Glory scattering in strongfield ionization
Abstract
Since Keldysh’s pioneering work fifty years ago, the theory of strongfield approximation has achieved great success in providing a solid foundation to strongfield atomic ionization. However, the neglect of Coulomb interaction remains a longstanding gap between the wellestablished opticaltunneling picture and the experimental observations, which has been highlighted in the past decade by a series of striking disagreements extending from the low energy regime near the ionization threshold to the intermediate energy around twice the ponderomotive energy, not only in longitudinal direction but also transverse direction. Here we report that the gap can be largely bridged with the aid of an effect named as glory, which is associated with a specific type of singularity concealed in Coulomb forward rescattering. We reveal that the trajectory lying exactly on the singular point dominates both the photoelectron longitudinal and transverse momentum distribution, which directly connects tunneling theory with photoelectron distribution after posttunneling propagation.

National Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

CAPT, HEDPS, and IFSA Collaborative Innovation Center of MoE, Peking University, Beijing 100871, China
email: lbfu@iapcm.ac.cn;
email: liu_jie@iapcm.ac.cn;
Introduction
As a beautiful phenomenon, the optical glory is a series of bright concentric rings surrounding the observer’s shadow when the light is backward scattered[1, 2]. Until 1959 its semiclassical counterpart in quantum scattering[3] was identified and connected with a specific singularity, i.e., the axial caustic structure[4], where a trajectory with nonzero impact parameter is scattered from a spherical object and emerges in the forward or backward direction. Similarly as other singularities from critical phenomena to black holes, since then the glory scattering has been explored and expected to be good probes of the world in a number of areas such as nuclear physics, atomic physics, and gravitation[5, 6, 7, 8, 9, 10]. In the paper, we report the emergence of glory scattering in strongfield ionization.
In the past half century the researchers have strived to understand the mechanism of electron ionization in laser fields. According to the wellestablished tunnelingionization scenario, many quantities including the rate and transverse momentum distribution have been clarified at the instant when the electron tunnels out of the potential trap. However, the neglect of Coulomb interaction of the tunneling photoelectron with its parent ion stands as a critical problem isolating the strongfield approximation (SFA)[11, 12, 13] or tunneling theory from the experimental observations even in the atomic single ionization in linearly polarized (LP) laser field. Typically, in the past decade various striking disagreements have been observed between SFA and experimental strongfield photoelectron spectra, such as the peak structure in the lowenergy structure[14, 15] near the ionization threshold and the ordersofmagnitude enhancement in the intermediate energy regime[16] around twice the ponderomotive energy (). Besides the discrepancies in the longitudinal direction parallel with the field polarization direction, the shape and width of the transverse momentum distribution after posttunneling propagation also exhibit obvious deviation[17, 18] from the theoretic Gaussian distribution. Hence, the gap prohibits severely the researchers from direct test of the ionization theory, and leads to critical problems in decoding the information and developing new technics such as photoelectron holography[19]. Interestingly, our study find that, because of the dominance of the special trajectory corresponding to the singularity in the glory scattering, the glory effect can be helpful to bridge the gap and provide deeper insight to the strongfield processes.
Results
0.1 Axial caustic and glory in rescattering
In the semiclassical picture, the photoelectron experiences a threestep process in the Coulomblaser field as shown in Fig. 1(a). At first, the electron tunnels out of the binding potential distorted by a strong electric field at time . After acceleration under the influence of laser field in the second step, the electron can be driven back and rescattered from the parent ion. Due to the uncertainty of the momentum, in trajectory perspective[20, 21, 22] the photoelectron can be estimated as launched with different initial transverse momentum and detected with the corresponding asymptotic momentum . In the mapping between and a special singularity structure known as axial caustic manifests itself. As shown in Fig. 1(c), the trajectories launched with initial momenta distributed on a specific circle[23] in the momentum plane can finally converge at the origin. So the Jacobian vanishes and the number of the trajectories along the laser polarization direction diverges. The singularity originates from the threedimensional geometry of scattering and gives rise to glory scattering. Here the trajectory lying exactly at the singular point is termed as glory trajectory (GT) as plotted in Fig. 1(a).
