Breakdown of the Dipole Approximation in Strong-Field Ionization

We report the breakdown of the electric dipole approximation in the long-wavelength limit in strong-field ionization with linearly polarized few-cycle mid-infrared laser pulses at intensities on the order of $10^{13}\mathrm{W}/{\mathrm{cm}}^2$. Photoelectron momentum distributions were recorded by velocity map imaging and projected onto the beam propagation axis. We observe an increasing shift of the peak of this projection opposite to the beam propagation direction with increasing laser intensities. From a comparison with semiclassical simulations, we identify the combined action of the magnetic field of the laser pulse and the Coulomb potential as the origin of our observations.

Figure 1

Illustration of the wavelength-intensity parameter space in strong-field ionization, taking the magnetic field component into account. The area where the dipole approximation is considered as valid (dipole oasis) is depicted as the green dotted region. The well-known short-wavelength dipole limit arises for wavelengths on the order of the atomic scale, i.e., for $\lambda =1\mathrm{a}.\mathrm{u}.$. The long-wavelength limit arises due to the laser magnetic field component, and is characterized by the ratio $U_p/2\omega c=1\mathrm{a}.\mathrm{u}.$ [4]. The experiment presented in this Letter was performed at a wavelength of $3.4\mu \mathrm{m}$ at intensities close to this limit (orange triangles). The radiation pressure limit arises for $U_p^2/2c^2=0.5\mathrm{a}.\mathrm{u}.$, and true relativistic effects start to occur around $2U_p/c^2=1\mathrm{a}.\mathrm{u}.$ [4]. The parameter ${\mathrm{\Gamma }}_R=\sqrt{U_p^3{I}_p}/3c^2\omega$ indicates the limit where the spatially spread electron wave packet essentially misses the ion under the influence of the magnetic field [6].

Figure 2

(a) Typical projected photoelectron momentum distribution (PMD) of xenon recorded at an intensity of $6\times 10^{13}\mathrm{W}/{\mathrm{cm}}^2$ with linear polarization using a VMIS at a center wavelength of $3.4\mu \mathrm{m}$. We show the plane spanned by the laser polarization (labeled $p_x$) and propagation (labeled $p_z$) direction. The orange arrow depicts the center spot resulting from field ionization of highly excited Rydberg states used as reference for $p_z=0\mathrm{a}.\mathrm{u}.$, and the dashed boxes indicate the areas taken for the momentum-offset analysis. (b) Projections of the PMD onto the beam propagation direction together with Lorentzian fits. The orange curve (squares) is used to set the $p_z=0\mathrm{a}.\mathrm{u}.$ reference and the offset of the maximum of the photoelectron distribution is extracted from the fit on the green markers (circles).

Figure 3

PMDs and their projections recorded in xenon and helium at different intensities. Upper figures: measured and simulated PMDs. Lower figures: projections of these PMDs onto the beam propagation, as used to extract the momentum offset (see Fig. 2). (a) Experimentally measured PMDs of xenon recorded at 3 and $6\times 10^{13}\mathrm{W}/{\mathrm{cm}}^2$. (b) Corresponding simulated PMDs at the same intensities as (a) reproducing the negative offset through the combined influence of the Coulomb potential and the full electromagnetic laser field. (c) Simulated PMD excluding the magnetic field. The projection exhibits no offset. (d) Measured PMD of helium at 800 nm and an intensity of $1.4\times 10^{14}\mathrm{W}/{\mathrm{cm}}^2$. Because of the shorter wavelength, the dipole approximation is valid and no offset is visible.

Figure 4

Extracted peak offsets along the laser propagation direction as a function of laser intensity for different target gases from experiment (filled markers) and semiclassical simulation (open markers). The points show a clear trend towards increasing negative offsets (i.e., opposite to the beam propagation direction) for increasing laser intensities. The uncertainties are indicated as error bars and the gray shaded area, respectively.