## Abstract

Relativistic continuum dynamics for electrons from the ionization of atoms in an ultraintense (10^{17} W/cm^{2} to 10^{20} W/cm^{2}) laser focus are analyzed using a semi-classical wavelet model. The results quantify the energy and angle resolved photoionization yields due to the developing relativistic dynamics in ultraintense fields. Using the final state momentum, the bremsstrahlung radiation yield is calculated and shows a linear relationship between the radiation cutoff and the laser intensity. At 10^{20} W/cm^{2} photons with energies out to 10MeV should be observed. The results are quantitatively comparable to the observed angle resolved photoelectron spectra of current ultraintense laser-atom experiments. The results show the azimuthal angular distributions becoming more isotropic with increasing intensity.

© 2004 Optical Society of America

## 1. Introduction

Ultraintense light matter interactions ushered in a paradigm shift for high field physics through the break down of the approximation v/c=0, the introduction of magnetic field effects, the suppression of non-sequential ionization, and other phenomena. Novel physical effects like the photo-generation of particles with thousand [1] to million [2] electron volt energies expanded our knowledge into previously uncharted territory, and led to questions about the excitation and radiation mechanisms in ultraintense fields. A comprehensive knowledge of ultraintense field processes, such as Larmor or bremsstrahlung radiation [3], and a more complete physical picture of the ultraintense regime requires an understanding of atomic ionization and continuum electronic dynamics. Atomic studies provide an opportunity to test models of ultraintense field interactions without collective plasma effects. The results may also be applied, for example, as an ultraintense field calibration standard and could play a role in understanding the atomic processes within laser plasmas.

The ultraintense barrier tunneling ionization process for an atom has been studied both experimentally and theoretically [4, 5, 6]. It is the relativistic continuum dynamics of the electron subsequent to tunneling that will be discussed in this paper. Classical trajectory and Monte-Carlo techniques have been used to model ultraintense field electron dynamics [7]. Some salient points, such as the electron energy gain [8] when electrons are introduced at the right phase of the field [9] and non-planar dynamics in a real laser focus [10], have been established by these methods. Fully quantum calculations using the Dirac or Klein-Gordon equation are difficult but have been used to gain insight into electron-ion scattering [11] and properties of the continuum wave functions [12]. Since the ionization process is quantum mechanical, incorporation of the quantum nature into the model is essential for an accurate representation of many phenomena. Keeping this in mind, we use a semi-classical, continuum wavelet model that incorporates the wave function probability from tunneling ionization. In this approach a classical relativistic trajectory is generated for each time interval where tunneling occurs. Every trajectory has an ionization probability associated with it, and is called a wavelet. When all wavelets are projected onto position or momentum spaces they create, albeit semi-classically, a dynamical view of the atomic response of a bound electron to the ultraintense field during the whole laser pulse. The wavelet approach allows us to model the spatially distributed electron probability in the continuum, calculate a probability distribution of electron continuum momentum states, and capture a clear change in the continuum dynamics from 10^{17}W/cm^{2} (Fig. 1) to 10^{19}W/cm^{2} (Fig. 2).

## 2. Continuum momentum states

For intensities below 10^{16} W/cm^{2} or weakly focused geometries, the paraxial approximation is valid and simplifications occur in the classical equations of motion. In the paraxial limit, |*E*_{x}
(* r,t*)|=

*c*|

*B*

_{y}(

*r,t*)|, with all other components of E and B set to zero. It is well known that when the initial momentum is zero (consistent with ionization in a strong field,) the relativistic equations of motion

(m is the rest mass of a particle with charge q and relativistic factor *γ*) confine the dynamics to the propagation-polarization (x-z) plane. In this case, a relativistically invariant relationship exists, i.e. p_{z}=px ^{2}/2mc or equivalently tanθ=(2/(*γ*-1))^{1/2} where θ is the polar angle from propagation (z) axis [13]. Corrections to the paraxial approximation change this relationship significantly. Even though a laser may be polarized along the x-axis at the focus, significant y and z components to the field can exist away from the center of the focus. For example, at the exp(-1) beam waist (*w*_{0}
) and one Raleigh length (z_{R}=k${w}_{\mathit{0}}^{2}$/2) away from the center of an f#/2.5 focus E_{y}/E=0.005 and E_{z}/E=0.14 in the |x|=|y| plane and E_{z}/E=0.29 in the x-z plane. It is the more complicated interaction region and dynamics for atoms in ultraintense fields that requires quantitative, angle resolved information to compare to experimental photoelectron yields.

