Capturing photoelectron motion with guiding fictitious particles
CCapturing photoelectron motion with guiding fictitious particles
J. Dubois, S. A. Berman,
1, 2
C. Chandre, and T. Uzer Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
Photoelectron momentum distributions (PMDs) from atoms and molecules undergo qualitativechanges as laser parameters are varied. We present a model to interpret the shape of the PMDs. Theelectron’s motion is guided by a fictitious particle in our model, clearly characterizing two distinctdynamical behaviors: direct ionization and rescattering. As laser ellipticity is varied, our modelreproduces the bifurcation in the PMDs seen in experiments.
Subjecting atoms or molecules to intense laser fieldsgives rise to a variety of non-perturbative and highlynonlinear phenomena, such as high-harmonic generation(HHG), non-sequential multiple ionization (NSMI), andhigh-order above-threshold ionization (ATI). All thesephenomena are based on the key mechanism of attosec-ond physics, namely the recollision [1–6]. A recollisionis obtained when (i) an electron tunnel-ionizes, (ii) freelytravels in the laser field, and then upon return to the ioniccore, (iii) either recombines into an atomic or molecularbound state, or undergoes inelastic or elastic scattering.Ionized electron rescattering has broad applications inatomic and molecular physics. By experiencing a strongion-electron interaction, rescattered electrons probe theatomic or molecular structure. This is the basis for imag-ing techniques, e.g., laser-induced electron diffraction [7–9] (LIED) for molecular imaging [10] and photoelectronholography [11]. These techniques exploit the fact thatphotoelectron momentum distributions (PMDs) encodeinformation on the structure of the atom or the molecule.Understanding the photoelectron dynamics and identi-fying the mechanisms responsible for the shape of thePMDs is an essential step towards predicting and con-trolling [12] these strong-field phenomena.As laser parameters are varied, the shape of the PMDsundergoes drastic changes. To assess these qualitativechanges in experiments [13, 14], the location of the peaksof the PMDs are followed as a function of the laser ellip-ticity. For low ellipticity, the shape of the PMD is a singlecloud peaked at the origin, as a signature of Coulomb fo-cusing [15, 16]. For larger ellipticities, the cloud splitsinto two lobes as the Coulomb focusing recedes. Alongthe major polarization axis, the lobes’ peaks are shiftedfrom the origin, which is a signature of Coulomb asym-metry [17, 18]. The hypothesis made in Ref. [13] is thatthere is a bifurcation when varying the ellipticity of thelaser field. This bifurcation translates into a bifurcationin the ATI spectrum –that is, the energy distribution ofthe ionized electrons–, as observed in the upper panel ofFig. 1. When the peak of the PMD is near the origin,the maximum of the ATI spectrum is near zero energy.When the PMD splits into two lobes, the energy at whichthe ATI is maximum increases (mostly linearly) with in-creasing ellipticity.Both classical [19] and quantum [20] simulations suc-cessfully reproduce the PMDs observed experimentally. However, the underlying dynamical mechanism leadingto the drastic changes of shape of these distributionsfor varying ellipticities is an open question. Standardand widely used methods for the interpretation of thePMDs, like the strong-field approximation [1, 2] (SFA)and the Coulomb-perturbed SFA [17], fail to predictthese changes at low ellipticities, in particular the bi-furcation observed in Ref. [13]. The SFA neglects theCoulomb field after tunnel-ionization so it cannot capturethe Coulomb asymmetry, and the perturbative treatmentof the SFA is not sufficient to capture well Coulomb fo-cusing. Our objective in this Letter is to explain thePMDs and their qualitative changes in terms of micro-scopic mechanisms given by the electron dynamics aslaser parameters are varied, using a method which fullytakes into account the Coulomb field.We begin by building a reduced classical model whichreproduces the PMDs and which clearly exhibits the bi-furcation in question. Analyzing this model in terms ofits trajectories allows us to uncover the mechanisms re-sponsible for the bifurcation. In a nutshell, we demon-strate that the bifurcation of the ATI spectrum is a con-sequence of the depopulation of the Rydberg states of theguiding fictitious particle after a critical ellipticity. Thesimplicity of our reduced model allows us to obtain anexplicit expression for the critical ellipticity as a functionof the parameters of the laser and the atom. With ourmodel in hand, we can predict the shape of the PMDs,thereby providing essential information for imaging tech-niques.Here, our