Non-dipole recollision-gated double ionization and observable effects
NNon-dipole recollision-gated double ionization and observable effects
A. Emmanouilidou, T. Meltzer, and P. B. Corkum Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom Joint Laboratory for Attosecond Science, University of Ottawa and National Research Council,100 Sussex Drive, Ottawa, Ontario, Canada K1A 0R6 (Dated: September 26, 2018)Using a three-dimensional semiclassical model, we study double ionization for strongly-drivenHe fully accounting for magnetic field effects. For linearly and slightly elliptically polarized laserfields, we show that recollisions and the magnetic field combined act as a gate. This gate favorsmore transverse—with respect to the electric field—initial momenta of the tunneling electron thatare opposite to the propagation direction of the laser field. In the absence of non-dipole effects,the transverse initial momentum is symmetric with respect to zero. We find that this asymmetryin the transverse initial momentum gives rise to an asymmetry in a double ionization observable.Finally, we show that this asymmetry in the transverse initial momentum of the tunneling electronaccounts for a recently-reported unexpectedly large average sum of the electron momenta parallelto the propagation direction of the laser field.
PACS numbers: 32.80.Rm, 31.90.+s, 32.80.Fb,32.80.Wr
I. INTRODUCTION
Non-sequential double ionization (NSDI) in driventwo-electron atoms is a prototype process for explor-ing the electron-electron interaction in systems drivenby intense laser fields. As such, it has attracted a lotof interest [1, 2]. Most theoretical studies on NSDI areformulated in the framework of the dipole approxima-tion where magnetic field effects are neglected [3]. How-ever, in the general case that the vector potential A de-pends on both space and time, an electron experiencesa Lorentz force whose magnetic field component is givenby F B = q v × B . We work in the non-relativistic limit.In this limit, magnetic-field effects are expected to arisewhen the amplitude of the electron motion due to F B becomes 1 a.u., i.e. β ≈ U p / (2 ω c) ≈ p isthe ponderomotive energy. Non-dipole effects were pre-viously addressed in theoretical studies of the observedionization of Ne n+ (n ≤
8) in ultra-strong fields [6], ofstabilization [7] and of high-order harmonic generation[8–10] as well as in experimental studies [11, 12]. In re-cent studies in single ionization (SI), non-dipole effects ofthe electron momentum distribution along the propaga-tion direction of the laser field were addressed in experi-mental [13] and theoretical studies [14–17].In this work, we show that in double ionization themagnetic field in conjunction with the recollision act asa gate. This gate selects a subset of the initial tunneling-electron momenta along the propagation direction of thelaser field. Only this subset leads to double ioniza-tion. This gating is particularly pronounced at inten-sities smaller than the intensities satisfying the criterionfor the onset of magnetic field effects β ≈ F B force (to first order). In the cur-rent formulation, the change in momentum due to the F B force is along the +y-axis. The tunneling electron is the electron that initially tunnels in the field-loweredCoulomb potential. When non-dipole effects are fully ac-counted for, we show that the y-component of the initialmomentum of the tunneling-electron is mostly negativefor events leading to double ionization. In the dipoleapproximation, the initial momentum of the tunneling-electron that is transverse to the direction of the electricfield is symmetric with respect to zero. The term non-dipole recollision-gated ionization is adopted to describeionization resulting from an asymmetric distribution ofthe transverse tunneling-electron initial momentum dueto the combined effect of the recollision and the magneticfield. Non-dipole recollision-gated ionization is a generalphenomenon. We find that it underlies double electronescape in atoms driven by linearly and slightly ellipticallypolarized laser fields.Moreover, we show that non-dipole recollision-gatedionization results in an asymmetry in a double ionizationobservable. Let φ ∈ [0 ◦ , ◦ ] denote the angle of the fi-nal (t → ∞ ) momentum of each escaping electron withrespect to the propagation axis of the laser field. The ob-servable in question is P DIasym ( φ ) =P DI ( φ )-P DI (180 ◦ − φ ),where P DI ( φ ) is the probability of either one of thetwo electrons to escape with an angle φ . P DI ( φ ) andP DIasym ( φ ) are accessible by kinematically complete ex-periments. In the dipole approximation, P DIasym ( φ ) = 0.When non-dipole effects are accounted for, it is shownthat P DIasym ( φ ) >
0, for φ ∈ [0 ◦ , ◦ ]. This is in accordwith the effect of F B . We also find that P DIasym ( φ ) has con-siderable values over a wide interval of φ at lower inten-sities. This latter feature is an unexpected one. For theintensities considered the F B force has small magnitudethat increases with intensity. Thus, one would expect thedistribution P DIasym ( φ ) to be very narrowly peaked around90 ◦ with values increasing with intensity.We finally show that non-dipole recollision-gated ion-ization is the mechanism underlying the surprisingly a r X i v : . [ phy s i c s . a t o m - ph ] M a y large average sum of the momenta of the two escapingelectrons along the propagation direction of the laserfield. This large average sum of the electron momentais roughly an order of magnitude larger than twice theaverage of the respective electron momentum in singleionization. We recently reported this in ref.[18] for in-tensities around 10 Wcm − for He at 800 nm (near-infrared) and around 10 Wcm − for Xe at 3100 nm(mid-infrared) [18]. If magnetic-field effects are not ac-counted for the average momentum along the propaga-tion direction of the laser field is zero. II. MODEL
We study ionization in strongly-driven He using athree-dimensional (3D) semiclassical model that fully ac-counts for the magnetic field during time propagation—3D-SMND model. We developed this model in ref.[18] byextending a previously developed 3D semiclassical modelin the framework of the dipole approximation—3D-SMDmodel [19, 20]. The Hamiltonian describing the interac-tion of the fixed nucleus two-electron atom with the laserfield is given byH = ( p + A (y , t)) p + A (y , t)) −− c Z | r | − c Z | r | + c | r − r | . (1) A is the vector potential given by A (y , t) = − E ω e − ( ct − yc τ ) (sin ( ω t − ky)ˆx + χ cos( ω t − ky)ˆz) , (2) ω , k, E are the frequency, wavenumber, and strengthof the electric component of the laser field, respec-tively, and χ is the ellipticity. c is the velocity of lightand τ = FWHM / √ ln4 with FWHM the full-width-half-maximum of the laser field. All Coulomb forces are ac-counted for by setting c = c = c = 1. The laser fieldsconsidered in the current work are either linearly polar-ized, χ = 0, or have a small ellipticity of χ = 0 .
