Angular distributions in two-colour two-photon ionization of He
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t Angular distributions in two-colour two-photon ionization of He
H.F. Rey ∗ and H.W. van der Hart Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and PhysicsQueen’s University Belfast, Belfast BT7 1NN, United Kingdom (Dated: August 9, 2018)We present R-Matrix with time dependence (RMT) calculations for the photoionization of heliumirradiated by an EUV laser pulse and an overlapping IR pulse with an emphasis on the anisotropyparameters of the sidebands generated by the dressing laser field. We investigate how these param-eters depend on the amount of atomic structure included in the theoretical model for two-photonionization. To verify the accuracy of the RMT approach, our theoretical results are compared withexperiment.
PACS numbers: 31.15.A-,32.80.Rm
I. INTRODUCTION
Over the last 15 years significant progress has beenmade in the development of light sources, capable of gen-erating ultra-short pulses lasting just a fraction of a fem-tosecond [1]. One of the key experimental challenges hasbeen the characterisation of these light pulses [2]. Byoverlapping the ultra-short pulse with an IR pulse duringan ionization process, sidebands are observed in photo-electron spectra. Examination of these sidebands allowsthe extraction of the phase differences between the dif-ferent harmonic constituents of the ultra-short pulse [3].Knowledge of these phase differences then allows the re-construction of the ultra-short light pulse.Interest in sidebands created when photoionization byhigh-frequency light occurs in the presence of an IR fieldis not just limited to the characterisation of ultra-shortlight pulses. With the advent of free-electron lasers op-erating at high photon energies [4–6], the interplay be-tween EUV photons and IR photons in the photoioniza-tion process [7] can give valuable information about thetime delay between the EUV and the IR pulse.The experimental interest in sidebands generated bythe addition of an IR field to an EUV field should becomplemented by theoretical investigation. A good de-scription of these sidebands is given by the soft-photonapproximation (SPA) [8], in which the angular distribu-tion of the ejected electron is modified by Bessel func-tions, depending on the number of IR photons absorbedand the angle at which the electron is emitted with re-spect to the polarization direction. This approximationapplies when the energy of the IR photon is much largerthan the energy of electrons emitted by the EUV field byitself. When the energies of the emitted electron and theIR photon are comparable, the ionic potential will affectthe motion of the outgoing electron, and the predictionsof the SPA may not be as accurate.In the present study, we investigate the effect of the ∗ Electronic address: [email protected] residual ionic potential on the angular distributions ofsidebands generated during EUV+IR photoionizationof He using the R-matrix including time-dependence(RMT) approach [9–11]. The He atom has been cho-sen, since the angular distributions of the photoelectronejected in two-colour fields have been the subject of ex-perimental investigation, either when the EUV pulse issufficiently energetic to eject an electron from He [12, 13],or when the pulse excites one of the 1s electrons to an ex-cite np state [14, 15]. In addition, for He, it is straightfor-ward to change the amount of atomic structure retainedin the calculations. A similar study of photo-electron an-gular distributions has recently been performed for Ar us-ing the time-dependent R-matrix (TDRM) approach [16],including a comparison with experiment [17], the SPA[8] and model potential anisotropy parameters [18]. Thepresent calculation thus also allows us to assess the com-putational efficiency of the two different time-dependentR-matrix approaches.The RMT approach was developed only a few yearsago[9–11], and has since been applied to study time de-lays in photoionization of Ne [19], to two-photon dou-ble ionization of He [20], and to IR-assisted photoioniza-tion of Ne + [21]. Similar to the TDRM approach [22], itemploys the standard R-matrix technique of separatingconfiguration space into an inner region and an outer re-gion. However, the wavefunction is propagated using anArnoldi approach [23, 24] in the RMT