Electron emission perpendicular to polarization direction in laser assisted XUV atomic ionization
EElectron emission perpendicular to polarization direction in laserassisted XUV atomic ionization
A. A. Gramajo and R. Della Picca
Centro At´omico Bariloche (CNEA) and CONICET, 8400 Bariloche, Argentina
D. G. Arb´o
Institute for Astronomy and Space Physics IAFE (UBA-Conicet), Buenos Aires, Argentina (Dated: November 8, 2018)
Abstract
We present a theoretical study of ionization of the hydrogen atom due to an XUV pulse in thepresence of an IR laser with both fields linearly polarized in the same direction. In particular, westudy the energy distribution of photoelectrons emitted perpendicularly to the polarization direc-tion. By means of a very simple semiclassical model which considers electron trajectories born atdifferent ionization times, the electron energy spectrum can be interpreted as the interplay of intra- and intercycle interferences. The intracycle interference pattern stems from the coherent superpo-sition of four electron trajectories giving rise to (i) interference of electron trajectories born duringthe same half cycle ( intrahalfcycle interference) and (ii) interference between electron trajectoriesborn during the first half cycle with those born during the second half cycle ( interhalfcycle interfer-ence). The intercycle interference is responsible for the formation of the sidebands. We also showthat the destructive interhalfcycle interference for the absorption and emission of an even numberof IR laser photons is responsible for the characteristic sidebands in the perpendicular directionseparated by twice the IR photon energy. We analyze the dependence of the energy spectrum onthe laser intensity and the time delay between the XUV pulse and the IR laser. Finally, we showthat our semiclassical simulations are in very good agreement with quantum calculations within thestrong field approximation and the numerical solution of the time-dependent Schr¨odinger equation.
PACS numbers: 32.80.Rm, 32.80.Fb, 03.65.Sq a r X i v : . [ phy s i c s . a t o m - ph ] M a r . INTRODUCTION More than twenty years have passed since the publication of one of the first theoreticalpredictions of sidebands in laser assisted photoelectric effect (LAPE) [1]. The simultaneousabsorption of one high-frequency photon and the exchange of several additional photonsfrom the laser field lead to equally spaced sideband peaks in the photoelectron (PE) spectra.Since this pioneer work, a lot of experiments have been performed in this area. Typically,the XUV+ IR field was firstly obtained through high-order harmonic generation using theoriginal IR laser field as its source [2–5]. In contrast to this kind of XUV radiation generation,the monochromaticity of the femtosecond XUV pulse from a free electron laser (FEL) enablesthe study of two-color multiphoton ionization without additional overlapping contributionsfrom neighboring harmonics [6–11].Several studies have been performed to analyze the PE emission in LAPE dependingon different features of the fields: Temporal duration, intensity, polarization state, etc.For example, the temporal overlap between the XUV and IR pulses establishes two welldistinguished regimes according to whether the XUV pulse duration is greater or less thanthe laser optical period [8, 12, 13]. Whereas in the former, the laser intensity is directlyrelated, in a non-trivial way, to the intensity of the appearing sideband peaks in the PEspectrum [8, 14, 15], the latter has been used to characterize the shape and duration of anIR laser pulse with a technique called “streak camera” [16–19]. Furthermore, the variationof the polarization states of each field gives rise to dichroic effects in the PE spectrum, whichopens the door to the control of the electronic emission [6, 9, 20–24].Hitherto, most of the PE spectra have been measured with angle integrated resolution.Only very recently it has been possible to measure angularly resolved PE spectra [11, 15, 23–26], which is fundamental to achieve a complete understanding of LAPE process. In contrastto experiments, theoretical analysis restricted to fixed emission angles do not present majordifficulties. Most of the theories of LAPE processes are based on the strong field approxima-tion (SFA) [27–29]. For example, the soft photon approximation (SPA) [14], derived fromthe SFA in the velocity gauge for infinitely long XUV and IR pulses, depicts satisfactorilythe experimental results [4, 6, 7, 10, 11, 30]. Besides, the analytic angle-resolved PE spectraderived by Kazansky et al. [31, 32] and Bivona et al. [33] are based on simplificationsof the temporal integration within the SFA. Furthermore, in our previous work [34], we2ave presented a semiclassical approach that describes the XUV+IR multiphoton ionizationalong the direction of polarization of both fields in very good agreement with the results bysolving ab initio the time dependent Schr¨odinger equation (TDSE). In that work, we haveinterpreted the PE spectrum as the coherent superposition of electron trajectories emittedwith the same optical cycle leading to an intracycle interference pattern that modulates