On the origin of the cusp in the transverse momentum distribution for the process of strong field ionization
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov On the origin of the cusp in the transverse momentum distribution for theprocess of strong field ionization.
I. A. Ivanov , ∗ Center for Relativistic Laser Science, Institute for Basic Science,Gwangju 500-712, Republic of Korea and Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200, Australia (Dated: October 11, 2018)We study the origin of the cusp-structure in the transverse or lateral electron momentumdistribution (TEMD) for the process of tunelling ionization driven by a linearly polarizedlaser pulse. We show that appearance of the cusp in the TEMD can be explained as follows.Projection on the set of the Coulomb scattering states leads to appearance of ”elementary”cusps which have simple structure as functions of the lateral momentum. This structure isindependent of the detailed dynamics of the ionization process and can be described analyt-ically. These ”elementary” cusps can be used to describe the cusp-structure in TEMD.
I. INTRODUCTION
The seminal paper by Keldysh [1] laid out the distinction between tunelling and multi-photonregimes in the photo-ionization process. Particularly fruitful the Keldysh’s paradigm proved forthe study of the tunelling ionization, a photo-ionization process characterized by the small valuesof the so-called Keldysh parameter g = w p | e | / E (here w , E and | e | are the frequency, fieldstrength and ionization potential of the target system expressed in atomic units). Subsequent de-velopments [2–5] elaborated on various aspects of this approach in the tunelling regime, making itan extremely useful and versatile tool for understanding tunelling photo-ionization. Comprehen-sive reviews of these developments (to which we will be referring below as tunelling theories) canbe found in [6, 7].A remarkable feature of the tunelling regime is that one may still use to some extent the classical ∗ Electronic address: [email protected] notions, such as that of the electron trajectory. This fact has been extensively used in the mod-eling of the tunelling photo-ionization. In this approach tunelling photo-ionization is regarded asa process in which electron first emerges into the continuum as a result of the under-the-barriertunelling. This part of the problem is described quantum mechanically, producing probabilisticdistributions of the electron’s characteristics (typically velocities), which can be used as initialconditions for the subsequent classical modeling describing electron motion after the ionizationevent. Such a procedure has been used with success to produce ionization spectra in good agree-ment with experiment [8] for complicated systems where truly ab initio treatment becomes hope-lessly complicated.These distributions which weight different initial conditions in the electron’s phase space arenot themselves observed in the experiment. The electron momentum distribution measured atthe detector can, however, provide an information about the distributions at the moment of theionization event, which offers an exciting possibility to look at this event experimentally [9, 10].Of course, from the strict quantum-mechanical point of view the notion of the electron escapingthe atom at a particular moment of time should be regarded with some caution [11]), nevertheless,this picture of electron escaping into the continuum proved extremely fruitful.Tunelling theories predict simple Gaussian-like structures for these initial distributions. If theafter-ionization-event motion is treated as guided by the laser field only (ionic core potential isneglected), the momentum distributions at the detector retains this Gaussian character, with apossible shift of the distribution in the momentum space due to the overall momentum electronacquires from the laser field after the ionization event [6]. Of particular interest, therefore, isthe so-called transverse or lateral electron momentum distribution (TEMD), which describes thedistribution of the electron momenta measured at the detector in the direction perpendicular to thepolarization plane of the driving pulse. In the simple picture when electron motion is guided bythe laser field only, the TEMD is unaffected by the motion subsequent to the ionization event.This prediction is not always true. While the TEMD measured at the detector is a Gaussian forthe driving pulse with close to circular polarization [12], it looks rather different for the case ofthe linear polarization. It has been found [13] that for the case of the linearly polarized laser pulsethe transverse electron momentum distribution exhibits a sharp cusp-like peak at zero transversemomentum. We studied this transition from the cusp-like to the Gaussian-like structure in TEMDnumerically using the ab initio solution of the time-dependent Schr¨odinger equation (TDSE) in[14].Study of the TEMD can provide other useful information. It has been demonstrated, bothexperimentally and theoretically [15], that the transverse electron momentum distributions in thetunneling and over the barrier ionization regimes (OBI) evolve in markedly different ways whenthe ellipticity parameter describing polarization state of the driving laser pulse increases. This factcan be used to make a a clear distinction between the tunneling and OBI regimes in the experiment.In the present work we study the origin of the cusp-structure in TEMD for the case of the lin-early polarized driving laser pulse in detail. In [13] this structure at zero transverse momentumhas been attributed to low-energy singularity of the Coulomb wave-function [13]. We show thatthough this interpretation is basically correct there is more to the story. Projection on the set of theCoulomb scattering states produces the ”elementary” cusps which have simple structure as func-tions of the lateral momentum. This structure can be described analytically. These ”elementary”cusps can be used to describe the cusp-structure in TEMD.
