polarized along $ẑ$ [Figs. 5(d) and 5(e), anti-respectively]. In doing so, we have simply rotated the momentum distribution by 90°, and, as expected, the p_z = 0 slice for $ŷ$ polarized light (b) is in good agreement with the p_x = 0 slice for $ẑ$ polarized light (d). The same can be said of Figs. 5(c) and 5(e). Note that this figure shows that the VMI lens does exhibit some aberrations, particularly in (c) and (d). From (b), we can see that the momentum distribution is not perfectly centered on the detector in $ŷ$⁠, indicating that the laser focus is not properly positioned along this axis. That being the case, we suspect that electrons with a sufficiently high momentum in −$ẑ$ can have trajectories that approach the V1 electrode too closely, altering their course. This results in a measurable effect on the measured y coordinate of the cluster, but is negligible in TOF: an apt description of what is seen in (c) and (d).

From Fig. 5, we can draw two conclusions: first and foremost, by comparing the results for laser polarization along $ŷ$ and $ẑ$⁠, we can see that the 3D momentum distribution can be accurately retrieved in a single measurement. Particularly, in this, we have verified the reliable measurement of p_z by the measurement of TOF. Second, as predicted by the SIMION simulations, the energy (or momentum) resolution along the TOF axis is notably less than that of the spatially resolved axes. For example, the splitting of the first ATI ring is shown in both Figs. 3(c) and 5(b), which is not observed in Fig. 5(d). This can also be seen by comparing the width of each ATI ring in Figs. 5(c) and 5(e). Despite this, having access to the 3D histogram provides information that was previously unavailable from the standard (projected) VMI image; e.g., the structures perpendicular to the polarization axis $(x̂)$ in the 2D image [Fig. 3(c)] are shown to be rings around $ŷ$ in Fig. 5(a). Furthermore, each ATI is a spherical shell with a preferential emission along the polarization axis.

Finally, to demonstrate the efficacy of the ion coincidence scheme, a dataset was taken for a mix of ambient air and Xe gas using the light polarized in $ẑ$⁠. To ionize molecular species in air, the pulse energy was raised to 48.8 μJ (as compared to 12.6 μJ for all other results presented). A relatively large dataset was taken, ∼240M laser shots over 3 days, to ensure good statistics for each ion species. Figure 6(a) shows the ion TOF spectrum for this run; of particular interest are the three dominating species: H₂O⁺, $N2+$⁠, and Xe⁺. The histograms of the total (3D) energy were produced for electrons in coincidence with each of these ion species and are shown in Fig. 6(b). Here, it is clearly seen that there is a shift in ATI peaks, due, in part, to differing ionization potentials (I_p). This can also be attributed to laser intensity effects; that is, ions with a lower I_p will be ionized from a larger volume and, thus, from a larger range of field intensities, which results in a change in the ATI structure. Furthermore, Figs. 6(c)–6(e) present slices of the 3D histogram for each ion at p_x = 0, and Fig. 6(f) presents additional slices for $N2+$ at p_x = ±0.2 amu and p_x = ±0.4 amu. This figure shows that not only are the ATI peaks shifted, but there is an entirely different angular momentum structure for each species. The cumulative result is three unique ATI spectra, measured within a single dataset, as expected for the three ionic species with vastly different electronic structures.

We have demonstrated the efficacy of a plano–convex electrostatic field in a thick-lens VMI design to enhance its 3D electron momentum resolution capabilities. SIMION simulations show that, with the simple addition of a mesh electrode to form the plano–convex field, the PCTL-VMI not only retains its high spatial energy resolution but also extends the detectable electron cutoff energy for a given repeller voltage. This is of particular importance for 3D measurements that rely on the electron TOF as they necessitate low repeller voltages to maintain a large enough TOF spread to resolve energetic features. Furthermore, the VMI design spreads the electron TOF distribution, thus requiring a less temporal resolution of the electronic detection scheme to attain a similar overall energy resolution. We have confirmed these simulations experimentally by demonstrating the PCTL-VMI capability to collect electrons up to ∼7 eV with a TOF spread of ∼30 ns, both being improved compared to a previous work by factors of ∼1.4 and ∼3.75, respectively. Additionally, the PCTL-VMI is equipped with a coincident ion TOF spectrometer that allows for the effective gating of electrons from different ionic species in a gas mixture and recovers the unique 3D electron momentum distribution for each. These techniques have the potential to lend themselves to more advanced measurements, particularly involving systems where the electron momentum distributions possess non-trivial symmetries.

This work was done under Air Force Office of Scientific Research (AFOSR) Grant No. FA9550-21-1-0387. T.S. was partially funded by the U.S. Department of Energy (DOE) under Grant No. DE-SC0019098. C.C., G.M., and T.W. gratefully acknowledge the support from the Department of Energy, Basic Energy Sciences Division, under Award No. DEF-G02-08ER15983. We would also like to thank Dr. Andrei Nomerotski for a useful discussion that led to this work.

The authors have no conflicts to disclose.

M.D. and E.M. performed the bulk of the experiment and analysis with assistance from T.S., M.D., C.A.T.-H., G.N.G., and T.W. conceived the experiment. N.G.H. designed the electrostatic lens and VMI chamber. N.G.H., M.D., Z.R., G.H., and K.W. built the apparatus. C.C., G.M., and T.W. aided with the implementation of the TPX3 camera. M.D. performed SIMION simulations. M.D. prepared the manuscript. All authors contributed in discussion and have reviewed the manuscript.

Michael Davino: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Edward McManus: Formal analysis (equal); Writing – review & editing (equal). Nora G. Helming: Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). Chuan Cheng: Supervision (equal); Writing – review & editing (equal). Gönenç Moǧol: Supervision (equal); Writing – review & editing (equal). Zhanna Rodnova: Methodology (equal); Writing – review & editing (equal). Geoffrey Harrison: Methodology (equal); Writing – review & editing (equal). Kevin Watson: Methodology (equal); Writing – review & editing (equal). Thomas Weinacht: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). George N. Gibson: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Tobias Saule: Methodology (equal); Writing – review & editing (equal). Carlos A. Trallero-Herrero: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.