thesidebands, which can be thought as a consequence of the intercycle interference of electrontrajectories born at different optical cycles.To the best of our knowledge, LAPE ionization has not been studied in detail for emissiondirections different from the polarization axis. Furthermore, Haber et al. have noted theneed for a more comprehensive theoretical description due to the poor agreement betweentheoretical and experimental PE angular distributions for the two-color two-photon abovethreshold ionization [25, 35]. Several theories, like SPA, predict no emission in the directionperpendicular to the polarization axis. However, Bivona et al. envisaged non-zero emissionfor XUV ionization of hydrogen by short intense pulses [33]. Therefore, in the present paper,we extend our previous work [34] for LAPE from H(1s) to study the emission in the directionperpendicular to the polarization axis of both XUV and laser fields. In contrast to thecase of forward emission, we find that transversal emission has relatively low probabilities,i.e., the PE energy range is highly reduced. However, we observe that the PE emission isnon-vanishing in agreement with Bivona et al. [33]. Moreover, the PE emission is due tothe absorption and emission of an odd number of IR photons following one XUV photonabsorption, whose photoionization line is absent in the PE spectrum. Hence, PE spectrain the perpendicular direction can hardly be observed for laser intensities lower than 10 W/cm . Experimental measurements with strong lasers would be highly desirable in orderto corroborate the present study. A recent work by D¨usterer et al. [11] shows that they canbe attainable now.The paper is organized as follows: In Sec. II, we describe the semiclassical model (SCM)used to calculate the photoelectron spectra for the case of laser-assisted XUV ionizationperpendicular to the polarization direction, which leads to simple analytical expressions. InSec. III, we present the results and discuss over the comparison among of the SCM and theSFA outcomes and the ab initio calculation of the TDSE. Concluding remarks are presentedin Sec. IV. Atomic units are used throughout the paper, except when otherwise stated.3 I. THEORY OF THE SEMICLASSICAL MODEL
We study the ionization of an atomic system interacting with an XUV pulse assisted byan IR laser. In the single-active-electron (SAE) approximation the TDSE reads i ∂∂t | ψ ( t ) (cid:105) = (cid:104) H + H int ( t ) (cid:105) | ψ ( t ) (cid:105) , (1)where H = (cid:126)p / V ( r ) is the time-independent atomic Hamiltonian, whose first termcorresponds to the electron kinetic energy, and its second term to the electron-core Coulombinteraction. The second term in the right-hand side of Eq. (1), i.e, H int = (cid:126)r. (cid:126)F X ( t ) + (cid:126)r. (cid:126)F L ( t ),describes the interaction of the atom with both time-dependent XUV [ (cid:126)F X ( t )] and IR [ (cid:126)F L ( t )]electric fields in the length gauge.The electron initially bound in the atomic state | φ i (cid:105) is emitted with final momentum (cid:126)k and energy E = k / | φ f (cid:105) belonging to the continuum. Then, thephotoelectron momentum distributions can be calculated as dPd(cid:126)k = | T if | (2)where T if is the T-matrix element corresponding to the transition φ i → φ f .Within the time-dependent distorted wave theory, the transition amplitude in the priorform and length gauge is expressed as T if = − i (cid:90) + ∞−∞ dt (cid:104) χ − f ( (cid:126)r, t ) | H int ( (cid:126)r, t ) | φ i ( (cid:126)r, t ) (cid:105) (3)where φ i ( (cid:126)r, t ) = ϕ i ( (cid:126)r ) e iI p t is the initial atomic state, I p the ionization potential, and χ − f ( (cid:126)r, t )is the distorted final state [36, 37]. The SFA neglects the Coulomb distortion in the finalchannel produced by the ejected-electron state due to its interaction with the residual ion.Hence, we can approximate the distorted final state with the Volkov function, which is thesolution of the Schr¨odinger equation for a free electron in an electromagnetic field [38], i.e., χ − f = χ Vf , where χ Vf ( (cid:126)r, t ) = (2 π ) − / exp (cid:20) i (cid:0) (cid:126)k + (cid:126)A ( t ) (cid:1) .(cid:126)r + i (cid:90) ∞ t dt (cid:48) (cid:0) (cid:126)k + (cid:126)A ( t (cid:48) ) (cid:1) (cid:21) (4)and the vector potential due to the total external field is defined as (cid:126)A ( t ) = − (cid:82) t dt (cid:48) [ (cid:126)F X ( t (cid:48) ) + (cid:126)F L ( t (cid:48) )]. 4e consider the atomic photoionization due to a short XUV pulse assisted by an IRlaser where both of them are linearly polarized in the same direction ˆ z . For simplicity,we consider a hydrogen atom initially in the ground state, however, the present study canbe easily generalized to any atom within the SAE approximation. In the present work,we restrict the photoelectron momentum (cid:126)k = k z ˆ z + k ρ ˆ ρ (in cylindrical coordinates) to thedirection perpendicular to the polarization axis, i.e., k z = 0 and k ρ ≥
