Spectral observations and settings
observable = StrongField.SfaEnergyDistribution(pi/2, 0.0, 200, 10*omega)
sfaSettings = StrongField.Settings([E1], "VelocityGauge", true, true, false, false, true)
comp = StrongField.Computation(observable, nuclearModel, grid, initialLevel, finalLevel,
                                              beam, envelope, polarization, volkov, sfaSettings)
StrongField.perform(comp, output=true)
```
**Figure 4.** Julia input for generating the black-solid ATI spectrum in the left panel of Figure 5 for a krypton target, if irradiated by an *np* = 8 cycle sin<sup>2</sup> laser pulse with a central wavelength of *λ* = 800 nm and intensity *I* = 10 14 W/cm2. The laser pulse is right-circularly polarized and has a carrier–envelope phase *φ* (CEP) = 0. This input describes the complete strong-field computation, but where the 4*p* 6 <sup>1</sup>*S*0 ground and the final 4*p* 5 <sup>2</sup>*P*1/2,3/2 levels of krypton are assumed to be generated before by the JAC toolbox. Although no attempt is made to explain this input in all detail, this figure nicely demonstrates how readily JAC can be utilized to generate rather different spectra and distributions. See text for further explanations.

In the input above, we finally also specify as observable an SfaEnergyDistribution(), and which is to be calculated for *ϑ* = *π*/2 and *ϕ* = 0 (i.e., along the *x*-axis), and for 200-electron energies in the interval 0 ≤ *ε* kin ≤ 10 *ω* ≈ 15 eV. All this input together determines the (strong-field) computation comp::StrongField.Computation and can be readily adopted to many other experimental scenarios. All that is needed in JAC is to perform(comp, output=true) this computation, and where the optional parameter output=true just tells JAC to return the calculated data (amplitudes) to the user for printing and post-processing.

Figure 5 displays the photoelectron energy spectra, emitted along the *x*-axis, for a krypton target and a right-circularly polarized laser pulse. The left panel shows the spectra as obtained for a computed with a hydrogenic 1*s* initial wave function with adopted ionization potential and for a plane-wave Volkov continuum (black-solid curve) as well as a Coulomb–Volkov continuum (red dashed curve). On the right panel, in contrast,

the spectra are computed for an initial 4*p* orbital of neutral krypton and a plane-wave Volkov continuum (black-solid curve), a Coulomb–Volkov continuum (red-dashed curve) as well as a distorted-Volkov continuum (blue-dotted curve). In all these computations, a right-circularly polarized sin<sup>2</sup> pulse of wavelength *λ* = 800 nm, intensity *I* = 10<sup>14</sup> W/cm2, carrier–envelope phase *φ*(CEP) = 0 and with just *np* = 8 cycles has been utilized.

**Figure 5.** Photoelectron energy spectra, emitted along the *x*-axis within the polarization plane, for a neutral krypton target and a right-circularly polarized laser pulse. The left panel shows the spectra as computed with a hydrogenic 1*s* initial wave function with adopted ionization potential and for a plane-wave Volkov continuum (black-solid curve) as well as a Coulomb–Volkov continuum (red-dashed curve). On the right panel, in contrast, the spectra are computed for an initial 4*p* orbital of neutral krypton and a plane-wave Volkov continuum (black-solid curve), a Coulomb–Volkov continuum (red-dashed curve) as well as a distorted-Volkov continuum (blue-dotted curve). In all these computations, a right-circularly polarized sin<sup>2</sup> pulse of wavelength *λ* = 800 nm, intensity *I* = 10<sup>14</sup> W/cm2, carrier–envelope phase *φ*(CEP) = 0 and with *np* = 8 cycles has been utilized.

Input quite similar to Figure 4 can be employed also for studying the angle and momentum distributions of photoelectrons for different laser pulses and targets. While no further input data will be shown below, we refer the reader for details to the User Guide and the online documentation of the JAC program. Moreover, rather moderate changes to this input will be sufficient in the future to expand the StrongField module to other gauges, amplitudes or many-electron features. While such an expansion of the code indeed appears straightforward, major effort will still be needed for its implementation and testing.