The ultraintense continuum model we develop here is subdivided as follows: tunneling ionization, semi-classical continuum dynamics, and extraction of the momentum distribution based on the probability. The sample in the laser focus was varied from 1 atom to a random distribution comparable to a density of 10^{15} atoms/cm^{3}. With regard to ionization, the excitation of the bound atomic system is treated in the adiabatic, DC limit since the photon energy is less than 1% of the bound state energy. Ultraintense [6] and intense [14] field experiments show the ionization yields from the 10^{15} W/cm^{2} to 10^{19} W/cm^{2} intensity range agree with the ionization yields calculated using a WKB tunneling ionization model. The magnetic field is approximated to have a negligible impact on tunneling ionization since the Zeeman energy and v/c for the electron within 0.5nm of the nucleus are both small. Once in the continuum, the ionization is modeled with the wavelets using a time step of order 10^{-18} s and probability assigned by the previously described tunneling ionization. Thus, the ionization wavelet may be thought of as the quantum mechanical electron flux from the ion in the quasi-static limit. In fact, the interaction is not static. At the peak of the electron momentum an error of 1% is introduced by using a constant momentum across the wavelet; i.e. the change in the deBroglie wavelength across each wavelet is in fact Δλ/λ≅0.01 though it is approximated as zero.

The interaction of a single ion with an 800nm, 11fs FWHM laser pulse generates approximately 10^{3} wavelets whose dynamics are solved numerically [15]. The calculations are propagated for fifteen cycles or until the electron is well out of the focal region. In the focus, the Coulomb field and radiation damping are neglected. Two movies of the continuum dynamics from an atom in an intense field (FWHM=11fs, 800nm, f/# 1.5) are shown in Fig. 1. For each figure, the electron probability from an atom at the origin (center of the focus) is projected on the x-z plane and followed in time. The eight level natural logarithmic white-yellow-red-black scheme for the electron probability is normalized to the peak and shows one exp(-1) order of magnitude per change in tone. For reference, the movie includes an adjacent frame for the electric field in the focus at the z axis, which accounts for the attenuation due to diffraction of the light beyond the focus. The ionization of Ne^{+7} to form Ne^{+8} at 2 10^{17} W/cm^{2} is shown in Fig. 1 from -7.7fs before to 10.6fs after the peak in the pulse. The movie shows the “bursts” of ionization near the peaks of the optical field and the oscillation of the electron driven by the laser field responsible for high field re-scattering [16], non-sequential ionization [17] and high harmonic generation [18]. One can see in Fig. 1 the electron is relativistic as shown by drift along z. However, as the extent of the probability distribution is near the center of the focus the continuum may be approximately described by p_{z}=${{\mathrm{p}}_{\mathrm{x}}}^{2}$/(2mc) dynamics with a spatially uniform electric and magnetic field. For the electron wavelets in Fig. 1, **k**·**r** is less than 1 and *γ* is less than 1.1. With these conditions the electron will oscillate in phase with the laser field and the parabolic relationship between p_{x} and p_{z} will result in a similar relationship between the x and z components of the drift velocity causing the wave fronts to become parabolic.

The ionization of Ar^{+15} to form Ar^{+16} at 1 10^{19}W/cm^{2} is shown in Fig. 2. from -6.5fs before to 8fs after the peak in the pulse. At 10^{19}W/cm^{2} the dynamics are clearly different with the electron probability from each tunneling ionization “burst” distributed over a 10^{-6}m length scale, which is comparable to the focus and wavelength dimension. There are several reasons for this drastic change. Even before an ionization wave front has experienced two cycles of field, it has traveled half a Raleigh length in the z direction. The large **k**·**r** shift allows the leading edge (ionization at phases before the optical peak of the pulse) of the electron wave front to “catch the wave” and “surf” out of the focus on the face of the field while later portions of the electron probability are too late and ride up on the wave and continue to oscillate, visibly lagging behind the field phase within a few femtoseconds after entering the continuum. Calculations with plane waves show the dynamics of the first cycles are not a strong function of the focus parameters; whether using a plane wave or f#/1.5 focus, the first one to two cycles of the field show wave fronts very similar to those in Fig. 2. Although near the center of the f#/1.5 focus the field is nearly paraxial, the large electron velocity at 10^{19} W/cm^{2} (γ≅2) destroys the parabolic relationship between the x and z components of the drift velocities. As shown in Fig. 2, a significant amount of electron probability approaches the Raleigh length while the field is still large, a scenario made possible by large **k**·**r**, the motion is dominated by non-paraxial fields, which are very different from the paraxial fields in phase and amplitude.

