1. Introduction

When an atom is ionized by an intense laser field, the final-state electron will have an energy of interaction with the field measured by the ponderomotive energy U_p. This quantity represents the energy of the oscillatory motion of the free electron immersed in the strong field. When U_p approaches the rest energy mc ² of the electron, then the velocity of the electron must be regarded as relativistic, and theoretical treatments of the ionization process must be done relativistically. This conclusion is borne out by fully relativistic calculations[1],[2] done for ionization by circularly polarized light. In those calculations it is found that the inequality

is satisfied in that U_p ≈ .01 mc ² leads to observable relativistic effects. The criterion in Eq. (1) is strongly frequency dependent because of the inverse quadratic dependence of U_p on the frequency. A factor 10^-2 inserted into Eq. (1) would correspond to about 10¹⁸ W/cm ² laser intensity for an excimer laser wavelength of 248 nm, 10¹⁷ W/cm ² for 800 nm, and 5 × 10¹⁴ W/cm ² for the CO ₂ laser at 10.6 μm. These intensities are achievable.

When the ionizing laser radiation is linearly polarized, an additional relativistic effect emerges. In contrast to the simple straight-line oscillatory motion of a free electron in a plane-wave field in the non-relativistic case, a free electron in a strong plane-wave field executes a figure-8 motion[3],[4] with the long axis along the polarization direction of the electric field, and the plane of the figure defined by that direction and the direction of propagation of the plane wave. This figure-8 motion arises from the magnetic field of the plane wave coupled to the electric field, and the usual non-relativistic dipole approximation neglects the magnetic field. A failure of the dipole approximation should have physical consequences when the short axis of the figure-8 is of the order of the atomic radius. On these grounds, the dipole approximation is expected to fail[5] when

where α is the fine-structure constant. The limit (2) is even more strongly frequency dependent than the condition (1) for most frequencies, and is reached at lower intensities. It is found herein that a factor of about 10^-2 is appropriate in Eq. (2), just as it was in Eq. (1), at least for the high frequencies examined here. However, the effects of this loss of the dipole approximation on total ion yields, angular distributions, or photoelectron spectra may not be of equal significance. All of this remains to be explored at the low frequencies (ω≪1 a.u.) for which very intense lasers are available.

The theoretical method to be employed here is the Strong Field Approximation (SFA). This has already been applied to the problem of atomic ionization under relativistic conditions, but results have been published[1, 2] only for circular polarization of the laser radiation, in the framework of the Dirac theory. We report here analogous Dirac results for linear polarization [6] of the ionizing laser.

We note that the SFA, by employing Volkov solutions for the final state, automatically includes all field-induced oscillations of the ionized electron. Rescattering in the final state is neglected. The neglect of rescattering is unimportant in the circular polarization case, which explains the great accuracy attainable with the SFA [7] for this case. With relativistic Volkov solutions employed, the emitted photoelectrons acquire the expected relativistic projection in the direction of propagation of the field[1], an effect that can be interpreted as a result of the momentum of the absorbed photons. The resulting increase in the distance from the atom of the field-induced trajectory of the photoelectron can be viewed as the cause of the relativistic strengthening of the stabilization effect that was found[2].

A brief overview of the SFA is helpful to understand why it is useful for the description of relativistic ionization in strong fields. In its original non-relativistic form[8, 5], the SFA approximates the exact time-reversed transition amplitude

by replacing the complete final-state wave function Ψ by the Volkov solution. Equation (3) is written in atomic units. These units are employed in the remainder of the paper. The Volkov solution Ψ^Volk is an exact solution for a free charged particle in an electromagnetic field, and so the nature of the SFA is that it assumes that the final-state motion of the ionized electron is dominated by the laser field, and effects on the detached electron of the binding potential are neglected.

We remark on the differences between the SFA and the well-known Keldysh approximation[9]. The Keldysh approximation can be formulated from Eq.(3), though it was not so expressed by Keldysh. The differences are: (a) The H^I used by Keldysh is in the “length” gauge, rather than the “velocity” gauge of Eq. (4); (b) Keldysh makes a low-frequency approximation from the outset; and (c) Keldysh assumes that the final state electron is produced with zero momentum. Extension to the relativistic case is not possible because of limitation (a); limitation (b) precludes extension to high frequencies, unlike the SFA, which works well[10] at high frequency; and (c) taken together with (a) make the Keldysh approximation into a tunneling method, which cannot be used to explore the stabilization phenomenon for which the SFA is well suited[11, 2].