0. The case of emissionparallel to the polarization axis, i.e., k ρ = 0, was studied recently in [34].With the appropriate choice of the IR and XUV laser parameters considered, we canassume that the energy domain of the LAPE processes is well separated from the domain ofionization by an IR laser alone. In other words, the contribution of IR ionization is negligiblein the energy domain where the absorption of one XUV photon takes place. Besides, if we setthe general expression of the XUV pulse of duration τ X as (cid:126)F X ( t ) = ˆ zF X ( t ) cos( ω X t ), where F X ( t ) is a slowly nonzero varying envelope function, the matrix element can be written as T if = − i (cid:90) t + τ X t dt d z (cid:0) (cid:126)k + (cid:126)A ( t ) (cid:1) F X ( t ) e iS ( t ) (5)with ( t , t + τ X ) the temporal interval where F X ( t ) is nonzero. S ( t ) is the generalizedaction S ( t ) = − (cid:90) ∞ t dt (cid:48) (cid:34) (cid:0) (cid:126)k + (cid:126)A ( t (cid:48) ) (cid:1) I p − ω X (cid:35) (6)and the z -component of the dipole element for the 1 s state is d z ( (cid:126)v ) = − iπ / (2 I p ) / ˆ z · (cid:126)v (cid:2) v + (2 I p ) (cid:3) . (7)In Eq. (5) we have used the rotating wave approximation which accounts, in this case, forthe absorption of only one XUV photon and neglects, thus, the contribution of XUV photonemission. As the frequency of the XUV pulse is much higher than the IR laser one, theXUV contribution to the vector potential can be neglected [39, 40], regarding that the XUVintensity is not much higher than the laser one. For the sake of simplicity, we restrict ouranalysis to the case where the XUV pulse duration is a multiple of half the IR optical cycle,i.e., τ X = N T L = 2 N π/ω L , where T L and ω L are the laser period and the frequency of theIR laser, respectively, and 2 N is an integer positive number. During the temporal lapse theXUV pulse is acting, the IR electric field can be modeled as a cosine-like wave, hence, thevector potential can be written as (cid:126)A ( t ) = A L sin ( ω L t )ˆ z with A L = F L /ω L and F L theamplitude of the laser electric field. 5he SCM consists of solving the time integral Eq. (5) by means of the saddle pointapproximation [41–44], wherein the transition amplitude can be thought of as a coherentsuperposition of the amplitudes of all electron classical trajectories with final momentum (cid:126)k over the stationary points t s of the generalized action S ( t ) in Eq. (6) T if = (cid:88) t s √ π F X d z ( (cid:126)k + (cid:126)A ( t s )) | ¨ S ( t s ) | / exp (cid:20) iS ( t s ) + iπ (cid:16) ¨ S ( t s ) (cid:17)(cid:21) , (8)where ¨ S ( t ) = d S ( t ) /dt = − (cid:104) (cid:126)k + (cid:126)A ( t ) (cid:105) · (cid:126)F ( t ) and sgn denotes the sign function. Then, fromthe saddle-point equation, i.e., ˙ S = dS ( t s ) /dt = 0, the ionization times fulfill the relation A ( t s ) + k ρ = v , (9)where v = (cid:112) ω X − I p ) is the initial velocity of the electron at the ionization time. In ion-ization by an IR laser alone, release times are complex due to the fact that the active electronescapes the core via tunneling through the potential barrier formed by the interaction be-tween the core and the external field, i.e., V ( r )+ (cid:126)r. (cid:126)F L ( t ). Contrarily, in LAPE, real solutionsof Eq. (9) correspond to real ionization times t s . From Eq. (9), the domain of allowed clas-sical trajectories perpendicular to the polarization axis is (cid:112) v − A L ≤ k ρ ≤ v whether v ≥ F L /ω L . Non-classical trajectories with complex ionization times have a momentum k ρ outside the classical domain. In this work, we neglect the small weight of non-classicaltrajectories with complex ionization times since its imaginary parts give rise to exponentialdecay factors.The ionization times that verify Eq. (9) are shown schematically in Fig. 1 for one IR opti-cal cycle. As we can observe, there are four ionization times per optical cycle and, therefore,the total number of interfering trajectories with the same final momentum perpendicularto the polarization axis is 4 N . A quick analysis of equations (8) and (9) indicates that theperiodicity for the solution of Eq. (9) is π/ω L , which is half of that corresponding to theparallel emission case [34]. Therefore, the sum over the emission times can be performedalternatively over 2 N half cycles with two emission times in each of them. They are theearly ionization time t ( m, and the late ionization time t ( m, corresponding to the m -thoptical half cycle, where t ( m,β ) = t (1 ,β ) + π ( m − /ω L with β = 1 ,
2. The expressions for the6