approach, com-pared to a Crank-Nicholson scheme in the TDRM ap-proach. Spatially, the TDRM approach applies a sequen-tial R-matrix propagation scheme, which restricts the de-gree of parallelisation to about 100-200 processors. Thissequential step does not occur within the RMT approach,and it can therefore be efficiently parallelised over sub-stantially larger numbers of processors [20].In the present report, we will first describe the RMTapproach and indicate the differences between the RMTapproach and the TDRM approach. We then apply theRMT approach to two-colour two-photon ionization ofHe, and present the associated anisotropy parameters.We briefly investigate how these parameters depend onthe theoretical description of He. We compare the ob-tained parameters to those obtained experimentally aswell as the predictions from the SPA. II. THE R-MATRIX INCLUDING TIMEAPPROACH
The R-matrix including time (RMT) approach is a new ab initio method to solve accurately the time dependentSchr¨odinger equation (TDSE) for multi-electron atomsin intense laser light [9–11]. The approach adopts thestandard R-matrix approach [25] of splitting configura-tion space into two distinct regions: an inner region andan outer region. In the inner region, all N + 1 electronsin the system are contained within a sphere of radius b around the nucleus of the atom. N therefore indicatesthe number of electrons left on the ion after a single elec-tron has been ejected. In the outer region N electronsstill remain within the sphere of radius b , but one of theelectrons has now left the sphere, so that its radial dis-tance from the nucleus, r N +1 , is larger than b .In the inner region, a standard R-matrix basis expan-sion is used to describe the wave function [25]. In thisexpansion, states of the system under investigation aredescribed in terms of antisymmetrised direct products ofresidual-ion states with a full continuum basis for theoutgoing electron. Additional correlation orbitals can beincluded to improve the description of the system. All in-teractions are accounted for in the Hamiltonian, includ-ing electron exchange and correlation effects between allpairs of electrons.In the outer region, the wavefunction is described interms of a direct product of residual-ion states, coupledwith the spin and angular momentum of the outgoingelectron, with the radial wavefunction for the outgoingelectron. Since the outer electron is separated from theother electrons, its wavefunction can be treated sepa-rately from the others, and hence exchange effects canbe neglected. In the RMT method, the radial wavefunc-tion for the outer electron is described in terms of a fi-nite difference grid, similar to the approach pursued inthe HELIUM codes [23]. This is the first difference com-pared to the TDRM approach [22], in which the wavefunction was described in terms of a very dense set ofB-spline functions.In sections II A and II B, we give a brief overview ofthe theory underpinning the RMT approach. A full de-scription of the approach is given in [10, 11]. To obtainthe properties of the system under investigation, we startwith the time-dependent Schr¨odinger equation (TDSE), i ∂∂t Ψ( X N +1 , t ) = H N +1 ( t )Ψ( X N +1 , t ) , (1)where H N +1 is the full Hamiltonian for the system, givenby: H N +1 = N +1 X i =1 − ▽ i − Zr i + N +1 X i>j =1 r ij + E ( t ) · N +1 X i =1 r i . (2) In this equation X N +1 ≡ x , x , . . . , x N +1 and x i ≡ r i σ i , with r i and σ i the position and spin vectors of the i th electron, respectively. Z indicates the nuclear chargeand E ( t ) is the electric field of the light pulse. Further-more, r ij = | r i − r j | . The nucleus has been taken at theorigin of the coordinate system. A. The outer region