framework is the three-step model. In stepone of the recollision process, the electron tunnel-ionizesthrough the barrier induced by the laser field on the ioniccore potential [21, 22]. We consider an elliptically po-larized electric field E ( t ) = f ( t ) E / (cid:112) ξ + 1[cos( ωt )ˆ x + ξ sin( ωt )ˆ y ], where E , ω , f and ξ are the field ampli-tude, frequency, envelope and polarization, respectively.After tunnel-ionization, the initial conditions of the elec-tron ( r , p , t ) are determined by t and p ⊥ , the ioniza-tion time and the initial transverse velocity, respectively.The electron is initially at the outer edge of the potentialbarrier, in the opposite direction of the electric field, i.e., r = − [ I p E ( t ) / | E ( t ) | ][1+(1 − | E ( t ) | /I p ) / ], where I p is the ionization potential of the atom. The initial lon-gitudinal velocity of the electron is zero, i.e., p = p ⊥ ˆ n ,for a unit vector ˆ n such that ˆ n · E ( t ) = 0. In step a r X i v : . [ n li n . C D ] A p r FIG. 1: Upper panel: ATI spectrum as a function of ellip-ticity, computed using CTMC from Hamiltonian (1). Thecolor scale indicates the probability distribution of photoelec-tron energies. The grey curve is the prediction of our model.Middle panel: The T-trajectory (see text) final momentum P = P x ˆ x + P y ˆ y as a function of the laser ellipticity, scaled by E /ω . The solid colored curves and circles are computed us-ing our model and Hamiltonian (1), respectively. The shadedarea is where the T-trajectory is rescattered in both our modeland Hamiltonian (1). The solid and dashed black curves arecomputed using the SFA and the perturbed SFA, respectively(the lower curves correspond to P x and the upper curves to P y ). Lower panels: The T-trajectory for ξ = 0 .
25 (left panel)and ξ = 0 . f = 1. The blue andcyan solid curves are the T-trajectory of Hamiltonian (1) andour model, respectively. The dashed cyan curve is the ficti-tious particle trajectory guiding the T-trajectory. The blackcrosses show the initial position of the T-trajectory. The axesuse atomic units unless stated otherwise. two, the trajectory of the electron is obtained classically.In classical trajectory Monte Carlo (CTMC) simulations,ensembles of trajectories are integrated, with each oneweighted by the Ammosov-Delone-Krainov [22] (ADK)ionization rate corresponding to the trajectory’s t and p ⊥ . The trajectory with the highest weight correspondsto the trajectory initiated with zero velocity ( p ⊥ = 0) atthe peak of the electric field, when the barrier width is thethinnest. We refer to this trajectory as the T-trajectory.Here, we take the ionization time of the T-trajectory tobe ωt = π . The final momentum of the T-trajectoryis denoted P = P x ˆ x + P y ˆ y . We assume that when the T-trajectory is not rescattered, the location of the peakof the PMDs is at P .In the SFA, the T-trajectory reaches the detector with-out experiencing a recollision with the ionic core for alllaser polarizations, with a final momentum equal to itsinitial drift momentum. In the middle panel of Fig. 1,we show the final momentum of the T-trajectory, whichin the SFA is P SFA = ˆ y ( E /ω ) ξ/ (cid:112) ξ + 1. The SFA so-lution does not exhibit a bifurcation for increasing ellip-ticity, in contradiction with the ATI spectrum depictedin the upper panel of Fig. 1 and the experimental re-sults [13, 14].In order to remedy this shortcoming, a Coulomb-perturbed SFA [17] is used in Ref. [13]. The correc-tion of the final electron momentum is given by ∆ P = − (cid:82) ∞ t r SFA ( t ) / | r SFA ( t ) | d t , where r SFA ( t ) is the SFA elec-tron trajectory. In the middle panel of Fig. 1, we seethat the Coulomb-corrected final momentum of the T-trajectory, i.e., P ≈ P SFA + ∆ P , does not exhibit a bi-furcation for increasing ellipticity either, nor does it pre-dict a change of dynamical behavior of the T-trajectory.In addition, it was noted in Ref. [13] that this methoddoes not predict correctly the location of the center of thePMDs for low ellipticities both in P x and in P y . Hence,for low ellipticities and this range of laser parameters,a perturbed SFA is not the adapted framework for in-cluding the Coulomb interaction in order to assess thePMDs.Instead of perturbing the SFA, we consider here anaveraging method over a fast timescale [23] to describethe photoelectron dynamics. In the dipole approximationformulated in length gauge, the dynamics of the electroninteracting with an electric field and an ionic core is gov-erned by Hamiltonian H ( r , p , t ) = | p | V ( r ) + r · E ( t ) , (1)where atomic units (a.u.) are used unless stated oth-erwise. Here, the atom is He, the field wavelength is λ = 780 nm ( ω = 0 . . u . ) and the laser intensityis I = 8 × W · cm − ( E = 0 .