II. THEORY AND RESULTS
We will be guided below to a considerable extent by the numerical results provided by thesolution of the TDSE for a hydrogen atom. We will briefly describe the procedure, therefore. Wesolve TDSE for a hydrogen atom in presence of a laser pulse: i ¶Y ( r ) ¶ t = (cid:0) ˆ H atom + ˆ H int ( t ) (cid:1) Y ( r ) . (1)Operator ˆ H int ( t ) in Eq. (1) describes interaction of the atom with the EM field. We use velocityform for this operator: ˆ H int ( t ) = A ( t ) · ˆ p , (2)with A ( t ) = − Z t E ( t ) d t . (3)The laser pulse is linearly polarized along the z -direction, which we use as a quantization axis: E z = E f ( t ) cos w t (4)For the base frequency of the pulse we use w = .
057 a.u. (corresponding to the wavelengthof 790 nm). The function f ( t ) = sin ( p t / T ) in Eq. (4) (here T = NT is a total pulse duration, N is an integer, T = p / w is an optical cycle of the field). We report below results for various pulsedurations T and field strengths E . TDSE is solved for a time interval ( , T ) . Initial state of thesystem is a ground state of the hydrogen atom.To solve the TDSE we employed the procedure described in the works [16, 17]. Solution of theTDSE is represented as a series in spherical harmonics: Y ( r , t ) = L max (cid:229) l = f l ( r , t ) Y l ( q ) . (5)The radial part of the TDSE is discretized on the grid with the step-size d r = . R max =
600 a.u. We consider below relatively short total pulse durations and moderatelystrong field intensities (not exceeding 6 optical cycles and 3 . × W/cm respectively). Weused L max =
50 in the calculations reported below. The necessary checks ensuring that for suchfield parameters calculation is well converged with respect to L max and R max have been performed.Substitution of the expansion (5) into the TDSE gives a system of coupled equations for theradial functions f l ( r , t ) . To solve this system we use the matrix iteration method [18]. Ionizationamplitudes a ( p ) are obtained by projecting solution of the TDSE at the end of the laser pulse onthe set of the ingoing scattering states y ( − ) p ( r ) of the hydrogen atom: y ( − ) p ( r ) = (cid:229) lµ i l e − i h l ( p ) Y ∗ lµ ( p ) Y lµ ( r ) R l p ( r ) . (6)For the linearly polarized laser pulse and the coordinate system we employ only the terms with µ =
0, of course, actually contribute to the projection. Differential photo-ionization cross-sectionis computed as P ( p ) = | a ( p ) | . We are interested in the transverse or lateral electron momentumdistribution, describing probability to detect a photo-electron with a given value of the momentumcomponent p ⊥ perpendicular to the polarization plane. Because of the symmetry of the problemdue to the linear polarization of the driving pulse any plane containing polarization vector can bechosen as a polarization plane. Choosing ( y , z ) -plane as a polarization plane, we obtain for TEMDas function of the lateral momentum p ⊥ = p x : W ( p ⊥ ) = Z P ( p x , p y , p z ) d p y d p z (7)TEMD obtained using this procedure are shown in Figure 1 for two sets of the driving pulseparameters. W ( p ^ ) ( a . u . ) p ^ (a.u.)CoulombPlane wavesYukawa 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.2 -0.1 0 0.1 0.2 W ( p ^ ) ( a . u . ) p ^ (a.u.)CoulombPlane wavesYukawa FIG. 1: (Color online) Left panel: TEMD for the laser pulse (4) with pulse intensity of 10 W/cm andtotal duration of 6 optical cycles. Right panel: the same for the field intensity of 3 . × W/cm andpulse duration of 4 optical cycles. Solid (red) line- projection on the ingoing Coulomb scattering states (6).Dash (green)- projection on the basis of plane waves. Short dash (blue)- projection on the set of the ingoingscattering states of the Yukawa potential. Distributions obtained following the prescription described above and using the Coulomb in-going scattering states (6) for the projection operation (solid lines in Figure 1) show the cusp-likebehavior at p ⊥ =