The limitations of SFA have been explored for circular polarization of the electromagnetic field by finding the principal Coulomb correction[12] to the SFA. When so corrected, the SFA is valid when either α ₀ ≳ 10 or U_p/E_B ≳ 10, where α ₀ is the radius of motion of a classical free electron in the circularly polarized field, and E_B is the binding energy of the field-free atom. Since either or both of these conditions are satisfied in recent strong-field experiments, a comparison of the predictions of the SFA with experimental measurements[13] of the photoelectron spectrum shows essentially perfect agreement [7] for circular polarization. For conditions leading to relativistic ionization by circularly polarized fields, the SFA has negligible error.

The situation is more complicated for linear polarization. The essential difference between the two polarization states is that the photoelectron spectrum for strong-field circular polarization is relatively narrow, and centered approximately at the U_p corresponding to the peak laser intensity laser in the laser pulse. All emitted electrons are thus energetic, and the Coulomb field has little effect. Furthermore, there is no “revisiting” of the ionized atom by the photoelectron. For linear polarization, a substantial portion of the spectrum is at low energy, and the ionized electron does oscillate back to the vicinity of the ionized atom. Although calculated linear polarization spectra [7] show all the features of the experimental spectra[13, 14], the agreement is not unqualified, as it is for circular polarization. However, as the laser intensity increases into the relativistic domain, field dominance of the photoelectron motion will be more strongly asserted, and a decreasing proportion of the spectrum is at low energy. Stated in other terms, the rescattering effects neglected in the SFA are of decreasing significance as the intensity increases. Therefore, the SFA should work well for linear polarization in the relativistic domain.

2. Relativistic Dirac calculation

The transition amplitude in a Dirac formulation of the ionization problem starts from the exact expression

replacing the non-relativistic Eq. (3). In place of the “natural units” with ħ = 1, c = 1 customarily employed in relativistic work, atomic units are retained here to be consistent with the non-relativistic expressions. The states Φ and Ψ are Dirac spinor states for the atom bound by the central potential V for, respectively, the system without and with the laser field present. The laser field is defined by the four-vector potential A^μ, and the γ^μ are the Dirac matrices. Conventions for quantities in the Dirac space are as described in the book of Bjorken and Drell[15].

As in the non-relativistic SFA, the Dirac form of the relativistic SFA comes from the replacement of Ψ representing an exact solution of Eq. (7) with the corresponding Dirac Volkov solution

where m is the mass of the electron, E is the relativistic electron energy, V is the volume of a box normalization, p^μ and k^μ are four-momentum vectors for the electron and the applied field of frequency ω, and scalar products like p ∙ k, k ∙ x, and A ∙ p are relativistic scalar products such that, for example,

The quantity u in Eq. (8) is a Dirac spinor that satisfies the equation (γ^μ p_μ - c) u = 0. The field-free solution Φ_i in Eq. (5) is taken to be the solution of the Dirac equation for the hydrogen atom as given, for example, in [15].

The procedure followed in proceeding from the transition amplitude in Eq. (5) to the expression for the transition rate follows closely that given for the circular polarization calculation in Ref. [1]. The differential transition rate with respect to the solid angle Ω is

where Z is the magnitude of the charge on the hydrogen-like atom, a is the amplitude of the vector potential, and ρ is the magnitude of the three-vector

where p and k are the three-vector parts of p^μ and k^μ, and η = a ²/4c ² p ∙ k. The summation index n can be identified as the number of photons participating in the process. Considerable physical content is represented by the 𝑈 parameters (that is, 𝑈_A, 𝑈_B, and 𝑈_C) stated below.

The non-relativistic limit of Eq. (9) is equivalent to the expression in Ref. [8]

where the angular dependence is contained in the Bessel function, J_n. For linear polarization J_n is the generalized Bessel function described in previous work [8]

In the polar coordinate system chosen for the linear polarization case, θ is the angle between the velocity vector of the emitted electron and the polarization vector. The quantity z is the intensity parameter defined by

A similar derivation was also performed [6] in the context of a scalar electron, where solutions to the Klein-Gordon equation were used for the Volkov solution and for the hydrogen-atom wave functions. Spin-averaged results for Dirac spin-1/2 electrons were found[6] to be very similar to those for Klein-Gordon scalar electrons. These results will not be presented here. Both derivations rely upon the SFA which is expected to become very accurate at high energies. That is, when U_P ≫ E_B and p ²/2 ≫ E_B then the solution using the SFA method should be very accurate.

The differential transition rate resulting from the derivation reported here follows from a Dirac treatment of all elements of the problem, including bound states, unbound states, and interaction terms. Hence, spin effects are included in full. As derived, the differential transition rate is a result of averaging over initial spin states of the electron, and summing over final spin states.