In the outer region the ( N + 1)-electron wavefunctionis expanded as follows [25]:Ψ( X N +1 , t ) = X p Φ p ( X N ; ˆ r , σ N +1 ) 1 r F p ( r, t ) , (3)where r ≡ r N +1 is the radial distance of the ( N + 1) th electron. The channel functions Φ p are formed by cou-pling relevant states of the residual N -electron ionic sys-tem Φ T ( X N ) with angular and spin components of theejected-electron wavefunction. The functions F p ( r, t ) de-scribe the radial wavefunction of the outer electron in the p th channel.By left-projecting the TDSE (1) onto the channel func-tions Φ p , and integrating over all spatial and spin coordi-nates except r , equation (1) can be rewritten in terms ofa set of coupled partial differential equations for F p ( r, t )[10, 11]: i ∂∂t F p ( r, t ) = h II p ( r ) F p ( r, t ) + X p ′ [ W E pp ′ ( r ) + W D pp ′ ( t )+ W P pp ′ ( r, t )] F p ′ ( r, t ) , (4)where h II p ( r ) describes the energy of the residual-ionstate, the kinetic energy, the screened nuclear attractionand centrifugal repulsion for the outer electron, h II p ( r ) = − d dr + l p ( l p + 1)2 r − Z − Nr + E p . (5)The other terms in equation (4) describe the couplingbetween the different channels. W E pp ′ ( r ) describes thecoupling due to long-range repulsion terms arising fromthe residual electrons, excluding the screening term, W E pp ′ ( r ) = * r − Φ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 | r − r j | − Nr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − Φ p ′ + . (6) W D pp ′ ( t ) describes the coupling due to interaction be-tween the light field and the residual-ion states, W D pp ′ ( t ) = * r − Φ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E ( t ) · N X i =1 r i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r − Φ p ′ + . (7)Finally, W P pp ′ ( t ) describes the coupling due to interac-tion between the light field and the outer electron, W P pp ′ ( r, t ) = (cid:10) r − Φ p | E ( t ) · r | r − Φ p ′ (cid:11) . (8) b R ! r r N+1
Inner Region R-matrix basis set Many-electron Outer Region FD grid methods Multi-channel single-electron b - ! r b + ! r b -2 ! r b +2 ! r FIG. 1: The R-matrix division-of-space concept. In the in-ner region a basis expansion of the wavefunction is chosen,while in the outer region a grid-based representation is con-sidered. The vertical dashes indicate the grid points used inthe calculation. The outer-region grid extrapolates into theinner region. At these points, the inner region wavefunctionis projected onto the outer-region grid associated with eachchannel. The boundary of the inner region is at r = b andthe outer region boundary is at r = R . Equation (4) describes a system of equations, whichcan be solved efficiently using finite-difference techniques[10, 11]. The radial wavefunction for the outer electron, F p ( r, t ), is discretised on a radial grid as shown in figure 1.The potential-energy terms and the angular-momentumrepulsion term can be determined directly at each gridpoint for each channel. The second derivative term in thekinetic energy operator is implemented using a five-pointfinite-difference rule. Near the boundary with the innerregion, wavefunction information from the inner regionis required. We therefore extend the finite difference gridinto the inner region by 2 N A points, where N A is the or-der of the Arnoldi propagator. Following the determina-tion of the inner-region field-free eigenstates, we projectthese inner-region eigenstates onto the channel functions.Using the time-dependent coefficients C k ( t ) associatedwith the field-free inner-region eigenstates (see sectionII B), we can evaluate the time-dependent inner-regionwavefunction on the finite-difference grid extension intothe inner region (eg. the points b − δr and b − δr in fig-ure 1). Once the wavefunction is known on these points,all required orders of the kinetic energy operator can beevaluated accurately in the outer region. To propagatethe outer region wavefunction in time from t to t + δt weemploy the Arnoldi propagator [23, 24].The most demanding part in the solution of equa-tion (4) in the present approach is the evaluation of theright-hand side. This is effectively a matrix-vector mul-tiplication, where the matrix will be sparse due to theangular-momentum constraints on the individual matrixelements. On the other hand, solution of the outer-regionequations in the TDRM approach requires the (compu-tationally demanding) determination of time-dependentGreen’s functions across the entire outer region [22]. Inaddition, the TDRM approach determines the updatedwavefunction through an R-matrix propagation scheme,which cannot be carried out in parallel. In the RMT ap-proach, the updated wavefunction is obtained through apurely local computational scheme, and can therefore beparallelised over very large numbers of cores. As a conse- quence, the RMT codes are substantially faster than theTDRM codes, especially for large-scale problems. B. The inner region