05. For A given by Eq. (2), E and B are along the x- and z-axis, respectively, with small components along the z-and x-axis, respectively, for laser fields with χ = 0 . F B are mainly along the y-axis. Unless oth-erwise stated, all Coulomb forces as well as the electricand the magnetic field are fully accounted for during timepropagation. To switch off a Coulomb interaction, theappropriate constant is set equal to zero. For example,to switch off the interaction of electron 1 with the nu-cleus, c is set equal to zero. Moreover, we address theCoulomb singularity by using regularized coordinates [21]which were also employed in the 3D-SMD model [19, 20].The initial state in the 3D-SMND model entails oneelectron tunneling through the field-lowered Coulombpotential. The electron tunnels with a non-relativistic quantum tunneling rate given by the Ammosov-Delone-Krainov (ADK) formula [22, 23]. A non-relativistic ADKrate results in this Gaussian distribution being centeredaround zero. In ref.[24] non-dipole effects were accountedfor in the ADK rate. It was shown that the most prob-able initial transverse momentum ranges from 0.33 I p /cto almost zero with increasing E / (2I p ) / , with I p theionization energy of the tunneling electron. In this work,the smallest intensity considered is 7 × Wcm − forHe. At this intensity, if non-dipole effects are accountedfor in the ADK rate, the transverse momentum of thetunneling electron is centered around 0.12 I Hep /c for Hewhich is 7.9 × − a.u. (I Hep =0.904 a.u.). In what fol-lows, we neglect this very small asymmetry and describethe distribution of the initial transverse momentum bya Gaussian distribution centered around zero [22, 23].We do so in order to clearly illustrate an important phe-nomenon, i.e. non-dipole recollision-gated ionization . Inaddition, we set the initial momentum along the direc-tion of the electric field equal to zero. The remainingelectron is initially described by a microcanonical distri-bution [25]. We denote the tunneling and bound elec-trons by electrons 1 and 2, respectively. The 3D-SMNDmodel is described in more detail in ref.[18].
III. NON-DIPOLE RECOLLISION GATEDIONIZATION
In this work, we discuss non-dipole effects in doubleionization. We do so in the context of He when driven byan 800 nm, 12 fs FWHM laser field that is linearly po-larized at intensities 1.3 × Wcm − , 2 × Wcm − and 3.8 × Wcm − and that is slightly elliptically po-larized with χ = 0 .