The divergence at the singularity is overcome by the wave property of particle[24]. Typically, in a fieldfree glory scattering it is known that the differential cross section can be approximated well by the square of a zeroorder Bessel function, i.e., when the scattered angle . denotes the angular momentum of GT[3]. The effect can be demonstrated by a model in Fig. 1(b), where a Gaussian wavepacket launched from with the expected momentum is scattered by a Coulomb potential[25]. Among all the possible paths, GT propagates along the forward direction, of which the emergent impact parameter can be obtained as by solving the Kepler problem[26]. Here atomic units are used unless otherwise specified. In panel (d) the glory is clearly illustrated by a bright central spot and a series of concentric rings in the numerical asymptotic transverse momentum distribution. Especially, in panel (e) the distribution along axis fits well with where . We notice that, in contrast with Rutherford formula for the plane wave, in the scattering of a localized wavepacket the divergence fades away. Instead, the property of the axial caustic as well as GT emerges .
Similarly, in simiclassical picture, the amplitude to detect an ionized electron with momentum near the polarization direction can be evaluated by integrating over all possible trajectories launched after tunneling:
(1) 
where . denotes the classical action of the trajectory which is launched from and approaches obeying Newton’s law of motion, is the corresponding prefactor of the trajectory and can be estimated as the weight of the trajectory, and is the ionization potential of the atom. It is known that so that in Eqn. (1) the saddle points where represent the semiclassical trajectories with the asymptotic momentum . Therefore, it seems reasonable to approximate the amplitude as by applying the steepest descent method (SDM) and summing over all the saddle points. However, as a result of the axial caustic in Fig. 1, around the polarization direction such an approximation is invalid because approaches 0 and diverges. Instead, we propose a semiclassical method termed as posttunneling pathintegral (PTPI) using the Feynman’s pathintegral in configuration space[27]. Its details have been stated in the Method section. The method tackles with the singularity in the forward direction, and reveals that the GT corresponding to the singular point dominates the photoelectron momentum distribution in both longitudinal and transversal direction by a simple equation,
(2) 
Here the asymptotic momentum and the emergent impact parameter of GT both are the functions of the corresponding tunneling time .
0.2 Longitudinal momentum distribution
Firstly, we concentrate on the photoelectron longitudinal momentum distribution. In order to testify the theory, we solve the timedependent Schrödinger equation (TDSE) of hydrogen (H) atom in LP fields with a generalized pseudospectral method[28]. In Fig. 2(a) the simulated longitudinal momentum distribution corresponding to is plotted with black dots. And the results of our theory are plotted in thick solid curves. During the implementation of PTPI method, the adiabatic tunneling theory suggests zero initial longitudinal velocity after tunneling. However, because of the oscillation of the field, SFA method gives the initial momentum[29, 30] where represents the saddle point satisfying . As a result, we apply adiabatic (A) and nonadiabatic (NA) initial condition to determine the GT. The predictions of Eqn. (2) are denoted as APTPI (blue line) and NAPTPI (red line), respectively. Interestingly, we find that all the GTs experience forward rescattering as shown in Fig. 3. No backward GT is detected. After forward rescattering the GTs in APTPI gain energy no more than , while in NAPTPI the GTs can achieve higher energy. In comparison, the result of simpleman model (SMM)[31] is plotted in green thin line. More details of the calculations are elaborated in Method section.
