1. Introduction

A deflection of the photoelectron angular distribution along the direction of laser light propagation has been observed in recent experiments at field intensities where the electron velocities approach the speed of light [1–3]. The relativistic drift motion in the forward direction is well known from both classical and quantum theoretical considerations of the electron dynamics in the field of a plane electromagnetic field. References to early papers can be found in Refs. [1–5] and in a number of recent publications [6–14].

The experiments [1–3] were performed in the long pulse regime when electrons escape from the laser focus through the side surface and its full theoretical description can not be given if the field is modeled as a pure plane wave [8,9]. The theory must on equal grounds account for both the longitudinal relativistic drift and the ponderomotive acceleration in the transverse direction. The electron motion in a focused field has been studied numerically for photoelectrons [10] and in the special case when the relativistic electron beam and laser pulse travel in the same direction [11]. On the other hand, the equations of the averaged motion valid at relativistic laser intensities were derived [12] (for review see [13]) and applied to calculate analytically the photoelectron momentum outside long and short focused laser pulses with circular and linear polarizations [14]. Below, the problem analogous to that treated in Ref. [14] is solved for the field with arbitrary elliptical polarization. The conditions for applicability of the model are discussed.

An elliptically polarized, focused laser pulse is described by the vector potential

where φ = t - z / c and ξ is the ellipticity. The first two arguments in A⃗ reflect a smooth dependence of the field amplitude F(x⃗,φ) on the coordinates and time, which is in fact scaled by the size of the focus R, the pulse duration τ, and the diffraction length kR ² as F(x⃗_⊥/R,z/kR ²,φ/τ) [12,13].

Ionization is treated along the lines of the semiclassical model [7], but the electron motion after ionization is described by the relativistic equations. The semiclassical model in the form relevant to the adopted velocity gauge [15,16] says that at the time when the phase of the field is φ ₀ the transition from the bound state occurs to the only continuum state ∣ p⃗⟩ labeled by the canonical momentum, for which the classical kinetic momentum (and the velocity) equals zero: π⃗(φ ₀) = p⃗ - e A⃗₀/ c = 0. Here A⃗₀ = A⃗(x⃗₀,φ ₀ ; φ ₀) is the vector potential at the position of the atom at the time of the transition. The velocity distribution at the moment of ionization [17] is neglected in such an approach, and so the released electron starts motion in the laser field with zero velocity. After ionization the electron does not interact with the parent ion. The probability of ionization at the field phase φ ₀ is governed by the tunneling factor exp(-2F_a /3∣ F⃗( φ ₀ ∣ ) and is largest when the instantaneous value of the electric field ∣ F⃗(φ ₀ )∣ corresponding to the potential Eq. (1) is maximal.

2. Short pulses

When the electron displacement after ionization remains small in comparison to the size of the focus, the field amplitude can be evaluated at the ion position, so that the field acting on the electron becomes that of a plane wave. Exact analytical solutions to the relativistic equations of motion in the field of a plane electromagnetic wave are known only in a parametric form. We use φ= t -z/c as the parameter, and with the initial condition of zero velocity at φ ₀, the kinetic momentum and the energy for φ > φ ₀ are equal to

In short laser pulses, the approximation of taking the field to be a plane wave holds during the entire interaction time. Then the limit φ → ∞ in Eq. (2) gives the final momentum and energy. As the field goes to zero, the transverse momentum of a free electron outside the field is π⃗(+∞) = e A⃗₀ / c, and π_z (+ ∞) and π ₀ (+ ∞) are found according to Eq. (2).

A different way to calculate that leads to the same result in short pulses but is more convenient for generalization to the case of arbitrary pulse duration includes averaging over the period 2π / ω in the parameter φ denoted below as ⟨…⟩ [12,13]. For the vector potential we have ⟨ A⃗⟩ = 0 and ⟨( e A⃗/c )²⟩ = 2mU where the ponderomotive potential U in an elliptically polarized field is

Here I is the laser intensity. In the field of a focused pulse the ponderomotive potential varies in space and time in the same way as the field amplitude does. The averaging in Eq. (2) reveals the drift motion after ionization. With the notation q_i = ⟨ π_i ⟩one gets at φ > φ ₀:

At φ → ∞ when the wiggling motion dies out, the kinetic and the average momenta coincide: π⃗(+∞) = q⃗(+∞) and equally represent the momentum of a free electron after interaction with the laser. The limit φ → ∞ in Eqs. (2) and (4) gives the same result.