05 at 2 × Wcm − . At these inten-sities the ponderomotive energy U p = E / (4 ω ) is 2.86a.u., 4.39 a.u. and 8.35 a.u., respectively. Thus, the max-imum energy of electron 1, 3.17U p , is above the energyneeded to ionize He + . A. Magnetic field asymmetry in P
DIi ( φ ) First, we show that the magnetic field causes an asym-metry in the double ionization probability of electron ito ionize with an angle φ , which is denoted by P DIi ( φ )with i = 1 , SI ( φ ) is the corre-sponding probability in single ionization. φ is the angleof the final momentum of electron i with respect to thepropagation axis of the laser field, i.e. cos φ = p i · ˆy / | p i | .The y-component of the electron momentum is parallelto the propagation direction of the laser field and to F B .To show this asymmetry, we plot P DI1 ( φ ) of the tun-nel electron and P DI2 ( φ ) of the initially bound elec-tron in Fig. 1(a) and (c), respectively, while we plotP SI ( φ ) in Fig. 1(e). These plots are at an intensityof 2 × Wcm − with the magnetic field switched-on and off. When the magnetic field is switched-on,we find that all probability distributions are asymmet-ric with respect to φ = 90 ◦ . This asymmetry is dueto the magnetic field. Indeed, when the magneticfield is switched-off all distributions are shown to besymmetric with respect to φ = 90 ◦ . The latter is ex-pected, since there is no preferred direction of electronescape on the plane that is perpendicular to the x-axis(polarization direction). Moreover, with the magneticfield switched-on, we find that P DIi ( φ ) > P DIi (180 ◦ − φ )and P SI ( φ ) > P SI (180 ◦ − φ ) for φ ∈ [0 ◦ , ◦ ]. Equiv-alently P DIi , asym ( φ ) =P DIi ( φ )-P DIi (180 ◦ − φ ) > SIasym ( φ ) =P SI ( φ )-P SI (180 ◦ − φ ) > φ ∈ [0 ◦ , ◦ ].This is consistent with the gain of momentum due tothe F B force being along the +y-axis. That is, an elec-tron is more likely to ionize with a positive rather thana negative y-component of the final momentum. FIG. 1. (a) P
DI1 ( φ ) of the tunneling electron, (c) P DI2 ( φ ) ofthe initially bound electron and (e) P SI ( φ ) in single ionizationare plotted as a function of φ at 2 × Wcm − , with themagnetic field switched-on and off. (b) P DI1 , asym ( φ ) of the tun-neling electron, (d) P DI2 , asym ( φ ) of the initially bound electronand (f) P SIasym ( φ ) in single ionization are plotted as a func-tion of φ at three intensities with χ = 0 and at one intensitywith χ = 0 . φ is binned in intervals of 18 ◦ and † denotes amultiplication factor of 10 Wcm − . The asymmetry with respect to φ = 90 ◦ is better illus-trated in Fig. 1. We plot P DIi , asym ( φ ) and P SIasym ( φ ) as afunction of φ at 1.3 × Wcm − , 2 × Wcm − and3.8 × Wcm − and at 2 × Wcm − with χ = 0 . SIasym ( φ ) is almost zeroat 1.3 × Wcm − . At the higher intensity of 3.8 × Wcm − , P SIasym ( φ ) is sharply centered around 90 ◦ reach-ing roughly 7%, see Fig. 1(f). These features of P SI ( φ )are in accord with the effect of the F B force. F B issmall for the intensities considered. Therefore, F B has an observable effect mostly when the y-component of theelectron momentum is small as well, i.e. for an angle ofescape φ = 90 ◦ . In addition, | F B | is three times largerfor the higher intensity compared to the smaller one. Asa result P SIasym ( φ ) has larger values at higher intensities.In double ionization, we plot P DI2 , asym ( φ ) of the initiallybound electron in Fig. 1(d). It is shown that P DI2 , asym ( φ )resembles mostly P SIasym ( φ ) rather than P DI1 , asym ( φ ) inFig. 1(b). Indeed, P DI2 , asym ( φ ) has larger values for higherintensities, as is the case for P SIasym ( φ ), reaching roughly5.5% at 3.8 × Wcm − . We also find that the distri-bution P DI1 , asym ( φ ) of the tunneling electron has differentfeatures from P SIasym ( φ ), compare Fig. 1(b) with Fig. 1(f).We find that P DI1 , asym ( φ ) is much wider than P SIasym ( φ ).Also, for φ ∈ [45 ◦ , ◦ ], P DI1 , asym ( φ ) has higher values atthe smaller intensities of 1.3 × Wcm − and 2 × Wcm − rather than at 3.8 × Wcm − —4% comparedto 2.5%. We have shown in ref.[18] that strong recolli-sions [27] prevail for strongly-driven He at 800 nm at in-tensities of 1.3 × Wcm − and 2 × Wcm − , whilesoft ones prevail at 3.8 × Wcm − . It then followsthat P DI1 , asym ( φ ) has higher values for strong recollisions.This is also supported by the small values of P DI1 , asym ( φ ) at2 × Wcm − for a laser pulse with a small ellipticity of χ = 0 .
05, see Fig. 1(b). We find (not shown) that the rec-ollisions are soft at 2 × Wcm − for a laser pulse with χ = 0 .
05. The times of recollision correspond roughly tozeros of the laser field for strong recollisions and extremaof the laser field for soft ones [18, 26]. Moreover, thetransfer of energy, compared to U p , from electron 1 toelectron 2 is larger for a strong recollision and smaller fora soft one [18, 26]. Later in the paper, we explain in de-tail why the width and the values of P DI1 , asym ( φ ) are largeat 2 × Wcm − , smaller at 3.8 × Wcm − and evensmaller at 2 × Wcm − for a laser pulse with χ = 0 . φ ,P DI1 ( φ ) (Fig. 1(a)) and P DI2 ( φ ) (Fig. 1(c)), respectively,are not experimentally accessible. However, in a kine-matically complete experiment, for each doubly-ionizedevent, the angle φ of each ionizing electron can be mea-sured. Then, the probability distribution for any one ofthe two electrons to ionize with an angle φ , P DI ( φ ), canbe obtained, for φ ∈ [0 ◦ , ◦ ]. We compute and plotthe distribution P DIasym ( φ ) as a function of φ in Fig. 2.P DIasym ( φ ) is found to have significant values at smallerintensities over the same wide range of φ as P DI1 , asym ( φ )does. However, P DIasym ( φ ) < P DI1 , asym ( φ ) at the smaller in-tensities. Moreover, at 3.8 × Wcm − and at 2 × Wcm − with χ = 0 .