IG. 1. Emission times (solutions of Eq. (9)) as the intersection of the two curves A ( t ) = A L sin ( ω L t ) in red solid line and v − k ρ in black solid line for one IR optical cycle. In thisparticular case, the XUV pulse starts when the potential vector vanishes. ionization times can be easily derived from Eq. (9), t (1 , = 1 ω L sin − (cid:104)(cid:113) ( v − k ρ ) /A L (cid:105) (10a) t (1 , = πω L − t (1 , . (10b)From Eq. (6), the generalized action and its second derivative at the time t s for electrontrajectories along the perpendicular direction can be written as S ( t s ) = (cid:16) k ρ I p + U p − ω X (cid:17) t s − U p ω L sin(2 ω L t s ) (11)and ¨ S ( t s ) = F L A L sin(2 ω L t s ) / , (12)respectively, where U p = ( F L / ω L ) is the ponderomotive energy of the oscillating electrondriven by the laser field.According to equations (7) and (10), the dipole elements d z evaluated at emission timesof consecutive half cycles differ in a sign, i.e., d z (cid:0) k ρ ˆ ρ + ˆ zA ( t ( m,β ) ) (cid:1) = √ A L iπω X sin( ω L t ( m,β ) )= − d z (cid:0) k ρ ˆ ρ + ˆ zA ( t ( m +1 ,β ) ) (cid:1) . (13)7ence, the odd and even half cycles have opposite contributions. Including equations (10)and (13) into Eq. (8), the ionization probability of Eq. (2) can be written as | T if | = Γ( k ρ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) m =1 2 (cid:88) β =1 ( − m exp (cid:20) iS ( t ( m,β ) ) + iπ (cid:16) ¨ S ( t ( m,β ) ) (cid:17)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (14)Eq. (14) can be interpreted as the coherent sum over interfering trajectories decomposed intothose associated with the two release times within the same half cycle (inner summation)and those associated with release times in the 2 N different half cycles (outer summation).The ionization probability Γ( k ρ ) contains all identical factors for all subsequent ionizationtrajectories which depend on the final momentum k ρ , i.e.,Γ( k ρ ) = 4 F X πω X ω L (cid:113) v − k ρ (cid:113) k ρ − v + A L . (15)In the same way as in previous works [34, 45, 46] and after a bit of algebra, it can beshown that N (cid:88) m =1 2 (cid:88) β =1 ( − m e [ iS ( t ( m,β ) )+ iπ sgn ( ¨ S ( t ( m,β ) ) )] = 2 N (cid:88) m =1 e i ( ¯ S m + mπ ) cos (cid:18) ∆ S m π (cid:19) (16)where ¯ S m = [ S ( t ( m, ) + S ( t ( m, )] / S + m ( ˜ S/
2) is the average action of the two tra-jectories released in the m -th half cycle, with ˜ S = (2 π/ω L )( E + I p + U p − ω X ) and S anunimportant constant that will be canceled out when the absolute value is taken in Eq. (14).The accumulated action between the two release times t ( m, and t ( m, within the same m -thhalf cycle, ∆ S m = S ( t ( m, ) − S ( t ( m, ) in Eq. (16), is given by∆ S = ˜ S (cid:26) π sin − (cid:104)(cid:113) ( v − k ρ ) /A L (cid:105) − (cid:27) − ω L (cid:113) v − k ρ (cid:113) k ρ − v + A L , (17)where we have omitted the subscript m , since it is independent of which half-cycle is con-sidered. Finally, due to the linear dependence of the average action ¯ S m on the cycle number m and the factorization of the cosine factor in the right side of Eq. (16), the ionization8robability can be easily written as | T if | = 4Γ( k ρ ) cos (cid:18) ∆ S π (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) intrahalfcycle (cid:34) sin ( N ˜ S/ S/ (cid:35) (cid:124) (cid:123)(cid:122) (cid:125) interhalfcycle . (18a)= 4Γ( k ρ ) 4 cos (cid:18) ∆ S π (cid:19) sin (cid:32) ˜ S (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) intracycle (cid:34) sin ( N ˜ S/ S/ (cid:35) (cid:124) (cid:123)(cid:122) (cid:125) intercycle (18b)Equations (18a) and (18b) indicate that the photoelectron spectrum can be factorizedin two different ways. On one hand, (i) the factorization in Eq. (18a) highlights thecontribution of the pair of electron trajectories within the same half cycle ( intrahalfcy-cle interference), governed by the factor G ( k ρ ) = cos (∆ S/ π/ interhalfcycle interfer-ence) described by the factor H ( k ρ ) = (cid:104) sin ( N ˜ S/ / cos ( ˜ S/ (cid:105) . On the other hand, (ii)the factor F ( k ρ ) = 4 cos (∆ S/ π/
4) sin ( ˜ S/
4) stemming from the contribution of thefour trajectories within the same optical cycle ( intracycle interference), and the factor B ( k ρ ) = sin ( N ˜ S/ / sin ( ˜ S/
2) stemming from trajectories released at different cycles ( in-tercycle interference, in correspondence with previous analysis of Eq. (23) in [34]). Whereasin (i) the interference of 2 N half cycles is highlighted giving rise to the intrahalf- and inter-halfcycle factors, in (ii) we think of the coherent contributions of N different optical cyclessplitting the contribution in intra- and intercycle interference patterns. Obviously, the twodifferent factorizations give rise to the same results, i.e., G ( k ρ ) H ( k ρ ) = F ( k ρ ) B ( k ρ ).In Fig. 2(a), we plot the intrahalfcycle function G ( k ρ ) and the interhalfcycle H ( k ρ ) for aXUV laser pulse duration τ X = 2 T L as a function of the energy. Whereas the intrahalfcyclefactor G ( k ρ ) exhibits a non-periodic oscillation, the interhalfcycle H ( k ρ ) is periodic in thefinal photoelectron energy with peaks at positions E (cid:96) = k ρ / E (cid:96) = ω X + (2 (cid:96) + 1) ω L − I p − U p . (19)with (cid:96) = ..., − , − , , , , ... . In fact, in the limit of infinitely long XUV and IR pulses,lim N →∞ H ( k ρ ) = (cid:80) l δ ( ˜ S/ π/ − (cid:96)π ), the ionization probability vanishes unless the finalenergy satisfies Eq. (19) which gives the positions of the different sidebands. We see that theenergy difference between two consecutive sidebands is 2 ω L and not ω L as for the emissionin the direction parallel to the polarization axis [34]. From Eq. (19), it is easy to see that9 -5 -4 -3 -2 -1 0 1 2 3 4 5 Number of absorbed / emitted photons H ( k ρ ) ( a r b . un it s ) G ( k ρ ) ( a r b . un it s ) InterhalfcycleIntrahalfcycleIntercycle (b) G ( k ρ ) H ( k ρ ) ( a r b . un it s ) E (a.u.)