0. While the TEMD remains continuous at this point, its derivative suffersdiscontinuity. It was suggested in [13] that cusp originates from the singularity of the Coulombcontinuum wave-function at zero energy. We subsequently found some numerical evidence [14]supporting this statement. What interests us in the present work is elucidating the nature of thecusp and the precise type of the discontinuity which the lateral distribution has at p ⊥ =
0. As weshall see, some analytical results describing the discontinuity can be obtained.We begin by presenting results of a few numerical experiments. We make sure first that thecusp is indeed due to the projection on the set of the Coulomb continuum wave-functions. InFigure 1 we present results obtained if the same solutions of the TDSE at the end of the laser pulseare projected on the set of the plane-waves and the ingoing states of the Yukawa potential V ( r ) = − e − . r / r instead of the Coulomb scattering states. The spectra obtained by using plane-wavesbasis and scattering stated of the Yukawa potential, though agreeing quantitatively rather well withthe spectrum obtained by projecting solution of the TDSE on the Coulomb scattering states, showno cusp-like behavior at p ⊥ =
0. The cusp arises, therefore, as a result of the projection operationusing the Coulomb scattering states as was surmised in [13]. The question which interests us isthe detailed mechanism responsible for the appearance of the cusp.We note first that cusp cannot be introduced by the integration procedure, when overlaps be-tween the TDSE solution and the Coulomb scattering states are computed. The amplitude func-tions f l ( r , T ) in the expansion of the TDSE solution (5) are square-integrable functions with typ-ical spatial extent corresponding to the distance the outgoing electron wave-packet can have trav-eled by the end of the laser pulse. Integration of such functions cannot introduce any low-energysingular behavior. Indeed, we can consider that to a good approximation the amplitude functions f l ( r , T ) have finite support, being non-zero only in the finite region of space (this is what theyare in the numerical calculation anyway). Integration of such functions cannot introduce any newsingularities which are not already present in the integrand. We must, therefore, look carefully atthe singularities present in the Coulomb scattering state (6).There are two factors in Eq. (6) we have to examine: the Coulomb scattering phase-shifts andthe radial Coulomb wave-functions. Explicit expression for the Coulomb phase-shifts reads [19]: h l ( p ) = arg G (cid:18) l + − ip (cid:19) , (8)and it exhibits a highly singular behavior at p =
0. On the other hand, the radial functions R l p ( r ) in Eq. (6) can be written (we use the d ( p − p ′ ) normalization) as [19]: R l p ( r ) = b ( p ) g ( p ) g l p ( r ) , (9)where g ( p ) = − e − p p , (10)and b ( p ) = √ p . (11)The function g l p ( r ) can be found as the solution of the radial Schr¨odinger equation satisfying aboundary condition g l p ( r ) → C l e l + when r → C l is a constant factor independent of energy.By the well-known Poincare theorem g l p ( r ) is, therefore, an entire function of energy, i.e. anentire function of p . The radial wave-function in (6) is singular at p = g ( p ) and b ( p ) given by Eq. (10) and Eq. (11).We have identified, thus, three potential culprits which may introduce singular low-energybehavior and which may be