05, P
DIasym ( φ ) > P DI1 , asym ( φ ) at φ = 81 ◦ but has non zero values for a wider range of φ com-pared to P DI2 , asym ( φ ). These features are expected sinceP DIasym ( φ ) accounts for both the tunneling and the initiallybound electron. However, the features of the experimen- FIG. 2. In double ionization, P
DIasym ( φ ) for the two escapingelectrons is plotted as a function of φ . Three intensities areconsidered and φ is binned in intervals of 18 ◦ . † denotes amultiplication factor of 10 Wcm − . tally accessible P DIasym ( φ ) still capture the main featuresof P DI1 , asym ( φ ). B. Asymmetric transverse electron 1 momentumat the tunnel time
In the following sections we discuss the mechanism re-sponsible for the features of P
DI1 , asym ( φ ) and therefore forthe features of the observable P DIasym ( φ ). We find that thismechanism is a signature of recollision exclusive to non-dipole effects. We adopt the term non-dipole recollision-gated ionization to describe it. We find that the magneticfield and the recollision act together as a gate that selectsonly a subset of transverse initial momenta of the tunnel-ing electron that lead to double ionization. For linearlypolarized light, this gating is illustrated in Fig. 3(a1) and(a2) at an intensity of 2 × Wcm − with the magneticfield switched-on and off, respectively. We plot the prob-ability distribution P(p , t ) of electron 1 to tunnel-ionizewith a y-component of the initial momentum equal top , t . We find that P(p , t ) is asymmetric when the mag-netic field is switched-on. Specifically, it is more likely forelectron 1 to tunnel-ionize with a negative rather than apositive y-component of the initial momentum. In addi-tion, P(p , t ) peaks around small negative values of themomentum of electron 1. Instead, P(p , t ) is symmetricwhen the magnetic field is switched-off. We also find thatP(p , t ) is symmetric around zero (not shown) when themagnetic field is switched-on and off. This is expectedsince there is no force acting along the z-axis due to thelaser field.We find that this asymmetry in P(p , t ) persists at ahigher intensity of 3.8 × Wcm − for linearly polar-ized light, see Fig. 3(a4). We find that an asymmetry inP(p , t ) is also present at 2 × Wcm − for a laser pulsewith a small ellipticity of χ = 0 .
05, see Fig. 3(a3). Incontrast, for all the above cases, we find that the initially
FIG. 3. In double ionization, the distributions of the y-component of the electron 1 momentum at two different timesare plotted; (a1)-(a4) at the time electron 1 tunnel-ionizes(momentum p , t ); (b1)-(b4) at the time just before the timeof recollision (momentum p , t r ).The distributions of the y-component of the electron 1 position are plotted (c1)-(c4) atthe time just before the time of recollision (position r , t r ).P DI1 , asym ( φ ) is plotted in panels (d1)-(d4) for better compari-son of its features with the features of P(p , t r ). Panels (d1),(d3) and (d4) are the same as the plots in Fig. 1(b). bound electron has a symmetric distribution P(p , t ) atthe time electron 1 tunnel-ionizes and just before thetime of recollision. Moreover, in single ionization we findthat the escaping electron has a symmetric distributionP(p y , t ) at the time this electron tunnel-ionizes. C. Asymmetric transverse electron 1 momentumshortly before recollision
In single ionization, to understand the features ofP
SIasym ( φ ) one must obtain the distribution P(p , t ) atthe time electron 1 tunnel-ionizes. Indeed, we computethe y-component of the escaping electron’s momentumboth with all Coulomb forces switched-off and with allCoulomb forces accounted for. In both cases we startthe time propagation from the instant electron 1 tunnel-ionizes. We have shown in ref.[18] that the average y-component of the electron 1 momentum is roughly thesame for both cases. At the time electron 1 tunnel-ionizes, we find that the y-component of the transversemomentum of electron 1 is roughly symmetric aroundzero. This initial momentum distribution combined withthe F B force give rise to the electron ionizing mostlywith φ slightly less than 90 ◦ or equivalently give rise toa sharply peaked distribution P SIasym ( φ ) (Fig. 1(f)).In double ionization, to understand the features ofP DI1 , asym ( φ ), first, we must obtain the distribution P(p , t r )of the y-component of the electron 1 momentum shortlybefore the time of recollision. Then, we must find theeffect of the recollision itself on the distribution P(p , t r ).The validity of these two steps is supported by the fol-lowing computations. We propagate the y-component ofthe momentum of electrons 1 and 2 from the time elec-tron 1 tunnel-ionizes up to the time of recollision. Wedo so using the full 3D-SMND model with all Coulombforces accounted for. Next, using as initial conditionsthe momenta of electrons 1 and 2 shortly after the timeof recollision, we propagate from the time of recollisiononwards with all Coulomb forces and the magnetic fieldswitched-off. The final average y-component of the mo-mentum of electron 1 is roughly equal in both cases andthe same holds for electron 2 [18]. Therefore, the decisivetime in double ionization is the time of recollision.Given the above, we first compute P(p , t r ) just be-fore the time of recollision. This is done by extractingfrom the full 3D-SMND model the distribution of the y-component of the electron 1 momentum at a time justbefore the recollision, for instance at t bef = t r − / bef such asto avoid the sharp change of the momenta which occursat t r , see ref. [18]. At all intensities considered, we findthat shortly before the time of recollision a positive overa negative y-component of electron 1 momentum is fa-vored. This is shown at intensities of 2 × Wcm − and3.8 × Wcm − for linearly polarized light in Fig. 3(b1)and (b4), respectively, and at 2 × Wcm − for ellip-tically polarized light with χ = 0 .