Comparing with other theory, NAPTPI prediction fits well with the simulated result extending from the lowenergy part near the ionization threshold to the intermediate energy regime exceeding . It has been demonstrated in panel (a) where the longitudinal momentum distribution can be divided into 5 regimes. In the first regime near the ionization threshold, the photoelectron distribution is enhanced by one to two orders compared to SMM prediction, which is originated from strong focusing effect of the ionic potential[32] represented by the large emergent impact parameter as shown in Fig. 3. In the second regime the distribution displays a plateau structure extending to 2, differently from the predictions of SMM and APTPI method which both experience rapid drop when the electron energy approaches . After the energy exceeds where the transition from the direct to the rescattered electrons is expected in SFA theory, the curve of simulation as well as NAPTPI method displays a slow decline without abrupt cutoff. In this regime, the predictions of SMM and APTPI disappear while the agreement between NAPTPI and simulation persists till about . In the later, the deviation between NAPTPI and the simulated result is broadened until the electron energy exceeds the classical cutoff energy . Since in our theory only forward glory scattering is detected during atomic single ionization, the deviation implies the dominant contribution of the backward rescattering to the high energy electron. Here the wavelength of the laser is 800 nm, and the peak value of the electric field is 0.05 a.u..
We confirm above observations by varying the laser parameters. As demonstrated in panel (b), the ab initio simulations and the NAPTPI predictions have been compared, where the laser parameters with 800nm, (pink) and (blue), and 1200nm (black) and (red) have been used. While the corresponding Keldysh parameter is ranging from 0.58 to 1.42, Eqn. (2) can correctly predict the trend of all the distribution curves up to several times of . The sharp enhancement in the low energy regime near the ionization threshold manifests itself in the simulated red curve in 1200nm laser field, while it is somehow obscured by the interference structures in the pink curve with shorter wavelength laser. In the intermediate energy regime around , in contrast to ordersofmagnitude discrepancy between strongfield approximation and simulation[16], NAPTPI predictions agree well with the simulated results with error in one order. Finally, we notice that our theory also reproduces a small plateau structure near emerging in simulation in 1200nm laser fields, which can be distinguished in the red curve. The feature is just the ”ionization surprise” of lowenergystructure recently observed in midIR fields, which disappears and degenerates into a rapid slope when the wavelength decreases[14, 33].
0.3 Transverse momentum distribution
Secondly, we focus on the photoelectron transverse momentum distribution. So weak is the radiation pressure of the laser field used here that we apply dipole approximation and obtain the updown symmetric twodimension (2D) spectra as shown in Fig. 2(d). It is obvious in the map that the transverse momentum distribution varies with the longitudinal momentum . Hence, in Fig. 2(c) we plot the transverse momentum distribution at different , i.e., and .The inter and intracycle interferences in these curves are smoothed by using the envelope of the distribution curves within a narrow interval around each . Notice that we find the GT and corresponding to each by varying the tunneling time , and scale the transverse momentum by the corresponding . Nonadiabatic initial condition is applied since no GT with can be found using adiabatic condition. While the peak amplitude of these distributions have been accounted of with Eqn. (2) in the above, we normalize these curves by their central peaks.
Comparing the simulated results in dots with the black solid curve of , it is clear that all the curves fall on the same theoretic profile, especially around the center peak. It confirms that the transverse distribution can be depicted by a Bessel function as suggested by Eqn. (2), where either the width of the Bessel function is determined by the emergent impact parameter of GT. In contrast, it is well known in tunneling theory that the transverse momentum of the electron tunneling at is subject to a Gaussian distribution, of which the width is determined by , where is the electric field strength at time [34]. For instance, the Gaussian distribution corresponding to has been plotted in black dashed line in panel (c). The comparison shows consistently with experimental observations[17, 18] that the profile of momentum distribution has been narrowed and distorted severely after posttunneling propagation in LP field. In the wings of the momentum distribution with high , the deviation between theory and simulation is enlarged, which is originated from the neglect of the difference in tunneling time and other approximations during treating Eqn. (1) in Method section.
