One of the issues facing the use of laser pulses to accelerate electrons is the relative difference in the velocity of the electrons and of the light wave. For direct field acceleration, injected electrons must be close to the phase velocity, while for wakefield acceleration and ponderomotive acceleration, it is the pulse group velocity relative to the electron velocity that is relevant. Typically the phase and group velocities are at or close to the speed of light, and fast electrons must be injected, and timed with the pulse. Spatio-temporal shaping of the pulses offers a route toward controlling the pulse group velocity over a wide range. In recent years, several groups have been pursuing the use of spatial chirp, the ordering of the pulse frequency components in position or angle, to control the spatio-temporal intensity to slow down the effective group velocity of the wave. Angular spatial chirp produces a pulse front tilt that presents unique opportunities to study and control electron dynamics, since the effective pulse velocity can be decreased substantially below the vacuum speed of light (c).

Tilted pulses have been applied in several areas of non-linear optics and laser-matter interactions. Control of the angular dispersion has been used to match pulse fronts in second harmonic generation of broadband pulses in thick non-linear crystals [1] and for optical parametric amplification [2]. Similarly, other groups have used tilted pulses in lithium niobate to obtain high energy THz pulses [3]. The spatially-dependent arrival time of a tilted pulse can also be used for traveling-wave pumping of an x-ray laser in an ablating plasma (e.g., [4]). The tilted pulse has also been shown to result in interesting asymmetric structures during micromachining on surfaces [5, 6]. When a pulse is focused with angular chirp, the intensity is localized along the optical axis due to simultaneous spatial and temporal focusing (SSTF) [7, 8]. This intensity localization is extremely useful for micromachining [9, 10] and laser surgery [11].

One of the primary mechanisms for a laser beam to couple to plasmas is the ponderomotive force. The optical pressure drives lighter electrons away from regions of high intensity. For relativistic intensities (where the average oscillation energy of the electron in the field is comparable to its rest mass), the ponderomotive force can drive most or all of the electrons out of the focal region, leaving a wakefield behind the pulse that can be used to accelerate electrons [12]. In this paper, we propose to use tilted laser pulses, like a snowplow, to knock the electrons out of the focus to one side. By slowing down the effective velocity of the pulse we predict that the intensity required to drive the electrons out is dramatically reduced compared to using a conventionally-focused beam. In this paper, we consider the acceleration at densities sufficiently low that space charge is not a factor; in later work we will consider higher densities, including the wakefield regime. While it is beyond the scope of this initial paper to fully evaluate this scheme for applications, an ultrafast source of moderate energy electron pulses can have application to ultrafast electron diffraction [13], generation of coherent light, and injection into wakefield accelerators.

Before presenting the details, we illustrate the acceleration scheme in Figure 1, which shows a simulation of the process using the particle-in-cell code OSIRIS 4.0 [14, 15]. In the top left frame, the tilted pulse (orange) is about to meet a stationary electron bunch at the origin (blue). As the pulse propagates to the right, the electron bunch is accelerated in the direction of the tilted intensity gradient. Because the effective interaction velocity along this direction is slower than c, the electron bunch is captured and accelerated. In section 2, we provide a brief overview of the optics of tilted pulses. In section 3, the simple non-relativistic theory of tilted pulse acceleration is presented, followed by the relativistic theory and modeling in section 4. Finally, in section 5, we offer a summary and conclusion.

In ultrafast laser systems, care is typically taken to ensure that all of the spectral components of the beam are spatially overlapped. However, in the past several years, more researchers have been seeking to exploit special properties of beams that have spatial chirp, where one or more of the beam parameters vary across the spectrum. For example, a beam with transverse spatial chirp exhibits the "lighthouse" effect, where the wavefront angle is a function of time. This effect has lead to the production of angularly separated attosecond pulses through high order harmonic generation [16]. Angular spatial chirp leads to a complementary effect, known as simultaneous spatial and temporal focusing (SSTF). These beams have unique properties that can be exploited to control and manipulate non-linear optical processes. SSTF leads to a localization of the intensity along the optical axis that is especially useful for micromachining [6, 10] and laser surgery [11], as well as non-linear microscopy [17, 18].