IntrahalfcycleIntracycleF(k ρ )B(k ρ )=G(k ρ )H(k ρ ) FIG. 2. Buildup of the interference pattern following the SCM for N = 2. (a) Intrahalfcycleinterference pattern given by G ( k ρ ) in red line, intercycle pattern given by the factor B ( k ρ ) in greendotted line and interhalfcycle pattern given by the factor H ( k ρ ) in blue line. (b) Intracycle patterngiven by the factor F ( k ρ ) in grey line and total interference pattern F ( k ρ ) B ( k ρ ) = G ( k ρ ) H ( k ρ ) inblack line. Vertical lines depict the positions of the SBs E (cid:96) of Eq. (19). The IR laser parametersare F L = 0 . ω L = 0 .
05, and τ L = 5 T L and the XUV frecuency is ω X = 1 . only odd numbers (2 (cid:96) + 1) of laser photons can be absorbed or emitted together with theabsorption of one XUV photon. Due to the lack of sidebands for the absorption or emissionof an even number of laser photons, the absorption of only one XUV photon alone (in theabsence of absorption or emission of IR photons) is forbidden. The intrahalfcycle patterndisplays few oscillations with maxima depending on the electron kinetic energy. These canbe easily calculated through ∆ S = (2 q − / π , with integer q . In Fig. 2(b), we plot the totalinterference pattern corresponding to an XUV pulse of duration τ X = 2 T L , and the intracyclefactor F ( k ρ ). For the sake of comparison, we reproduce in Fig 2(b) the intrahalfcycle factor G ( k ρ ) of Fig 2(a). The multiplication of both intrahalfcycle and interhalfcycle factors, i.e., G ( k ρ ) H ( k ρ ), is displayed in Fig. 2(b), where we observed how the intrahalfcycle interference10attern [ G ( k ρ )] works as a modulation of the intracycle interference pattern [ F ( k ρ )] and thelatter does the same with the sidebands (intercycle interference pattern).On the other hand, Eq. (18b) shows that the photoelectron spectrum can be thought asthe intercycle pattern with peaks at positions E n = nω L + ω X − I p − U p modulated by theintracycle interference pattern given by the factor F ( k ρ ). Therefore, the lack of even-ordersidebands stems from the factor sin ( ˜ S/
4) into the intracycle factor F ( k ρ ) [see Eq. (18a)].The factor sin ( ˜ S/
4) reflects the fact that the dipole element has opposite signs for thetwo different half cycles into the same optical cycle [see Eq. (13)] giving rise to destructiveinterference between the contribution of the two electron trajectories of the first half cyclewith the corresponding to the second half cycle of every optical cycle during the time intervalthat the XUV pulse is on. Contrarily, for emissions in the parallel direction, whereas theionization during one of the two half cycles contributes to emissions in one direction (forwardor backward), the other half cycle will contribute to the opposite direction [34]. Therefore,no interference is produced for parallel emissions allowing to all peaks separated by ω L . III. RESULTS AND DISCUSSION
At the time of probing the general conclusion of the SCM that the ionization probabilityof electrons emitted perpendicularly to the polarization axis of the XUV and the laser pulsecan be factorized in two different contributions in two different ways: (i) intrahalfcycle and interhalfcycle interferences [Eq. (18a)] and (ii) intracycle and intercycle interferences [Eq.