#### *3.2. Photoelectron Angular Distribution for Elliptically-Polarized Laser Pulses*

In the electric-dipole (E1) approximation, the angular distribution of the photoelectrons is restricted to the *x* − *y* polarization plane and just reflects at fixed photoelectron energy the ionization probability in Equation (1) for different azimuthal angles 0 ≤ *ϕ* ≤ 360◦. If, moreover, the laser field dominates the electron dynamics in the continuum, the observed photoelectron angular distribution (PAD) should also reflect the symmetry of the vector potential of the laser beam. In practice, however, a *Coulomb asymmetry* in the PAD was (first) observed by Goreslavski et al. [33] in the ATI of xenon gas targets and, since then, has been found to be a valuable testbed for improving theory. For lithium, argon and xenon, for example, the SFA theory was shown to reproduce this asymmetry, if a target-specific initial orbital function is chosen along with a distorted-Volkov continuum for the active electron [34].

Figure 6 displays different photoelectron angular distributions in the polarization plane (*ϑp* = *π*/2) for a krypton target. Angular distributions are shown for ellipticallypolarized laser pulses with  = 0.36 (left panel) and *ε* = 0.56 (right panel) at fixed photoelectron energy *ε p* ≈ 2.9 *ω* according to the third ATI peak in Figure 5. Different approximations are compared for these angular distributions: a hydrogenic 1*s* initial orbital together with a plane-wave Volkov continuum (black-solid curves); a self-consistent initial 4*p* orbital of neutral krypton together with a Coulomb–Volkov continuum (red long-dashed curves); the same initial 4*p* orbital but together with a distorted-Volkov continuum (blue-

dashed curves). All these distributions are normalized on their maximum, while all other laser parameters are the same as in Figure 5. Indeed, a self-consistent 4*p* orbital of neutral krypton together with a Coulomb–Volkov continuum (red long-dashed curves) leads to a clear rotation of the PAD as mentioned above. Moreover, the PAD no longer exhibits an inversion symmetry with regard to the origin because of the short duration of the laser pulse. If, in addition, the Coulomb–Volkov continuum is replaced by an distorted-Volkov continuum (blue-dashed curves), and which accounts for an outgoing electron in the potential of the Kr<sup>+</sup> photoion, the rotation angle still changes rather remarkably. In Ref. [19], it was shown that such a distorted-Volkov continuum (often) leads for different targets to better agreemen<sup>t</sup> with experiment.

**Figure 6.** Photoelectron angular distributions in the polarization plane (*ϑp* = *π*/2) for a krypton target. Angular distributions are shown for elliptically-polarized laser beams with  = 0.36 (**left panel**) and  = 0.56 (**right panel**) at fixed photoelectron energy *ε p* ≈ 2.9 *ω* according to the third ATI peak in Figure 5. Different approximations are compared for these angular distributions: a hydrogenic 1*s* initial orbital together with a plane-wave Volkov continuum (black-solid curves); a self-consistent initial 4*p* orbital of neutral krypton together with a Coulomb–Volkov continuum (red long-dashed curves); the same initial 4*p* orbital but together with a distorted-Volkov continuum (blue-dashed curves). All distributions are normalized on their maximum, while all other laser parameters are the same as in Figure 5.

#### *3.3. Photoelectron Momentum Distribution for Few-Cycle Laser Pulses*

Theoretical photoelectron momentum distributions (PMD) have been calculated in the literature by means of quite different methods, and by making use of even a larger number of case-specific modifications to these methods. Generally, the experimentally observed symmetries of the PMD cannot be explained so readily by just applying a planewave Volkov continuum [33], but can be improved further if the Coulomb potential of the residual ion is taken into account. In our implementation of the SFA direct amplitude, this is achieved by replacing the plane-wave Volkov continuum by either Coulomb–Volkov or distorted-Volkov states. For the low-energy photoelectrons with (say) *ε p* - 2*ω*, the ionization probability is then often enhanced by up to an order of magnitude, if the ionic charge just increases from *Z* = 0 to 1. This has been explained by the attractive Coulomb potential of the residual ion that pulls the electron back to the ion and hence reduces its kinetic energy. The low-energy part of the ATI spectra can be further improved by adding a short-range potential to the (long-range) Coulomb potential and by making use of distorted-Volkov states.