05 in Fig. 3(b3). Incontrast, when the magnetic field is switched-off the dis-tribution P(p , t r ) is symmetric with respect to zero as il-lustrated for an intensity of 2 × Wcm − in Fig. 3(b2).We have now established that the shift towards nega-tive momenta of P(p , t ) at the time electron 1 tunnel-ionizes maps to a shift towards positive momenta ofP(p , t r ) just before the time of recollision. The widthis another interesting feature of the distribution P(p , t )and, by extension, of the distribution P(p , t r ). FromFig. 3(b1)-(b4), we find that the width of P(p , t r ) at2 × Wcm − with linearly polarized light is the largestone, while the width of P(p , t r ) at 2 × Wcm − with χ = 0 .
05 is the smallest one. Moreover, the width ofP(p , t ) is comparable with the width of P(p , t r ) for boththe linear and the elliptical laser field at 2 × Wcm − .However, the width of P(p , t ) is larger than the widthof P(p , t r ) at an intensity of 3.8 × Wcm − .We find that the widths of P(p , t ) and P(p , t r ) areconsistent with Coulomb focusing which mainly refers tomultiple returns of electron 1 to the core [28, 29]. In addition, for the larger intensity of 3.8 × Wcm − ,the widths are also consistent with a larger effect of theCoulomb potential of the ion on electron 1. The lattereffect is not due to multiple returns of electron 1 to thecore but rather due to electron 1 tunnel-ionizing closerto the nucleus at higher intensities.Indeed, we compute in Table I the number of timeselectron 1 returns to the core before it finally escapes.We find that electron 1 returns more times to the coreat 2 × Wcm − . Namely, electron 1 escapes withonly one return to the core in 27% of doubly-ionizedevents. Also, it returns roughly the same number oftimes at 2 × Wcm − with χ = 0 .
05 and at 3.8 × Wcm − . In these two latter cases, electron 1 escapeswith only one return to the core in more than 50% ofdoubly-ionized events. We also find (not shown) that thewidth of the distributions P(p , t ) and P(p , t r ) increaseswith increasing number of returns to the core for all laserfields considered. The above features are consistent withCoulomb focusing. Thus, Coulomb focusing explains whythe width of P(p , t ) at 2 × Wcm − is larger than thewidth of P(p , t ) at 2 × Wcm − with χ = 0 . . † . † . † . † ( χ = 0) ( χ = 0 , M . F . off) ( χ = 0 .
05) ( χ = 0)1 Return 27% 24% 53% 57%2 Returns 24% 24% 10% 16%3 Returns 27% 27% 19% 14% > † Intensities given in units of 10 Wcm − TABLE I. Number of returns to the core of electron 1 fordoubly-ionized events.
Moreover, at the larger intensity of 3.8 × Wcm − electron 1 exits the field-lowered Coulomb barrier closerto the nucleus. Indeed, we find that the average dis-tance of electron 1 from the nucleus at the time electron 1tunnel-ionizes is 2.4 a.u. at 3.8 × Wcm − comparedto roughly 3.5 a.u. at 2 × Wcm − with χ = 0 and χ = 0 .
05, see Table II. This is consistent with the widthof P(p , t r ) being significantly smaller than the width ofP(p , t ) at 3.8 × Wcm − . That is, the Coulomb po-tential of the ion has a large effect on electron 1 from thetime electron 1 tunnel-ionizes onwards. This is not thecase for the smaller intensities of 2 × Wcm − with χ = 0 and χ = 0 .
05 where the widths of the distributionsP(p , t ) and P(p , t r ) are similar. D. Glancing angles in recollisions
We have shown in the previous section that effectivelythe only force that could result in a change of the mo-menta of the two electrons between the final time and thetime shortly after recollision is due to the electric field. . † . † . † . † ( χ = 0) ( χ = 0 , M . F . off) ( χ = 0 .