(18b)], we need to compare the outcome of SCM calculations with quantum ones. We haveperformed calculations within the SFA and TDSE methods, which have been extensivelycovered in the literature and in our previous work [34] and we do not repeat here. For theSFA calculating method, please refer, for example, to Refs. [31–33, 37, 45, 46], and for the abinitio numerical solutions of the TDSE we employ the generalized pseudospectral methodcombined with the split-operator representation of the time-evolution operator, which isthoroughly explained in the literature (see, for example, [47–49]). For the computationalfeasibility of the SFA and TDSE calculations, we take the XUV pulse and the IR laser fieldmodeled as (cid:126)F i ( t ) = F i ( t − t ib ) cos (cid:104) ω i (cid:16) t − ∆ i − τ L (cid:17)(cid:105) ˆ z, (20)11here i =L and X denote the IR laser and XUV pulses, respectively. The envelopes of theelectric fields in Eq. (20) were chosen as F i ( t ) = F i t/T i if 0 ≤ t ≤ T i T i ≤ t ≤ τ i − T i ( τ i − t ) /T i if τ i − T i ≤ t ≤ τ i (21)and zero otherwise, where T i = 2 π/ω i and τ i are the i − field period and pulse duration,respectively. It describes a central flattop region and linear one-cycle ramp on and rampoff. For the sake of simplicity, we suppose that the duration of both laser fields compriseinteger number of cycles, i.e., τ i = N i T i where N i is a positive integer. In addition, as wehave mentioned before, we also consider the case where τ X = N T L . We choose the originof the time scale as the beginning of the IR laser pulse, i.e., t Lb = 0, with no displacementof the laser pulse ∆ L = 0. In this way, the IR laser field is a cosine-like pulse centeredin the middle of the pulse, t = τ L /
2. In Eq. (20), the time delay of the XUV pulse withrespect to the laser pulse is ∆ X and t Xb = ∆ X + τ L / − τ X / t Xb . Therefore, for the sake of comparison of the ionization yield for different XUV pulsedurations, the active window should be in phase with the vector potential. For that, wedefine the module 2 π optical phase φ ≡ ω L t Xb = ω L ∆ X + ( N L − N ) π as the phase of thestarting time of the XUV pulse with respect to the vector potential (cid:126)A ( t ) [50].In the following, we probe the results of the SCM by comparing them to quantum sim-ulations. We consider the IR and XUV frequencies as ω L = 0 .
05 and ω X = 30 ω L = 1 . τ L = 5 T L and three different XUV pulses with durations τ X = T L / T L , and 2 T L ( i.e. N = 1 /
2, 1 and 2). In Figs. 3 and 4 we consider the cor-responding time delays ∆ X = T L / T L /
2, and T L , so that the optical phases are the same φ = π . In Fig. 3(a) and 3(b) we show results of the SFA and the numerical solution ofthe TDSE, respectively, for the same XUV and IR pulse parameters used in Fig. 2 with F X = F L = 0 .
05. The agreement among the SCM [Fig. 2], the SFA [Fig. 3(a)], and TDSE[Fig. 3(b)] energy distributions is very good since the effect of the Coulomb potential onthe energy spectrum for electron emission in the perpendicular direction is very small if notnegligible. However, the analysis of the effect of the Coulomb potential of the remaining core12
Number of absorbed / emitted photons
E (a.u.) d P / d E ( a r b . un it s ) (a) SFA τ x =T L /2 τ x =T L τ x =2T L (b) TDSE τ x =T L /2 τ x =T L τ x =2T L -2 0 2 0 200 400 600 A L ( t ) ( a r b . un it s ) F L ( t ) ( a r b . un it s ) t (a.u.) ∆ X FIG. 3. Photoelectron spectra in the perpendicular direction calculated within (a) the SFA and(b) the TDSE, for different XUV pulse durations τ X = T L / , T L , and 2 T L and respective timedelays ∆ X = T L / T L /
2, and T L . The XUV and IR parameters are the same as in Fig. 2 and F X = 0 .