Figure 7 shows the photoelectron momentum distributions in the polarization plane (*ϑp* = *π*/2) for the strong-field ionization of a krypton target. Momentum distributions are shown for circularly-polarized laser beams with three different CEP phases: *φ*(CEP) = 0 ◦ (left panel), *φ*(CEP) = 45 ◦ (middle panel) and *φ*(CEP) = 90 ◦ (right panel) and by applying a self-consistent initial 4*p* orbital of neutral krypton together with a plane-wave Volkov

continuum. All further laser parameters are the same as in Figure 5. Obviously, the PMD in this figure exhibits a (very) clear rotation since the photoelectrons are preferably emitted in the polarization plane along the maximum of the vector potential [6], and which changes with the carrier envelope phase *φ*(CEP). It will be interesting to explain with JAC in future work how the Coulomb asymmetry and the CEP dependence act together upon the angular or momentum distributions, and, especially of the initial-bound and distorted-Volkov continuum states of different atomic targets, are taken properly into account. In these studies, both the Coulomb and short-range interactions can be easily incorporated into the continuum by just replacing the radial wave functions of the active electron.

**Figure 7.** Photoelectron momentum distributions in the polarization plane (*ϑp* = *π*/2) for a krypton target. Momentum distributions are shown for circularly-polarized laser beams with three different CEP phases: *φ*(CEP) = 0 ◦ (**left panel**), *φ*(CEP) = 45 ◦ (**middle panel**) and *φ*(CEP) = 90 ◦ (**right panel**) and by applying a self-consistent initial 4*p* orbital of neutral krypton together with a plane-wave Volkov continuum. All further laser parameters are the same as in Figure 5.

## **4. Conclusions and Outlook**

Up to the present, the SFA has been found as perhaps the most powerful method for predicting or analyzing the electron dynamics in strong-field ionization. Often, this approximation helps describe features in the observed electron distributions even *quantitatively*, if the initial-bound and final-continuum states of the photoelectron are well adopted to the target atoms, and if combined with a proper parameterization of the laser field. With the present implementation of the direct SFA amplitude into the JAC toolbox, this method can now be applied to different targets and strong-field scenarios. In particular, the implementation of the SFA in the partial-wave representation enables us to readily control (and replace) the wave functions and various details about the laser–electron interaction. It also enables us to extend this implementation for incorporating further interactions and mechanisms into the modeling.

Detailed calculations are performed for a krypton target as well as for different ATI spectra and PMD. These examples clearly show how the target potential affects the photoelectrons on their way to the detector and, hence, all the observed spectra. In particular, we have demonstrated how the electronic structure of the atomic targets can be taken into account in the representation of the active electron and how the dynamics of the outgoing electron can be readily controlled by applying different approximations for the Volkov continuum. Moreover, the use of partial waves and spherical tensor operators facilitates a simpler comparison of different pulse shapes and how they influence the observed ATI spectra and PMD.

Several extensions to the SFA are still desirable and appear feasible within a framework, which is based on a partial-wave representation of the associated strong-field amplitudes. While further effort will be needed to decompose these amplitudes into a form, suitable for computations, a few useful extensions concern:

• **Non-dipole interactions:** For spatially-structured light fields, non-dipole contributions to the Volkov continuum usually arise from the spatially dependent Volkov phase [35–37], and which need first to be expressed into a partial-wave representation

in order to become applicable within JAC. These *non-dipole* terms beyond the widely used E1 approximation capture the combined-electric *and* magnetic-fields upon the electron dynamics [38,39]. Their implementation into the JAC toolbox will help predict the energy and momentum shifts at long wavelengths of the driving fields.


For all these desirable extensions, the partial-wave representation of the SFA [15], and its implementation in JAC provides a straight and conceivably the best way to advance theory and the light–atom interaction in strong fields.

**Author Contributions:** Methodology, S.F. and B.B.; software, S.F. and B.B.; writing—review and editing, S.F. and B.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—440556973.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
