When an atom interacts with intense laser field^[1], electrons will be ionized from the parent ion^[2,3]. After tunneling ionization, some electrons leave the parent ion and reach the detector directly, and some electrons are driven back to the parent ion by the laser field and then rescatter off the parent ion. In fact, these two kinds of electrons will give rise to interference in the final photoelectron momentum distributions (PMDs) when their final momenta are equal, and these electrons are also named the reference beam and the signal beam, respectively. The interference process has also been referred to as strong-field photoelectron holography (SFPH)^[4], a novel quantum analog of optical holography containing structural and dynamic information in PMDs^[5–7]. Being an SFPH, the spiderlike structure in PMDs has been extensively investigated both experimentally and theoretically^[8–16]. In 2012, Hickstein et al.^[13] intuitively demonstrated that the shape and feature of the spiderlike structure depend on the number of times an electron passes its parent ion before rescattering. Because the spatial and temporal information of both the parent ions and electrons is encoded, the interference patterns have been used to extract structural and dynamic information in recent investigation^[17–21]. Particularly, the scattering amplitude, f(p,θ)=|f(p,θ)|eiα(p,θ), has been used to characterize the strength of a scattering process, where α(p,θ) is the phase of the scattering amplitude induced by the rescattering of the photoelectron with the parent ion. This phase can be of great significance in atomic physics and elementary particle physics^[22,23]. In strong-field physics, the phase of the scattering amplitude is related to the structural information encoded in SFPH. In 2016, Zhou et al.^[14] utilized a screened Coulomb potential model to extract this phase, and little following research was reported.

In the paper, we present our numerical results on the role of the scattering-amplitude phase in the spiderlike structure by deploying the semiclassical rescattering model (SRM) for hydrogen atoms. With this model, we show that the primary spiderlike structure arises from the interference between the reference electrons and the signal electrons scattered on their first revisit. For comparison, we also present the spiderlike structure simulated by solving the time-dependent Schrödinger equation (TDSE). The simulations show that the interference patterns obtained from the SRM do not agree with the TDSE results if the phase of the scattering amplitude is neglected, and these patterns agree with each other if this phase is included. We find that the time difference between electron ionization and rescattering extracted from SRM agrees fairly well with that calculated from TDSE.

In order to study the role of phase in the scattering amplitude of the spiderlike structure, we deploy SRM, which is based on the classical recollision three-step model^[24–29]. In our simulation, the laser field is linearly polarized along the x axis (in atomic units, a.u.):E(t)=E0sin(ωt).(1)

In SRM, we assume that the initial velocity of the signal electron is zero (i.e., vxsig=vysig=0), while the initial velocity of the reference electron is assumed to be vxref=0 and vyref≠0. At the rescattering time, tc, the signal electron is driven back to its initial position [i.e., x(tc)=y(tc)=0]. The rescattering time, tc, can be obtained by solving the equation of motion:sin(ωtc)−sin(ωt0sig)−ω(tc−t0sig)cos(ωt0sig)=0.(2)

The signal electron is elastically scattered by the parent ion at an angle θc. In our simulation, the scattering angle θc is within the range of −90° to 90°. The final momentum of the signal electron can be written aspx=vccosθc−E0ωcos(ωtc),(3)py=vcsinθc,(4)where vc is the velocity of the signal electron at the rescattering time tc. When the final momentum of the signal electron is equal to that of the reference electron, interference will occur. Given the final momentum value, the ionization time and the initial velocity of the reference electron can be obtained ast0ref=1ω[arccos(−ωpx/E0)],(5)vy=py.(6)

The phase of each trajectory is given by the classical action along the path:S=∫0∞[v2(t)2+Ip]dt.(7)

In our simulation, the phase difference is given byΔθ=12∫t0reftcvx2dt+12vy2(tc−t0ref)−12∫t0sigtcvx2dt+Ip(t0sig−t0ref)+α,(8)where α is the phase of the scattering amplitude, as suggested by the theory deciphering the spiderlike structure^[14,15]. Equation (8) can be rewritten in a simple form^[14,15]:Δθ=12py2(tc−t0ref)+α.(9)

From Eq. (9), it is clear that information of tc−t0ref and α is encoded in the interference patterns of the PMDs.

