Dissecting Sub-Cycle Interference in Photoelectron Holography
Nicholas Werby, Andrew S. Maxwell, Ruaridh Forbes, Philip H. Bucksbaum, Carla Figueira de Morisson Faria
DDissecting Sub-Cycle Interference in Photoelectron Holography
Nicholas Werby,
1, 2, ∗ Andrew S. Maxwell,
3, 4, ∗ Ruaridh Forbes,
1, 2, 5
Philip H. Bucksbaum,
1, 2, 6 and Carla Figueira de Morisson Faria † Stanford PULSE Institute, SLAC National Accelerator Laboratory2575 Sand Hill Road, Menlo Park, CA 94025, USA Department of Physics, Stanford University, Stanford, CA 94305, USA Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain Department of Physics & Astronomy, University College London,Gower Street, London, WC1E 6BT, United Kingdom Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA Department of Applied Physics, Stanford University, Stanford, CA 94305, USA (Dated: February 25, 2021)Multipath holographic interference in strong-field quantum tunnel ionization is key to revealingsub-Angstrom attosecond dynamics for molecular movies. This critical sub-cycle motion is oftenobscured by longer time-scale effects such as ring-shaped patterns that appear in above-thresholdionization (ATI). In the present work, we overcome this problem by combining two novel techniquesin theory and experimental analysis: unit-cell averaging and time-filtering data and simulations.Together these suppress ATI rings and enable an unprecedented highly-detailed quantitative matchbetween strong-field ionization experiments in argon and the Coulomb-quantum orbit strong-fieldapproximation (CQSFA) theory. Velocity map images reveal fine modulations on the holographicspider-like interference fringes that form near the polarization axis. CQSFA theory traces this tothe interference of three types of electron pathways. The level of agreement between experimentand theory allows sensitive determination of quantum phase differences and symmetries, providingan important tool for quantitative dynamical imaging in quantum systems.
I. INTRODUCTION
In the study of attosecond (10 − s) science, probingmatter with a strong laser field has emerged as a promi-nent tool for revealing internal dynamics of atoms andmolecules [1–4]. The photoelectron emitted in strong-field ionization (SFI) can follow a wide variety of field-driven trajectories depending on the phase of the laserfield at the time of ionization. Photoelectron vector mo-mentum distributions (PMD) encode these trajectories asintricate interference patterns displayed in angularly re-solved photoelectron measurements. Significant work hasbeen applied towards isolating and disentangling thesepatterns in order to determine the electron [2, 5–17] andsometimes the core [18–20] dynamics.The interference of photoelectron trajectories containsinformation about the structure of the underlying par-ent ion, and a breakthrough in disentangling PMDs toprobe the parent atom or molecule came in the formof photoelectron holography [1, 2, 7]. Ultrafast photo-electron holography brings together high electron cur-rrents, coherence, and subfemtosecond resolution, andallows the retrieval of quantum phase differences. Thismakes it a popular alternative to pump-probe interfero-metric schemes such as the Reconstruction of AttosecondBurst By Interference of Two-photon Transition (RAB-BITT) technique [21], the Spectral Phase Interferome-try for the Direct Electric Field Reconstruction (SPI- ∗ These two authors contributed equally † [email protected] DER) [22] and the Frequency Resolved Optical Gating(FROG) [23] (for a review see Ref. [24]). The pat-terns visible in experiment are produced by the interfer-ence of different electronic pathways to the detector (seeFIG. 1 (a)). These pathways undergo varying degrees ofinteraction with the parent ion and so they pick up dif-ferent phases. The interference between the trajectories,recorded by the detector, can reveal these phases andbe employed for imaging. Many interference patternshave been identified as the combination of two photo-electron pathways which have been used to probe andimage the target. Among these two-trajectory interfer-ence patterns are the fan-like structure [9, 25–28] (seeFIG. 1 (b)), the result of the interference between directand forward deflected trajectories; the spider-leg struc-ture [2, 5, 7] (see FIG. 1 (c)), the result of the interferencebetween forward scattered and forward deflected trajec-tories; and the fishbone-like structure [6, 8], which oc-curs in the same region as the spider but has fringes thatare nearly orthogonal to the polarization axis. All theholographic structure and analysis to date has relied ontwo-trajectory interference. However, many of the above-stated patterns overlap, with some models predicting atleast four relevant trajectories (see FIG. 1 (a)) [12], whilepatterns like the fish-bone structure require elaborate ex-perimental methodologies [8] to extract and differentiatefrom more dominating features. More preferable wouldbe to use a multi-trajectory analysis [10]. In this workwe do just that presenting a three-trajectory pattern thatleads to a ‘modulated spider’.A prominent technique to disentangle different typesof quantum interference is to simulate combinations of a r X i v : . [ phy s i c s . a t o m - ph ] F e b FIG. 1: (a) The four CQSFA trajectories found bysolving the saddle point equations Eq. (7) and Eq. (8).The arrows mark the direction of travel and passage oftime: between each arrow head 0 . p = ( − . , . × W/cm and wavelengthof λ = 800 nm. Panel (a), bottom half shows the CQSFA with unit-cell averaging, while the top half shows without(the unit start is defined by taking φ = 0 in Eq. (1)). The top half of panel (b) shows the time-filtered experimentalresults, while the bottom half presents the unit-cell averaged CQSFA calculation after receiving the same filteringtreatment as the experimental data. Panels (c) and (d) respectively show the lineouts along close to the parallel axisand along the first spider leg as indicated on panel (b).cell of ionizations are averaged over. This not only re-moves the aforementioned asymmetries but also ensuresall combinations of trajectories that were present in theexperiment are accounted for.By bridging the gap between experiment and calcula-tion, many previously unexplored subtle sub-cycle fea-tures are revealed. In this paper, we present a high fi-delity PMD of argon gas photoionized by a multi-cyclelaser