Electron acceleration in direct laser-solid interactions far beyond the ponderomotive limit

In laser-solid interactions, electrons may be generated and subsequently accelerated to energies of the order-of-magnitude of the ponderomotive limit, with the underlying process dominated by direct laser acceleration. Breaking this limit, realized here by a radially-polarized laser pulse incident upon a wire target, can be associated with several novel effects. Three-dimensional Particle-In-Cell simulations show a relativistic intense laser pulse can extract electrons from the wire and inject them into the accelerating field. Anti-dephasing, resulting from collective plasma effects, are shown here to enhance the accelerated electron energy by two orders of magnitude compared to the ponderomotive limit. It is demonstrated that ultra-short radially polarized pulses produce super-ponderomotive electrons more efficiently than pulses of the linear and circular polarization varieties.

that ultra-short radially polarized pulses produce super-ponderomotive electrons more efficiently than pulses of the linear and circular polarization varieties.

Introduction
Generation of energetic electrons by laser interaction with matter has witnessed considerable development over the past four decades. In the interaction of a laser pulse with an under-dense plasma (ambient electron density c e n n 03 . 0  , where c n is the critical density) (1) GeV-energy electron beams can be obtained via wakefield acceleration (2). Larger numbers of electrons can be accelerated in denser plasmas ( c e n n 1 . 0  ) where direct laser acceleration is dominant, to energies below the ponderomotive limit (3)(4)(5)(6). The term ponderomotive limit refers to the maximum energy gain,   2 2 0 2 1 mc a  , by a free electron during interaction with a plane-wave laser pulse (7,8). Here, ) is a dimensionless intensity parameter, in which 0 E is the electric field amplitude and  its frequency, while m and e  are the mass and charge, respectively, of the electron, and c is the speed of light in vacuum. The ponderomotive scaling (3)(4)(5)(6)(7)(8) is widely quoted within the context of discussions of direct laser acceleration (9,10) in the interaction of intense laser fields with dense matter ( c e n n  ), including solid-density plasmas. Generation of electrons which gain over 7 times the ponderomotive energy in near-critical plasmas ( c e n n  ) has recently been reported (11)(12)(13)(14)(15)(16)(17)(18) . This process is associated with important effects, such as the enhancement of ion acceleration in the near-critical plasmas (19). Electron energy beyond the ponderomotive limit is achieved by anti-dephasing acceleration (ADA). Electrons reach such energies in near-critical plasmas by self-injection into the acceleration phase of the laser field, aided by stochastic motion (11,12), transverse electric fields (13), and longitudinal electric (14,15) or magnetic fields (16,17). In the absence of anti-dephasing, energies of the electrons generated in laser-solid interactions hover, generally, around the ponderomotive limit (3)(4)(5)(6). Only MeV electrons are generated when linearly-polarized (LP) and circularly-polarized (CP) terawatt laser pulses are used. However, an experiment in which a micro-wire, used as an advanced solid target to generate and transport hot electrons over millimeters (20)(21)(22), has recently demonstrated reaching several times the ponderomotive energy, when LP laser pulses are used (23,24). Near-critical plasmas can be generated in the vicinity of solid targets for some applications (13,15). Here, ADA is put forward as a new mechanism for electron acceleration directly from a solid target. It promises to deliver significantly higher electron energies, and to simplify experimental implementation. Previous work on applications associated with electron acceleration to ponderomotive energies, from laser-solid interactions, include ion acceleration (25), fast ignition (26), laser hole-boring (27), high-order harmonic generation (28), half-cycle XUV pulse generation (29), Bremsstrahlung x-ray generation (30), and generation of terahertz radiation (31). Progress in these applications, stands to be advanced by availability of more energetic, shorter and denser electron bunches.
In this article, the novel mechanism of ADA, in the interaction of radially-polarized (RP) laser pulses of ultrashort duration with solid wires, is conceived, intuitively explained and backed up by particle-in-cell (PIC) simulations. Our investigations demonstrate unprecedented attosecond picocoulomb electron bunches with energies around one hundred MeV, i.e., two orders of magnitude higher than the ponderomotive limit. In particular, employing RP pulses, electrons are extracted from a wire target, and accelerated by the laser fields, see Fig. 1 for a schematic. Self-injection with a small dephasing rate is caused by the collective motion of the plasma electrons and the complex laser field variations.

Results
Mechanism of anti-dephasing acceleration (ADA) Direct laser acceleration by LP pulses leads to the generation of periodic electron bunches (32) and CP pulses generate spiral currents (22). In these cases, the azimuthal electron motion (17) makes it difficult to define an acceleration phase. By contrast, the acceleration phase in an RP laser field is determined directly by the regions of negative axial electric fields, with their troughs at phases   where N is an integer (33). The radial and axial electric field components ( r E and z E ) of the unperturbed laser pulse are shown schematically in Fig.  1(A). When the RP laser pulse propagates along the wire of radius smaller than its own waist radius at focus, electrons get knocked out and subsequently accelerated by the laser field. In each laser cycle, distortion to the electron distribution, and the process of electron injection, can typically be described as follows, with the help of Fig. 1. The force associated with r E pulls electrons into the vacuum around phase1, and pushes them backwards relative to the target over phase 3. Meanwhile electrons get accelerated by z E around phase 2 of a half cycle, and decelerated during interaction with the subsequent half cycle. Due to the acceleration around the trough of z E , dephasing around 3 is weaker than that about phase 1. As shown schematically in Fig. 1

