RPLBs are the free-space analogs of the transverse-magnetic (TM) modes produced in the superconducting cavities used in conventional linear accelerators. However, acceleration with high-power lasers differs considerably from that with radio frequencies. In this section we discuss how acceleration in RPLBs proceeds and explain how attosecond electron pulses can be produced.

#### 3.1. Free-Space Electron Acceleration in Laser Beams

The classical motion of a charged particle in a laser beam is given by the Lorentz force equation [

35]:

where the particle’s attributes are its mass, charge, velocity, kinetic momentum, and energy defined by the symbols

m,

q, v,

**p** =

γm**v**, and

W =

γmc^{2}, respectively, with

γ = (1 − v

^{2}/c^{2})

^{−1}^{/}^{2}. E and B are the electromagnetic field components of the laser beam [e.g., from

Equations (1) or

(2)]. For a non-relativistic motion in a weak harmonic electric field of angular frequency

ω_{0}:

It is thus common to define a dimensionless parameter

${a}_{0}^{2}={(q/mc{\omega}_{0})}^{2}{E}_{0}^{2}$ whose value relative to 1 characterizes distinct regimes of particle dynamics (see

Table 1).

The non-relativistic regime is often referred to as the ponderomotive regime. Effectively, charged particles in a relatively weak laser beam

$({a}_{0}^{2}\ll 1)$ drift away from the high-intensity regions with an average motion that is independent of the laser polarization [

44]. The mathematical form of the ponderomotive force (PF)—the net force resulting from the temporal average of the quivering particle motion associated with the rapid field oscillations—was initially proposed in the 1960’s [

45,

46]. Relativistic

$({a}_{0}^{2}\sim 1)$ formulations came some 30 years later from independent authors [

47–

49]. In ponderomotive acceleration models, it is often assumed that the effect of the longitudinal electric field component cancels out when averaging over a full laser field cycle. For ultra-relativistic laser intensities

$({a}_{0}^{2}\gg 1)$, this assumption fails [

16]. To effectively reveal the phase-sensitive sub-cycle dynamics that characterizes longitudinal acceleration in RPLBs, it is necessary to take into account all the field components and their oscillations, as it is done when working directly with the Lorentz force equation.

Despite the controversy that has surrounded the experimental observation of ponderomotive electron acceleration in Gaussian laser beams [

50–

53], the mechanism by which the particles acquire a significant (relativistic) momentum is now well understood and accepted [

54]. The relativistic generalization of the original PF model [

45–

49] helped to explain and understand experimental observations in terms of relativistic ponderomotive scattering (RPS), where particles are expelled out of the beam focus within only a few laser cycles [

55,

56]. Typically, RPS proceeds in three steps. First, electrons are released by ionization (predominantly tunnel and above-threshold ionization) and accelerated toward the low-intensity regions of the laser beam by the transverse electric field component. Second, the trajectories are bent in the k ∝ (E × B) direction by the magnetic field (

via the v × B contribution of the Lorentz force). Third and finally, the longitudinal electric field component takes over and provides a final extra push. Although the amplitude of the longitudinal component is usually tiny (∝

E_{0}λ_{0}/w_{0}), including it into calculations changes the result from vanishingly small to considerable acceleration [

54,

55,

57]. This can be explained by the fact that an electron moving longitudinally can possibly stay in phase with the longitudinal electric field over longer distances.

During RPS the electrons leave the focal region of an ultra-intense Gaussian beam with a large transverse momentum, and very few of them remain close to the beam propagation axis [

51,

58,

59]. To limit the transverse excursion, it was proposed using a combination of a Gaussian beam and higher-order modes to create an intensity minimum at beam center that acts as a confining potential [

60]. It was later shown that the use of RPLBs improves even further upon longitudinal acceleration and electron beam confinement [

61,

62]. The particular geometry of RPLBs forces a significant proportion of the particles to move toward the beam propagation axis and remain there while they are accelerated by the longitudinal electric field component. Authors usually refer to this phenomenon as to relativistic ponderomotive trapping (RPT) [

61], in analogy to RPS where electrons are pushed toward the beam periphery.

#### 3.2. Direct Longitudinal Electron Acceleration with Radially Polarized Laser Beams

By RPS and RPT, free electrons gain a substantial energy due to the combined action of the transverse and longitudinal electric field components. However, strong acceleration can take place at the center of RPLBs without the action of the transverse field components [

16,

17,

63,

64]. Direct longitudinal acceleration with RPLBs allows for matching the electrons with the longitudinal field oscillations and effectively offers the possibility of producing well collimated quasi-monoenergetic relativistic attosecond electron pulses [

16,

18,

19].

The fundamental issue with longitudinal acceleration with RPLBs is the superluminous axial phase velocity. In fact, RPLBs have an axial variation of the carrier phase of (

p + 2)

π from

z = −

∞ to

∞, independently of the beam spot size and pulse duration [

30,

39,

65]. This is (

p + 2) times that of the fundamental Gaussian beam, twice for the lowest order (

p = 0) RPLB. This higher value of the Gouy phase shift is due to the fact that RPLBs diffract more rapidly than the fundamental Gaussian mode [

66]. When considering a point of constant phase along

r = 0, the phase velocity is

where it was assumed that

w_{0} ≫

λ_{0} and

p = 0, for simplicity. Because of the velocity mismatch, a particle moving in the beam will inevitably drift with respect to the carrier oscillations. Within a given time interval Δ

t, the net energy gain is optimal if that drift is less than half the laser wavelength,

i.e., if Δ

t|υ

_{phase} − υ

_{z}| ≲

λ_{0}/2, where

υ_{z} is the particle’s longitudinal velocity. During that same time interval, the particle has traveled over a distance

which defines the dephasing length. An interaction with the laser field for exactly two dephasing lengths—corresponding to a complete cycle drift—results in no acceleration, on average. For the special case of an electron with a relativistic longitudinal velocity (

v_{z} ≃

c), the dephasing length is:

It is thus observed that around the beam waist (

z ≃ 0), where the longitudinal field is the most intense, the distance over which a relativistic electron remains in phase with the laser field is only about the Rayleigh distance [Δ

z_{dph} ≃ (

πz_{R})/2]. This sets a fundamental limit on the energy that can be transferred from the laser field to particles (see also [

14]).