responsible for the formation of a cusp. Let us study them one by one.Consider first the effect of the Coulomb scattering phase-shift. In Figure 2 we present resultsof a simple numerical experiment obtained if in the expression for the Coulomb ingoing scatteringstate (6) we put h l ( p ) = R l p ( r ) ). W ( p ^ ) ( a . u . ) p ^ (a.u.)Coulomb h =0 g (p)=1 FIG. 2: (Color online) Left panel: TEMD for the laser pulse (4) with pulse intensity of 3 . × W/cm and pulse duration of 4 optical cycles. Solid (red) line- projection on the correct ingoing Coulomb states(6). Dash (green)- projection on the set of states (6) with h l ( p ) =
0. Short dash (blue)- projection on set ofthe states (6) with g ( p ) = One can see that removal of the scattering phase-shifts h l ( p ) form the Eq. (6) hardly producesany effect on the lateral spectrum. We could, in fact, anticipate this. Indeed, from the expressionfor the Coulomb phase-shifts (8) and elementary properties of the Gamma-function one can easilydeduce the relation: h l + ( p ) = h l ( p ) − p + O ( p ) and hence: h l + ( p ) = h ( p ) − ( l + ) p + O ( p ) , (12)valid when p →
0. We see, thus, that at low energies the effect of the Coulomb phase-shifts h l ( p ) in Eq. (6) reduces to introducing an energy-independent phase-factors for different termsin the sum in Eq. (6), and overall dependence of the photo-ionization amplitude on the h ( p ) ,which cancels out when we compute the squared modulus of the amplitude. On the other hand,with increasing energy Coulomb phase-shifts decrease fast, which explains the fact that Coulombphase-shifts have virtually no effect on the TEMD. The role of the factor g ( k ) is equally insignifi-cant as can be seen from Figure 2, where we show the spectrum obtained if we put g ( p ) = p =
0, does notblow up at this point and tends to be one with increasing energy.We are left, therefore, with the only factor in the Eq. (6) which blows up at p =
0, the factor b ( p ) in Eq. (11). There are, of course, other factors in Eq. (6) which are singular at p =
0. The sphericalharmonics Y lµ ( p ) are, strictly speaking, singular functions of the components of the vector p at p = p , and, in a strict mathematical sense the function p is singular at p = p . These singularities are, however, only mildones, in particular they do not lead to the unbounded growth of the function. The only singularitywhich does lead to such a growth is the one due to the factor b ( p ) .We are now in a position to elucidate the nature of the cusp in the lateral distribution. Tothis end, let us note that because of the symmetry of the problem the differential photo-ionizationcross-section P ( p ) is, in fact, a function of two variables only: P ( p ) = P ( p , cos q ) , where q is theangle between electron momentum p and the z -axis. Expanding this expression in powers of q wemay write for the differential cross-section: P ( p ) = ¥ (cid:229) n = P n ( p ) cos n q , (13)where P n ( p ) are functions of of p only. Coefficients of this expansion can be computed numer-ically from the known solution of the TDSE by re-expanding products of the spherical harmonics Y lµ ( p ) occurring in the expression for the squared modulus of the amplitude | a ( p ) | in series ofspherical harmonics with the help of the well-known formulas, and re-expanding in turn the result-ing spherical harmonics in powers of cos q . Important point here is that coefficients P n ( p ) dependonly on p and inherit from the amplitudes the singular behavior at p = b ( p ) inEq. (11). The p − singular behavior of the amplitudes at p = p − singularbehavior of the coefficients P n ( p ) at p =
