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Ionization in intense laser fields beyond the electric dipole approximation: concepts, methods, achievements and future directions

May 13, 2021

During the ionization process with strong laser fields the photoelectrons can reach large enough velocities such that the magnetic field component of the laser field becomes significant and the dipole approximation breaks down. The ionization dynamics and the final momentum of the electron is therefore modified by the entire Lorentz force (Fig. 1 and 2). In contrast the magnetic field interaction is neglected in the dipole approximation. Rapid developments in laser technology and advancements in the accuracy of the measurements techniques have enabled the observation of the influence of such non-dipole effects on the final angular photoelectron momentum distributions. More recently the number of studies on ionization beyond the dipole approximation has increased significantly, providing more important insight into fundamental properties of ionization processes. For example we have shown that the final three dimensional photoelectron momentum spectra is significantly affected by the non-dipole drift with the parent–ion interaction (Fig. 1-3), the linear multiphoton momentum transfer on a sub-cycle time scale (Fig. 3) and the sharing of the transferred linear photon momenta between the electron and the ion.

In this review article [1] we present an overview of the underlying mechanisms and we review the experimental techniques and the achievements in this field. We focus on ionization in strong laser fields in the regime where the dipole approximation is not valid but a fully relativistic description is not required. In this review, we summarized the recent results and showed not only the important effects beyond the dipole approximation, but we also demonstrated its potential to obtain additional information about fundamental processes in light–matter interaction. We put our research results in a broader context with additional discussions about other efforts and the connection to measurements of single photon ionization.

Ever since our initial publication of reference [1] in 2014 there has been an increasing number of new observations of non-dipole effects, leading to the fact that during the preparation of this review we even had trouble to keep up with the new publications on this topic.

Despite of all these new articles the exploration has not yet saturated. We have suggested more exciting experiments and discussed still open issues that need to be resolved. In particular, time-dependent measurements have the potential to provide deeper insight into fundamental mechanisms that is not only restricted to single ionization but can also be extended to multi-ionization processes.

The emergence of novel light sources operating at high intensities and long wavelengths will enable more exciting experiments in this field. We hope that with this review we can make the atomic, molecular and optical science community more aware of the effects beyond the dipole approximation and in particular its fundamental implications and possible applications.
  1. Maurer, J., and Keller, U. (2021). Ionization in intense laser fields beyond the electric dipole approximation: concepts, methods, achievements and future directions. J. Phys. B: Atom. Mol. Opt. Phys. 54, 094001 (10.1088/1361-6455/abf731)
  2. Ludwig, A., Maurer, J., Mayer, B.W., Phillips, C.R., Gallmann, L., and Keller, U. (2014). Breakdown of the Dipole Approximation in Strong-Field Ionization. Phys Rev Lett 113, 243001. (
  3. Maurer, J., Willenberg, B., Danek, J., Mayer, B.W., Phillips, C.R., Gallmann, L., Klaiber, M., Hatsagortsyan, K.Z., Keitel, C.H., and Keller, U. (2018). Probing the ionization wave packet and recollision dynamics with an elliptically polarized strong laser field in the nondipole regime. Phys Rev A 97. (://WOS:000419702700005)
  4. Willenberg, B., Maurer, J., Mayer, B.W., and Keller, U. (2019). Sub-cycle time resolution of multi-photon momentum transfer in strong-field ionization. Nat Commun 10, 5548. (

Fig.1: Isosurface of a reconstructed photoelectron momentum distributions (PMD) recorded at a wavelength of 3.4 μm and an intensity of 6 × 1013 Wcm−2 with elliptical polarization 0.11. The PMD shows the lobes due to direct electrons and the ridge that stems from Coulomb-focused electrons induced by the entire Lorentz force in the regime of the breakdown of the dipole approximation (according to Ref. 3).

Fig. 2: The peak of the PMD as a function of the ellipticity. The peaks transitions with increasing ellipticity from negative photoelectron momentum along beam propagation direction z (i.e. opposite to the beam propagation direction) to positive values (along the beam propagation direction) of pz. The opposite beam propagation is counterintuitive with the photon pressure argument (according to Ref. 3).

Fig. 3: Ionization yield and the pz-shift as a function of the ionization time. The measurement shows a clear offset in the ionization phase between the minimal pz-shift and the maximal ionization yield. This offset translate to a time delay between the most probable electron trajectory and the one with minimal momentum transfer. In addition to the results from CTMC calculations, we show the curves from the pure classical propagation (‘model’) and from two models with an ellipticity-dependent initial momentum along the beam propagation axis (according to Ref. 4).

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