pulse and filtered to remove the ATI dependence.We introduce the idea of unit-cell averaging in CQSFAcalculations and demonstrate how it matches the exper-iment. Unit-cell averaging employs an ansatz which in-coherently averages over ensembles of trajectories withdifferent time ordering. Variations in the time order-ing results from different initial conditions of the laserfield, which accurately approximates the incoherent av-eraging that will occur in an experiment. We then com-pare our experimental PMD to unit-cell averaged andfiltered CQSFA calculations, and explore the newly re-vealed holographic features which are well matched be-tween calculations and experiment.This article is organized in the following way. In Sec. IIwe compare experiment and theory with the methodsof time-filtering and unit-cell averaging which enable ef-fective comparison between sub-cycle features. Next, inSec. III and IV the methodology of time-filtering andunit-cell averaging, respectively, is outlined. Followingthis, in Sec. V we demonstrate with the CQSFA that themodulations on the spider legs are a three-trajectory in-terference pattern. In Sec. VI this interference pattern isused to demonstrate the existence of Gouy and bound-state phases for the photoelectrons. Finally, in Sec. VIIwe state our conclusions. II. SUB-CYCLE INTERFERENCECOMPARISON: BRIDGING THE GAP
The result of our experimental and theoretical efforts,with specific emphasis on the sub-cycle interference isshown in FIG. 2. In FIG. 2 (b) we show a high resolu-tion, time-filtered experimental PMD of argon and com-pare with computations using the CQSFA. In general wefind very strong agreement between the experiment andthe CQSFA. The main features of the spider and fan-like structures are all clearly visible. Particularly goodagreement is found near the polarization axis for the ax-ial fringes and first spider leg. Notable features in theexperimental spectra are modulations on the spider legs(see dotted and dashed lines), which are visible due tothe exceptionally high resolution of the experiment, whilethe time-filtering technique separates and highlights themodulations with fringes that are broader than the ATIrings. These modulations are well-matched by the unit-cell averaged and filtered CQSFA calculation, FIG. 2 (b).In panel (a) of FIG. 2 we present the CQSFA resultswith and without unit-cell averaging and without anyfiltering. Without the unit-cell averaging the CQSFA re-sults are asymmetric (see Sec. IV and Appendix A formore details) and the modulations along the spider legsare not correctly reproduced. However, on the lowerright-hand [left-hand] side of the panel broad [fine] modu-lations on the spider legs can be seen. It is a combinationof both the broad and fine modulations (unit-cell aver-aging incoherently mixes both sides of the PMD) thatleads to the modulation seen in experiment. Fine mod-ulations are visible in the inverted experimental data;(see FIG. 3 and Sec. III for more details) however, it isnot clear whether these interferences trace back to theseCQSFA fine modulations or to the ATI rings. FilteringFIG. 3: The application of the time-filtering technique to both the experimental data and the CQSFA calculations.(a) The top half shows the raw experimental data after it has been inverted via polar onion-peeling. (b) The tophalf shows the CQSFA calculation with just unit-cell averaging applied (see Sec. IV). The bottom halves of bothpanels display the result of the time-filtering upon the top halves. The ATI rings and the fine modulations of theCQSFA have both been removed, without disrupting underlying structure.both the experiment and the CQSFA data removes thefine modulations and the ATI rings and thus allows foran unambiguous comparison of the two.In panels (c) and (d) we plot lineouts near the paral-lel axis (i.e. along the laser polarization direction) andalong the first spider leg, respectively, from both the fil-tered experimental and filtered CQSFA results in panel(b) (see dotted and dashed lines). In both panels broadmodulations along the axial and first spider leg lineoutare observed. Both the period of modulation as well asthe overall signal amplitude are in good agreement be-tween the experimental and theoretical results, except athigher momenta. Only the modulation depth is not sowell matched, which could be explained by incoherent ef-fects such as variation of the laser intensity over the focalvolume.To fully analyze these results we must understand somefurther details on the time-filtering and unit-cell averag-ing techniques as well as certain details about the exper-imental and theoretical methods.
III. EXPERIMENTAL METHODS ANDTIME-FILTERING
We employ common techniques for the strong-field ion-ization of argon atoms. Argon gas is pulsed through anEven-Lavie [37] valve before being strong-field ionized byan 800 nm, 40 fs, linearly polarized Ti:sapphire laser pulse with 200 TW/cm peak intensity. The intensitywas determined by fitting the signal drop-off predictedby the CQSFA along the axial lineout (FIG. 2 (c)) tothe experiment. Fits were performed at 25 TW/cm in-tervals. In this way we conclude that our intensity isdetermined to no better than about 7%. This value isconsistent with a calculation based on measured focalparameters for the setup.The photoelectrons are extracted in a velocity mapimaging (VMI) spectrometer [38], impact a micro-channel plate detector and phosphor screen, and arerecorded by a CCD camera. On-the-fly peak finding [39]is employed to increase the fidelity of the final spectrum.For the experimental results shown here, 63 billion elec-tron impacts are recorded.The laser pulse is linearly polarized in the detectorplane so the VMI records an axial and perpendicular pro-jection of the cylindrically symmetric vector momentumfor each electron. This may be inverted to generate the p (cid:107) − p ⊥ cross section of the ionized Newton sphere. Here, p (cid:107) refers to the momentum along the polarization axis ofthe laser, and p ⊥ to be the momentum perpendicular toboth the polarization axis and the spectrometer axis. Weemploy the polar onion-peeling algorithm [40] to invertour raw spectrum, see the top half of FIG. 3 (a).In photoelectron spectra generated through SFI, theATI rings tend to dominate and obscure other featurespresent in the spectra. This is especially problematicfor an analysis of holographic