0. Integrating Eq. (13) over the ( p y , p z ) -plane (only termswith even n give nonzero contributions, of course) we obtain for the TEMD: W ( p ⊥ ) = ¥ (cid:229) n = , ,... W n ( p ⊥ ) , (14)where W n ( p ⊥ ) = ¥ Z dq p Z d f qP n ( p ) (cid:18) q cos f p (cid:19) n = p n ! G ( n / + ) n − ¥ Z dqP n ( p ) q n + (cid:0) p ⊥ + q (cid:1) n , (15)where p = q p ⊥ + q , and we used a cylindrical coordinate system ( q , f ) in the ( p y , p z ) -plane.Using Eq. (15) we can obtain asymptotic behavior of W n ( p ⊥ ) for p ⊥ →
0. As we mentionedabove P n ( p ) behave as p − for small energies. Let us choose some small positive Q and represent P n ( p ) in the interval ( , Q ) as P n ( p ) = C n / p + P ′ n ( p ) where P ′ n ( p ) is non-singular at p =
0. Singularbehavior of the integrals in Eq. (15) at p ⊥ = C n / p in P n ( p ) over the interval ( , Q ) . Indeed, the integrands in both integral overthe interval ( , ¥ ) with P ′ n ( p ) and the integral over the interval ( Q , ¥ ) with C n / p contain smoothregular functions as integrands which cannot lead to a small- p ⊥ singular behavior. We can write,therefore: W n ( p ⊥ ) = C n I n ( p ⊥ , Q ) + W reg n ( p ⊥ ) , (16)where I n ( p ⊥ , Q ) = Q Z dq q n + (cid:0) p ⊥ + q (cid:1) n + , (17) C n is a constant, and W reg n ( p ⊥ ) are non-singular at p ⊥ =
0, behaving near this point as W reg n ( p ⊥ ) ≈ u n + u n p ⊥ with some constant u n and u n .It is clear from Eq. (17) why this contribution becomes singular at p ⊥ =
0. While the integralhas a finite value if we put p ⊥ =
0, an attempt to calculate the derivative with respect to p ⊥ bydifferentiating under the integral sign leads to a divergent expression. We have to be more carefulin evaluating asymptotic of the integral. Using well-known formulas for the integral representationand asymptotic properties of the hypergeometric function F ( a , b ; c ; z ) [20] we obtain:0 I n ( p ⊥ , Q ) = Q n + | p ⊥ | n + G (cid:0) n + (cid:1) G (cid:0) (cid:1) G (cid:0) n + (cid:1) F (cid:18) n + , n + n +
32 ; − Q p ⊥ (cid:19) ≈ A n + B n | p ⊥ | p ⊥ → , (18)where we do not give explicit expressions for the unimportant constant factors.From Eq. (16) and Eq. (18) we conclude that W n ( p ⊥ ) behave for small p ⊥ as linear functionsof | p ⊥ | : W n ( p ⊥ , Q ) = g n + h n | p ⊥ | + O (cid:0) p ⊥ (cid:1) p ⊥ → , (19)where g n , h n are some constants. This implies the following cusp structure of W n ( p ⊥ ) at p ⊥ = W n ( p ⊥ ) with respect to p ⊥ is discontinuous at p ⊥ =
0, the second derivative is,therefore, infinite. That this formula indeed reproduces asymptotic behavior correctly, can be seenfrom Figure 3. The contributions W n ( p ⊥ ) as functions of lateral momentum obtained from theTDSE calculation are shown in Figure 3. W n ( p ⊥ ) vary considerably with n in magnitude and neednot be positive, to facilitate the comparison we present the scaled contributions: W n ( p ⊥ ) / | W n ( ) | .In Figure 3 we present also the results of the linear fits: W n ( p ⊥ ) / | W n ( ) | = A + B | p ⊥ ) | .Two features are apparent from Figure 3. First, for small values of p ⊥ the contributions W n ( p ⊥ ) are indeed linear functions of | p ⊥ | in agreement with the asymptotic estimate (19) we made above.Second, the region where this asymptotic estimate represents W n ( p ⊥ ) accurately shrinks with n .There is, of course, nothing unusual in such behavior. Asymptotic estimates give us asymptoticbehavior for small values of a parameter, but they do not necessarily tell us how small the param-eter should be for the estimate to be accurate. A glance at the behavior of the integrals in Eq. (17)as functions of lateral momentum may help us to understand what is happening. We show in Fig-ure 4 integrals I n ( p ⊥ , Q ) as functions of p ⊥ for a fixed value of Q = .