05. Vertical lines depict the positions of the sidebands according to Eq. (19). on the electron yield deserves a thorough study, which is beyond the scope of this paper.As predicted in Eq. (18a), the intrahalfcycle interference pattern, calculated as the energydistribution for a XUV pulse duration of half a laser cycle, i.e., τ X = T L /
2, modulates theintracycle interference pattern, calculated as the energy distribution for a XUV pulse dura-tion of one laser cycle, i.e., τ X = T L . In the same way, the latter modulates the sidebandsin the energy distribution for a longer XUV pulse, i.e., τ X = 2 T L , as shown in Fig. 3(a) andFig. 3(b). For the latter case, (when the XUV pulse duration involves several periods of thelaser, i.e., τ X = 2 T L ), the positions of the sidebands obtained by the quantum calculations(SFA and TDSE) in Figs. 3(a) and 3(b) agree with the SCM expressed in Eq. (19). Asexpected, the energy spectra for the quantum SFA and TDSE calculations extend beyondthe classical limits E low = v / − U p = 0 . E up = v / FIG. 4. Photoelectron spectra in the perpendicular direction (in arbitrary units) calculated atdifferent laser field strengths within the SCM (a, d, and g), the SFA (b, e, and h), and the TDSE(c, f, and i). The XUV pulse durations are τ X = T L / τ X = T L (d-f), and τ X = 2 T L (g-i).The other XUV and IR parameters as in previous figures. In green dotted line we show the classicalboundaries and in black dotted line the E (cid:96) values given by Eq. (19). We have investigated over the dependence of the energy distribution for photoelectronsemitted in the direction perpendicular to the polarization axis on the intensity of the XUVpulse. We have checked that the total (angle- and energy-integrated) ionization probabilityis essentially proportional to the intensity of the XUV pulse whereas the overall shape ofthe energy distribution in the transversal direction (Fig. 3) remains rather unchanged whenvarying the intensity of the XUV pulse (not shown). Contrarily, the intensity of the IR laserhas a strong effect on the shape of the energy distribution. In Fig. 4 we show calculationsof the energy distribution in the perpendicular direction within the SCM (a), (d), and (g),the SFA in (b), (e), and (h), and the TDSE in (c), (f), and (i), for laser field intensities14rom I L = 0 up to 8 . × W/cm ( F L = 0 . τ X = T L / T L , and 2 T L inFigs. 2 and 3 are cuts of Fig. 4 at I L = 8 . × W/cm . The classical boundaries E low and E up drawn in dotted lines exactly delimit the SCM spectrogram of Fig. 4(a), (d) and(g), as expected. For the case where τ X = T L / S = (2 q − / π with q = − , − , , , ... [see Eq. (17)]. For the cases τ X = T L (secondcolumn of Fig. 4), we observed in Figs. 4(d), (e), and (f) that the intrahalfcycle interferencepatterns are flanked by stripes of zero or near zero probability distribution corresponding tothe zeros of the factor sin( ˜ S/
4) in the intracycle factor F ( k ρ ), i.e., ˜ S/ nπ , which gives E = ω X − I p − U p + 2 nω L . The slope of these minima is − U p /I L = − (2 ω L ) − and the energydifference between consecutive minima (and maxima) is 2 ω L . For the case of τ X = 2 T L (third column of Fig. 4), we see in Fig. 4 (g), (h), and (i) that the stripes of the probabilitydistribution become even thinner due to the effect of the destructive intercycle interferencefor energy values much different from the conservation energy for absorption of one XUVphoton and an odd number of IR laser photons [Eq. (19)]. Moreover, when we compare theposition of the maxima with Eq. (19), marked as black dotted lines in Fig. 4 (g), we seean excellent agreement (see also Fig. 3). The domain of the SFA and TDSE spectrograms(second and third row of Fig. 3) extend beyond the classical boundaries with smooth edges.The characteristic intrahalf- and intracycle stripes with negative slope reproduce very wellthe SCM predictions. In Figs. 4 (c), (f), and (i), the TDSE calculations exhibit a strongprobability distribution for high values of the laser intensity I L > ∼ . × W/cm in thelow energy region which almost does not overlap with the laser assisted XUV ionization forthe longer XUV duration cases, but for the τ X = T L / ω L (cid:28) I p . For this reason, we can confirm that the SFA is a morereliable method to deal with laser assisted photoemission copared to ATI by IR lasers [34].Therefore, except for the region where ionization by the laser field alone becomes important,SFA and TDSE spectrograms exhibit a very good agreement between them and resemble15he SCM calculations qualitatively well. The resulting energy stripes become thinner andmore pronounced as the duration of the XUV pulse increases, exhibiting the fact that theintrahalfcycle interference pattern modulates the intracycle pattern, which, at the sametime, modulates the sidebands (intercycle interference pattern). FIG. 5. Photoelectron spectra in the perpendicular direction (in arbitrary units) calculated at as afunction of the time delay ∆ X within the SCM [(a), (d), and (g)], the SFA [(b), (e), and (h)], andthe TDSE [(c), (f), and (i)]. The XUV pulse durations are τ X = T L / τ X = T L [(d)-(f)],and τ X = 2 T L [(g)-(i)]. The other XUV and IR parameters as in previous figures. In dotted linewe show the energy values of Eq. (22). So far, we have performed our analysis of the electron emission in the transverse directionfor optical phase φ = π (since N L is odd). In order to reveal how the intracycle interferencepattern changes with the time delay, we vary ∆ X in an optical cycle, so that φ varies from 0to 2 π . In Fig. 5(a) we show the intrahalfcycle interference pattern calculated for τ X = T L / π .The horizontal stripes show the independence of the intracycle interference pattern with the16ime delay, except for the discontinuity for energy values equal to E disc = 12 (cid:2) v − A ( t Xb ) (cid:3) . (22)For φ = 0 the discontinuity is situated at E disc = v / X = 3 T L / N = 1 / A ( t Xb = 3 T L ) = 0], which coincides with the classical boundary. Fig.5(a) shows us that as φ (and ∆ X ) varies, the discontinuity follows the shape of the squareof the vector potential, which means that the discontinuity is π -periodic in φ , contrarilyto the 2 π − periodicity in the case of parallel emission [34]. For phase values φ = 0 , π, and 2 π , the discontinuity situates at E up = v / φ = π/
2, and 3 π/
2, itdoes at E low = v / − U p = 0 .