The time dependent ( N + 1)-electron wavefunctionin the inner region Ψ I ( X N +1 , t ) is expanded over field-free eigenstates of the Hamiltonian in the inner region ψ k ( X N +1 ) as follows [25]:Ψ I ( X N +1 , t ) = X k C k ( t ) ψ k ( X N +1 ) , (9)where all r i ≤ b and the C k ( t ) are the time-dependentexpansion coefficients associated with the field-free eigen-states.The TDSE in the inner region is given by: i ∂∂t Ψ( X N +1 , t ) = H N +1 ( t )Ψ( X N +1 , t ) , (10)where the time-dependent Hamiltonian is the same asgiven in expression (2). However, in the inner region,the Hamiltonian H N +1 ( t ) cannot be hermitian. Ioniza-tion requires an electron to escape to infinity, and forthis to occur, some part of the wavefunction must haveleft the inner region. Hence the total population in theinner region cannot be conserved. The non-hermiticityarises in the evaluation of the kinetic energy operatorat the boundary. In the determination of the field-freeinner-region eigenstates, however, this non-hermiticity iscompensated for through the addition of a Bloch opera-tor [25], L N +1 = 12 N +1 X i =1 δ ( r i − b ) (cid:18) ddr i − g − r i (cid:19) , (11)where the value of g can, in principle, be chosen freely.To propagate the inner-region wavefunction in time us-ing the correct time-dependent Hamiltonian, we thereforehave to remove the Bloch operator [10, 11]: i ∂∂t Ψ I ( X N +1 , t ) = H I ( t )Ψ I ( X N +1 , t ) −L N +1 Ψ I ( X N +1 , t ) , (12)where H I is the inner-region Hamiltonian including theBloch operator. However, since the Bloch operator onlyacts on the wavefunction at the boundary, we can applythe Bloch operator upon the outer-region wavefunctioninstead of the inner-region wavefunction: i ∂∂t Ψ I ( X N +1 , t ) = H I ( t )Ψ I ( X N +1 , t ) −L N +1 Ψ( X N +1 , t ) . (13)We now expand the inner-region wavefunction in terms ofthe field-free eigenstates ψ k ( X N +1 ), and project the innerregion TDSE (12) onto these eigenstates. This projectionthen provides a set of equations for the time evolution ofthe coefficients C k ( t ): ddt C k ( t ) = − i X k ′ H I kk ′ ( t ) C k ′ ( t ) + i X p ω pk ∂F p ( r, t ) ∂r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = b , (14)where H I kk ′ ( t ) is the time-dependent Hamiltonian matrixelement between field-free states ψ k and ψ k ′ , and ω pk arethe surface amplitudes of eigenstate ψ k with respect toouter region channel p [25].Using the approximation that the time-dependentHamiltonian is constant within the time interval [ t, t + δt ],we can now obtain an approximate solution to the inner-region TDSE (10) in terms of the so-called φ functions[26]. In matrix notation [10, 11, 27], C ( t + δt ) ≈ e − iδt H I C ( t ) + n j X j =1 ( δt ) j φ j ( − iδt H I ) U j ( t ) , (15)where U ( t ) = C ( t ) , U j ( t ) = i d j − dt j − S ( t ) , (16)with S ( t ) = 12 X p ω pk ∂F p ( r, t ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) r = b . (17)The φ j functions can be regarded as “shifted” exponen-tiation functions, as can be appreciated through theirTaylor series [11], φ j ( z ) = ∞ X k =0 z k ( k + j )! . (18)For the propagation of the inner region wavefunction,we again use an Arnoldi method [23, 24]. However, wenow need separate propagators to evaluate each of the φ -functions as well as the exp( − iδt H I ) term. Thus each ofthe U j terms in equation (15) is propagated separately,and the final wavefunction at time t + δt is obtained bycombining all terms. C. Description of He and field parameters
Our interest focuses on the anisotropy parameters intwo-colour photoionization of He and the amount ofatomic structure included in the theoretical description.Hence we describe the He atom using three different basisexpansions. These basis expansions can be characterisedby the residual He + -ion states retained in the calcula-tions. The He basis is built by combining these residualion states with a complete set of continuum and bound-state functions for a single outgoing electron. The sim-plest basis expansion included employs only the 1s state of He + as a residual-ion state. Hence the He basis con-tains all 1s n/εℓ states. The second basis set we employedincluding the 1s, 2s and 2p states of He + as residual-ionstates, so that the He basis contains all 1s n/εℓ , 2s n/εℓ and 2p n/εℓ states. For the third basis set, we use pseudo-orbitals, 2 s and 2 p , as residual-ion states rather than