0.4 Glory trajectory and emergent impact parameter
In our theory GT is determined by a classical MenteCarlo method with the details listed in Method section. Briefly, given the nonadiabatic initial conditions according to tunneling theory, we obtain the classical trajectories by solving the Newton’s equation. Among these trajectories the ones finally propagating along the polarization direction are GTs. Because of the zero asymptotic transverse momentum of GT, the emergent impact parameter is well defined as in Fig. 3(a). By choosing the tunneling time and initial momentum as recorded in the inset of Fig. 3(b), a specific forward rescattered GT as well as can be obtained for every possible longitudinal momentum . In Fig. 3(a) the GTs during the strong field ionization corresponding to and are plotted from up to down.
The red curve in Fig. 3(b) illustrates clearly that the emergent impact parameter decreases as increases. In Eqn. (2) it is the large inducing the enhancement of electron along the polarization direction near the ionization threshold as in Fig. 2(a). It implies that the slow photoelectron experiences strong Coulomb effect, which agrees with previous experimental data[18]. Hence can be a measurement of the focusing effect of ionic potential.
0.5 Forward holographic interference fringe
It is demonstrated by Fig. 2(c) that the Bessel function in Eqn. (2) can depict the width of the transverse momentum central peak. The position of the first minimum of the oscillated distribution agrees well with the theoretical curve. As a result, in Fig. 2(d) we can determine the border of the central brightest lobe of the 2D momentum spectrum by corresponding to the first zero of for any longitudinal momentum . The nonadiabatic theoretic separation is plotted in Fig. 2(d) with red solid curve, which coincides with the ab initio calculation from to more than . Notice that the fringe structure in both the simulation and theory have exceeded the vertical white dashed line corresponding to 2, i.e., the cutoff energy in SMM, but the adiabatic theory in white line fails because no GT can be found when energy increases.
Above separation between the main lobe with the sidelobe is often obscured in nearIR laser field by other interference structures. Fortunately, it has manifested itself in the photoelectron forward holographic fringes using midIR laser fields[19]. In Fig. 4(a) and (b) we reproduce the experimental results[35] in 7000 nm and 16000 nm laser fields and plot our theoretic separation in red and blue curve, respectively, for best contrast. In comparison, the positions predicted by SFA and Coulombcorrected SFA (CCSFA) are plotted in orange open circles and purple solid triangles, respectively, which both evidently deviate from the experimental data. Specifically, SFA prediction locates near the experimental secondary dark fringe, and the CCSFA prediction shifts to the secondary bright fringe. In contrast, the NAPTPI method can depict the separation satisfactorily. The predictions of APTPI are plotted in black dashed curves which either fail similarly as in Fig. 2(d) when the longitudinal momentum approaches .
Interestingly, the above pattern can be understood by a virtual ring source in Fig. 1(a), which is formed by the virtual focus of the nonparallel trajectories adjacent to each branch of GT. According to HuygensFresnel principle the ring source just gives rise to the zerothorder Bessel oscillation[1]. We notice that the phase difference between the two semiclassical trajectories with the same momentum, i.e., the upward and downward in the same color, can be approximated as [36]. Hence Our theoretic pattern suggests that the contribution beyond the semiclassical trajectories in Fig. 1(a) apparently pushes the first dark fringe in holography from to due to the breakdown of SDM. It is in sharp contrast with other theoretical interpretations.
Discussion
In the work, we report the emergence of the forward glory scattering in laserassisted photoionization, and reveal that the GT corresponding to the singularity dominates the photoelectron momentum distribution by its angular momentum as well as the emergent impact parameter. It can be seen just as the reflection of a fundamental principle, i.e., the uncertainty principle, in quantum world. The uncertainty of position in glory scattering can be estimated as the distance between the GTs, i.e., , so the width of the momentum distribution . Therefore, we can anticipate that such effect is pervasive in strongfield processes. It is expected to provide a valuable window to probe the strong field phenomena such as the tunneling process and holographic interference structure.