The angle of the electron momentum outside the field relative to the direction of laser propagation θ is given by

To get the final result it is necessary to specify explicitly the vector potential at the time of ionization A⃗₀ or, according to Eq. (4), the transverse drift momentum e A⃗₀/ c that is an integral of the motion in short pulses.

For intermediate ellipticities the tunneling factor is maximum when the electric field reaches its maximum at ωφ ₀ = ±π / 2 and directed along the major axis of the polarization ellipse (x -axis). The vector potential at that time is along the smaller axis and hence the most probable initial drift momentum has only a y-projection and is equal to ξeF / ω. That estimate of the value of the drift momentum is valid as well for circular polarization ξ = 1. For very small ellipticities (in fact for $\xi <\gamma \sqrt{\frac{F}{F_a}}$, [12,13]), the vector potential at the maximum field is mainly along the x-axis and strictly at ωφ ₀ = ±π/2 turns to zero. But the ionization occurs during the interval δφ ₀ around the maximum, which is easily estimated from the tunneling exponential factor as $\mathit{\omega \delta }{\phi }_0=\sqrt{\frac{F_0\left(1-{\xi }^2\right)}{F_a}}$. Within that interval the typical drift momenta are of the order of F ₀ δφ ₀. Now combining these two estimates one can interpolate the value of the drift momentum acquired by the electron in the process of ionization as

Equation (6) gives the correct results in both limits ξ → 1 and ξ → 0. As was mentioned above in laser fields satisfying the inequality F ₀ / F_a << 1, the second term is essential only for small ellipticities. The condition F ₀ / F_a << 1 is compatible with the definition of the relativistic intensity U ≈ mc ² for ionization of multiple-charged ions with large ionization potentials I ∝ Z ². The atomic field F_a = (2I)^3/2 is proportional to Z ³.

From Eqs. (5,6) and (4) or (2), the estimate for the forward deflection of the photoelectron distribution in short laser pulses with arbitrary polarization is given by

Here U ₀ = U(x⃗₀,φ ₀) is the ponderomotive potential at the time and at the place of ionization. At a fixed intensity the forward deflection is maximum for circular polarization and decreases monotonically with decreasing ellipticity. At ξ → 0 the momentum distribution flattens against the plane perpendicular to the direction of laser field propagation. The reason is that the longitudinal drift decreases faster with decreasing ellipticity than does the transverse drift. It is worth noting that variation of the ellipticity, in addition to the changes in the forward deflection, simultaneously leads to a strong rearrangement of the angular distribution in the polarization plane [15,16,18,19].

3. Long pulses

In laser pulses that are not short, the electron displacement from the place of birth cannot be ignored, and the dependence on coordinates of the field amplitude must be properly accounted for. The following scenario could be used to calculate the final electron momentum and energy outside a focused laser pulse of arbitrary duration. Firstly, the initial drift after ionization is found just by putting φ = φ ₀ in Eq (4). Secondly, the initial conditions are used to integrate the averaged equations of motion [12,13]. Unfortunately in the general case when all the dependencies of the ponderomotive potential U = U ( x⃗_⊥/ R,z/ kR ², φ /τ) are essential, only a numerical integration is possible. Some progress in derivation of an analytical solution is achieved if the longitudinal gradient of the ponderomotive potential is neglected in the averaged equation [7]. That procedure corresponds to the limit of an infinite diffraction length kR ² → ∞ but R remains constant. In other words the focal volume is considered as an infinite round cylinder so that U = U(x⃗_⊥ / R,φ / τ). In that simplified model of the focused pulse there exists an approximate integral of motion [12,13]

Its particular value mc is found from the initial conditions as is explained above. The average momentum in an arbitrary laser field satisfies the equation

where m _*, is the electron effective mass in the light field. From Eqs. (8,9) in the limit φ → ∞ when the electron is in any case out of the field and the ponderomotive potential goes to zero, we get two equations for the final energy and momentum

