Gouy’s Phase Anomaly in Electron Waves Produced by Strong-Field Ionization

Abstract

Ionization of atoms by linearly polarized strong laser fields produces cylindrically symmetric photoelectron momentum distributions that exhibit modulations due to the interference of outgoing electron trajectories. For a faithful modeling, it is essential to include previously overlooked phase jumps occurring when trajectories pass through focal points. Such phase jumps are known as Gouy's phase anomaly in optics or as Maslov phases in semiclassical theory. Most importantly, because of Coulomb focusing in three dimensions, one out of two trajectories in photoelectron holography goes through a focal point as it crosses the symmetry axis in momentum space. In addition, there exist observable Maslov phases already in two dimensions. Clustering algorithms enable us to implement a semiclassical model with the correct preexponential factor that affects both the weight and the phase of each trajectory. We also derive a simple rule to relate two-dimensional and three-dimensional models. It explains the shifted interference fringes and weaker high-energy yield in three dimensions. The results are in excellent agreement with solutions of the time-dependent Schrödinger equation.The π phase shift of an electromagnetic wave as it passes through a focus is an astonishing effect. Even though it was already observed by Gouy [1] more than 100 years ago, recent advances in laser technology have shone new light on Gouy's phase [2][3][4][5]. Analogous phenomena have also been found in other types of waves such as acoustic waves [6], standing microwaves [7] and phonon-polariton wave packets in Raman scattering [8].In the past years, various experiments have been proposed and conducted for measuring the Gouy phase in matter waves [9][10][11]. In the present work, we demonstrate that Gouy's phase anomaly appears in electron wave packets produced by strong-field ionization leading to a significant imprint on interference structures in photoelectron momentum distributions (PMDs).Strong-field ionization may be viewed as a two-step process consisting of: (i) release of an electron from the target; (ii) acceleration of the electron by the electromagnetic field in presence of the potential of the parent ion [12,13]. Depending on the system geometry, different parts of the emitted wave packets are mapped to the same final momentum, creating interference structures [14][15][16]. As the positions of the interference fringes are determined by the phase difference between the wave packets, the emerging PMDs may be viewed as "phasometer". For linearly polarized laser pulses, the PMDs are dominated by photoelectron holography [15,17]. Since these holograms [18] may be qualitatively explained by the interference of a non-scattered "reference" wave packet and a scattered "signal" wave packet [19,20], they have been applied for ultrafast imaging [21][22][23].While PMDs obtained by numerical solution of the time-dependent Schrödinger equation (TDSE) in full dimensionality (3D) agree well with experimental data [15,17], PMDs obtained in reduced dimensi...

Dissecting subcycle interference in photoelectron holography

“…In recent work [46] it was demonstrated that additional Maslov phases (the semiclassical equivalent of Gouy phases) must be included to employ a two-dimensional model for a three-dimensional system, the additional phase for each trajectory is dependent on the number of sign changes of the perpendicular momentum p ⊥ (t ). In this case of the CQSFA these phases can be included by shifting the phase of orbits 3 and 4 by −π/2.…”

Semiclassical Trajectory Perspective of Glory Rescattering in Strong-field Photoelectron Holography

“…For the small scattering angle, the scattering amplitude predicted by the two-trajectory interference models (even with Gouy's phase modification) diverges because prefactor det( ∂ ps ∂ p f ) there tends to infinite [37]. In our UGRT and GRT, the presence of the term of the square root of the phase difference in the prefactor (see Eq.…”

“…The two-trajectory interference is used to explain for the SFPH where the modulation fringe can be expressed in the cosine function of the phase difference between the two trajectories. Recently, the Gouy's phase [35] is introduced to the two-trajectory strong-field interference picture in a 3D model to compensate for the divergence of the preexponential factors of the semiclassical propagator at focal points [37]. As a result, phase difference in the cosine function will be modified by a νπ/2 phase, where ν is the Maslov index [52][53][54][55].…”

“…Theoretically, quantum scattering theory based on strong field approximation (SFA) [29][30][31] as well as the Coulomb corrected strong field approximation (CCSFA) [32][33][34] has been successfully exploited to understand many interesting structures of PMDs. More recently, Gouy's phase [35] anomaly or Maslov phase [36] in strong field rescattering trajectories are addressed [37]. In the above discussions, two branches of trajectories, one is directly ionized trajectory, the other is rescattering trajectory, are considered.…”

Non-adiabatic Strong Field Ionization of Atomic Hydrogen

“…Our NACTS simulations are equivalent to the scheme presented in Ref. [37], but neglect the interference of trajectories. The result of our NACTS simulation is shown in Fig.…”