trajectory interferences inthe direct ionization regime below 2U p [41], where U p isthe ponderomotive energy of a free electron in the laserfield [42]. The ATI rings are a signature of multiple lasercycles, formed due to the interference of photoelectronpathways across these cycles. By removing these ATIrings from the spectrum, we can isolate the spectral fea-tures resulting from sub-cycle dynamics only.After inversion, we apply a time-filtering techniquethat effectively suppresses the contribution of inter-cycleinterferences, particularly ATI rings, to the experimentalPMD. The motivation and methodology for this tech-nique is outlined in significantly more detail in a previ-ous work [17]. In brief, the inversion process generatesa set of one-dimensional anisotropy parameters depen-dent on the radial momentum p r which contain the full3D information of the PMD [43]. These parameters canbe resampled to be functions of energy, which causes theATI rings to be periodic. The reciprocal space of energyis time, so we are able to perform a low-pass Fourier fil-ter on these resampled anisotropy parameters to suppressfeatures caused by interfering electron trajectories whichionize at least one field cycle apart from each other. Theresult is shown in the top half of FIG. 2 (b), where it isclear that ATI rings have been removed. A comparisonbetween the inverted experimental data and the time-filtered data is shown in panel (a) of FIG. 3.We also apply this filtering procedure to the results ofthe CQSFA calculations. We first generate the photo-electron angular distribution (PAD) Legendre decompo-sitions without onion-peeling to determine the anisotropyparameters for the CQSFA calculations. Then we can fil-ter the parameters using an identical filter to the one usedfor the experimental data to remove rapidly changing mo-mentum features. Throughout the paper, everywhere wecompare the CQSFA calculations directly to the experi-mental data, we filter them in this way. See FIG. 3 (b) fora comparison of the CQSFA calculations with and with-out the filtering. We note here that this paper serves asthe first application of the time-filtering technique as atool to make explicit measurements supporting quantumSFI theory. IV. UNIT-CELL AVERAGING
Here we discuss the key aspects of the CQSFA requiredto understand the unit-cell averaging methods employed.This method has been explored in detail in previous pub-lications [1, 9–12, 15, 16, 32, 33] (see Refs. [1, 10, 32] forkey details and a review), therefore, only a brief overviewrelated to the present work is provided.In the CQSFA to model the electron dynamics withina single-cycle unit cell we employ a monochromatic fieldgiven by the vector potential A ( t ) = 2 (cid:112) U p cos( ωt + φ ) , (1)where ω is the angular frequency of the laser and the electric field is given by E ( t ) = − ∂ A ( t ) /∂t . Note weemploy atomic units throughout unless otherwise stated.The variable φ is only important when the times are re-stricted to a single-cycle unit cell, where it controls the‘starting position’ of the laser field in the unit cell. Im-portantly, all the electron dynamics are contained withinthe action. This is achieved by applying Feynman pathintegral formalism [44] to the exact formalism of the tran-sition amplitude given in Ref. [41]. With the applicationof the saddle point approximation this leads to the fol-lowing expression for the ATI transition amplitude M ( p f ) ∝− i lim t →∞ (cid:88) s (cid:26) det (cid:20) ∂ p s ( t ) ∂ r s ( t s ) (cid:21)(cid:27) − / C ( t s ) e iS ( p s , r s ,t,t s )) (2)where C ( t s ) = (cid:115) πi∂ S ( p s , r s , t, t s ) /∂t s (cid:104) p + A ( t s ) | H I ( t s ) | Ψ (cid:105) , (3) s denotes the quantum orbits that solve the saddle pointequations (see Eq. (7) and 8), which are summed over.There are four distinct types of orbits in the CQSFA,which will be described in more detail in Sec. V. Thecombination of these orbits leads to the interference pat-terns observed in FIG. 2. The interaction Hamiltonian isgiven by ˆ H I ( t ) = − ˆ r · E ( t ). The action along each orbitreads S ( p , r , t, t (cid:48) ) = I p t (cid:48) − (cid:90) tt (cid:48) [ ˙ p ( τ ) · r ( τ ) + H ( r ( τ ) , p ( τ ) , τ )] dτ, (4)where I p is the ionization potential, the Hamiltonian H ( r ( τ ) , p ( τ ) , τ ) = 1 / p ( τ )+ A ( τ )) + V ( r ( τ )) and V ( r )is given by the effective potential for argon previouslyemployed in Refs. [15, 45]. An additional − π/ p ⊥ ( τ ), as detailedin Ref. [46], see Sec. VI for more details. The momentum p and coordinate r have been parameterized in terms ofthe time τ . In this monochromatic field approximation,the actions are periodic in the variable t (cid:48) . Thus, for anytime of ionization t (cid:48) = t s , there are additional solutions t (cid:48) = t s + nT , where T is the period of the laser field and n is any integer. Visualizations of the repeated trajectoriesare shown in FIG. 4. The periodic ionization times acrossmany cycles lead to the well-known ATI peaks/intercycleinterference [10, 47–49], which using this approach, is de-scribed by an analytic formula and can be completelyfactored out [10]. The inter-cycle interference is not ofinterest for photoelectron holography as it does not addany extra information on the target. In fact, the ATIring interference acts to obfuscate the holographic inter-ference, so in these results we restrict the CQSFA ion-ization times to a single-cycle unit cell. Restricting the orbit 1 orbit 2 orbit 3 orbit 4 ( a ) - - E ( t )( a . u . ) ϕ = ( b ) ϕ = π ( c ) ϕ = π ( d ) ϕ = π ( e ) - - p f || ( a . u . ) ( f ) ( g ) ( h )( i ) Re [ t ] ( cycles ) r ( a . u . ) ( j ) Re [ t ] ( cycles ) ( k ) Re [ t ] ( cycles ) ( l ) Re [ t ] ( cycles ) FIG. 4: The periodic unit cell is exemplified in three ways: 1) In the top row by the monochromatic electric fieldover three cycles for four starting positions [(a)–(d)], denoted by φ , with the time of ionization marked on the fieldfor each CQSFA orbit. 2) The middle row displays the time of ionization vs the parallel final momentum, at a fixedperpendicular momentum of p ⊥ = 0 .
13 a.u. , for all four orbits for the same four starting positions of the unit cell[(e)–(h)]. 3) The bottom row plots the distance from the parent ion over time for each CQSFA orbit for the samefour starting positions [(i)–(l)]. The trajectories all have the final momentum of p = ( − . , .
