Chirped pulse amplification (CPA) has revolutionized laser technology, elevating the output powers to the petawatt regime in the past thirty years or so [1, 2] and possibly beyond that soon [3]. Chirped optical pulses have many applications, especially in fiber-optic communication [4]. More recent applications include cooling of a mirror in cavity optomechanics [5]. Chirped pulses in acoustic and radar signal processing [6, 7] have a much longer history.

Chirped laser pulses have been employed in theoretical laser-acceleration investigations for some time now [8–11], but only a few experiments have shown some evidence of vacuum laser acceleration using few-cycle laser pulses [12, 13], let alone chirped few-cycle pulses. In any case, results based on the one-dimensional model, to be introduced in this paper, may be compared to those stemming from the full three-dimensional calculations, only qualitatively.

Popular models of chirping the frequency, ω, of a wave include letting ω = ω₀ + bt (a linear chirp) in which the parameter b has the unit of s⁻², t is the time, and ω₀ is the initial frequency (at t=0) [14–17]. Likewise, a quadratic chirp is obtained by taking ω = ω0+bt2, in which b has a unit of s⁻³, or ω = ω0+b1t+b2t2, with b₁ and b₂ having the units s⁻² and s⁻³, respectively, and so on.

In some recent applications, chirping the frequency of a laser pulse, of frequency ω₀ and wavenumber k₀ and propagating along the z-direction, has been modeled by replacing η = ω₀t − k₀z with η + bη² (linear, with b a dimensionless parameter) and quadratic (η → η + bη³). This work adopts the approach based on the variable η [18–21].

So, to model the relevant experimental situation and account for the chirp, time dependence in the frequency has been altered by hand, directly by adding terms which contain the time t and chirp parameters, or indirectly through changing η. This seems to work quite well, for a linear chirp, to model what transpires when a laser pulse is, for example, passed through a linear medium of the right properties, including the right index of refraction. The model we put forward in this work accounts for all orders of chirp in a unified way and employing a single parameter, thus lowering the degree of arbitrariness in describing the chirp. It allows for the use of non-linear media to produce chirps of orders above the linear, which stands a good chance of opening the door for more novel applications in the field. Precise control over the temporal chirp of ultrashort pulses is possible today employing devices such as acousto-optic programmable dispersive filters, which are capable of tailoring the spectral phase- and amplitude-phase to several orders [22].

Some background material, of direct relevance to this work, is briefly reviewed in section 2, before the fields of a chirped laser pulse are introduced in section 3. Dynamics of a single electron subjected to the chirped pulse will be discussed in section 4. Finally, a summary and our main conclusions will be given in section FF 5.

In a typical textbook exercise on electromagnetic waves, one is asked to show that any well-behaved function f(η) satisfies the full set of Maxwell's equations in a vacuum, exactly [23]. Based on this, the following qualify for adequately representing the linearly-polarized transverse fields of an un-chirped pulse (in SI units) [24] (1)E(η)=x^E0sin(φ0+η)sin4(η2N), (2)B(η)=ŷE0csin(φ0+η)sin4(η2N),

where φ₀ is an initial phase, E₀ is a constant amplitude, c is the speed of light in vacuum, and x^ and ŷ are unit vectors in the (transverse) x− and y−directions, respectively. For a few-cycle pulse, φ₀ serves as the carrier envelope phase (CEP). The fields in Equations (1, 2) are non-zero only for η ∈ [0, 2Nπ], in which N (giving the number of η−cycles in the pulse) is an integer chosen to ensure that the fields vanish at η = 0 and η = 2Nπ. The intensity of such a pulse will be calculated as the time-average of the Poynting vector in the focal plane. With μ₀ and ε₀ standing, respectively, for the permeability and permittivity of free space, the pulse intensity may be calculated from I0=ω02πN∫02πN/ω0(S·z^|z=0)dt; S=1μ0E×B, (3)       =3528[1-3225π(2N-5)!(2N+4)!sin(4πN)]cε0E02.

For our purposes in this work, the number of field cycles (in η) will be N ≥ 3, for which the electric field amplitude will be given by (4)E0=16I035cε0.

Our aim in this paper is to introduce a simple model for chirping the frequency of the fields given by Equations (1, 2) and to employ the chirped fields in investigating the dynamics of a single electron subjected to the pulse they represent. The chirped fields will be discussed briefly in the next section.

The starting point is the observation that, when one lets (5)η→η1-bη=η+bη2+b2η3+⋯,

in the argument of the sin oscillation in Equations (1, 2) one gets what resembles a linear chirp, a quadratic chirp, and so on, if the series is truncated at the second term, at the third term, …etc., respectively. The expansion in (5) is valid so long as |bη| < 1. So, provided this convergence condition is adhered to faithfully, the fields (6)E(η)=x^E0sin(φ0+η1-bη)sin4(η2N), (7)B(η)=ŷE0csin(φ0+η1-bη)sin4(η2N),

can be employed as chirped counterparts for (1) and (2). These fields also satisfy the full set of Maxwell's equations, exactly. Note that the model depends upon a single dimensionless parameter, b. A chirp, modeled in this fashion, destroys the symmetry of the un-chirped oscillation merely slightly, but sensitively enough to circumvent the Lawson-Woodward theorem [25–28] and result in net electron acceleration, as will be shown in the next section.