05 a.u. Figure 4 showsqualitatively the behavior similar to what we observe in Figure 3, the region where the asymptoticexpression represents I n ( p ⊥ , Q ) accurately shrinks with n . Thus, the behavior of W n ( p ⊥ ) is a con-sequence of the property of the integrals I n ( p ⊥ , Q ) that the region where linear in | p ⊥ | asymptotictakes over shrinks progressively with n .This feature of W n ( p ⊥ ) is quite important for understanding how the cusp in TEMD is pro-duced. As Eq. (19) shows the terms of the series (14) behave as linear functions of | p ⊥ | for smallenough p ⊥ . At first glance, that would suggest that the sum of the series (14), the TEMD W ( p ⊥ ) ,would exhibit the same behavior linear in | p ⊥ | near p ⊥ =
0. That would imply the following1 W n ( p ^ ) / | W n ( ) | p ^ (a.u.)n=0 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12-0.01 -0.005 0 0.005 0.01 W n ( p ^ ) / | W n ( ) | p ^ (a.u.)n=2 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02-0.01 -0.005 0 0.005 0.01 W n ( p ^ ) / | W n ( ) | p ^ (a.u.)n=4 -1.006-1.004-1.002-1-0.998-0.996-0.994-0.992-0.99-0.988-0.01 -0.005 0 0.005 0.01 W n ( p ^ ) / | W n ( ) | p ^ (a.u.)n=6 FIG. 3: (Color online). Solid (red) line: scaled terms of the series (14) W n ( p ⊥ ) / | W n ( ) | as functions oflateral momentum p ⊥ . Dash (green): linear fit: W n ( p ⊥ ) / | W n ( ) | = A + B | p ⊥ ) | . Field intensity 3 . × W/cm , pulse duration four optical cycles. cusp structure: TEMD would have discontinuous first and infinite second derivative at p ⊥ = p ⊥ with infinite firstderivative. In other words W ( p ⊥ ) grows visibly faster than | p ⊥ | near p ⊥ =
0. This apparent con-tradiction is resolved when one realizes that W ( p ⊥ ) is a sum (14) of the terms W n ( p ⊥ ) . Each of W n ( p ⊥ ) behaves as a linear function of | p ⊥ | in some vicinity of p ⊥ =
0, but, as we saw above, theinterval of p ⊥ on which linear dependence is a good approximation shrinks with n . If, as Figure 3suggests, outside the interval of applicability of the asymptotic law the W n ( p ⊥ ) grow slower with p ⊥ (we see this behavior for the integrals I n ( p ⊥ , Q ) in Figure 4), the sum of all W n ( p ⊥ ) in (14) willexhibit precisely the behavior seen in Figure 1- the growth which is faster than linear for p ⊥ → p ⊥ → n switch progressively from the relatively slow growth outside2the asymptotic region to a faster growth linear in | p ⊥ | , once p ⊥ is inside the region of the validityof the asymptotic law (19) for a particular n . I n ( p ^ ) ( a r b . un it s ) p ^ (a.u.)n=0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1-0.01 -0.005 0 0.005 0.01 I n ( p ^ ) ( a r b . un it s ) p ^ (a.u.)n=2 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01 -0.005 0 0.005 0.01 I n ( p ^ ) ( a r b . un it s ) p ^ (a.u.)n=4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01 -0.005 0 0.005 0.01 I n ( p ^ ) ( a r b . un it s ) p ^ (a.u.)n=6 FIG. 4: (Color online) Solid (red) line: Integrals I n ( p ⊥ , Q ) as functions of lateral momentum p ⊥ for Q = .