13) a.u. which is alsomarked by the horizontal line in the middle row. The solid trajectories indicate the ones that begin in/belong to thefirst unit cell, while the dashed trajectories belong to different unit cells. The unit cells are marked in all panels byvertical dashed lines. The line colors and markers correspond, in all panels, to the legend at the top.ionization times but not the final propagation time allowsphysical processes that would be present in a real laserpulse and require multiple cycles, such as recollisions [50],to be approximated by the monochromatic theory, whileremoving the inter-cycle effects. This approach is an ap-proximation to a real laser pulse, which neglects the laserenvelope effects, but still can give very good agreementwith experiment in the long-pulse case [15, 16]. Note thisis not the same as using a single-cycle top-hat laser pulse,which would introduce radical switch on/off effects in theelectron dynamics. A top hat pulse would also limit thepossible processes, e.g. no electrons ionized in the secondhalf cycle would return. Furthermore, it is not a realisticpulse to implement in the lab.The periodic nature of the monochromatic CQSFA canbe seen in FIG. 4, where in panels (a)–(d) the laser fieldis plotted for different starting positions φ and the re-sulting times of ionization are marked on the field foreach CQSFA orbit. The same ionization times are plot-ted directly below, panels (e)–(h), where the vertical axisdisplays the parallel final momentum to which each pointcorresponds. The perpendicular final momentum is fixedat p ⊥ = 0 .
13 a.u. The periodic nature is very clear over the 3 cycles plotted. As the ‘starting position’, φ , is in-creased the laser field and the times of ionization all shiftto the left. This leads to earlier times of ionization leav-ing the first unit cell (marked by vertical dashed lines),while other times of ionization from the second unit cellmove into the first. Thus, a different subset of trajecto-ries are selected. This is shown explicitly in FIG. 4 (i)–(l), where the distance of each trajectory from the parention is plotted over time. The trajectories that have theirstarting time (i.e. ionization time) in the first unit cellare denoted with solid lines. These clearly change as the φ increases and different trajectories have their startingpoint in the first unit cell.The variable φ has no bearing on the physics and dif-ferent values will lead to the same symmetric momen-tum distribution if the full (infinite) duration of themonochromatic field is considered. Each unit cell rep-resented in FIG. 4 contains the same information on theelectron dynamics regardless of the value of φ . However,if considering only a single-cycle unit cell, the differentordering of the orbits [see FIG. 8 (a)-(d)] and discon-tinuous cuts through the ionization times of the orbits inmomentum space [see FIG. 8 (e)-(h)] leads to asymmetryFIG. 5: The origin of the modulations on the spider legs. The parameters are the same as in FIG. 2. Panel (a) showsthe combined CQSFA calculation for three electron trajectories corresponding to orbits 1, 2, and 3 as presented inFIG. 1. The bottom half of panel (a) shows the effect of unit-cell averaging as discussed in the text. Panel (b)displays the CQSFA computations including only orbits 1 & 2 and 2 & 3, in the top and bottom half, respectively.and discontinuities (where the unit cell ‘cuts’ an orbit)in the final momentum distributions that change with φ .As previously stated we wish to focus only on a singleunit cell in order to examine the holographic sub-cycleeffects, while disposing of the non-holographic intercycleinterference. In the experiment we would like to simulate,the laser has a relatively long and gradually changing en-velope. Furthermore, the CEP will vary from pulse topulse, which will lead to different ordering of ionizationpathways just like when φ is varied in the CQSFA. In theexperiment we therefore expect that the measured pho-toelectron spectrum results from an incoherent averageover all the allowed ordering (in time) of the ionizationpathways. Thus, an incoherent average of the momentumdistribution with respect to φ in the CQSFA will combinethe trajectories in different orders, as will be the case inthe experiment, which will result in the removal of theasymmetries and discontinuities.In the appendix we describe in detail how this can beachieved via integration over φ . Here we present the unit-cell averaged probability P rob ( p ), in terms of a ‘correc-tion’ to P rob ( p ,
0) = | M ( p f ) | , the probability for φ = 0 P rob ( p f ) = P rob ( p f ,
0) + 2 ωπ sin (∆ S/ × (cid:88) i Now that we understand how the experiment and the-ory can be brought together to disentangle sub-cycle in-terference, we exploit the ability of the CQSFA to turnon/off interference pathways to demonstrate the originof the modulations on the spider-like interference pat-terns. To do this, we present some additional details ofthe CQSFA. Specifically, it is important to understandthe four CQSFA trajectories, examples of which are givenin FIG. 1. The equations of motion of the CQSFA tra-jectories are derived from the action via the applicationof the saddle point approximation, which leads to thesaddle point equations[ p ( t (cid:48) ) + A ( t (cid:48) )] / p = 0 , (7)˙ p ( τ ) = −∇ r V [ r ( τ )] and ˙ r ( τ ) = p ( τ ) + A ( τ ) . (8)The first of these, Eq. (7), provides the ionization times,while the pair of equations given by Eq. (8) describesthe propagation in the continuum. The result is the fourorbits shown in FIG. 1. These have been explained indetail in Refs. [1, 9–12, 15, 16, 32, 33] (see Ref. [12] forthe first implementation of all four orbits and Ref. [1] fora review) but a brief description follows.The four orbits were originally classified in Ref. [51],and they are shown