Collective effects of the perturbed electrons
We consider an ultrashort RP laser pulse with 2 0  a irradiating a wire target. The laser peak , the radial electric field r E is negative, so it works to knock electrons out of the target wire and the negative z E acts to accelerate them forward, while the opposite happens over the interval is the initial phase of the laser pulse. Figure 2(A) shows the positive r E around the wire is strengthened, due to perturbation of the electrons, while negative r E is weakened. Figure 2(C) shows many electrons are collected beyond the cylindrical boundary of the wire, around the phase 0  , at which the unperturbed trough of is expected (34). The negative electric field gradient, induced by electrons in the annular bunch, shifts the trough of z E to 2 0    as shown in Fig. 2(B), where the trough of r E also sits. The near-wire and off-axis z E are shown by solid-red and dashed-red curves in Fig. 2(D). While the off-axis z E is unperturbed, the near-wire z E is determined by the time-averaged field, as a contribution from the space-charge of the perturbed electrons.
Pre-acceleration of the electrons in the combined fields around the wire is enhanced by the large surface-to-volume ratio of the wire target, as illustrated in Fig. 3, where snapshots of the accelerating fields are shown, respectively, at times Prior to entering their respective acceleration phases, the electrons undergo radial and axial oscillations inside the target, forced by the laser field. Subsequently, i.e., when the field structure with forward-shifted negative z E and weakened negative r E are acting and the right phase is reached, they get pulled out of the target slowly and accelerated forward with velocities approaching c . Figures 3(E) and (F) show evolution in time of their phases and scaled energies, with the gray shades representing the unperturbed acceleration phase. It is shown, for example, that particles following these typical trajectories can stay in the region with negative z E , as they slowly drift in weakened negative r E , and can be injected into the accelerating phase with an injection (pre-acceleration) energy of 1 0   . Energy gain beyond the ponderomotive limit ADA occurs after the pre-accelerated electrons are injected into the laser fields. The dephasing rate is given by . Using the relativistic equations of motion, one gets with z p the axial electron momentum component, and The message of Eq. (2) is quite simple: for an electron initially at rest 1  R , but subjecting the electron subsequently to the action of a symmetrically oscillating z E alone will cause R to exhibit symmetric oscillations. However, the laser z E , combined with the z E generated by the plasma electrons [especially when comparable to 0 0 a E (14)] will reduce the dephasing rate significantly.
is clearly greater than c for tightly focused lasers. Electrons launched into phase II achieve higher energy gains, because the target becomes hotter with time, allowing more electrons to be knocked out and injected. In this regime, stronger space-charge and pinch effects (22) enhance acceleration by the on-axis axial component z E and boost the pre-acceleration to smaller R (13). The dephasing rates shown in Fig. 3(H) hover around 01 . 0 R and can be as small as 0.005.
The energy gain receives a boost from the reduction in R via Eq. (1) and approaches . Unfortunately, the optimal anti-dephasing conditions, which lead to maximum acceleration, cannot be sustained by all electrons. For example, the dephasing rates of electrons accelerated in phase I do not decrease monotonically, as shown in Fig. 3(H) due to the fact that they move radially off-axis, into regions for which beyond position of the peak of r E . By contrast, electrons in phase II sustain small R values and acquire higher energy, as shown in Fig. 3(G).

Discussion
The self-injection regime of electrons for ADA has been discussed through examples involving RP pulses in interaction with wire targets. The RP laser pulse with a discrete accelerating phase, as illustrated schematically in Fig. 1, generates an electron bunch with FWHM of about 481 attosecond, as shown in Fig. 4(A). More discrete attosecond electron bunches with super-ponderomotive energy can be generated with longer driving pulses. Energy spectra, of electrons driven by laser pulses of amplitudes 2 0  a and higher, are shown in Fig. 4(B), which shows evolution of the beam energy and charge with the driving laser intensity. Further PIC simulations show that the cutoff energies depend on 0 r only weakly, and that the results converge as the resolution is increased. Although the accelerating phase is hard to define in LP and CP pulses, electrons are also gathered at azimuthally-dependent phases. Therefore, the ADA regime works also with LP and CP pulses, resulting in the electrons getting accelerated by a continuous stable phase. 3D-PIC simulations, similar to the above, have been carried out, in which the RP pulses are replaced by LP and CP pulses of the same amplitudes 0 a , focal radii 0 . An RP pulse is evidently superior to the CP and LP pulses for generating high-energy electrons. In the RP case, electrons can be launched into discrete phases, as opposed to being injected into continuous phases using the other polarization modes. Thus, the beam charge in the former is smaller than in the latter. Total charges of the electron beams driven by ultrashort pulses, shown in Fig. 4, are of the order of tens of pC, and increase with laser duration (22,23). The ratio of the beam charge to the laser duration is similar to that in a laser-wire experiment of approximately the same intensity (36).  Fig. 4(D), the energy gain does not get affected drastically, at least for CP pulses.

Methods
In our calculations, the wire is assumed to be a cylinder, of length L and radius 0 r , lying with its axis along z  , with . Fields of the incident RP pulse are derived from the normalized vector potential, , where a is given by (37)