thephysical 2s and 2p orbitals. These pseudo-orbitals areconstructed as Sturmian-type orbitals r i e − αr with thesame exponential decay as the 1s orbital, minimal powerof the polynomial term, and orthogonality with respectto the 1s orbital. These basis sets are similar to the onesemployed in the investigation of the choice of gauge forthe laser field for time-dependent R-matrix theory [28].RMT calculations require a choice for the maximumangular momentum to be included within the calcula-tions. The calculations using the 1s residual-ion stateonly were performed with 3 different values for the max-imum angular momentum, L max = 5, L max = 7, and L max = 9, with the last two calculations producing verysimilar results. We therefore report results for all threebasis sets investigated using a maximum angular momen-tum L max = 7.In the generation of the He basis, we have used an R-matrix inner region radius of 20 a . This inner regionis sufficiently large to contain the He ground state. Thebasis used for the description of the continuum electronwithin the inner electron contains 70 B-splines with or-der k = 11. The knot point distribution varies from anearly quadratic spacing near the nucleus to a nearly lin-ear spacing near the inner region boundary. Additionalknot points are inserted further inward to improve thedescription of functions close to the nucleus. The outerregion radial wavefunction for the ejected electron is de-scribed on a finite difference grid extending to 1200 a.u.with a grid spacing δr = 0.075 a . The time step in thepropagation is set to δt = 0.005 a.u. The order of theArnoldi propagator is 14.We investigate ionization of helium irradiated by acombination of two laser pulses: an EUV pulse corre-sponding to the 17 th − st harmonic of the fundamen-tal laser field and an overlapping fundamental dressingfield. The wavelength of the fundamental field rangesfrom 790 −
810 nm and were chosen closely to resem-ble those used in the experiment [13]. The IR laser fieldconsidered is linearly polarized in the z -direction withan intensity of 5 × W cm − at peak. The IR pulseprofile is given by a 3-cycle sin ramp on, followed by 2cycles at a peak intensity and a 3-cycle sin ramp off (3-2-3). The EUV laser field considered is linearly polarizedin the z -direction with a peak intensity of 1 × Wcm − . The EUV pulse profile is given by a 3 n -cycle sin ramp on, followed by 2 n cycles at a peak intensity and a3 n -cycle sin ramp off (3-2-3), where n indicates the or-der of the EUV harmonic. Following the end of the pulse,the wavefunction is propagated in time corresponding tojust over 2.8 cycles of the IR field to ensure that the out-going electron is well separated from the residual ion andto ensure that the ejected electron is indeed a continuumelectron. III. RESULTS
In this report, we investigate the asymmetry parame-ters of two-colour two-photon ionization of He irradiatedby a combination of a short EUV pulse and a short IRpulse using the RMT approach. These parameters canbe obtained from the final-time wavefunction obtainedat the end of the calculations. This wavefunction is de-scribed in terms of total angular momentum, whereas theexperimental observations relate primarily to the ejectedelectron. In order to compare the results from our inves-tigations to experiment, we therefore first have to decou-ple the wavefunction of the outer electron and the wave-function of the residual ion states using Clebsch-Gordancoefficients [29]. Following this decoupling, we can con-struct the spatial wavefunction for the outgoing electron.This spatial wavefunction is then transformed into a mo-mentum distribution for the outgoing electron under theassumption that the Coulomb potential of the residualion can be neglected.
FIG. 2: (Colour online) Photoelectron momentum spectrumin the k x k z plane for helium irradiated by an IR field withwavelength of 790 nm and its 17 th harmonic. Figure 2 shows the ejected-electron momentum distri-bution of the ejected electron in the k x k z -plane for thecase that He is irradiated by an IR laser field with awavelength of 790 nm and an EUV field given by the17 th harmonic of the IR field. The IR field is relativelyweak, and we therefore show the momentum distributionon a logarithmic scale. The photoionization spectrum isdominated by the central EUV photoionization peak at | k | ≈ .