in real space in FIG. 1 for a specificfinal momentum. Orbit 1 (direct): the electron tunnelstowards the detector and reaches it directly. Orbit 2(forward deflected) and orbit 3 (forward scattered): theelectron tunnels away from the detector and then thelaser drives them back towards the detector. For orbit3 the electron’s transverse momentum changes sign, fororbit 2 it does not. Orbit 4 (backscattered): the electronis freed towards the detector, but backscatters off thecore.Using combinations of these CQSFA orbits we can fur-ther investigate the interferences presented in FIG. 2.The modulations on the spider legs (interference betweenorbits 2 and 3) can be traced to the fan-like interferencepattern (interference between orbits 1 and 2). In FIG. 5(a) we plot all three of these orbits (1, 2, and 3); in thebottom half of the panel we have applied unit-cell averag-ing and the modulations are clearly reproduced withoutrequiring the inclusion of orbit 4. Unit-cell averaging hasnot been applied in the top half of the panel. We findthis separates the modulations into fine modulations onthe left and broad modulations on the right. On the leftand right sides of the top half of FIG. 5 (b) we investigatethe different modulations by plotting the fan, which is theinterference between the two direct-like CQSFA orbits 1and 2. The figure is asymmetric as no unit-cell averaginghas been used ( φ = 0). The two different interferencetypes, seen in panel (a), are present on each side. In aprevious publication [10] we have referred to this as typeA and B interference. Type A [B], relating to the broad[fine] fringes on the right [left], occurs when there is less[more] than half a cycle difference between the times ofionization of the two interfering electron pathways.We have fixed the laser field ‘starting position’ φ = 0such that, in FIG. 5 (b), both type A and B occur on theright and left of the fan, respectively. Unit-cell averagingwill incoherently mix both interference types, howevertype A will tend to dominate. If both interference typeson the left and right of the top half FIG. 5 panel (b)are added onto the spider legs in the bottom half of thepanel then we get the results shown in FIG. 2 and FIG. 5(a). This shows clearly that the modulation effect is dueto the interference of three electron trajectories (CQSFAorbits 1, 2, and 3) as well as an incoherent mix of dif-ferent interference types A and B. Thus, in theory, weare able to see both sides of the fan imprinted in the spi-der. This has interesting consequences for photoelectronholography. For the spider it is known that the two in-terfering electron trajectories leave from the same side ofthe target but take different routes to the detector (with FIG. 6: PMD computed using the CQSFA examiningthe effect of the bound state. The same parameters areused as in FIG. 2. Panel (a) displays the CQSFA withand without the effect of the bound state in alternatingquadrants to enable the phase shift to be identifiedalong both axes. The CQSFA PMDs have been filteredto remove high frequency structures. Panels (b) and (c)compare the lineouts along close to the parallel axis andalong the first spider leg respectively as in FIG. 2. Thesame filtered experiment lineouts from FIG. 2 arereproduced. The goodness of fit metric R comparingthe experiment lineouts to each of the other two in eachplot is displayed. Note the bound state is ‘switched off’by setting the matrix element in Eq. (3) to 1.opposite transverse momentum components), while forthe fan the two interfering trajectories leave from oppo-site sides and have opposite longitudinal momenta. Thus,the fan and the spider probe in opposite directions, so thethree-trajectory combination has the capacity to probein both directions simultaneously. This could allow forholographic imaging of the bound state in both these di-rections. VI. REVEALING GOUY AND PARITY PHASES Previously, holographic interference has been used toprobe parity in the bound state [15]. This is possible asphotoelectron trajectories that leave the ion from oppo-site sides will acquire an additional π phase difference ifthe bound state orbital has odd parity, while there willbe no additional phase difference for even parity. So forthese trajectories the interference fringes will shift outof phase between odd and even parity. In Ref. [15] onlytwo trajectories were considered to extract the parity,primarily from the spiral-like structure, orbits 3 and 4.Furthermore a reference ‘atom’ was required to use dif-ferential holographic measurements to extract the parity.The imprint of the fan in the spider-like structure allowsus to see phase shifts between three trajectories. Herethere will be a π phase difference picked up between bothorbits 1 and 2 as well as orbits 1 and 3.We demonstrate the ability of probing the parity ofbound state in FIG. 6 by adding and removing the effectof the odd parity p -state of argon. In panel (a) we plotwith and without the effect of the bound state in alter-nating quadrants. Along the parallel axis it is clear thatthe fan-like modulations along the spider legs undergoa π phase shift. This is due to the odd parity of the p -state of argon, so that trajectories leaving in oppositelongitudinal directions pick up a π phase difference. Thesame π phase shift is also visible near the transverse axisat higher momenta via the spiral-like interference pat-tern, which occurs between forward- and back-scatteredtrajectories [16].Lineouts traced along the parallel axis and the firstspider leg are plotted in FIG. 6 (b) and (c), along withthe same experimental lineouts from FIG. 2. This com-parison with experiment enables a direct corroborationof the π phase difference due to the bound state. The R ‘goodness of fit’ is calculated in both cases of theCQSFA (with and without the bound state phases) vsthe experiment. Along