05. Dash (green): linear fit: I n ( p ⊥ , Q ) = A + B | p ⊥ ) | . To see this quantitatively we introduce a function: f ( z ) = ¥ (cid:229) m = C m z m , (20)where C m are the coefficients in Eq. (16) (the definition (20) takes into account that only theeven-order coefficients C n occur in Eq. (14)). The function f ( z ) in Eq. (20) is defined in termsof the set of the coefficients C n and encapsulates, therefore, information about the solution of theTDSE and, ultimately, information about the dynamics of the ionization process. For the sum ofthe series (14) we can write then (we omit the contributions of the regular parts W reg n ( p ⊥ ) since3they do not lead to the singular behavior): I ( p ⊥ ) = ¥ (cid:229) n = , ,... C n Q Z dq q n + (cid:0) p ⊥ + q (cid:1) n + = Q Z q q p ⊥ + q f (cid:18) q p ⊥ + q (cid:19) dq . (21)Substituting y = p ⊥ q p ⊥ + q we can rewrite Eq. (21) as: I ( p ⊥ ) = p ⊥ Z y ( Q ) dyy f (cid:0) − y (cid:1) , (22)where y ( Q ) = p ⊥ q p ⊥ + Q .Assuming that in this expression f ( z ) = z n , which corresponds to all but one coefficients C n inEq. (16) having zero values, reproduces, of course, the asymptotic law (19) we obtained above forthe individual terms W n ( p ⊥ ) of the series (14). More realistic assumption about f ( z ) can be basedon the observation we made above that the coefficients C n are ultimately related to the partial waveexpansion of the solution of the TDSE equation. It is clear that to establish small- p ⊥ behaviorof the sum of the series (14) we actually need to know only the large- n asymptotic behavior of C n . Partial wave expansions in the TDSE calculations converge, as a rule, rather slowly [18]. Aplausible assumption about the large- n asymptotic behavior of the sequence of C n , reflecting thisslow convergence, would be a power-like asymptotic behavior C n (cid:181) n − l with some positive l . Bythe well-known theorems of the complex analysis this implies presence of at least one singularpoint of f ( z ) on the circle of convergence | z | =
1. Let us assume, for example, that this singularityis a simple branch point at z =
1, so that f ( z ) = √ − z . We obtain then from Eq. (23): I ( p ⊥ ) = p ⊥ Z y ( Q ) dyy , (23)which, as one can easily see, leads to the following formula for the asymptotic behavior of I ( p ⊥ ) for p ⊥ → I ( p ⊥ ) = A + B | p ⊥ | ln | p ⊥ | p ⊥ → , (24)with some constants A and B . We see, thus, that while terms of the series (21) all have | p ⊥ | -cusps at p ⊥ =
0, the sum of the series (21) can exhibit more singular cusp-like behavior at thispoint. For the terms of series (21) the first derivatives with respect to p ⊥ are discontinuous at4 p ⊥ = p ⊥ =
0. The cusp-singularity of the sum of theseries (21), with the choice of the function f ( z ) we used above as an example, is more severe, thefirst derivative with respect to p ⊥ is infinite at p ⊥ = A + B | p ⊥ | ln | p ⊥ | ( A and B considered as fitting parameters) gives actually better results than the three-parameter fit basedon the equation: W ( p ⊥ ) = A + B | p ⊥ | a ( A , B and a as fitting parameters). We cannot, of course,claim that A + B | p ⊥ | ln | p ⊥ | is the actual behavior of the TEMD W ( p ⊥ ) for small p ⊥ . As we saw,to describe the cusp in the TEMD we rely on two ingredients. We need first to describe small- p ⊥ behavior of the ”partial” distributions W n ( p ⊥ ) in the (14). Eq. (16) and Eq. (18) provide an answerto this problem. To find small- p ⊥ behavior of the sum of the series (14) we also need to knowthe weights with which different I n ( p ⊥ ) in Eq. (16) contribute to the sum- the coefficients C n inthis equation. The logarithmic behavior in Eq. (24) obtains, in particular, assuming that f ( z ) inEq. (20), which encapsulates information about the coefficients C n , has a simple branch point at z =
1, an assumption leading to apparently satisfactory results, which we, however, did not prove.We may regard this formula, therefore, as a plausible but only a tentative expression.