Since fully relativistic Dirac formalisms are used, many physical processes are implicitly accounted for, including negative energy state processes such as pair production. With suitably stated S matrices, the rates for any of the processes can be derived. The process of interest in this treatment is the ionization of an electron from a ground state.

The polar coordinate system used in this derivation is defined such that Θ is the angle between p, the spatial part of the momentum four-vector for the emitted electron, and k, the spatial part of the em field propagation four-vector. A coordinate system is adopted where the direction of propagation of the laser field is along the x ³ axis, and the field is polarized along the x ¹ axis. Furthermore, Φ is defined as the angle between the projection of the momentum in the spatial plane transverse to k and the x ¹ axis.

The 𝑈 quantities are

and

Auxiliary quantities used in these expressions are

Upon reduction to the non-relativistic limit, the differential transition rate becomes

which is identical to the expression derived using the non-relativistic state vectors in the SFA (see Reiss[8]).

3. Numerical examples

Results for the dependence of ionization rate on field intensity are given here for two frequencies, both large. The computation of low frequency rates is very computer intensive, and will be presented in a later publication. Angular distributions require extensive graphics, and will not be reported here. Sample results for angular distributions and photoelectron spectra may be found in Ref. [6].

Figures 1 and 2 show ionization rates as a function of intensity for frequencies of ω = 8 a.u. and ω = 2 a.u., respectively. Some common features are exhibited in both figures:

 

Figure 1. Transition rate as a function of intensity for a frequency ω = 8 a.u. Both scales are logarithmic. One hundred points were computed for each decade in intensity.

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Figure 2. Transition rate as a function of intensity for a frequency ω = 2 a.u. Both scales are logarithmic. Twenty five points were computed for each decade in intensity.

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4. Discussion and conclusions

The linear polarization problem has no azimuthal symmetry about the propagation vector axis as in the circular polarization case. Furthermore, with relativistic effects, the linear polarization differential transition rate loses the symmetry about the polarization axis that exists in the non-relativistic calculation. Also, the relativistic differential transition rate is shifted in the direction of the propagation vector due to the momentum of the absorbed photons, thereby destroying symmetry about this axis.

The numerical predictions, appropriately, show that the relativistic ionization rate approaches the non-relativistic ionization rate at low intensities. At intensities somewhat beyond the pre-stabilization maximum in the rate, the relativistic ionization rate is depressed by an order of magnitude or more when compared to the non-relativistic rate, and its oscillations are suppressed. At still higher intensities, the relativistic rate begins to oscillate in similar fashion to the non-relativistic case, but always retains a smaller value - that is, the stabilization phenomenon remains more pronounced in the relativistic domain.

From Figs. 1 and 2, one can appraise the applicability of Eq.(2). If the factor 10^-2 is incorporated in Eq.(2) as was found to be true for Eq. (1), then the dipole approximation (which is a type of non-relativistic approximation) should fail when the intensity is about I ≈ 10ω ³. In fact, the figures show major differences between relativistic and non-relativistic results when

which can be regarded as good agreement.

An interesting correspondence between the dipole limit in Eq. (17) and the predicted onset of stabilization can be found. In Ref. [16] it was found that, at high frequencies, a maximum in the transition rate as a function of intensity (i.e., stabilization) will occur when

for the case of hydrogen, where the binding energy E_B is E_B = -5 a.u. The last element of Eq. (18) follows from the presence of high frequencies. In fact, Eq. (18) is found to be well satisfied in both figures. Furthermore, the ω ³ behavior found for both the onset of relativistic effects as found in Eq. (17) and for the onset of stabilization as in Eq. (18) means that the near concurrence of the two effects found in Figs. 1 and 2 is a general feature of strong-field ionization at high frequency.

We note that, in the long-pulse, fixed-intensity problem treated here, there is no field-induced drift of the photoelectron beyond its simple circular (circular polarization) or figure-8 (linear polarization) motion. This motion occurs in a frame of reference fixed with respect to the atom. “Revisiting” of the atom as described by the Volkov solution replicates itself on each cycle of the field. Quantum calculations involve no statement of initial conditions as in the case of classical problems. (One may say that quantum calculations represent an average over the possible initial conditions arising in the corresponding classical calculations.) These statements must be modified to some degree when the temporal intensity profile of the laser pulse is included, so that some net drift of the classical orbit with respect to the atom might occur. This effect should appear in a full laser-pulse calculation (as in Ref. [7]), where the spatial and temporal profile of a laser brought to a Gaussian focus in included, and where even the acceptance angle of the electron spectrometer is contained in the calculation. This type of completely detailed calculation has not yet been performed with relativistic expressions.