both the axial and first spider leglineout including the bound state phases gives higher R value. It is also evident that the peaks shift out of phasewhen the bound state phases are not included. Thus, wehave demonstrated that this methodology can be used todetermine phase inherent in the target.In recent work [46] it was demonstrated that additionalMaslov phases (the semi-classical equivalent of Gouyphases) must be included to employ a 2-dimensionalmodel for a 3-dimensional system, the additional phasefor each trajectory is dependent on the number of signchanges of the perpendicular momentum p ⊥ ( t ). In thiscase of the CQSFA these phases can be included by shift-ing the phase of orbits 3 and 4 by − π/ isagain shown, comparing the experiment to the CQSFAwith and without the phase correction. θ is measuredequivalently for each lineout in panel (a) with the 0 atthe axis and advancing along the lineout. Panel (c)displays the normalized residual of the CQSFAcomputation without the Gouy phase correctionsubtracted from the computation with the correction.This residual highlights the modification in the pitch ofthe spider-leg structure.0It is particularly noticeable that the spider legs and axialfringes shift towards higher p ⊥ momentum. This leads tothicker fringes along the polarization axis and a steepergradient along the spider legs, better matching experi-ment. The overall shift of the spider legs is exemplifiedin FIG. 7 (c), in which the normalized residual differ-ence plot between the CQSFA with and without the ad-ditional Gouy-related phases is shown. In FIG. 7 (b)lineouts are shown for the CQSFA with and without theGouy-related phases as well as the experiment. A muchbetter match can be observed for the CQSFA with theGouy phases, where the peaks almost line up with theexperiment. In the case of the CQSFA without the Gouyphases, there is a constant phase shift away from the ex-periment. This provides further experimental verificationof the additional phases predicted by Ref. [46]. VII. CONCLUSIONS We present two new methods for bringing together ex-perimental and theoretical results enabling an unprece-dented quantitative match between the two. We haveovercome two major obstacles to the interpretation ofholographic strong-field ionization data: artificial de-fects present in theoretical models with restricted ion-ization times and strong inter-cycle ATI interference inexperimental data. This enables the identification ofthe first three-trajectory interference pattern in photo-electron holography, which has the capacity to stronglyenhance current protocols. Such strong agreement alsoenables experimentally driven determination of intricatephases inherent within the system. Using a goodness-of-fit to the experiment we confirm that the bound stateimparts a phase shift of π on the CQSFA orbits 2 and3, while the recently investigated Gouy phases [46] (pre-viously missing from the CQSFA computation) imparta phase shift of − π/ analyt-ically produces PMDs without ATI rings and takes intoaccount all combinations of ionization pathways that oc-cur in experiment. An alternative approach would mostlylikely require two steps: firstly to model a host of laserpulses with different carrier envelope phases and then,secondly, to remove the ATI rings via the time-filteringtechnique in post processing. Not only would this takesignificantly more time to compute but it would be muchharder to trace the origin of the final mixture of inter-ference patterns. Our method is, of course, an approxi-mation, which neglects the idea of a laser envelope; how-ever, it yields precise agreement with experiment. Asargued in the introduction, for the long pulses employedin this work, this will be a very good approximation tothe electron dynamics. This argument of long pulses hasbeen made before (e.g. Refs. [16, 17, 57, 58]) but in thiswork we significantly improve on this idea. Finally, themethods presented in this work are applicable to low andintermediate photoelectron energies, in which there is anintricate interplay of the binding potential, the externalfield, and the core dynamics. This, together with the highsensitivity of the methods, opens a wide range of possi-bilities for dynamical imaging of correlated multielectronsystems in the attosecond regime. ACKNOWLEDGMENTS NW, RF, and PHB thank James P. Cryan for usefuland fruitful discussions. NW, RF, and PHB are sup-ported by the U.S. Department of Energy, Office of Sci-ence, Basic Energy Sciences (BES), Chemical Sciences,Geosciences, and Biosciences Division, AMOS Program.ASM and CFMF are supported by funding from the1UK Engineering and Physical Sciences Research Coun-cil (EPSRC). ASM acknowledges grant EP/P510270/1,which is within the remit of the InQuBATE Skills Hubfor Quantum Systems Engineering. CFMF would like toacknowledge EPSRC grant EP/T019530/1.ASM also acknowledges support from ERC AdG NO-QIA, Spanish Ministry of Economy and Competitive-ness (“Severo Ochoa” program for Centres of Excellencein R&D (CEX2019-000910-S), Plan National FIDEUAPID2019-106901GB-I00/10.13039/501100011033, FPI),Fundaci´o Privada Cellex, Fundaci´o Mir-Puig, and fromGeneralitat de Catalunya (AGAUR Grant No. 2017 SGR1341, CERCA program, QuantumCAT U16-011424, co-funded by the ERDF Operational Program of Cat-alonia 2014-2020), MINECO-EU QUANTERA MAQS(funded by State Research Agency (AEI) PCI2019-111828-2/10.13039/501100011033), EU Horizon 2020FET-OPEN OPTOLogic (Grant No 899794), and theNational Science Centre, Poland-Symfonia Grant No.2016/20/W/ST4/00314. Appendix A: Unit-Cell Averaging in the CQSFA In Sec. IV we outlined the ideas and main equationsbehind the new unit-cell averaging. In this section ofthe appendix we will fully derive the equation. We startby considering what happen when the starting phase φ ,introduced in Sec. IV, is increased from 