The tc and t0ref can be approximately determined from the saddle-point equations^[30]. The saddle-point equations for the signal electron are[k+A(t0sig)]22=−Ip,(10)(tc−t0sig)k=−∫t0sigtcdt′A(t′),(11)[k+A(tc)]22=[p+A(tc)]22,(12)and the saddle-point equation for the reference electron is[p+A(t0ref)]22=−Ip.(13)

In Eqs. (10)–(13), k and p are the electron drift momentum and final momentum, respectively. Ip is the ionization potential of the hydrogen atom. A(t) is the vector potential. Equations (10) and (13) stand for energy conservation in the process of tunneling, Eq. (11) is the return condition of signal electron, and Eq. (12) stands for energy conservation when the signal electron rescatters off the parent ion.

In order to obtain the PMDs, we also numerically solve the TDSE. In the length gauge, the TDSE is written in the following form (in a.u.):i∂ψ(r,t)∂t=[−12∇2+V(r)+r·E(t)]ψ(r,t),(14)where V(r)=−1/r2+b is the Coulomb potential softened to avoid singularity and to match the ionization potential of hydrogen. The soft parameter b is set to be 0.9.

We use the wavefunction-splitting technique to obtain the PMDs. The wavefunction-splitting technique allows us to reconstruct the external wavefunction in the momentum space and to calculate the photoelectron momentum spectra accurately^[31]. According to the wavefunction-splitting technique, the total simulation space is divided into the inner region (|r|<rin), outer region (rex<|r|<rM), and overlapping region (rex<|r|<rin). In the overlapping region, we use an absorbing function to obtain the internal wavefunction ψin and the external wavefunction ψex. The absorbing function is chosen to beVabs(r)=11+exp(r−r0Δr).(15)

The wavefunctions are related by the following form:ψin(r,t)=Vabs(r)ψ(r,t),(16)ψex(r,t)=[1−Vabs(r)]ψ(r,t).(17)

The final PMDs are obtained from the accumulated momentum-space external wavefunction.

In our numerical simulation, the intensity and wavelength of the laser pulses are varied around I=4×1013W/cm2 and λ=2000nm, respectively. The absolute phase of the laser pulse is zero.

We use the Monte Carlo algorithm in the simulation by sampling electrons ionized within two optical cycles (o.c.), i.e., [0,2T]. Our extensive simulation gives typical spiderlike interference structures, and we choose to show one with α=0 in Figs. 1(a)–1(c). In order to demonstrate the feasibility of SRM, we also numerically solve the TDSE to obtain the spiderlike structure, as given in Figs. 1(d)–1(f). One can discern three types of interference patterns in the momentum distributions^[32–35]: co-circular rings, short arcs, and spiderlike features. Corresponding to Figs. 1(a)–1(c), we show the reference trajectory and the signal trajectory contributing to the spiderlike feature in Fig. 1(g). We focus on the spiderlike structure in our discussion. One can observe that the spiderlike features in Figs. 1(a)–1(c) resemble those of the TDSE result in Figs. 1(d)–1(f), proving that the overall features obtained by the TDSE are reproduced by the SRM. Therefore, we can use SRM to retrieve the time and spatial information encoded in the holographic spiderlike structure. For making a quantitative comparison, we show, in Fig. 1(h), the cut-plot curves taken along the transverse (p_y axis) direction at a given longitudinal momentum (at fixed p_x) in the spiderlike patterns of Figs. 1(a)–1(f). One sees that these two curves do not overlap with each other, which may be attributed to the absence of the Coulomb potential in SRM^[27,28]. However, they will overlap if a proper non-zero value of phase α is used in the SRM simulation.

In order to reveal the role of the phase of the scattering amplitude, α, in the spiderlike structure and possibly in other interference patterns, we assume that α is a constant up to a zeroth-order approximation. In Figs. 2(a)–2(g), we chose to show some typical spiderlike patterns for α=0, π/20, π/10, 3π/20, π/5, π/4, and 3π/10, and several cut-plot curves at px=4.0 a.u. in Fig. 2(h). These cut-plot curves are the transverse momentum distributions manifesting that the spiderlike patterns are gradually narrowing in the transverse direction, and the intensities of the zeroth-order interference maxima (central maxima) are decreasing when α is increased from zero to 3π/10. This observation is in agreement with the theoretical prediction of Eq. (9). We take the first interference minimum in the transverse momentum distributions (cut-plot curves) for discussion. Equation (12) can be solved for the transverse momentum values corresponding to the first transverse minimum as py=2(π−α)/(tc−t0ref). It can be seen that the position of the first interference minimum depends on the value of α, and py will decrease when α increases. As a result, the first interference minimum unambiguously shifts to positions with smaller transverse momentum values. These analyses demonstrate that the phase α induced by Coulomb interaction of the signal electron with the parent ion plays an important role in the spiderlike patterns. In addition, Fig. 2(h) shows that the intensities of the zeroth-order interference maxima decrease when α increases. However, the reduction of the zeroth-order interference maxima was not reported before, possibly because the phase α is small at low momentum and small scattering angle, and the fact that the SRM does not consider the amplitudes of the signal and the reference electrons.