45 atomic units. Since the central peak is given bya p outgoing electron, a node is clearly visible at k z = 0for this central peak. The sidebands generated by ab-sorption or emission of an additional IR photon can alsobe identified easily in figure 2. As will be illustrated later in more detail, these sidebands show no node at k z = 0.The relative magnitudes between the central EUVpeak and the sidebands can be better assessed in figure 3,which shows the photoelectron momentum spectra alongthe laser polarization axis for a 790-nm IR field and its17 th harmonic. The figure shows that for the presentlaser parameters (short IR pulse, weak intensity), thephotoelectron momentum spectrum is dominated by thecentral photoemission peak, whereas the sidebands aresignificantly weaker in intensity. Two sidebands can beseen immediately by the side of the main EUV photoelec-tron peak. These peaks originate from the pulse shapeof the EUV laser pulse, which introduces additional fre-quency components. The sidebands associated with ab-sorption or emission of an additional IR photon are wellseparated from the main EUV peak, and we can thus eas-ily obtain a total intensity for each of the sidebands at agiven emission angle by integration over the sideband. −4 Photoelectron momentum (atomic units) | k φ ( k ) | ( a r b . un i t s ) FIG. 3: (Colour online) Photoelectron momentum spectrumfor helium irradiated by a 790-nm IR field and its 17 th har-monic along the polarization axis of the two laser fields. Themomentum spectrum shows that, for an eight-cycle IR-pulsewith peak intensity of 5 × W/cm , the spectrum is dom-inated by the central EUV peak. Two further sidebands canbe seen directly by the side of the EUV pulse. These originatefrom the EUV pulse shape. The asymmetry parameters for the two-colour two-photon ionization process describe the variation of thesideband intensity with emission angle, relative to thelaser polarization axis, θ . Figure 4 shows the variation ofthe integrated sideband intensity with cos θ for the caseof a 790-nm IR field and its 17 th harmonic. The figuredemonstrates that the positive sideband (absorption ofan IR photon) and the negative sideband (emission of anIR photon) show significant differences. This differencebetween positive and negative sidebands, and hence dif-ferences in the asymmetry parameters, has already beenobserved before in studies of Ar [16, 17]. Figure 4 showsthat the angular distribution for the higher sideband ismore peaked along the laser polarization axis, whereasthe negative sideband shows a more constant behaviouras a function of angle.This difference between positive and negative sidebandis not too unexpected. The present choice of laser pa-rameters means that the lower sideband corresponds toemission of electrons with an energy of only 0.8 eV, lessthan the IR photon energy. At this small energy, in-teractions with the residual ion can become important.Furthermore, the centrifugal repulsion potential can sig-nificantly affect the angular distribution. Whereas s elec-trons do not see this repulsive potential, d electrons do.The classical turning point for a d electron with an en-ergy corresponding to the negative sideband is about 2.8 a , whereas it is about 1.8 a at an energy correspondingto the positive sideband. Hence, the overlap between the d continuum and the He ground state will be significantlygreater for the higher sideband than for the lower side-band. The emission in the lower sideband should then bemore isotropic than the emission in the higher sideband.Once we have obtained the angular distributions as-sociated with the sidebands, we can use them to derivethe anisotropy parameters. It is also possible to derivethe anisotropy parameters using a perturbative approach,in which one can derive the anisotropy parameters fromthe magnitude and relative phase between the excita-tion of the s and the d continuum. In the present time-dependent calculation, we have chosen not to use the lat-ter method, as g and ℓ = 6 channels are also populatedduring the calculations. Since these channels can receivepopulation, they should be included in the determinationof the anisotropy parameters. We also include possiblecontributions from p , f and ℓ = 5 and 7 channels, eventhough, for long weak pulses, they do not contribute totwo-photon ionization. −1 −0.5 0 0.5 100.511.5x 10 −3 cos θ I n t en s i t y ( a r b . un i t s ) higher sidebandlower sideband FIG. 4: (Colour online) Intensities of the negative (lower)and positive (higher) sidebands as a function of cos θ for Heirradiated by the 17 th harmonic and overlapping fundamentalof the 790 nm pulse. The photoelectron angular distributions for two-photon ionization are given in terms of the anisotropyparameters by [17] I ( θ ) = σ π [1 + β P (cos θ ) + β P (cos θ )] , (19) where θ is the angle between the laser polarization vectorand electron velocity vector, σ is the total cross section,and β and β are the anisotropy parameters associatedwith the second and fourth order Legendre polynomials,respectively. We can thus obtain these parameters by fit-ting this formula to the theoretical angular distribution.For the data in figure 4, this gives anisotropy parametersof β = 1 .
83 and β = 0 .
48 for the negative sideband,and β = 3 .
03 and β = 1 .