0. As previ-ously demonstrated some trajectories will move outsidethe unit cell. The time of ionization for an arbitrary φ can be written as t (cid:48) + φ , where t (cid:48) is the time of ionizationfor φ = 0. Thus, a trajectory will move out of the unitcell if the real part of the time is less than zero, whichleads to the condition ω Re[ t (cid:48) ] < φ . When this conditionis satisfied the trajectory must be delayed by a field cyclein order for it to occur in the first unit cell. The delayamounts to including an additional phase ∆ S given by∆ S = 2 πω (cid:18) I p + U p + 12 p f (cid:19) . (A1)With this in mind, we can now write an expression forthe transition amplitude that is valid for any φM i ( p f , φ ) = M i ( p f ) exp (cid:2) iH ( φ − ωt Rei )∆ S (cid:3) , (A2)where M i ( p f ) is the transition amplitude, i ∈ [1 , 4] de-notes the CQSFA orbit, t Rei is the real part of the timeof ionization for the CQSFA orbit at φ = 0 and H is theHeaviside step function. The φ dependent probabilitydistribution can be computed via P rob ( p f , φ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 M i ( p f , φ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A3)With the definition given by Eq. (A3) we can plot thePMDs for the CQSFA at different values of φ , this hasbeen done in the top row of FIG. 8. The values φ = 0 and π do not exhibit discontinuities in the PMDs. All valuesof φ in between these values will have a discontinuity,which will occur when a trajectory moves outside of theunit cell. The values φ = 0 and π result in asymmetricmomentum distributions, which contradict the symme-try of the experiment, and are related by flipping the p || axis. They exhibit two types of broad and fine interfer-ence, previously dubbed type A and B, respectively [10].The value φ = 0 . π is nearly symmetric, with a curveddiscontinuity near p || = 0, but it contains almost exclu-sively type A interference. For the case of φ = 0 . π a diagonal discontinuity can be seen on the left of thepanel.In order to perform unit-cell averaging the probabilitydistribution given by Eq. (A3) must be integrated overall possible values of φP rob ( p f ) := 12 π (cid:90) π dφP rob ( p f , φ )= 12 π (cid:90) π dφ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 M i ( p f ) exp (cid:2) iH ( φ − ωt Rei )∆ S (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A4)which may be written as P rob ( p f ) = 12 π (cid:88) i,j =1 M i ( p f ) M j ( p f ) I φ . (A5)Here I φ is given by I φ = (cid:90) π dφ exp (cid:2) i ( H ( φ − ωt Rei ) − H ( φ − ωt Rej ))∆ S (cid:3) = 2 π + ω | ∆ t ij | (cid:0) e − is ij ∆ S − (cid:1) , (A6)where ∆ t ij = t Rei − t Rej and s ij = sign(∆ t ij ). Insertingthis into the probability distribution yields P rob ( p f ) = P rob ( p f , ω π (cid:88) i,j =1 M i ( p f ) M j ( p f ) | ∆ t ij | (cid:0) e − is ij ∆ S − (cid:1) . (A7)With some algebra this becomes P rob ( p f ) = P rob ( p f , 0) + 2 ωπ sin (∆ S/ × (cid:88) i 0. We showthe spiral-like structure in FIG. 8 (g); unit-cell averagingleads to the carpet-like structure [16, 59, 60] without re-quiring the addition of ATI rings. Finally, in FIG. 8 (h)all orbits with unit-cell averaging are shown as in FIG. 2. [1] C. F. d. M. Faria and A. S. Maxwell, Reports on Progressin Physics , 034401 (2020), publisher: IOP Publishing.[2] Y. Huismans, A. Rouz´ee, A. Gijsbertsen, J. H. Jung-mann, A. S. Smolkowska, P. S. W. M. Logman, F. L´epine,C. Cauchy, S. Zamith, T. Marchenko, J. M. Bakker,G. Berden, B. Redlich, A. F. G. v. d. Meer, H. G. Muller,W. Vermin, K. J. Schafer, M. Spanner, M. Y. Ivanov,O. Smirnova, D. Bauer, S. V. Popruzhenko, and M. J. J.Vrakking, Science , 61 (2011), publisher: AmericanAssociation for the Advancement of Science Section: Re-port.[3] T. Zuo, A. Bandrauk, and P. Corkum, Chemical PhysicsLetters , 313 (1996).[4] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. P´epin,J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, Nature , 867 (2004).[5] X. B. Bian, Y. Huismans, O. Smirnova, K. J. Yuan,M. J. J. Vrakking, and A. D. Bandrauk, Phys. Rev. A , 043420 (2011).[6] X.-B. Bian and A. D. Bandrauk, Phys. Rev. Lett. ,263003 (2012).[7] D. D. Hickstein, P. Ranitovic, S. Witte, X.-M. Tong,Y. Huismans, P. Arpin, X. Zhou, K. E. Keister, C. W.Hogle, B. Zhang, C. Ding, P. Johnsson, N. Toshima,M. J. J. Vrakking, M. M. Murnane, and H. C. Kapteyn,Physical Review Letters , 073004 (2012), publisher:American Physical Society.[8] M. Haertelt, X.-B. Bian, M. Spanner, A. Staudte, andP. B. Corkum, Physical Review Letters , 133001(2016).[9] X. Y. Lai, S. G. Yu, Y. Y. Huang, L. Q. Hua, C. Gong,W. Quan, C. F. d. M. Faria, and X. J. Liu, Phys. Rev. A , 013414 (2017).[10] A. S. Maxwell, A. Al-Jawahiry, T. Das, and C. Figueirade Morisson Faria, Phys. Rev. A , 023420 (2017),arXiv:1705.01518.[11] A. S. Maxwell, A. Al-Jawahiry, X. Y. Lai, and C. Figueirade Morisson Faria, J. Phys. B: At. Mol. Opt. Phys. ,044004 (2018).[12] A. S. Maxwell and C. Figueira de Morisson Faria, J. Phys.B: At. Mol. Phys. , 124001 (2018), arXiv:1802.00789.[13] M. He, Y. Li, Y. Zhou, M. Li, W. Cao, and P. Lu, Physi-cal Review Letters , 133204 (2018), publisher: Amer-ican Physical Society.[14] M. K¨ubel, Z. Dube, A. Y. Naumov, D. M. Villeneuve,P. B. Corkum, and A. Staudte, Nature Communications , 1042 (2019).[15] H. Kang, A. S. Maxwell, D. Trabert, X. Lai, S. Eckart,M. Kunitski, M. Sch¨offler, T. Jahnke, X. Bian, R. D¨orner,and C. F. d. M. Faria, Phys. Rev. A , 013109 (2020).[16] A. S. Maxwell, C. F. d. M. Faria, X. Lai, R. Sun, andX. Liu, Phys. Rev. A , 033111 (2020).[17] N. Werby, A. Natan, R. Forbes, and P. Bucksbaum,arXiv:2008.09712 [physics] (2021), arXiv: 2008.09712.[18] A. von Veltheim, B. Manschwetus, W. Quan,B. Borchers, G. Steinmeyer, H. Rottke, and W. Sandner,Phys. Rev. Lett. , 023001 (2013).[19] Y. Mi, N. Camus, L. Fechner, M. Laux, R. Moshammer,and T. Pfeifer, Phys. Rev. Lett. , 183201 (2017).