05) ( χ = 0) (cid:10) r (cid:11) (a.u.) 3.5 3.5 3.7 2.4 † Intensities given in units of 10 Wcm − TABLE II. Average distance from the nucleus of electron 1 atthe time electron 1 tunnel-ionizes, (cid:10) r (cid:11) . Between the time shortly after recollision takes place andthe asymptotic time, the electric field mainly affects thex-component and not the y-component of the momen-tum of electron 1. Moreover, at the smaller intensitiesconsidered, in the time interval following recollision, theelectric field does not affect significantly the magnitudeof the x-component of the electron 1 momentum; thiscomponent is mainly determined by the vector potentialat the recollision time. Given the above, it is enoughto find how the momentum of electron 1 (roughly equalto the x-component) and its y-component change fromjust before to just after the time of recollision due to therecollision itself. This will allow us to understand thefeatures of the distribution P
DI1 , asym ( φ ) of the final angle φ . A measure of the strength of a recollision is the an-gle θ , where cos θ = p , bef · p , aft / ( | p , bef || p , aft | ). Thatis, θ , is the angle between the momentum of electron 1just before and just after the time of recollision. Themomentum just before the time of recollision is roughlyalong the x-axis (polarization axis). Thus, θ is the angleof the momentum of electron 1 after the time of recolli-sion with respect to the x-axis. θ = 180 ◦ corresponds to a“head on” collision and complete backscattering. θ = 0 ◦ corresponds to forward scattering and thus to almost nochange due to the recollision.In Fig. 4, we show for each laser field considered inthis work, what is the probability for electron 1 to es-cape with a final angle φ and a scattering angle θ . It isshown that very strong recollisions, i.e. θ = 180 ◦ , takeplace only when electron 1 escapes with a momentumthat has a very small y-component, i.e. φ is around90 ◦ . However, even when electron 1 escapes with φ around 90 ◦ , it is more likely that a weak recollisiontakes place, i.e. θ = 0 ◦ , rather than a strong one with θ = 180 ◦ . Moreover, when electron 1 ionizes with mo-menta that have larger y-components with φ ∈ [45 ◦ , ◦ ]and φ ∈ [90 ◦ , ◦ ] the scattering angles θ are on averagesmaller than 90 ◦ . That is, in most cases, electron 1 ion-izes at glancing angles θ following recollision. Moreover,a comparison of the values of θ at 2 × Wcm − andat 3.8 × Wcm − , see Fig. 4(a) and (d), clearly showsthat overall the recollision is stronger at the smaller in-tensity. In addition, a comparison of the values of θ at2 × Wcm − and at 2 × Wcm − with χ = 0 . χ = 0. Thus, electron 1 escapesat glancing angles following recollision. FIG. 4. Double differential probability of electron 1 to have ascattering angle θ and a final angle φ at (a) 2 × Wcm − ,(b) 2 × Wcm − with the magnetic field switched-off, (c)2 × Wcm − with χ = 0 .
05 and (d) 3.8 × Wcm − . E. Asymmetric transverse electron 1 positionshortly before recollision
We first explain how the y-component of the momen-tum of electron 1 changes from just before to just afterthe time of recollision, due to the recollision itself, whenthe magnetic field is switched-off at 2 × Wcm − . Wefind that doubly-ionized events are equally likely to havepositive or negative y-component of the momentum andof the position of electron 1 just before the time of recol-lision, see Fig. 3(b2) an (c2). Moreover, we find that they-component of the final momentum of electron 1 is pos-itive (negative) depending on whether the y-componentof the position of electron 1 is negative (positive) just be-fore the time of recollision. The reason is that electron 1ionizes at glancing angles. The direction of the Coulombattraction of electron 1 from the nucleus just before thetime of recollision determines whether just after the timeof recollision, as well as at asymptotically large times, they-component of the momentum of electron 1 is positiveor negative. Indeed, we find that doubly-ionized eventswith r , t r > DI1 , asym , whiledoubly-ionized events with r , t r < DI1 , asym and cancel each other out.However, when the magnetic field is switched-on at2 × Wcm − , most (59%) doubly-ionized events haver , t r < non-dipole recollision gated ionization , since out of theselatter events 61% have both p , t r > , t < × Wcm − with χ = 0 and χ = 0 .