62 for the positive sideband.In the recent experimental investigation of theanisotropy parameters [13], six combinations of IR wave-length and associated harmonic were investigated. For anIR wavelength of 790 nm, and an EUV pulse given by its17 th harmonic, anisotropy parameters of β = 1 . ± . β = 0 . ± .
19 were obtained for the negative side-band, and anisotropy parameters of β = 2 . ± .
08 and β = 1 . ± .
03 were obtained for the positive sideband.Although the theoretical calculations lie slightly outsidethe experimental error bars for β for the negative side-band and β for the positive sideband, the overall agree-ment for this choice of laser wavelengths is very good. Amore detailed comparison of the theoretical and experi-mental anisotropy parameters is given in table I, whereanisotropy parameters are presented for all combinationsreported experimentally.For the case discussed above, an IR wavelength of 790nm, and an EUV pulse given by its 17 th harmonic, wealso show in table I, a comparison of the anisotropy pa-rameters obtained using the different He basis sets. Thetable shows that the differences in the anisotropy param-eters with respect to basis size are minimal in the presentcalculations, with a change of 0.01 in β for the negativesideband. This level of change is typical: the largestdifference is seen for case A, where β for the negativesideband increases to 1.45 when only the 1s state of He + is included. The changes between the different basis setslie thus within the experimental uncertainty. We havetherefore chosen to present results only for the case wherethe He basis is constructed using the He + β (+)2 is very similar in all cases studied, witha value very close to 3. β ( − )2 shows greater variation: itincreases with increasing photon energy of the harmonic.The negative sideband straddles the ionization thresholdat a wavelength of 810 nm. Just above the ionizationthreshold, ionization will be dominated by s electrons,and therefore the closer the negative sideband gets tothe ionization threshold, the emission process should be-come more and more isotropic. However, it may be diffi-cult to obtain accurate anisotropy parameters very closeto threshold in the present calculations, as the wavefunc-tion may need to be propagated for long times to sepa-rate population in high Rydberg states from populationin the low-energy continuum. The β parameters showvariation in both cases: we find a relatively small de- TABLE I: Experimental [13] and theoretical anisotropy parameters for sidebands generated by the 17 th to the 21 st harmonicsof the IR pulse overlapped by the fundamental pulse in helium. Anisotropy parameters for the lower sidebands are denoted bythe superscript ( − ) whereas positive sidebands are denoted by (+).Case Wavelenghts β ( − )2 β ( − )4 β (+)2 β (+)4 A 810 nm 17 th HH experiment 2.00 ± ± ± ± th HH RMT (3 states with pseudo-orbitals) 1.42 0.41 3.02 1.66B 801 nm 17 th HH experiment 1.04 ± ± ± ± th HH RMT (3 states with pseudo-orbitals) 1.65 0.45 3.02 1.64C 794 nm 17 th HH experiment 1.52 ± ± ± ± th HH RMT (3 states with pseudo-orbitals) 1.77 0.47 3.03 1.62D 790 nm 17 th HH experiment 1.73 ± ± ± ± th HH RMT (1 state) 1.83 0.48 3.03 1.62790 nm 17 th HH RMT (3 states with real orbitals) 1.84 0.49 3.03 1.62790 nm 17 th HH RMT (3 states with pseudo-orbitals) 1.84 0.49 3.03 1.62E 810 nm 19 th HH experiment 2.07 ± ± ± ± th HH RMT (3 states with pseudo-orbitals) 2.32 0.67 3.03 1.49F 790 nm 21 st HH experiment 2.06 ± ± ± ± st HH RMT (3 states with pseudo-orbitals) 2.55 0.82 2.99 1.38 crease in β (+)4 with increasing EUV energy, whereas the β ( − )4 shows a steady increase.We can compare our anisotropy parameters with pre-dictions using the Soft Photon Approximation (SPA) [8].In this approximation, it is assumed that the absorptionof the IR photon does not significantly affect the energy ofthe outgoing electron. This approximation may not workwell for the 17 th harmonic due to its proximity to the ion-ization threshold, but the approximation should be moreappropriate for higher harmonics. If it is assumed thatthe IR intensity is weak, then simple analytic expressionscan be derived for the anisotropy parameters, in termsof the single-photon anisotropy parameter, estimated atthe energy of the sideband [13]: β ± = 57
14 + 11 β (0)2 β (0)2 ! , β ± = 367 β (0)2 β (0)2 ! . (20)For single-photon ionization of the He ground-state tojust above the He +
1s threshold, only p electrons canbe ejected, and therefore the parameter β (0)2 = 2 at allphoton energies considered here. Substitution of thisvalue into the SPA estimates for the anisotropy parame-ters gives the following values, β ± = 20 / ≈ .