While the details of the spatio-temporal structure of these pulses can be found elsewhere [19], there are several features of these tilted pulses that are relevant to the current discussion. A simple understanding of temporal focusing and pulse front tilt can be obtained by considering a group of plane waves crossing at Z = 0 where the propagation angle (θ) relative to the Z−axis depends on frequency ω. The central frequency component ω₀ is defined to be along the Z−axis, so that θ(ω₀) = 0. In the spatial-spectral domain, this beam can be written as E0exp[-(ω-ω0)2/Δω2]exp[i(ω/c)(Xsin(θ(ω))+Zcos(θ(ω)))], where E₀ is the electric field amplitude and Δω is the spectral bandwidth. Derivatives of the spectral phase φ(X, Z, ω) = (ω/c)(X sinθ(ω) + Z cos θ(ω)) lead to an understanding of the spatio-temporal properties of the beam. Along the Z−axis, the group delay dispersion, φ2(Z)=∂ω2(ωZcosθ)/c|ω0 leads to a pulse that is fully compressed only at Z = 0. This longitudinal compression combines with the focusing and the spectral overlap to give a shorter depth of focus than would be expected for a conventionally focused beam of the same spot radius.

Along with intensity localization, the angular chirp leads to a strong pulse front tilt (PFT). At Z = 0, we can find the arrival time of the pulse by calculating the group delay: φ₁(X) = ∂_ωφ|_ω0 = ∂_ωθ|_ω0Xω₀/c. The pulse arrives earliest for X < 0, and the pulse sweeps across the target Z = 0 plane with a velocity v_xPF = X/φ₁(X). Noting that the pulse travels at a speed c in the Z direction, the pulse is seen to have a tilt in the pulse front with an angle given by tan(θ_PF) = c/v_xPF, where θ_PF = 0 when there is no PFT.

PFT control has been applied in previous work in non-linear optics such as THz generation in crystals [3], non-collinear optical parametric amplification (e.g., [20]) and achromatic phase matching [1]. In these examples the group velocity control makes the non-linear processes more efficient. PFT has also been exploited in traveling-wave pumping of x-ray lasers (e.g., [4]).

Spatially-chirped pulses can be made by converting the standard double pass pulse compressor to single pass by increasing the separation of the diffraction gratings by a factor of two. Normally, the separation of the gratings is fixed to optimize pulse compression, but we have shown a pulse compressor that allows us to vary the degree of spatial chirp while maintaining optimal pulse compression [21]. Rather than using a single pass compressor, the beam passes through the grating pair twice, but with a grating separation that is longer for one pass compared to the other. Our current design allows us to tune the PFT from zero all the way to θPF<88°.

The essential concepts behind the tilted pulse acceleration scheme can be most easily illustrated by considering acceleration in the non-relativistic limit. For simplicity, we will consider a tilted pulse propagating in the Z−direction with a tilt angle θ_PF where we also neglect limits on the transverse width of the beam. This non-physical idealized framework will serve as a reference to which more complete calculations can be compared.

An approximate form for the envelope of the pulse intensity, I, as

Here, I₀ is the peak intensity, and τ is the pulse duration. While this simple representation of a tilted pulse does not capture the full spatio-temporal evolution of the intensity through the focus, we will use it for conveying the essential dynamics of tilted pulse acceleration. We have developed an analytic expression for the evolution of the spatio-temporal pulse intensity following the method outlined in an earlier paper [18], and we find that the approximation of Equation (1) is best for electrons starting near the focal plane (Z = 0). Three dimensional effects will be discussed later in the paper.

The common definition of the non-relativistic ponderomotive potential is the cycle-averaged quiver energy of a free electron in an electromagnetic wave:

where e is the electron charge, E₀ is the electric field amplitude, m_e is the electron rest mass, r_e is the classical electron radius, λ is the laser wavelength, and I is the time-averaged beam intensity. In the non-relativistic limit, we can calculate the force on an electron from F = −∇U_p. Equation (2) can be derived from the linearized equations of motion of a single charged particle in the Lorentz force of a high-frequency electromagnetic field [22]. The force comes out of equations of motion for the guiding center of the fast oscillations of the particle. A more general form of the ponderomotive force can be derived through Vlasov theory that includes interactions with particles that are resonant with the field [23]. Another approach using the stress tensor and fluid theory addresses the time-dependent force that emerges from the longitudinal gradients of the field [24]. At relativistic intensities, the trajectories of the electron become anharmonic, and approximate forms of the ponderomotive force must be derived that account for the relativistic mass increase. In a later section we will use the approximation developed by Mora and Antonsen [25]. Owing to the complicated nature of these approximations, full-field calculations that do not rely on the ponderomotive approximations can be used to verify that the results of the simpler theory hold up. For this we turn to the particle-in-cell calculations that include the fast electron oscillations.