Methods
0.6 Derivation of the momentum distribution
The starting point is the transition amplitude as . In SFA approach is replaced by , then . is the saddle point corresponding to the asymptotical momentum . Here we recover the complete evolution operator U from , and use the positiontoposition propagator, so that
(3) 
When ,
(4) 
Without loss of generality, we set . Replacing the Feynman path integral propagator by the JWKB approximation[10] as , we have
(5) 
where denotes the classical action of the trajectory which is launched from and approaches obeying Newton’s law of motion. Instead of the integration of all the paths launched from everywhere, in PTPI we simplify it by only retaining the major part in tunneling scenario, i.e., those trajectories launched from the tunneling exit . Therefore we abbreviate as . The motion in the image time mainly contributes the weight of the trajectory, so that we only remain the real part of , i.e., , and use to represent the weight of the trajectory. Hence, we obtain Eqn. (1) as that
(6) 
Notice that the saddle points in Eqn. (6) represent the classical trajectories with the asymptotic momentum . So the application of SDM seems reasonable to reproduce the supposition of all the semiclassical trajectories in Fig. 1(a), which is similar as other trajectorybased Coulombcorrected methods. Unfortunately, it leads to the divergence of and wrong position of the fringe.
Actually, in trajectory perspective of laser assisted ionization, the GTs are launched with different initial transverse momenta distributed continuously on a specific circle in the momentum plane, and finally converge into the same asymptotic momentum after Coulomb forward rescattering[23]. Then any observer in Fig. 1(a) along the direction of the trajectories in yellow or magenta can receive only two branches sharing the same momentum, while the one along z axis direction can observe infinite GTs from all azimuth angles. These infinite GTs imply that the saddle points in Eqn. (6) are not well isolated, so that the quadratic approximation in SDM breaks down.
Hence, it is invalid to apply SDM to the integration in Eqn. (6) over the azimuth angle in cylindrical coordinates. Instead, we have
(7)  
where only the integration over and is calculated by SDM. The dependence of on has been neglected as . denotes the saddle point , i.e., and , which corresponds to the GT in Fig. 1(a). The emergent impact parameter of GT determines the frequency of Bessel function. Using shooting algorithm, we can obtain as well as the classical GTs with the asymptotic momentum , where the initial conditions are given by the tunneling theory. Finally, Eqn. (2) is obtained by inserting Eqn. (4)[33].
0.7 Implementation of PTPI
In order to calculate the photoelectron spectra near the polarization direction with Eqn. (2), it is needed to determine the GT as well as its weight and tunneling time corresponding to the final longitudinal momentum . It is implemented by using a semiclassical tunnelingionization model[21], where the atomic ionization consists of two essential physical processes, i.e., an electron tunnels through the Coulomb field that has been dramatically suppressed by the laser field, and the released electron is driven back by laser field to scatter with its parent ion. According to Landau’s effective potential theory, Schrödinger’s equation written in parabolic coordinates can be separated into two onedimensional equations, and there is a potential barrier along direction. Suppose that the electron tunnels only along the direction of electric field, the coordinate of the tunnel exit, i.e., , can be calculated from . The tunneled electrons at the tunneling exit have initially a Gaussian transverse velocity distribution. Each trajectory is weighted by the ADK ionization rate[34, 37] , where is the distribution of initial transverse velocity, and , depending on tunneling time as well as the ionization potential . In adiabatic PTPI method, the initial longitudinal momentum is set zero. And in nonadiabatic setting, , where .In the posttunneling process, the electron evolution in the combined oscillating laser field and Coulomb field is traced via the classical Newtonian equation . A sevencycle pulse is applied and the electric field along z axis is written as . Because of the strong dependence of the tunneling rate on the electric field, only the GTs launched at the peak cycle are considered in the calculation of Eqn. (2).
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This work is supported by the National Natural Science Foundation of China (Grants No. 11404027, No. 11274051, No. 11374040, No. 11475027, No. 11575027, and No. 11447015) and the National Basic Research Program of China (973 Program) (Grants No. 2013CBA01502 and No. 2013CB834100).

The authors declare that they have no competing financial interests.