[20] S. G. Walt, N. Bhargava Ram, M. Atala, N. I. Shvetsov-Shilovski, A. von Conta, D. Baykusheva, M. Lein, andH. J. W¨orner, Nature Communications , 15651 (2017),number: 1 Publisher: Nature Publishing Group.[21] S. Haessler, J. Caillat, W. Boutu, C. Giovanetti-Teixeira, T. Ruchon, T. Auguste, Z. Diveki, P. Breger, A. Maquet,B. Carr´e, R. Ta¨ıeb, and P. Sali`eres, Nature Physics ,200–206 (2010).[22] E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wy-att, L. Corner, and L. F. DiMauro, Phys. Rev. Lett. ,033905 (2005).[23] J. Gagnon, E. Goulielmakis, and V. S. Yakovlev, AppliedPhysics B , 25 (2008).[24] I. Orfanos, I. Makos, I. Liontos, E. Skantzakis, B. F¨org,D. Charalambidis, and P. Tzallas, APL Photonics ,080901 (2019), https://doi.org/10.1063/1.5086773.[25] A. Rudenko, K. Zrost, C. D. Schr¨oter, V. L. B. de Jesus,B. Feuerstein, R. Moshammer, and J. Ullrich, J. Phys.B: At. Mol. Opt. Phys. , L407 (2004), arXiv:0408064[physics].[26] C. M. Maharjan, A. S. Alnaser, I. Litvinyuk, P. Rani-tovic, and C. L. Cocke, J. Phys. B: At. Mol. Opt. Phys. , 1955 (2006).[27] R. Gopal, K. Simeonidis, R. Moshammer, T. Ergler,M. D¨urr, M. Kurka, K.-U. K¨uhnel, S. Tschuch, C.-D.Schr¨oter, D. Bauer, J. Ullrich, A. Rudenko, O. Herrw-erth, T. Uphues, M. Schultze, E. Goulielmakis, M. Uib-eracker, M. Lezius, and M. F. Kling, Phys. Rev. Lett. , 053001 (2009).[28] D. G. Arb´o, E. Persson, and J. Burgd¨orfer, Phys. Rev. A , 063407 (2006).[29] W. Quan, Z. Lin, M. Wu, H. Kang, H. Liu, X. Liu,J. Chen, J. Liu, X. T. He, S. G. Chen, H. Xiong, L. Guo,H. Xu, Y. Fu, Y. Cheng, and Z. Z. Xu, Physical ReviewLetters , 093001 (2009), publisher: American Physi-cal Society.[30] C. I. Blaga, F. Catoire, P. Colosimo, G. G. Paulus, H. G.Muller, P. Agostini, and L. F. DiMauro, Nature Physics , 335 (2009), number: 5 Publisher: Nature PublishingGroup.[31] M. Spanner, O. Smirnova, and P. B. Corkum, J. Phys.B: At. Mol. Opt. Phys. , L243 (2004).[32] X.-Y. Lai, C. Poli, H. Schomerus, and C. F. d. M. Faria,Phys. Rev. A , 043407 (2015).[33] A. S. Maxwell, S. V. Popruzhenko, and C. Figueirade Morisson Faria, Phys. Rev. A , 063423 (2018),arXiv:1808.00817.[34] M. F. Kling, J. Rauschenberger, A. J. Verhoef,E. Hasovi´c, T. Uphues, D. B. Miloˇsevi´c, H. G. Muller,and M. J. J. Vrakking, New Journal of Physics , 025024(2008), publisher: IOP Publishing.[35] B. Bergues, M. K¨ubel, N. G. Johnson, B. Fischer, N. Ca-mus, K. J. Betsch, O. Herrwerth, A. Senftleben, A. M.Sayler, T. Rathje, T. Pfeifer, I. Ben-Itzhak, R. R. Jones,G. G. Paulus, F. Krausz, R. Moshammer, J. Ullrich, andM. F. Kling, Nature Communications , 813 (2012).[36] S. Yu, X. Lai, Y. Wang, S. Xu, L. Hua, W. Quan, andX. Liu, Phys. Rev. A , 023414 (2020).[37] U. Even, EPJ Techniques and Instrumentation , 17(2015).[38] A. T. J. B. Eppink and D. H. Parker, Review of Scien-tific Instruments , 3477 (1997), publisher: AmericanInstitute of Physics.[39] B.-Y. Chang, R. C. Hoetzlein, J. A. Mueller, J. D. Geiser,and P. L. Houston, Review of Scientific Instruments ,1665 (1998).[40] G. M. Roberts, J. L. Nixon, J. Lecointre, E. Wrede, and J. R. R. Verlet, Review of Scientific Instruments ,053104 (2009).[41] W. Becker, F. Grasbon, R. Kopold, D. Miloˇsevi´c,G. Paulus, and H. Walther, Above-Threshold Ionization:From Classical Features to Quantum Effects , edited byB. Bederson and H. Walther, Adv. At. Mol. Opt. Phys.,Vol. 48 (Academic Press, 2002) pp. 35 – 98.[42] P. H. Bucksbaum, R. R. Freeman, M. Bashkansky, andT. J. McIlrath, Journal of the Optical Society of AmericaB , 760 (1987).[43] K. L. Reid, Annual Review of Physical Chemistry ,397 (2003).[44] H. Kleinert, Path Integrals in Quantum Mechanics,Statistics, Polymer Physics, and Financial Markets (World Scientific, 2009).[45] X. M. Tong and C. D. Lin, J. Phys. B: At. Mol. Opt.Phys. , 2593 (2005), arXiv:1205.0519.[46] S. Brennecke, N. Eicke, and M. Lein, Phys. Rev. Lett. , 153202 (2020).[47] R. R. Freeman, P. H. Bucksbaum, H. Milchberg,S. Darack, D. Schumacher, and M. E. Geusic, Phys. Rev.Lett. , 1092 (1987).[48] D. G. Arb´o, K. L. Ishikawa, K. Schiessl, E. Persson, andJ. Burgd¨orfer, Phys. Rev. A , 021403 (2010).[49] D. G. Arb´o, K. L. Ishikawa, E. Persson, andJ. Burgd¨orfer, Nuclear Instruments and Methods inPhysics Research Section B: Beam Interactions with Ma-terials and Atoms , 24 (2012), proceedings of theFifth International Conference on Elementary Processesin Atomic Systems Belgrade, Serbia, 21-25 June 2011.[50] W. Becker, S. P. Goreslavski, D. B. Miloˇsevi´c, and G. G.Paulus, Journal of Physics B: Atomic, Molecular and Op-tical Physics , 162002 (2018).[51] T.-M. Yan, S. V. Popruzhenko, M. J. J. Vrakking, andD. Bauer, Phys. Rev. Lett. , 253002 (2010).[52] M. Meckel, A. Staudte, S. Patchkovskii, D. M. Vil-leneuve, P. B. Corkum, R. D¨orner, and M. Spanner, Na-ture Physics , 594 (2014), number: 8 Publisher: Na-ture Publishing Group.[53] G. Porat, G. Alon, S. Rozen, O. Pedatzur, M. Kr¨uger,D. Azoury, A. Natan, G. Orenstein, B. D. Bruner,M. J. J. Vrakking, and N. Dudovich, Nature Commu-nications , 2805 (2018).[54] D. V. Else, B. Bauer, and C. Nayak, Phys. Rev. Lett. , 090402 (2016).[55] U. D. Giovannini and H. H¨ubener, Journal of Physics:Materials , 012001 (2019).[56] V. Roudnev and B. D. Esry, Phys. Rev. Lett. , 220406(2007).[57] X. Liu and C. Figueira de Morisson Faria, Phys. Rev.Lett. , 133006 (2004).[58] X. Xie, T. Wang, S. Yu, X. Lai, S. Roither, D. Kartashov,A. Baltuˇska, X. Liu, A. Staudte, and M. Kitzler, Phys.Rev. Lett. , 243201 (2017).[59] P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski,W. Becker, G. G. Paulus, B. Feti´c, and D. B. Miloˇsevi´c,New Journal of Physics , 055019 (2012).[60] P. A. Korneev, S. V. Popruzhenko, S. P. Goreslavski,T. M. Yan, D. Bauer, W. Becker, M. K¨ubel, M. F. Kling,C. R¨odel, M. W¨unsche, and G. G. Paulus, Phys. Rev.Lett.108