048 a . u . ). The fieldenvelope f consists of a two laser-cycle plateau followedby a two laser-cycle linear ramp-down, unless stated oth-erwise. The position of the electron is r , and its canoni-cally conjugate momentum is p . We use a soft Coulombpotential [24] V ( r ) = − ( | r | + 1) − / to describe the ion-electron interaction.Averaging Hamiltonian (1) over the fast timescale, setby the period of the laser field, using Lie transformperturbation theory [23] reveals that a fictitious parti-cle guides the electron, as shown in the lower panels ofFig. 1. We build a hierarchy of models using an itera-tive method for computing the n -th order perturbativeexpansion from the ( n − FIG. 2: Polar plots of the PADs for ξ = 0 .
25 (left panel) and ξ = 0 . electron phase space coordinates are of the form r = r g + E ( t ) /ω , (2a) p = p g + A ( t ) , (2b)where ( r g , p g ) are the canonically conjugate variables ofthe guiding fictitious particle, and A ( t ) is the vector po-tential. Here, it is straightforward to see that p g is theelectron drift-momentum. The guiding fictitious particledynamics is governed by the averaged Hamiltonian H ( r g , p g ) = | p g | V eff ( r g , p g ) . (3)We notice that this Hamiltonian no longer depends ontime, as a result of averaging. Consequently, its energy E = H ( r g , p g ) is conserved. At the lowest order in theperturbative expansion, V eff ( r g , p g ) = V ( r g ). Thus, theangular momentum of the guiding fictitious particle isalso conserved, and the system is integrable in the Liou-ville sense. At higher order in the perturbative expan-sion, the effective potential corresponds to the first non-trivial order of the Kramers-Henneberger potential [25],and depends on the laser parameters. In this case, theangular momentum is no longer conserved, and as a con-sequence, the averaged system is no longer integrable.Our reduced model is valid for all positive energies andwhen ω (cid:29) ω g = (2 |E| ) / for negative energy, where ω g is the approximate frequency of the guiding fictitiousparticle trajectory, provided the electron and the guidingfictitious particle are outside the ionic core region. Here,we focus on the lowest-order model.In our model, after tunnel-ionization, the electron isdriven by a guiding fictitious particle. The initial con-ditions of the fictitious particle guiding the electron aredetermined by substituting the initial conditions of theelectron ( r , p , t ) in Eqs. (2). Then, the guiding ficti-tious particle dynamics is governed by Hamiltonian (3).When the electric field is turned off, the vector poten-tial vanishes and the electron coordinates, in particular the momenta, become the same as that of the guidingfictitious particle. Figure 2 shows photoelectron angulardistributions (PADs) computed using CTMC methodsfrom Hamiltonian (1), which is compared with CTMCfrom the SFA, the perturbed SFA [17] and our model. Inthe left panel, we observe excellent agreement betweenthe prediction of our model and the full system beforeaveraging. Moreover, since only direct electrons, i.e., theones that do not undergo rescattering, reach the detec-tor in our model, it becomes possible to locate rescat-tered electron contributions in the full system. For ex-ample, we observe two peaks around π/ π/ E T . If E T > P . However, the T-trajectory is deflected due tothe effective Coulomb interaction in the averaged Hamil-tonian (3). The Coulomb asymmetry observed in Fig. 2is the direct consequence of this deviation. If E T < E T depends on thelaser parameters, and in particular on the field elliptic-ity, through the change of initial coordinates (2). Thereexists a critical polarization ξ c such that E T ( ξ c ) = 0. Anapproximation of the critical ellipticity is ξ c (cid:39) √ ω E / (cid:0) γ / (cid:1) − / , (4)where γ = (cid:112) I p / p is the Keldysh parameter [21]and U p = E / ω is the ponderomotive energy. Wehave assumed that V ( r g ) (cid:39) − / | r g | , ξ c (cid:28)
1, and r (cid:39) I p (cid:112) ξ c + 1 /E . If ξ < ξ c then E T <
0, and theT-trajectory is rescattered. In our model, the observ-able P does not exist because the T-trajectory does notreach the detector. If ξ > ξ c then E T >
0, and the T-trajectory reaches the detector without recolliding. For I = 8 × W · cm − , the critical field polarization ob-tained from Eq. (4) is ξ c ≈ .
32, in agreement with upperpanel of Fig. 1. For I = 8 × W · cm − , the criti-cal field polarization obtained from Eq. (4) is ξ c ≈ . I = 1 . × W · cm − , a wavelength of 790 nm and anAr atom, it is given by ξ c ≈ .