05, respec-tively, and in (c4) at 3.8 × Wcm − . As for the casewhen the magnetic field is switched-off, when the mag-netic field is switched-on electron 1 ionizes at glancingangles. Therefore, we find that doubly-ionized eventswith r , t r > DI1 , asym anddoubly-ionized events with r , t r < DI1 , asym . However, the latter events are 59%of all doubly-ionized events and thus overall P DI1 , asym haspositive values.From the above it also follows that a small width ofP(p , t r ) affects only doubly-ionized events with small y-components of the electron 1 final momenta. On theother hand, a large width of P(p , t r ) just before thetime of recollision affects doubly-ionized events with y-components of the electron 1 final momenta ranging fromsmall to large. However, doubly-ionized events with largey-components of the electron 1 momenta correspond tosmaller φ in P DI1 , asym . Indeed, this is clear in Fig. 5 whencomparing the distribution of the y-component of the fi-nal momentum of electron 1 in the smaller φ of 27 ◦ withthe one in the larger φ of 81 ◦ . The large y-componentsof the momenta of electron 1 are a result of the recolli-sion. This is the case since for each intensity, P(p , t r ) aresimilar for all φ bins, see Fig. 5. P(p , t r ) is depicted inFig. 3(b1), (b3) and (b4) for three different laser fields.Most doubly-ionized events just before the time of rec- FIG. 5. Double differential probability of the tunneling elec-tron to have a y-component of the momentum equal to p , t r at the time of recollision and equal to p at the asymptotictime for two different φ s at 2 . × W / cm . φ is binned inintervals of 18 ◦ . ollision have positive y-component of the momentum ofelectron 1. Thus, the width of P(p , t r ) is also a measureof the shift towards positive y-components of the finalmomenta of electron 1.Finally, we compare the width of the distribution of thex-component of the momentum of electron 1 at intensi-ties where strong recollisions prevail with the width atintensities where soft ones do. We find that this width islarger just before and just after the time of recollision aswell as at asymptotically large times at intensities wheresoft recollisions prevail. Indeed, for strong recollisions,the x-component of the momentum of electron 1 is de-termined mainly by the vector potential at the time of recollision. However, for soft recollisons this is not quitethe case since electron 1 only transfers a small amount ofits energy to electron 2. In addition, for soft recollisions,the time of recollison has a much broader range of values,see ref.[18].We now combine the width of the asymmetry of P(p )with the width of the distribution of the x-componentof the momentum of electron 1. It then follows thatP DI1 , asym ( φ ) should have higher values over a larger rangeof φ at 2 × Wcm − (stronger recollision) and smallervalues at 2 × Wcm − with χ = 0 .
05 (softer recolli-sion). Indeed, this is the case as shown in Fig. 3(d1)-(d4).
IV. AVERAGE SUM ELECTRON MOMENTAIN DOUBLE IONIZATION
In ref.[18], we have shown that in double ionizationthe ratio (cid:10) p + p (cid:11) DI / (cid:104) p y (cid:105) SI is maximum and roughlyequal to eight at intensities 1.3 × Wcm − and 2 × Wcm − , see Fig. 6. (cid:10) p + p (cid:11) is the average sum of thetwo electron momenta along the propagation direction ofthe laser field, while (cid:104) p y (cid:105) SI is the corresponding averageelectron momentum in single ionization. In Fig. 6(a), (cid:10) p + p (cid:11) DI / (cid:104) p y (cid:105) SI is shown to decrease with increas-ing intensity, for the intensities considered. Moreover,in Fig. 6(b), it is shown that it is (cid:10) p (cid:11) DI of the tun-neling electron that contributes the most to (cid:10) p + p (cid:11) DI for the intensities considered. The ratio (cid:10) p (cid:11) DI / (cid:104) p y (cid:105) SI has surprisingly large values at intensities smaller thanthe intensities satisfying the criterion for the onset ofmagnetic-field effects β ≈ (cid:104) p y (cid:105) SI increases from 0.0035 a.u. at 0.7 × Wcm − to 0.028 a.u. at 4.8 × Wcm − , see Fig. 6(a)and ref.[18]. The small values of the average electronmomentum in single ionization and the increase of thisaverage with intensity are in accord with the effect of the F B force. The latter increases with intensity, since themagnetic field increases. The increase of (cid:104) p y (cid:105) SI with in-tensity has been addressed in several experimental andtheoretical studies [13–17].In what follows, we show that non dipole recolli-sion gated ionization accounts for the large values of (cid:10) p (cid:11) DI / (cid:104) p y (cid:105) SI and thus for the large average sum of thetwo electron momenta along the propagation direction ofthe laser field at smaller intensities of 1.3-2 × Wcm − .To do so, we express (cid:10) p iy (cid:11) DI as (cid:10) p iy (cid:11) DI = (cid:90) ◦ ◦ (cid:10) p iy ( φ ) (cid:11) DI P DIi ( φ ) dφ, (3)with i = 1 , non-dipole recollision gated ion-ization accounts for the asymmetry in P DI1 ( φ ). Next,we investigate the influence of the magnetic field on (cid:10) p iy ( φ ) (cid:11) DI . To do so, (cid:10) p ( φ ) (cid:11) DI of the tunnel electron, FIG. 6. (a) The average sum of the two electron momenta (cid:10) p + p (cid:11) DI in double ionization (black dot-dashed line withopen circles), twice the average electron momentum (cid:104) p y (cid:105) SI in single ionization (black solid line with circles) and the ra-tio (cid:10) p + p (cid:11) DI / (cid:104) p y (cid:105) SI (red dotted line with triangles) as afunction of the intensity of the laser field. (b) The averagemomentum of the tunneling electron (cid:10) p (cid:11) DI (grey dot-dashedline with open circles) and