In the non-relativistic limit, we can calculate the force on an electron from F = −∇U_p, obtaining:

where U_P0 is the peak ponderomotive potential. By looking at the vector components of Equation (3), we can see that with a simple rotation of the reference frame we can align the force along a z′ axis that is normal to the pulse front. In this rotated frame, the laser pulse has a ponderomotive potential that is traveling at a reduced speed v_PF = c cos θ_PF:

Now consider a reference frame that is moving in the z′ direction at the pulse front velocity. If an electron is at rest in the lab frame, then it is moving toward the pulse front with velocity −c cos(θ_PF) in the moving frame. Upon reflection from the potential, the velocity changes to + c cos(θ_PF). Therefore, from simple energy considerations, the electron will be “reflected” from this potential hill if the peak ponderomotive potential is greater than the capture threshold:

where the superscript “cap0” reminds the reader that the initial electron velocity is zero.

Now suppose the electron has an initial velocity in the lab frame v₀, where v₀ > 0 corresponds to an electron moving in the same direction as the pulse front in the rotated lab frame, i.e., along the z′ axis. In the moving frame, the electron approaches the stationary potential at the speed −(c cos(θ_PF) − v₀) leading to a new capture threshold:

It is important to note that with strong angular chirp approaching the laser focus it is straightforward to produce pulse front angles as large as θPF=88°. In such cases, capture can take place with a ponderomotive potential that is approximately 3 orders of magnitude less than the electron rest mass energy.

When the electron is captured and accelerated (reflected in the moving frame), the reversal of the electron velocity in the moving frame corresponds to a net acceleration in the lab frame to a velocity of 2c cos(θ_PF) − v₀, or an output kinetic energy of

Here the minus sign is taken if v₀ > 0, which corresponds to a seed electron velocity in the same direction as the pulse front. Note that even if UP0>UPcap, the output energy is determined only by the pulse front velocity (v_PF) and the seed electron energy (v₀), not the value of U_P0. If v₀ = 0, KEout=4UPcap0. We will see later how a finite beam size affects this result.

In the first form of Equation (7), we see that the output kinetic energy is smaller if the initial direction is co-moving with the pulse (v₀ > 0) compared to if it is moving into the pulse (v₀ < 0). This is perhaps counter-intuitive, but consider the situation in the pulse frame. If the particle is initially at rest in the lab frame, the change in momentum after reflection is 2m_ev_PF. Any additional momentum toward the pulse front will increase the momentum transfer, while if the particle is initially moving in the same direction as the pulse front in the lab frame, the momentum increase from the interaction is lower since the relative velocities are smaller. This is similar to the situation in American baseball, where the batter can hit the ball farther off a fast ball than if the ball is hit off a tee (i.e., starting from rest). One might conclude from this that if electrons are injected into the field, it is best to seed the electrons in the direction opposite the pulse front. However, this is only true if the pulse intensity is sufficient to reflect the electrons, i.e., UP0>UPcap(v0). Since the capture threshold, Equation (6), is higher for v₀ < 0, we can consider whether a combination of smaller tilt angle and co-moving injection would be a more efficient acceleration scheme.

If the available input intensity is fixed, and we have the ability to seed the acceleration with electrons, we can adjust θ_PF so that UP0≥UPcap(v0). We can then calculate the KE that is added to the input KE: ΔKE_out = KE_out − KE_in, assuming the tilt angle is always adjusted so that UP0=UPcap as v₀ is varied. Using Equation (6) to solve for cos (θ_PF), we find that

Here the upper sign is taken for co-moving injection (v₀ > 0). As an example, suppose we accelerate an electron bunch from rest using a peak ponderomotive potential of U_P0 and the angle is adjusted to be at the capture threshold (UP0=UPcap0). Then, the electrons of kinetic energy 4U_P0 are injected into a second stage with a pulse of the same intensity, and θ_PF for this stage is adjusted to be at the threshold for capture. At the end, KE_out = 12U_P0, which is 1.5x higher than if all the energy were put into a single stage. For a fixed total energy, the optimum fraction of energy to use in the first stage is just above 0.25, which yields an improvement of 1.62x over the single stage case. We will see in section 4.1 that when the initial electron velocity is near zero, variations in the output kinetic energy are very sensitive to input energy perturbations, however, when the acceleration is seeded with some average initial velocity, this sensitivity is dramatically reduced.