The final momentum distributions for atoms in a focus with a peak intensity of 2 10^{17}W/cm^{2}, 1 10^{19}W/cm^{2}, and 1 10^{20}W/cm^{2} are shown in Fig. 3. The normalized probability is shown in an eight color logarithmic scale with each change in tone representing 10^{-0.5} in scale. Figures 3(a, b) is for the ionization of Ne^{+7}, Figs. 3(c, d) is for Ar^{+15}, and Fig. 3(e, f) is for the ionization of Na^{+10}. In Figs. 3(b, d, f) the relativistic paraxial solution is also shown. Several features in the p_{x}-p_{y} momentum distribution can be seen in Figs. 3(a, c, e). The non-zero p_{y} results primarily from a nonzero B_{z} with some contribution from the nonzero E_{z}, E_{y} whose absolute magnitudes become significant enough in high fields to alter the final state momentum [8]. As the intensity increases from Figs. 3(a–c), the electron velocity along x increases. The larger v_{x} combined with B_{z} increases the p_{y} momentum as one moves from 10^{17} W/cm^{2} to 10^{20} W/cm^{2} until in Fig 3c the distribution almost appears to be elliptical along p_{y} rather than p_{x} as shown in Fig. 3(a). All the p_{x}-p_{y} plots exhibit left-right (x) and up-down (y) reflection symmetries as expected from a laser pulse with several cycles. In the ionization of atoms by ultraintense fields, the final state momentum is affected by two compounding occurrences: an ionization mechanism that limits the “birth” of the electron probability to near the peak of the field and an electron displacement magnitude - across the laser focus - that increases the interaction with non-paraxial fields. The bulk of the electron probability, which does not surf out near the center of the focus traverses back and forth within the focus for a few cycles, gaining significant py momentum due to the strong B_{z} fields at the edge of the focus and deviating significantly from the paraxial solution (see Figs. 3(a,c,e).) Ionization from atoms placed away from the center of the focus produce low probability momentum states that also deviate from the paraxial solution. The result can be seen most clearly in Fig. 3(d), and Fig. 3(f) by the increase in the range of momentum beyond the paraxial solution.

The final states are projected onto kinetic energy-angle plots in Fig. 4. The electron trajectory polar angle from the z-axis, θ, is plotted for two different azimuthal angles from the electric field. The E-θ plot in the plane containing the electric field (x-z plane) is plotted in Figs. 4(a, b, c) while the E-θ plot for the y-z plane is plotted in Figs. 4(d, e, f). Superimposed on the Figs. 4(a, b, c) plots is the paraxial E-θ function. The probability color scale is as shown in Fig. 3. The features in the E-θ plots are related to correlations between the momentum coordinates. For example, the apparent paradox in the angular features of electron kinetic energy distribution for 2 10^{17} W/cm^{2} where a nearly isotropic p_{x}-p_{y} momentum yields an energy spectrum that is highly aligned along x as shown in Fig. 3(a) compared to Fig. 3(d). This alignment is due to a correlation whereby electrons in the plane of polarization (x) gain significantly more p_{z} than electrons out of the plane, i.e., those with a significant p_{y} component. To aid in the comparison of the energy spectrum to recent experiment [1] and theory [7], the azimuthally resolved electron emission is shown in Fig. 5. These plots give azimuthal angle resolved photoelectron spectra (Table 1, azimuthal angle measured from E_{x} into B_{y}) into a range of polar angles for representative low and high energy regions.

In the case of low energies for all three intensities, the emission is nearly isotropic. However, in the high energy limit, for 2 10^{17}W/cm^{2}, 1 10^{19}W/cm^{2}, and 1 10^{20}W/cm^{2} the full-width-half-maximum azimuthal emission angle for the results in Fig. 5, i.e., the full ϕ angle about ϕ=0 degree (x-axis), is 18 degree, 22 degree, and 112 degree, respectively. For the highest electron energies, the angular spread in the emission increases rather than decreases. This effect is due to the large vx/c, and consequent *v*_{x}
×B_{z} force, for ultraintense intensities. The decreased directionality at higher intensities must be considered when proposing electrons from an ultraintense intensity focus as an injector for particle accelerators.

Electron dynamics both in and out of the focus are important for analyzing Larmor radiation, plasma effects, and bremsstrahlung radiation. It is well known in the non-relativistic case that electrons have a speed from 0 to 2q*E/mω* inside the focus while exiting electrons have a speed range from 0 to q*E/mω*. With increasing intensity the electron may remain in phase with the field due to p_{z} (or more generally **k**·**r**) and the exiting electron speed may approach the speed of the electron inside the focus. Consistent with this observation across the intensity range studied here, the ratio of the peak final electron kinetic energy to the peak kinetic energy in the focus increases from 0.25 for the non-relativistic case to 0.26, 0.35, and 0.81 for 2 10^{17} W/cm^{2}, 10^{19} W/cm^{2}, and 10^{20} W/cm^{2}, respectively. The electron energies produced by the interaction of ultraintense fields with atoms have an upper limit of 12 MeV at 10^{20} W/cm^{2} (Fig. 4(c).) Consistent with lower intensity studies, the highest energy final states correspond to ionization from phases off the peak of the field. This maximum energy can be increased by using a weaker focus [8] thereby suppressing the role of the non-paraxial components and the Guoy phase shift (tan^{-1}(z/z_{R})) mismatch between the electron and the laser field.

