Strong-field photoelectron holography in the subcycle limit

Strong-field photoelectron holography is promising for the study of electron dynamics and structure in atoms and molecules, with superior spatiotemporal resolution compared to conventional electron and X-ray diffractometry. However, the application of strong-field photoelectron holography has been hindered by inter-cycle interference from multicycle fields. Here, we address this challenge by employing a near-single-cycle field to suppress the inter-cycle interference. We observed and separated two distinct holographic patterns for the first time. Our measurements allow us not only to identify the Gouy phase effect on electron wavepackets and holographic patterns but also to correctly extract the internuclear separation of the target molecule from the holographic pattern. Our work leads to a leap jump from theory to application in the field of strong-field photoelectron holography-based ultrafast imaging of molecular structures.

electron wavefunction is split into the inner spatial region ( ∈ (,   )) and outer spatial regions ( >   ): (, ) =  %& (, ) +  '() (, ).The wavefunction in the inner region is numerically propagated under the full Hamiltonian, while in the outer region, the electron-core interaction is much smaller than the electron-laser interaction; hence, the Coulomb potential can be neglected.Thus, the wavefunction in the outer region is analytically propagated using the Volkov Hamiltonian:  * = p ! !− E() • r.At each time step, we first evolve the inner wavefunction, and part of the wave packet will enter the outer region.Then we transform the wavefunction in the outer region  '() to the momentum space: Thus, the outer wavefunction is propagated from  % to the end of the pulse as where After the end of the laser pulse, the final photoelectron momentum distribution is obtained as where  is the direction of the final momentum of the photoelectron.To compare with experimental results, the focal volume averaging has also been done.Through the focus, the laser intensity changes.Hence, the photoelectron angular distributions have been calculated for different laser intensities along the propagation direction.These photoelectron distributions have been integrated over the spatial profile of a Gaussian laser focus 6,7 .

S2. Coulomb quantum-orbit strong-field approximation (CQSFA) theory
To simulate the two kinds of PH patterns, a CQSFA theory is used [8][9][10][11]  (S6) where  = (′) = − • (′) is the electron-core interaction and (,  < ) is the time-evolution operator.In our work, the electron wavefunction | " ⟩ is written as a weighted sum of the constituent atomic orbitals of each core, according to the theory of the linear combination of atomic orbitals: (S8) where | `⟩ =   + (′) denotes the velocity of the electron at the initial time.Note that the bound states of the system have been neglected in the above-stated closure relation.Physically, this approximation means that we are excluding the processes involving the excited bound states at the instant ′ and focusing on the bound-continuum transitions.Furthermore, by employing the Feynman path integral formalism and saddle-point approximation, the transition amplitude can be rewritten as: where the index  denotes different quantum trajectories with the ionization time which are solved using an iteration scheme for any given final momentum.Once the saddle points are solved, we can proceed to solve Newton's equations to obtain the trajectories.By substituting these trajectories into the derived transition formula (S9), we can determine the corresponding photoelectron amplitude.
In our work, we also take into account the Gouy phase effect on SFPH within the CQSFA method.Specifically, when incorporating the Gouy phase, we manually add a phase shift of −  2 ⁄ to the action of the forward-and backward-scattered trajectories in Eq. (S9).
Conversely, in simulations where the Gouy phase is not considered, we omit this additional phase adjustment.Similar approaches have been discussed in prior literature for the spider PH 12 .

