The Sub-Cycle Electron Momentum Spectrum in Strong-Field IonizationNicholas Werby,1, 2 Adi Natan,1, 2 Ruaridh Forbes,1, 2 and Philip H. Bucksbaum1, 2, 31)Stanford PULSE Institute, SLAC National Accelerator Laboratory2575 Sand Hill Road, Menlo Park, CA 940252)Department of Physics, Stanford University, Stanford, CA 943053)Department of Applied Physics, Stanford University, Stanford, CA 94305

(Dated: 25 August 2020)

We present a simple and effective technique to recover sub-cycle features in the momentum distributions for atoms andsmall molecules subject to strong-field ionization. Quantum calculations predict complex sub-cycle interferences thathave attracted wide interest. Experimental searches for these features must overcome the significant challenge that ahighly controlled 1- to 2-cycle pulse is difficult to realize in the laboratory, whereas sub-cycle features produced bylonger pulses are dominated by above-threshold ionization (ATI), a signature of multi-cycle interference. We overcomethese difficulties by first decomposing the momentum distributions in a Legendre basis, and then directly filtering theone-dimensional radial coefficients. This method reveals interference structures in unprecedented detail.

When an atom or molecule is photoionized by a stronglaser field, the photoelectron undergoes a series of com-plex field-driven dynamics1–10. Electron vector momen-tum distributions obtained using angle-resolving techniquessuch as velocity map imaging (VMI) contain detailed in-formation on rapidly evolving molecular geometries of theparent11–13, holographic structures from the interference ofelectron trajectories4–7,10,14–17, and nonlinear electron inter-actions with the strong field18. Each of these processes canproduce patterns in the spectrum with characteristic features;however, patterns from different processes usually overlap onthe electron detector, which impedes straightforward analy-sis. Indeed, experiments in the strong-field ionization (SFI)regime where the ionization during each field cycle is signifi-cant must either carefully choose running parameters such asellipticity or field shape to emphasize specific dynamics orperform significant analysis on the observed spectrum to dis-entangle these patterns to make the desired measurement.

A primary way to disentangle these processes is to comparethe measured spectrum to quantum SFI calculations. In thestrong-field regime, electron spectra are largely determined bythe phase of the laser field at the moment of photoionization,and the subsequent evolution in the field. This is known asthe strong-field approximation (SFA)19,20. Calculations oftenassume strong-field conditions that are practically unattain-able, such as a few uniform strong-field cycles that turn onand off instantly15,16. Experiments cannot mimic this. Ultra-short laser pulses have a time-varying field envelope, whichgives rise to a carrier envelope phase (CEP) parameter thatgoverns the electron dynamics but is absent from calculationsthat assume uniform cycles21,22. Thus to complement this ex-perimental approach and more closely produce experimentalspectra comparable to these few-cycle calculations, we requirea different approach.

We are interested in extracting sub-cycle information fromspectra generated with a standard 30-50 fs multi-cycle laserpulse from a commercial Ti:Sapphire laser system. This re-quires identifying the key features that differentiate multi-cycle spectra from single-cycle spectra. Operating in the typ-ical SFI regime, the primary multi-cycle dynamics appear asa comb of above threshold ionization (ATI) peaks, which areequally spaced by the energy of the driving laser photons23,24.

Under ordinary conditions, this comb dominates the momen-tum spectrum, and frequently obscures patterns generated bysub-cycle dynamics. This is most noticeable in the direct ion-ization regime of the spectrum below 2Up, where Up is theponderomotive energy of a free electron in the laser field25.Here, holographic features resulting from the interferences ofelectron trajectories after ionization are prevalent15. Observ-ing these intra-cycle dynamics would be greatly improved ifthe ATI contribution to the spectrum could be mitigated.

In this letter, we present a filtering technique that effectivelyeliminates the energy-periodic background ATI comb in pho-toelectron momentum spectra. This removes the multi-cyclecontributions to the spectrum, thus leaving only the single-and sub-cycle dynamics. Importantly, since we do not requirea short laser pulse, the resultant spectrum closely resemblesthat due to a single cycle of a steady-state laser field, whichmore closely models common SFI calculations15,16. If thismethod is performed on a sub-10 fs optical pulse, the sub-cycle CEP-dependent effects will also be visible26. Further-more, the filtering technique is broadly applicable to angle-resolved SFI electron spectra and is independent of experi-mental parameters, making it a simple tool for extracting sub-cycle structures from multi-cycle data. To illustrate the fil-tering, we demonstrate it on a high fidelity VMI spectrum ofargon gas photoionized with an approximately 50 fs, 800 nmcommercial Ti:sapphire laser at an intensity of 200 TW/cm2.

The raw VMI detector image requires several layers ofprocessing to reach the effective single-cycle spectrum. Thesteps are outlined below and in Fig. 1. The two-dimensionalVMI raw detector image is first mapped and calibrated tomeasure the transverse momentum of the electrons strikingit28. The resulting image is a projection on (Pρ ,Pz) of athree-dimensional Newton sphere of electron momenta (Pρ ,Pz, φ)29. Here Pρ is the momentum magnitude transverse topolarization axis of the ionizing laser field, Pz is the momen-tum magnitude parallel to the polarization axis, and φ is theazimuthal angle about the polarization axis. Since photoion-ization with linearly polarized light is cylindrically symmetricabout the polarization (z) axis, reconstructing just the Pz−Pρ

slice of the Newton Sphere from its full projection is sufficientto display the full three-dimensional information.