## 3. Bremsstrahlung radiation

When electrons with energies less than 100MeV traverse through a low Z material (e.g., nitrogen) the primary interaction is the collision excitation and ionization of the material [19] [20] with the accompanying loss of the electron’s kinetic energy over thousands of such processes. Even for high Z materials, such as lead, the energy below which collision losses exceed radiation losses is 10MeV. Therefore, based on the energy spectra in Fig. 4, one should expect the electron products from ultraintense field ionization of atoms will predominantly lose energy by collision interactions. The products from the collision interaction include, for example, e-2e, Auger processes and photons from the decay of the excited states. The magnitude, angular distributions, and energies of the secondary electron and decay products for the collision excited material depend upon the cross sections that are unique for every atom or molecule. The details of these secondary products are not described here. In this section, we calculate the bremsstrahlung radiation spectrum from the photoelectrons emitted from the focus as they collide with a nitrogen target. Naturally, electron energy loss due to the bremsstrahlung radiation and inelastic collisions are included. The loss due to inelastic collisions is treated by neglecting spin and exchange effects and the effective collision excitation energy for nitrogen is taken to be 80.5eV. This approximation introduces an error of less than 6% in the collision cross section. The pair-production phenomenon, which can occur when a photon interacts with a nucleus, is not considered here due to the low yield of photons with energy greater than 1 MeV. We use the Born approximation since v/c> Z/137 is satisfied for our conditions and the electron is deflected into small angles after scattering [21]. In general, the bremsstrahlung radiation is within an angle θ=mc^{2}/E_{0} of the initial direction of the electron with energy E_{0}. For low energy electrons (E_{0}≅mc^{2}) the angular distribution of the radiation should become isotropic.

The bremsstrahlung scattering process is modeled with an energy dependent screening of the nuclear charge by the bound electrons. In bremsstrahlung radiation, an electron (p_{0}, E_{0}) incident upon an atom is scattered into a new state (p, E) and emits a photon (k) to conserve energy and momentum. Following Bethe and Heitler [21] for the unscreened potential, the cross-section for emission of a photon of wave number in the range *k* to *k*+*dk* is

where,

and

Here *p* is the momentum times *c* and *k*=*E*_{0}*-E*. With screening, the cross section is given by,

where *ϕ*
_{1} and *ϕ*
_{2} are functions of *χ*=*100µk*/(*E0*
*E Z*^{1/3}
), which is obtained from the atomic screening length. In the case where *χ*<1.5, i.e., large energy collisions with large impact parameters and lower energy radiation, screening effects are included as the interaction involves the outer ranges of the atomic potential. In the case of higher frequency radiation relative to the electron energy, the interaction is short range and the unscreened potential is used. The equations for the radiation spectrum and the fractional energy loss of the electron as it propagates are solved numerically using a finite difference method. The bremsstrahlung radiation energy spectrum is then integrated over all time, i.e., until the electron energy is less than 500eV. The calculation evolves the electron energy spectrum; the electron energy spectrum for the ionization leaving the focus is used as the initial condition. The bremsstrahlung radiation spectrum, normalized to give the total energy radiated by a single electron, i.e. one ionization event averaged over the electron spectrum, is shown in Fig. 6. The spectrum is relatively flat out to the cutoff energy, which scales linearly with intensity. Small features in the yield, most prominent in the 10^{19} W/cm^{2} case, are the result of features in the energy distribution of the photoelectrons.

Since the high energy cutoff scales linearly with intensity and is relatively easy to measure with the highest energy photons directed forward with the electrons, it may be possible to use the bremsstrahlung radiation as a simple calibration of the focus for ultraintense lasers. Such measurements would avoid the problems of space charge and Coulomb explosion, which plague traditional ion charge state calibrations of peak laser intensities beyond 10^{17}W/cm^{2}.

In conclusion, we have presented the electron momentum, energy, and bremsstrahlung radiation expected from the ionization of atoms by ultraintense fields in a focused geometry. The results quantify, within the approximations of the model, the evolution of the electron dynamics from the edge of the relativistic intensity range at 10^{17}W/cm^{2} up to 10^{20}W/cm^{2} where the interpretation of the ionization dynamics, final energies, and radiation processes all depend upon a relativistic perspective. Several notable findings include: the increasing azimuthal isotropy of the high energy photoelectron spectra with higher intensity, the change in the peak final/peak focus kinetic energy from ¼ as the intensity exceeds 10^{17} W/cm^{2}, and the bremsstrahlung spectra with a high energy cutoff scaling linearly with intensity over the 10^{17} W/cm^{2} to 10^{20} W/cm^{2} range.

## Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 0140331.

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