5, losing entity in both cases. The SFA and TDSE energydistributions, in the respective Fig. 5(b) and (c), exhibit similar characteristics to the SCM,but with a richer π − periodic structure. Interestingly, the discontinuity at E disc is reflectedas a jump of the probability distributions for the same energy values. The remarkableresemblance between the computationally cheap SFA and the ab initio solution of the TDSEresults shows, once again, that the SFA is very appropriate to explain and reproduce theelectron yield in LAPE processes. Low energy contributions in TDSE calculations shown inFig. 5(a) are due to IR ionization as described before in Fig. 4(c), (d) and (e).For τ X = T L in Fig. 5(d), the SCM spectrum displays horizontal lines corresponding tothe intracycle interference or, what is the same, to the interplay between the intrahalfcyclefactor G ( k ρ ) and the factor sin ( ˜ S/ τ X = 2 T L in Fig. 5(g) the SCM spectrum displays horizontal lines corresponding tothe intercycle interference modulated by the intracycle pattern of Fig. 5(d). We note thatthere is no discontinuity in factor G ( k ρ ) at the energy values E (cid:96) given by Eq. (19). Hence,as the sidebands get narrower, discontinuity of the intracycle modulation blurs. Continuityin the intra- and intrahalfcycle factors is related to the fact that the accumulated action atboth sides of the discontinuity verifies that ∆ S | E>E disc + ∆ S | E 2, where ∆ S | E>E disc (∆ S | E 4) gives exactly the same result at E (cid:96) independently on φ . Once more, from the SFA spectrograms displayed in Fig. 5(e) and (h)and the corresponding TDSE calculations in Fig. 5(f) and (i), we can see, once again, thatthe agreement between the SFA and TDSE spectrograms is very good, with the exceptionof a contribution at low energies due to the ionization by the IR laser pulse alone, which17s strongly suppressed in the SFA calculations. By comparing the intrahalfcycle pattern for τ X = T L / τ X = T L on the center column [Figs. 5 (d), (e), and (f)] and the whole interference pat-tern for τ X = 2 T L on the right column [Figs. 5 (g), (h), and (i)], we corroborate the SCMprediction that the intrahalfcycle interference pattern (spectrogram for τ X = T L / 2) worksas a modulator of the intracycle pattern (spectrogram for τ X = T L ), whereas, the latter doesthe same with the intercycle interference pattern or sidebands. IV. CONCLUSIONS We have studied the electron emission produced by atomic hydrogen in its ground statesubject to an XUV pulse in the presence of an infrared laser pulse in the direction perpendic-ular to the common polarization axis of both pulses. The previously developed SCM [34] forLAPE (XUV + IR) in the forward direction has been extended for perpendicular emission.In accordance to our recent study of LAPE in the forward direction [34], the PE spectrumcan be factorized as two contributions: One accounting for sidebands formation and theother as a modulation. Whereas the former can be interpreted as the intercycle interferenceof electron trajectories from different optical cycles of the IR laser, the latter corresponds tointracycle interference stemming from the coherent superposition of four electron trajectoriesborn in the same optical cycle. Contrarily to parallel emission, the intracycle interferencepattern for transversal emission can be decomposed as the contribution of the two interfer-ing trajectories born within the same half optical period ( intrahalfcycle interference) andthe Young-type interference between the contributions of the two half cycles into the sameoptical cycle ( interhalfcycle interference). We have shown that the electron trajectories borninto the two half cycles within the same optical cycle interfere destructively for the absorp-tion and/or emission of an even number of IR photons, which leads to the exchange of onlyan odd number of laser photons in the formation of the sidebands. Therefore, the absorptionline of the XUV photon alone (with no exchange of laser photons) is forbidden. We showthat the intrahalfcycle interference pattern modulates the intracycle pattern, which, in thesame way, modulates the sidebands. We have observed a very good agreement of our SCMenergy spectrum with the corresponding one to the SFA and the ab initio solution of theTDSE. 18y studying the dependence of the electron emission on the laser intensity, we haveobserved that as the IR field increases the spectra becomes wider and approximately boundedwithin the classical energy domain. We can conclude that the SFA is accurate to describe thePE spectrum perpendicular to the polarization direction, especially for low and moderatelaser intensities so that the electron ionization by the IR laser alone is low compared toLAPE. Finally, by analyzing the electron spectrum as a function of the time delay betweenthe two pulses ∆ X , we have shown that the intrahalfcycle pattern is π − periodic in theoptical phase with a probability jump that reproduces the profile of the square of the laservector potential. 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