86 and β ± = 8 / ≈ .
14. The values shown in table I showthat with increasing EUV-photon energy the anisotropyparameters change towards the SPA predictions, but thathigher EUV-photon energies than studied here need to beconsidered for the SPA to provide close agreement.Table I also provides a comparison of the experimen-tally obtained anisotropy parameters with the presentones. The agreement between theory and experiment isbest for the case studied earlier: an IR field of 790 nmand the 17 th harmonic, case D in table I. For the othercases, significant differences are seen between the exper-imental results and the present theoretical results. Thebest agreement is seen for β (+)2 , but for β ( − )2 , other than case D, the smallest difference is 0.25. For the β param-eters, reasonably good agreement is obtained for all casesinvolving the 17 th harmonic, but the differences becomepronounced for the cases involving the 19 th and 21 st har-monic.The origin of the differences between the experimen-tal results and theory is unclear. The β parameters canbe related directly to the relative magnitude between theemission of a d electron and emission of an s electron[13]. The present results, in particular those for the neg-ative sideband, are consistent with a picture in which theemission of d electron is reduced when one approachesthe ionization threshold. Extrapolation of the numericalresults to higher photon energy gives anisotropy param-eters consistent with the predictions of the SPA. On theother hand, the present calculations use very clean laserpulses for both the IR and the EUV pulse. It is unre-alistic to expect such a clean pulse for the EUV pulseexperimentally, as it is obtained through harmonic gen-eration. These differences in laser parameters may wellbe the root origin for the differences seen between theoryand experiment. IV. CONCLUSIONS
We have demonstrated the application of the RMT ap-proach to the investigation of photoelectron angular dis-tributions and anisotropy parameters derived from theseangular distributions. The negative sidebands tend tobe more isotropic than the positive sidebands. The ob-tained anisotropy parameters differ noticeably from thepredictions of the SPA due to the proximity of the EUVphoton energy to the ionization threshold. For increas-ing EUV photon energy, the anisotropy parameters movecloser to the predictions of the SPA. The agreement withexperiment is good for the case of an IR pulse with awavelength of 790 nm and an EUV pulse given by its17 th harmonic, but notable differences are seen for othercombinations of laser pulse.The RMT approach has been developed recently forthe investigation of atomic processes in intense ultra-short light fields. The present calculations demonstratethat the approach can be used to obtain photoelectrondistributions. In the present study, we compare these dis-tributions for the case that the IR field is weak, but theapproach should also be capable of treating more intenseIR fields. In these cases, the sideband structure becomessignificantly more complicated [7], and it would be inter-esting to see how the approach compares to, for exam-ple, the SPA when more IR photons are absorbed by theejected electron. Whereas calculations at low IR intensitycan be carried out using either the TDRM or RMT ap-proach, at high IR intensities, the RMT approach wouldbe strongly preferred, as the RMT approach is more suit-able for large-scale parallelisation. This increase in par-allelisation scale becomes particularly important whenmany angular momenta need to be included in the cal- culations. Acknowledgements
This research has been supported by the EuropeanCommission Marie Curie Initial Training Network COR-INF and by the UK Engineering and Physical SciencesResearch Council under grant no. EP/G055416/1. Theauthors would like to thank J.S. Parker and K.T. Taylorfor assistance with the RMT codes and valuable discus-sions. The main development of the RMT codes wascarried out by M.A. Lysaght and L.R. Moore. This workmade use of the facilities of HECToR, the UK’s nationalhigh-performance computing service, which was providedby UoE HPCx Ltd at the University of Edinburgh, CrayInc and NAG Ltd, and funded by the Office of Scienceand Technology through EPSRC’s High End ComputingProgramme. [1] M. Ivanov and F. Krausz,
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