For investigating the time information pertaining to ionization and rescattering, we use a window function^[14] to average the interference fringes over px in a narrow interval px±Δpx for a better signal-to-noise ratio and a curve-fitting procedure for the cut-plot curves. By choosing the cut position of px=0.4 a.u. and a suitable window width of Δpx=0.08 a.u., we obtain several cut-plot curves of the interference structures at the fixed px. Then, we use the curve-fitting procedure to extract the interference term cos(Δθ) from these curves. The quantities tc−t0ref and α encoded in these spiderlike structures are extracted from the interference term cos(Δθ) by using Eq. (9). Figure 3(b) presents quantity tc−t0ref extracted by fitting the cut-plot curves [like those in Fig. 2(h)]; it is clear that the extracted tc−t0ref decreases with the increasing final momentum px. Then, we compare the extracted tc−t0ref with the calculated tc−t0ref by the saddle-point equations^[30] and show them in Fig. 3(b). Their excellent agreement further demonstrates that this time difference is not the main reason for the disparity of the interference patterns obtained by SRM and TDSE. Quantities tc−t0ref extracted by fitting to Eq. (9) and calculated from the saddle-point equations both deviate obviously from the quantity tc−t0ref obtained directly by the SRM. Nevertheless, the time difference obtained by these three methods manifests a similar law of linear variation with the final momentum px. This monotonic variation of tc−t0ref with px can explain the shift of the spiderlike feature: with increasing px, tc−t0ref decreases, leading to an increase of the py value at which the first interference minimum is located. As a result, the spiderlike structures become thinner for smaller longitudinal momenta, which are in agreement with the appearance of the interference patterns in Figs. 1(a)–1(f).

We also monitor variations of higher-order interference fringes, or more spider legs, with the phase by studying how the positions, or the transverse momentum values at which the first, second, and third interference minima are located, shift with α. To that end, we obtain the cut-plot curves at a cut position of px=0.4 a.u. in the spiderlike patterns [as shown in Fig. 4(a)], then we compare the interference minimum positions, pymin, extracted by fitting to Eq. (9) with that calculated from the SRM, and the result is shown in Fig. 4(b). It is obvious that these two kinds of pymin values are in good agreement because the phase α is taken into account. With the increase of the phase of the scattering amplitude, the position of the interference minimum decreases to smaller transverse momenta, and thus the shift of the spiderlike structure becomes appreciable, which corroborates our previous analysis on the first interference minimum. These observations also reveal that SRM is powerful in elucidating SFPH.

To further study the impact of the phase of the scattering amplitude in the spiderlike structure, we use a fitting algorithm^[14] to extract the value of phase α from the cut-plot curves of the spiderlike patterns simulated by TDSE, and the results are presented in Fig. 5(a). One can see that α is close to zero when py=0 a.u., and thus the zeroth-order interference maximum will not decrease in intensity. Then, we use the value of phase α extracted from the TDSE results to obtain a modified cut-plot curve of the spiderlike structure simulated by SRM combining with Eq. (9), and this curve is given in Fig. 5(b). Clearly, the modified curve agrees well with that obtained by the TDSE, simply because we have included both the phase difference between the signal electron and the reference electron accumulated during their propagation and the phase of the scattering amplitude.

In summary, by deploying SRM for hydrogen atoms, we have successfully simulated the spiderlike structure in PMDs and proven that the spiderlike structure well reproduces that by TDSE upon considering the phase of the scattering amplitude. Analyses on the cut-plot curves taken from the spiderlike patterns for interrogating the role of the phase have demonstrated their significance in deciphering the spiderlike interference patterns. Our simulation shows that the spiderlike feature shifts to positions of smaller transverse momentum values with increasing α. The time differences in electron ionization and rescattering calculated by SRM and the saddle-point equations are either in agreement or show similar laws of linear variation, thus strengthening the reliability of the SRM and corroborating the significance of the phase of the scattering amplitude in spiderlike PMDs.