III. CONCLUSION
We considered in detail the formation of the cusp in the TEMD for the process of strong fieldionization. As we saw, one can push analytic approach to this problem quite far. Our startingpoint was the series (14) resulting from the expansion of the differential probability in the powersof cos q - the angle between the polarization vector and the electron momentum. We were able toshow that the terms W n ( p ⊥ ) of this series behave as linear functions of | p ⊥ | for small p ⊥ . Thisbehavior is a consequence of the properties of the Coulomb continuous spectrum wave-functions,and is present, therefore, for any system regardless of what the actual Hamiltonian is, as long as thewave-function of the system after the end of the laser pulse is projected on the set of the Coulombwave-functions. If this were the whole story the TEMD would have the | p ⊥ | -cusp at p ⊥ =
0. Forsuch a cusp the first derivative at p ⊥ = W n ( p ⊥ ) in (14) with the small- p ⊥ asymptotic which we established in Eq. (19), constitute thebuilding blocks from which TEMD can be build by summing up the expansion (14). It is at this5 -0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.3 -0.2 -0.1 0 0.1 0.2 0.3 W ( p ^ ) ( a . u . ) p ^ (a.u.) FIG. 5: (Color online) TEMD for the field intensity of 3 . × W/cm and pulse duration of 4 optical cy-cles. Solid (red) line: TDSE calculation. Dash (green): fit based on the equation: W ( p ⊥ ) = A + B | p ⊥ | ln | p ⊥ | (fitting parameters A and B ). Short dash (blue): fit based on the equation: W ( p ⊥ ) = A + B | p ⊥ | a (fitting pa-rameters A , B and a ). stage, where dynamic information, i.e. the information about particular details of the ionizationprocess, becomes important. The ”partial” lateral distributions W n ( p ⊥ ) considered as functions of n enter the series (14) with different weights. Mathematically, it is reflected in the Eq. (16) whichrepresents W n ( p ⊥ ) as a product of the integral I n ( p ⊥ ) and a coefficient C n which is a functionof n only. Coefficients C n depend, of course, on the dynamics of the system, since they resultultimately from the projection of the TDSE wave-function at the end of the laser pulse. Thefunction we introduced in Eq. (20) conveniently encapsulates this information. As we saw, thesummation procedure can make the character of the cusp for the sum of the series (14) differentfrom the linear | p ⊥ | -cusp which each of the terms of the series exhibits at p ⊥ =
0. The reason forthis is, roughly speaking, the fact that for the terms of the series (14) with higher n the region oflateral momenta for which linear in | p ⊥ | asymptotic law holds for W n ( p ⊥ ) shrinks, or in strictermathematical language, the fact that the small- p ⊥ asymptotic (19) for W n ( p ⊥ ) is non-uniform in n .6To summarize, we demonstrated that the cusp in the TEMD arises as a consequence of twofactors: The singularity of the Coulomb wave-function produces a simple cusp of the A + Bp ⊥ type. The view expressed in [13] that cusp is due to the singularity of the Coulomb scatteringstate is, therefore, basically correct, the Coulomb wave-function is responsible for the presence ofthe cusp. This fact has nothing to do whatsoever with dynamics of the photo-ionization process.The character of the cusp we observe in the TEMD, however, may differ from the A + Bp ⊥ -typecreated by the Coulomb wave-function. The origin of this difference lies in the dynamics, it isultimately due to the properties of the coefficients of the expansions (13), (14), which depend onthe wave-function at the end of the laser pulse. IV. ACKNOWLEDGMENTS
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