the bound electron (cid:10) p (cid:11) DI (blackdot-dashed line with circles) in double ionization and the ra-tio (cid:10) p (cid:11) DI / (cid:104) p y (cid:105) SI (red dotted line with squares) as a functionof the intensity of the laser field. (cid:10) p ( φ ) (cid:11) DI of the bound electron and (cid:104) p y ( φ ) (cid:105) SI are plot-ted in Fig. 7(a), (b) and (c), respectively, at 2 × Wcm − with linearly polarized light and with the mag-netic field switched-on and off. It is shown that themagnitude of the average electron momentum increasesas a function of φ , both in double and in single ioniza-tion. This is evident mostly for φ around 0 ◦ and 180 ◦ .Moreover, it is clearly seen that the magnetic field hasno influence on any of the average electron momentaconsidered and that (cid:10) p iy ( φ ) (cid:11) DI = (cid:10) p iy (180 ◦ − φ ) (cid:11) DI and (cid:104) p y ( φ ) (cid:105) SI = (cid:104) p y (180 ◦ − φ ) (cid:105) SI . This is expected when themagnetic field is switched-off, since there is no preferreddirection of electron escape on the plane that is per-pendicular to the polarization direction (x-axis) of thelaser field. We find that similar results hold at 1.3 × Wcm − and at 3.8 × Wcm − . FIG. 7. (a) (cid:10) p (cid:11) DI of the tunneling electron, (b) (cid:10) p (cid:11) DI ofthe initially bound electron and (c) (cid:104) p y (cid:105) SI in single ionizationare plotted as a function of φ at 2 × Wcm − with themagnetic field switched-on and off. φ is binned in intervals of18 ◦ . † denotes a multiplication factor of 10 Wcm − . We also find (not shown) that the effect of the magneticfield on (cid:10) p ( φ ) (cid:11) DI is very small even at a more differ-ential level. Specifically, the dependence of (cid:10) p ( φ, t) (cid:11) DI on time is very similar both with the magnetic fieldswitched-on and off. The only difference is a small os- cillation due to the magnetic field. It is noted that (cid:10) p ( φ, t → ∞ ) (cid:11) DI = (cid:10) p ( φ ) (cid:11) DI . We find that similar re-sults (not shown) hold for the bound electron.Using Eq. (3), we can now explain why (cid:10) p (cid:11) DI is muchlarger than (cid:104) p y (cid:105) SI at intensities around 2 × Wcm − .When the magnetic field is switched-on, (cid:10) p ( φ ) (cid:11) DI doesnot change but P DI1 , asym ( φ ) does. P DI1 , asym ( φ ) is much widerthan P SIasym ( φ ) and than P DI2 , asym ( φ ) and has higher val-ues at 2 × Wcm − rather than at 3.8 × Wcm − .The higher values of P DI1 , asym ( φ ) over a wider range of φ compared to P SIasym ( φ ) and P DI2 , asym ( φ ) result in smaller φ and thus larger (cid:10) p ( φ ) (cid:11) DI (Fig. 7) having a significantweight in Eq. (3). V. CONCLUSIONS
We account for non-dipole effects in double ionization.We show that the recollision and the magnetic field acttogether as a gate. This gate gives rise to a distribution ofthe y-component of the tunneling electron initial momen-tum, P(p , t ), which is shifted towards negative values;negative values in the y-axis are opposite to the propa-gation direction of the laser field. The term non-dipolerecollision-gated ionization was adopted to describe thiseffect. We show that this asymmetry in P(p , t ) mapsin time to an asymmetry of the transverse electron 1momentum just before the time of recollision, i.e. toan asymmetry in P(p , t r ). Namely, the y-component ofthe momentum of the tunneling electron is shifted to-wards positive values just before the time of recollision.Moreover, the asymmetry in P(p , t ) maps in time to anasymmetry of the transverse electron 1 position just be-fore the time of recollision. Namely, the y-component ofthe position of the tunneling electron is shifted towardsnegative values just before the time of recollision.The above asymmetries combined with the tunnelingelectron escaping at glancing angles following a recolli-sion give rise to an asymmetry in P DI1 ( φ ) with respectto φ = 90 ◦ . The latter is the probability distribution ofelectron 1 to escape with an angle φ . We find the asym-metry in P DI1 ( φ ) to be more significant, i.e. higher valuesof P DI1 , asym ( φ ) over a wider range of φ , in double ioniza-tion compared to single ionization. Moreover, we findthat higher values of P DI1 , asym ( φ ) over a wider range of φ result from larger widths of P(p , t r ) just before the timeof recollision. The latter width depends on the intensityand the ellipticity of the laser pulse and we show thatit is related to Coulomb focusing. We also show that itis the asymmetry in P DI1 ( φ ) over a wide range of φ thataccounts for the large values of the average transverseelectron 1 momentum and thus of the large average sumof the two electron momenta at smaller intensities. Eventhough not as pronounced, we find these features of theprobability distribution P DI1 ( φ ) of the tunneling electronto also be present in an experimentally accessible observ-able. Namely, the probability distribution for electron 1or 2 to escape with an angle φ . This observable effectof the non-dipole recollision-gated ionization can be mea-sured by future experiments.Finally, in the current work we show that magneticfield effects cause an offset of the transverse momentumand position of the recolliding electron just before recol-lision takes place. We show how these asymmetries leadto asymmetries in observables experimentally accessible.We conjecture that this is a general phenomenon not re-stricted to magnetic field effects. Namely, these observ- able asymmetries will be present in any process that hastwo delayed steps and allows an electron to gain an offsetbefore recollision takes place. VI. ACKNOWLEDGMENTS
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