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FIG. 1. The application of the novel filtering technique to our argon spectrum. Here, we’ve zoomed into the spectrum to emphasize the regionwithin 2Up where the filtering has the greatest effect. Following the red arrows outlines the order in which the data processing was performed.(a) The top half of the image shows the reconstructed Pz−Pρ momentum slice from polar onion-peeling.27 (b) Transforming the onion-peeledspectrum to an energy-spaced radial grid yields the top half of the image. In this representation the ATI rings are equally spaced, and the radialaxis counts the ATI order. (c) The bottom half shows the filtered energy-spaced spectrum as detailed in the text on the same scale. (d) Thebottom half shows the filtered momentum slice on the same scale as the original spectrum. Importantly for (c,d) we note the filtering has onlyremoved the ATI bands, and all other structures remain unaffected.

Reconstructing this momentum slice begins by symmetriz-ing the VMI data. Each raw spectrum is centered and rotated.We impose four-quadrant symmetry on the raw data by aver-aging each pixel in the raw image with its counterparts in theother three quadrants.

We then invert the projection through the procedure of polaronion-peeling27. Beginning from the outermost radius of theraw data, we construct the Legendre decomposition fit to thephotoelectron angular distribution (PAD) at that radius usingLegendre polynomials of even orders Pn(cos(θ)).

I(θ ,Pr(R)) = ∑neven

βn(Pr(R))Pn(cos(θ)) (1)

Here, θ is the angle around the center of our 2-dimensionalslice and I(θ ,Pr(R)) is the PAD fit at the detector radius Rwith corresponding momentum magnitude Pr(R). In our de-composition, we set the maximum Legendre order to 42, wellbeyond the required angular resolution for holographic predic-tions in SFI15. We then convolve this fit about the z-axis andproject the resulting spherical shell back to the image plane.This generates the contribution from that spherical shell of theNewton sphere. Subtracting this contribution from the spec-trum "peels" this shell of the Newton sphere off the spectrum.Repeating this procedure for successively smaller radii yieldsthe fully inverted spectrum.

This process generates the cylindrical momentum slicePz−Pρ as a superposition of Legendre polynomials whose co-efficients, βn(Pr), are one-dimensional functions of Pr. All ofthe information in the two-dimensional momentum distribu-tion is now encoded in a one-dimensional βn(Pr) coefficientfor each included order n30.

The actual method of inversion to generate the Pz−Pρ mo-mentum slice is not critical to the filtering method, althoughthere are specific advantages to any polar inversion technique.The one-dimensional βn(Pr) are simpler to analyze than a fulltwo-dimensional momentum distribution, which improves thefidelity and lowers the computational burden. Additionally,a polar basis is natural for ATI rings, which are concentricwith the momentum origin. Direct two-dimensional numeri-cal integration techniques31 are just as effective as an inver-sion technique, but manipulating and filtering the resultanttwo-dimensional slice directly can prove computationally in-tensive and less accurate. Another nearly equivalent alterna-tive is pBaseX32, which also generates one-dimensional polardata sets.

At this point in the analysis the sub-cycle features are ob-scured by highly prominent ring structures of ATI23. The ATIappears as a prominent comb of peaks in each order of βn(Pr)spaced by the photon energy of the ionizing laser, so we trans-form the radial grid from momentum to energy by direct re-sampling βn(Pr)→ βn(Er). The result is shown in panel (b)of Fig. 1. The Fourier transforms of each βn(Er) contains theATI ring frequency (Fig. 2). We apply a finite impulse re-sponse low-pass filter to block this entirely, without affectingany sub-cycle structures in the spectrum. The inverse trans-form returns the original momentum spacing and restores theimage according to Eq. 1 without the ATI features, and pre-serves the scattering features that are caused by sub-cycle in-terferences in field ionization and rescattering, as shown inFig. 3.

In conclusion, we have presented a procedure to filter outthe ATI comb in a strong-field ionization spectrum, to more

3

FIG. 2. (a) This plot shows the β6(Er) coefficient of our samplespectrum. The energy axis Er is presented in units of the energy of asingle photon at the central laser frequency, hω . Shown are the samecoefficient before (blue) and after (orange) filtering. Prior to filter-ing we see the evenly spaced ATI comb quite prominently. After weapply the low-pass filter we see that the filtering only removes theATI comb, and does not disrupt the underlying shape of the coeffi-cient. (b) This shows the Fourier transform of the coefficient. Wesee a large spike at a time of 1 inverse-cycle frequency exactly as weexpect for an ATI comb. The shape of this peak contains sidebandsthat reflect the slow modulations of the ATI amplitude. Applying alow-pass filter with a cut-off frequency just below this peak success-fully removes only the ATI comb. The sixth order coefficient waschosen arbitrarily to highlight the form of the filtering technique foran arbitrary coefficient, and the other orders can be analyzed in anidentical way.

clearly reveal the underlying features of the strong-field ion-ization process. This filtering is not simply an aestheticchange. The filtered momentum specrum shown in Fig. 3contains holographic features from sub-cycle trajectory inter-ferences that are normally obscured by the ATI rings15. Thefiltered spectrum displays subtle wiggling modulations in thespider-leg structures that have not been seen previously14,33,as well as periodic minima along them. Clear observations ofthese holographic features can help validate strong-field the-ories and lead to a clearer understanding of the electron dy-namics that generate them.

ACKNOWLEDGMENTS

This work is supported by the U.S. Department of En-ergy, Office of Science, Basic Energy Sciences (BES), Chem-ical Sciences, Geosciences, and Biosciences Division, AMOSProgram.

DATA AVAILABILITY

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

FIG. 3. Magnified image of a single quadrant of the filtered spec-trum shown in Fig. 1 on the same logarithmic color scale. This viewclearly shows previously unexplored modulations along the spider-leg structures, fan-like structures close to the origin of the spectrum,and periodic minima along the polarization axis. Each of these fea-tures are normally obscured by the ATI rings.

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