Assuming a perfect synchronization between an electron moving along the

z axis and the laser field, the maximum variation in the total electron energy from

z_{i} to

z_{f} is

where

Ẽ_{z} is the complex envelope of the field component given at

Equation (2b). Above, −

e is the electron charge,

${E}_{z0}=[2\sqrt{2}\mathrm{exp}(1/2)/({k}_{0}{w}_{0})]{E}_{0}$ is the amplitude of the longitudinal electric field component,

A(

z) = [

w_{0}/w(

z)]

^{2}, and

B(

z) =

z_{R}/R(

z). Respectively,

w(

z) =

w_{0}[1 + (

z/z_{R})

^{2}]

^{1}^{/}^{2} and

$R(z)=z+{z}_{R}^{2}/z$ are the beam waist size and radius of curvature of the wavefront at

z. According to

Equation (11b), there are two situations where the energy gain is optimal. One is when the acceleration occurs between two axial positions where the wavefront curvature radius is minimum but of opposite sign. This happens when [

z_{i}, z_{f}] = [−

z_{R}, z_{R}]. The other situation is when acceleration takes place between two positions where the beam spot size is minimal and infinite, respectively. This corresponds to [

z_{i}, z_{f}] = [0

, ±∞]. For these two cases Δ

W =

ez_{R}E_{z}_{0}. This defines the following theoretical limit to the energy gain (see also [

14,

67,

68]):

where

P [TW] is the laser power in terawatts.

Different longitudinal acceleration scenarios were reported in the literature ; a selected list is given in

Table 2. The first case is that of an electron that travels from

z = −

∞ to

∞ at ultrarelativistic speed. Because of the Gouy phase, it drifts by 2

π in the wave and, on average, the energy is zero. This is a typical example that illustrates the Lawson-Woodward theorem [

14]. To break the symmetry and allow for substantial electron energy gains, Scully and Zubairy [

69], Esarey

et al. [

14], and Liu

et al. [

15] suggested different avenues to effectively limit the electron interaction to −

z_{R} ≲

z ≲

z_{R}. However, the proposed schemes require optical materials close to the high field intensity regions. Such a configuration is inevitably limited by the destruction of the device and does not take full advantage of the high peak power delivered by actual ultraintense lasers.

As a matter of fact, material destruction—where free electrons are released due to ionization by the intense laser field—can be considered as a part of the acceleration process. A target should thus be composed preferably of a material with deeply bound inner shells so that most electrons remain bound during the rise time of the laser pulse, but released near the peak [

17,

19,

70,

71]. In that situation, electrons experience an optimal acceleration from

z ≃ 0 to ~

∞. In that range, the carrier phase shifts only by

π. Here, the Lawson-Woodward theorem is not violated because the electron is initially at rest and the interaction is maintained over a semi-infinite distance. According to works by some of us, where preionized targets were considered [

16,

18,

67], half of Δ

W_{lim} comes from acceleration outside the Rayleigh zone,

i.e., between

z_{R} and

∞. In this region, the longitudinal electric field is weaker than at focus but the Gouy phase evolves much more slowly (which considerably increases the dephasing time).

The integration of

Equation (11a) to get Δ

W_{lim} represents ideal acceleration scenarios. Rigorous numerical simulations show that, instead, the maximum energy gain is much less unless the beam parameters are carefully optimized [

67,

68,

73–

75]. We will see in Section 4 that exceptional conditions are provided by ultrashort and tightly focused pulses.

#### 3.3. Threshold for Sub-Cycle Acceleration and Attosecond Bunching

We now proceed with the evaluation of the threshold for sub-cycle acceleration. To account for the contribution of the longitudinal electric field component, the normalized field parameter

a_{0} is split into radial (

a_{r}) and axial (

a_{z}) components. These two new parameters are defined as follows:

where

|Ẽ_{r}|_{peak} and

| Ẽ_{z}|_{peak} represent the peak values of the field envelope of the radial and longitudinal components, respectively.

With

Equations (1b) and

(4), the threshold power corresponding to

${a}_{z}^{2}=1$ is found to be

In the paraxial limit [

18,

67]:

On the other hand, the analytical evaluation of

${a}_{r}^{2}=1$, and of the corresponding threshold power, is not straightforward and best obtained numerically. Results are shown in

Figure 2, where the different acceleration regimes in RPLBs are summarized.

To illustrate the different regimes of longitudinal electron acceleration in RPLP, we performed a series of simulations, where the time-dependent Lorentz force equation was integrated. Results are shown in

Figure 3. It is observed that above threshold

$({a}_{z}^{2}\gtrsim 1)$ the longitudinal electric component of the laser field is strong enough to accelerate an electron initially at rest at focus to a relativistic velocity within a half laser period. In that regime, electrons released in the same half-cycle are naturally bunched together. This is shown in

Figure 4, where three-dimensional simulations were done in the single-electron approximation. More realistic simulations that include self-consistent particle interactions and ionization dynamics indicate that, in most conditions, a train of attosecond electron packets—each separated by a laser wavelength—is formed [

19]. A single attosecond electron pulse could be produced if a single-cycle RPLP is used.