Although the force is only in the direction normal to the pulse front, there is motion along the pulse front at the speed vx′=csin(θPF). Of course in this ideal case of infinite transverse width, this motion has no bearing on the kinematics. For a finite beam width however, there will be some electrons that will slide off the pulse front before being fully accelerated. We can estimate the time, ΔT, the electron is accelerated by the pulse by considering an infinite width (in X) beam with a triangular temporal profile and a duration of τ_tri from peak to toe. Here, the force is constant, and since the momentum change is 2m_ec cos(θ_PF), then ΔT = 4τ_tri/r_Up, where rUp=UP0/UPcap0 is the measure of how far the focused intensity is above the capture threshold. The acceleration time is shorter for shorter pulses and larger r_Up. The distance the electron travels in the rotated frame in the x′ direction is Δx′=cΔTcos(θPF). To get the transverse displacement in the un-rotated X direction, we integrate the equation of motion in the z′ direction, then rotate back to obtain ΔX=2cτrUpsin(2θpf)((1+rUp4)2+ 1). For θPF=60°, τ_tri = 20fs, and r_Up = 2, the focused spot diameter must be a minimum of ΔX = 17μm. However, for a steeper angle, θPF=85°, ΔX = 3.4μm, which is considerably smaller.

Finally, we note that in the stationary lab frame (Z, X) the output velocity vector is {vz,vx}={2ccos2(θPF),csin(2θPF)} and the output angle (tan-1(vx/vz)) is simply in the direction of the normal to the pulse front, consistent with the direction of the applied force.

Several very important observations come from this simple analysis. First, all electrons will be captured, provide the capture condition is satisfied. Second, the output kinetic energy is dependent only on the PFT angle, not on the intensity of the laser. Therefore, even when the laser pulse has a transverse intensity profile, the electrons that are fully captured will have the same output energy. These two properties of the tilted pulse acceleration scheme show promise for creating an electron bunch with high brightness. The analysis later in this paper will explore the robustness of the scheme, accounting for the effects of relativity and the non-constant transverse intensity profile.

In this work, we have proposed an efficient laboratory-scale method of accelerating electrons from rest to relativistic energies using tilted ultrafast laser pulses. The tilt of the pulse front extends the interaction of the short pulse with the electrons such that the process acts at multiple time and length scales and thus requires a less energetic pulse for ponderomotive capture. At a given time within the pulse, an electron interacts with a beam with a width that is reduced by the pulse front tilt; this leads to larger ponderomotive force since the intensity gradient is larger. At the same time, the electron can interact with the full width of the beam as it accelerates. Correspondingly, the local pulse duration is transform limited, but the tilt of the pulse allows for an extended total duration of the pulse. This is in contrast to a non-tilted ponderomotive schemes where a captured electron does not have an extended period of time to interact with the pulse.

We have described this acceleration process with a simple ponderomotive model in one- and three-dimensions. We also demonstrated that this effect can be seen in more rigorous relativistic and full field simulations. Our analysis and simulations show that when the beam is engineered to a flat transverse profile, the method has promise to produce narrow energy and angular distributions for the accelerated electrons. Since the energy threshold can be modest with large tilt angle and short pulses, tilted pulse ponderomotive acceleration can be achieved with table-top laser systems, leading to possible applications in ultrafast electron diffraction. Our calculations indicate that the scheme can be staged to accelerate at least up to the 10MeV level (see Figure 4). In ongoing work we are considering other pulse structures that could go to higher energy for seeding laser wakefield electron accelerators.

CD conceived and derived the original non-relativistic theory and began the original manuscript draft. AW ran the particle-in-cell simulations and contributed to the manuscript. Single particle calculations were performed and verified independently by both authors.