27, also in agreement withexperimental measurements [14]. We notice that the adhoc criterion used in Ref. [13] based on the perturbedSFA theory [17] does not provide a correct estimate of ξ c for intensities smaller than 5 × W · cm − .The final momentum of the fictitious particle guid-ing the T-trajectory is P = √ E T (ˆ x cos Θ + ˆ y sin Θ) for ξ ≥ ξ c , where Θ is the scattering angle of the ficti-tious particle guiding the T-trajectory. Assuming that V ( r g ) (cid:39) − / | r g | , the scattering angle is Θ = π/ − (2 E T (cid:96) + 1) − / , where (cid:96) is the guiding fictitiousparticle angular momentum. Close to the bifurcation, theguiding fictitious particle energy is E T ≈ p ξ c ( ξ − ξ c ),and we have P x ≈ − (cid:112) ξ c ( E /ω )( ξ − ξ c ) / ,P y ≈ E /ω )( ξ − ξ c ) . We notice that the bifurcation is observed for both P x and P y . Consequently, we show that Coulomb focusingbreaks down when Coulomb asymmetry becomes signifi-cant, as experimentally observed [13].Two kinds of photoelectrons coexist –direct and rescat-tered electrons– and contribute to the PMDs, and bothare essential for probing the ion-electron interaction.However, the chaotic behavior of the rescattered electrontrajectories, as shown in high-energetic part of ATI spec-tra [27], reduces their local contribution in the PMDs.Figure 3 shows the scattering angle of the electron asa function of the initial conditions ( t , p ⊥ ), computedfrom the trajectories of Hamiltonian (1). We observechaotic regions which are the signature of the highly non-linear interactions driving the electrons during rescatter-ing. Two main chaotic regions, centered at ωt = π and ωt = 3 π/
2, are surrounded by initial conditions leadingto electrons trapped into Rydberg states. We refer tothis set of domains as the rescattering domain. In ourmodel, the rescattering domain is determined by E < E = 0.We notice the very good agreement between the region E < ξ < ξ c , the initial conditions of the T-trajectory belong tothe rescattering domain. Hence, even if the rescatteredtrajectories are heavily weighted by the ADK ionizationrate, their local contribution in the PMDs is relativelyweak. Consequently, the electrons that contribute themost are the ones close to the boundaries of the rescat-tering domain, corresponding to electrons reaching thedetector with energy E = 0. Therefore, the maximum of FIG. 3: Scattering angle of the electron of Hamiltonian (1)as a function of the initial conditions ( t , p ⊥ ), for ξ = 0 . ξ = 0 . f with an eight laser-cycle plateau and a two laser-cycle ramp-down. The final electron energy is negative in grey areas. Theblack lines show where the guiding fictitious particle energyis E = 0 according to our model (3). The crosses show theinitial conditions of the T-trajectory. The momentum p ⊥ isin atomic units. the ATI spectrum is at zero energy. As the laser param-eters are varied, particularly the ellipticity, the rescatter-ing domain moves in the plane of initial conditions aftertunnel-ionization. For ξ > ξ c , the T-trajectory no longerbelongs to the rescattering domain, as seen in the bottompanel of Fig. 3, so the ATI spectrum is peaked at E T andthe PMDs are dominated by direct electrons. Thus, wepredict that the peak of the ATI spectrum is located at E = max { , E T } , as shown in the upper panel of Fig. 1.We notice that when ξ increases further away from ξ c , therescattering domain moves to regions of initial conditionswith very low ADK ionization rate. Consequently, thecontribution of rescattered electrons and electrons withenergy E = 0 in the PMDs becomes very weak. Hence,we observe a lack of electrons in the neighborhood of theorigin of the PMDs.In summary, we determined the microscopic mecha-nisms responsible for the shape of PMDs from the analy-sis of Hamiltonian (1), and in particular, we showed thatthe change of shape observed in Ref. [13, 14] as ellipticityis varied corresponds to a bifurcation. Our approach isbased on a fictitious particle guiding the photoelectronmotion. This model provides several predictions on thephotoelectron motion and the shape of the PMDs, andallows the control of the ratio between the yield of rescat-tered and direct electrons, elements which are essentialfor imaging techniques.We acknowledge Cornelia Hofmann and Ursula Kellerfor helpful discussions. The project leading to this re- search has received funding from the European Union’sHorizon 2020 research and innovation program under theMarie Sk(cid:32)lodowska-Curie grant agreement No. 734557.S.A.B. and T.U. acknowledge funding from the NSF(Grant No. PHY1602823). 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