Some additional assumptions are necessary to get the missing third relation necessary to calculate explicitly the final momentum. If we suppose that the laser pulse is short then the ponderomotive potential does not depend on x⃗_⊥ and hence q⃗_⊥ is a constant equal to the initial value calculated from Eq. (4). Then Eq. (10) brings us back to the same results as the ones derived for short pulses from Eq. (2) or (4). New results are available in the opposite case of a long pulse when the ponderomotive potential does not depend on φ and the average energy q ₀ and as well as q_z become integrals of the motion. Accounting for the initial values following from Eq. (4), we find the photoelectron momentum outside a long laser pulse to be

In the case of long laser pulses we have from Eqs. (5, 6, 11)

The last term in the denominator accounting for the small transverse initial drift momenta for polarizations close to the linear case is not very essential as the electron gains large transverse momentum due to the ponderomotive acceleration. If that term is neglected, Eq. (12) yields the results for ξ = 0 and ξ = 1 in Ref. [14].

The forward deflection in long pulses is maximal for circular polarization and decreases with decreasing ellipticity, but not so drastically as in short pulses. For the two extreme polarizations the angle satisfies the equation tg θ_lin / tg θ_cir = √2 [7].

4. Conditions of applicability

It was assumed above that the electron starts motion in the laser field being at rest. Then the energy and momentum outside the field represent the changes between the initial and final states, and also could be interpreted as the energy and momentum gained by the electron during the interaction with the field. Introducing the notations for the changes Δπ_z = π_z (∞), Δπ ₀ = π ₀ (∞) - mc ² and so on, and remembering that π⃗(∞)= q⃗(∞), we have from Eq. (10)

and the relation between the longitudinal and transverse momenta

Sometimes [10] Eqs. (13,14) are considered as following without any additional assumptions only from energy-momentum conservation in the electron-photon interaction. If so, they must be valid for arbitrary electromagnetic field and any initial conditions for the electron motion. But Eq. (9) is not true, for example, in the case of the Kapitza-Dirac effect: the electron crossing the field of a standing light wave gains the momentum Δπ_z = 2k_z but its energy does not change Δπ ₀ = 0. If the calculations leading to Eq. (10) are repeated with arbitrary initial value π_ ≠ mc, the result Δπ_z = Δ( ${\vec{\pi }}_{\perp }^2$ )/2π_ is different from Eq. (14).

In our calculations the additional condition on the electron momentum in Eqs. (13, 14) is imposed by the conservation law for the averaged momentum given in Eq. (8). The law itself is a consequence of the model of the focused pulse with infinite diffraction length. In the short pulse limit the field is treated as a plane wave and in that case there exists an integral of the motion relating instantaneous values of the kinetic momentum

According to the Noether theorem, conservation of π_ is the consequence of the ( t - z / c )-dependence specific to a plane wave field. Averaging of Eq. (15) over oscillations turns it into Eq. (8). The quantum theory sheds some light on the conservation of π_. From that point of view an arbitrary plane wave propagating in the positive direction of the z -axis consists of photons with different frequencies but strictly parallel wave vectors k_z = ω / c > 0. Matrix elements corresponding to absorption and/or emission of several photons of the wave relate the initial electron state with the energy π ₀ and the momentum π_z to the virtual states characterized by the energy π̃₀ and the momentum π̃_z equal to

By subtracting these equations one gets π̃₀ - cπ̃_z = π ₀ - cπ ₀. That selection rule for transitions to the virtual states originates due to the specific dispersion relation for photons with parallel wave vectors. Real processes with energy-momentum conservation as in Eq. (16) are forbidden.

If the focused pulse is represented as a superposition of plane monochromatic waves, then in the approximation of an infinite diffraction length the small transverse projections of the wave vectors are accounted for only in the lowest (linear) order. This means that the dispersion relation for a photon with frequency ω and wave vector k⃗ = ( k⃗_⊥, k_z ) has the form k_z ≈ ω / c, as in the plane wave field. Accounting for the term quadratic in k _⊥ in the decomposition of the exact formula $k_z=\sqrt{{\left(\frac{\omega }{c}\right)}^2-{\overrightarrow{k}}_{\perp }^2}$ would have lead to a finite diffraction length.

In conclusion we repeat that the relations for the energy and momentum gains in the form of Eq. (13,14) are valid for electrons starting with zero velocity in a cylindrical focal volume in laser pulses of arbitrary duration. The formulas describing the forward deflection of electron trajectories in an elliptically polarized field are applicable under the same conditions except the pulse duration: the laser pulse is assumed to be short (Eq. (7)) or long (Eq. (12)).