S3. Experimental setup
A set of strong-field photoelectron holography experiments was carried out using a linearly polarized carrier-envelope-phase (CEP)-stabilized 3.3-fs near-single-cycle laser at 723 nm and a 25-fs multicycle laser at 800 nm.In both cases, the laser polarization direction was parallel to the detector plane (see Fig. S1(b)).The experimental setup is schematically shown in Fig. S1.It should be noted that the specific wavelengths used in our experiment were determined by the inherent capabilities of our laser source systems.Driving laser wavelength could influence the spacing between above-threshold ionization peaks in the photoelectron momentum distribution.The photon energy difference between the 800 nm (1.55 eV) and 723 nm (1.71 eV) wavelengths, on the other hand, is 0.16 eV (or 0.005 a.u.) and has no significant effect on the experimental observables.For the multicycle experiment, Ti:sapphire amplifier output was picked up and directed to the experimental chamber.The intensity of the multicycle laser pulse is controlled with a half-wave plate (HWP) and a linear polarizer (LP).
For the near-single-cycle experiment, we begin with CEP-stabilized 1.5 mJ, 25 fs nearinfrared (NIR) pulses from the output of the Ti:sapphire amplifier.The amplifier output pulse is directed and focused on a Ne gas-filled hollow-core fiber (inner diameter 400 µm, 1 m long) to achieve spectral broadening.The spectrally broadened pulses are then compressed using 7 pairs of chirped mirrors and a fused silica wedge pair (50 × 25 mm, wedge angle 4 degree).
One of the wedge pairs was installed in a motorized linear stage for fine control of the wedge position for the CEP and dispersion scan experiments.To achieve a near-single-cycle laser pulse 13,14 , we further compress the third-order dispersion (TOD) using a 1.5-mm-thick ammonium dihydrogen phosphate (ADP) crystal and another pair of chirped mirrors.For the pulse characterization, we implemented a dispersion scan 15 setup, and experimental details will be discussed in Section S4.In the case of a near-single-cycle pulse, laser intensity was adjusted using a broadband reflective neutral density filter.Inside the experimental chamber, the laser pulses were focused on the gas target using an off-axis parabolic (OAP) mirror with a  = 150 mm focal length (see Fig. S1(b)).For the near-single-cycle experiment, estimated laser peak intensities at the interaction point were ~2.5 × 10 FL W cm -2 for N2 and ~1.7 × 10 FL W cm -2 for O2.A gas target was injected via an effusive nozzle (50 µm) integrated into the repeller plate of the velocity map imaging (VMI) spectrometer 16 .The resulting photoelectron momentum distributions were focused on a MCP-phosphor screen assembly detector using electrostatic lenses and recorded using a CMOS camera.The energy calibration of the VMI spectrometer was carried out by measuring the above-threshold ionization peaks of Xe gases with multicycle laser pulses.The laser peak intensities were estimated by analyzing the 2 C classical cutoffs from the photoelectron energy distribution, where the  C is ponderomotive energy.

S4. Dispersion compensation and pulse characterization
For the near-single-cycle pulse characterization, we utilized the dispersion scan method and the schematic layout depicted in Figure S2.After proper dispersion control, the generated near-single-cycle laser pulses are sent to the dispersion scan setup and focused on the beta barium borate (BBO) crystal for second harmonic generation (SHG).Generated second harmonics and fundamental NIR beams propagated collinearly after the BBO, and the second harmonic signal was separated from its fundamental using a wedge reflection at the Brewster angle and detected by a spectrometer (HR4000, Ocean Optics).
A dispersion scan trace is collected when scanning a wedge with a motorized linear stage with a home-made Labview program.The pulse characteristics are retrieved with the differential evolution algorithm 17 , which is displayed in Fig. S3.
S7) where  > ( +  2 ⁄ ) and  > ( −  2 ⁄ ) denote bound states of one nitrogen atom and another nitrogen atom, respectively and | > | is the weighted coefficients with the phase of  > .By introducing the closure relation ∫  ^"| ^"⟩ ⟨ ^"| = 1, Eq. (S6) is written as () = − lim )→; C ′ C  ` T  U(,  < )| `⟩⟨ `| = ( < )U " ( < )O ) +; i is the action of the quantum trajectory, and the index  represents the component atomic orbitals | > ⟩ of the electron wavefunction with the expansion coefficients  > according to the theory of the linear combination of atomic orbitals.The last term  %• !⁄ +  >  +%• !⁄ corresponds to the two-center interference from the two cores of N2 molecule, where the coefficient  > = (−1) K % for the molecular orbital with g symmetry, and  > = (−1) K % .F for the molecular orbital with u symmetry.The coefficient  > denotes the orbital quantum number of the atomic orbitals.Because the ionization of the HOMO of N2 molecule is dominant, the initial state | " ⟩ is assumed to be in the HOMO state in our work.The different quantum trajectories are obtained from three saddle-point equations:

Figure S1 .
Figure S1.Schematic representation of the experimental setup.Sources for near-single-cycle and multicycle laser pulse generation are shown in (a).(b) depicts the VMI setup.The laser polarization direction was parallel to the detector plane.HWP: half-wave plate, LP: linear polarizer, CM: chirped mirror, GVD: group velocity dispersion, TOD: third-order dispersion, ADP: ammonium dihydrogen phosphate, ND: neutral density, OAP: off-axis parabolic mirror, MCP: microchannel plate

Figure S2 .
Figure S2.Schematic of dispersion scan setup.Ultrashort pulses are sent to an ultrathin (5 μm) BBO crystal, and the generated SHG signal is separated from the fundamental by wedge reflection at the Brewster angle and measured with a spectrometer while the dispersion is varied with wedge insertion.

Figure S3 .
Figure S3.Dispersion scan measurement of a generated near-single-cycle laser pulse.(a) Measured trace (b) Retrieved trace.(c) The corresponding supercontinuum output spectrum (black) and the reconstructed spectral phase (blue).(d) Reconstructed near-single-cycle pulse intensity envelope (solid curve) exhibiting a pulse duration of 3.3 fs.The reconstructed envelope is nearly identical to the Fourier transform-limited envelope (dashed curve), which is 3.2 fs.
. The CQSFA theory is derived from the exact transition amplitude of an electron from an initial state | " ⟩