Equations (6, 7) have not been produced rigorously. Rather, they have been constructed with the aim of encompassing linear chirp along with higher-order chirps in a single equation. In principle, they also lead to electron dynamics that can be handled analytically, and to results that may be interpreted easily. The choice expressed by Equation (5) is not unique. The above-mentioned goals can be achieved by other alternative, but not totally unrelated, models such as exponential, logarithmic, and power chirps [29].

Next the dynamics of a single electron (mass m and charge −e) injected along +z, the propagation direction of the pulse, is investigated. Without any loss of generality, the assumption is made here that the front of the pulse will catch up with the electron exactly at t = 0, and precisely at the instant it is at the origin of coordinates. This assumption may be difficult to realize experimentally, in which case the delay between the particle and the front of the pulse must be taken into account. This merely results in a set of initial conditions different from the ones employed below, if the above assumption is made. In this model, we assume no laser-electron timing jitter. That is, we select an electron injection phase for the chirped electric field to evidence the potential theoretical energy gain a single electron may experience. In practical terms, if we consider optical wavelengths (e.g., λ ~ 1 μm) to achieve the high intensity regime presented here, we would require the injection phase error to be ~ λ/10 or smaller, or equivalently, lower than ~ 300 attoseconds for a single electron. These timing error levels are currently achievable using any general self-injected scheme because the electron originates from a process triggered by the same accelerating laser fields.

Assuming no delay, the electron will have the relativistic injection kinetic energy K0=(γ0-1)mc2, where γ0=(1-β02)-1/2 and β₀ is the injection (initial) speed scaled by c. The relativistic Newton-Lorentz equations of motion (or energy-momentum transfer equations) of the electron may be combined to give (8)dβdt=eγmc[β(β·E)-(E+cβ×B)],

where γ = (1 − β²)−1/2 is the Lorentz factor of the electron and β is its velocity vector scaled by c. Equation (8) possess analytic, but admittedly cumbersome, solutions for linear and quadratic chirps, in terms of Fresnel cosine and sine integral functions [20, 21]. Beyond that, resorting to numerical methods seems inevitable, which will be done in this work. The vector Equation (8) is equivalent to three component (coupled) differential equations. Our numerical calculations employ a Runge-Kutta-based Mathematica code developed by one of us and used over the years [18–21] to solve problems involving laser-matter interactions. Note that, in principle, a single integration of Equation (8) yields the Lorentz factor γ of the electron (or its energy scaled by the rest energy mc²) which then may be used to find the kinetic energy K = (γ − 1)mc². A second integration, employing the appropriate initial conditions, then yields the particle trajectory.

Some examples are taken up next. Specifically, dynamics of a single electron injected with initial kinetic energy K₀ ~ 2 MeV (γ₀ = 5) and subsequently submitted to a laser pulse of N cycles in η, and whose intensity is I₀, will be discussed. The tacit assumption is made here that the electrons are produced, by whatever means, outside the region of interaction with the laser and subsequently guided to where they interact as described.

Behavior of the exit kinetic energy of the electron, following interaction with the laser pulse, as a function of the chirp parameter, is shown in Figure 1. The rows are for N = 3, 4, and 5, respectively, and the columns are for I0=1020 and 10²¹ W/cm², respectively. Note first that the range of b values decreases with increasing N, reflecting the role played by the convergence condition |bη| < 1. Second, the exit kinetic energy K_exit → K₀ in the limit of b → 0, implying zero gain by the electron, as predicted by the Lawson-Woodward theorem [25–27]. Due to the fact that K ∝ I₀, the values of K_exit in (d)-(f) are 10 times the corresponding values in (a)-(c). Finally, note that the exit kinetic energy values are sensitive to variations in b.

In Figure 2, the un-chirped and chirped normalized electric fields are shown together for a pulse of 5 cycles in η. It is clear that the symmetry of the pulse is partially destroyed by chirping. Corresponding positive and negative parts, of each oscillation in η, are no longer identical. Thus, for example, the acceleration which results from interaction of an electron with a negative part of an oscillation, does not get canceled by the deceleration resulting from interaction with the corresponding, but no longer identical, positive part. The end result is, therefore, one of net acceleration or, equivalently, net energy gain by the electron from the pulse.

The example of single-electron acceleration by a 5-cycle chirped pulse, with a chirp parameter b = −0.038, is analyzed further by studying Figures 3, 4. Evolution of the kinetic energy of the electron, with η, is shown in Figure 3. The interaction results in a gain of 2.482 GeV by the electron. The comments presented in the previous paragraph are further supported by Figure 3B, which displays the normalized electric field seen by the electron during interaction with the pulse, including the correlation between the lack of symmetry of the field and the net energy gain by the electron. Note that, viewed as a function of η, the time t = η/ω₀ + z(η)/c. This is obviously a transcendental equation that can best be shown graphically. The t vs. η relationship is shown in Figure 3C. From this figure, the total laboratory time of laser-particle interaction is less than 0.446 ns, for the parameters used. This interaction time is equivalent to about 133796 T, where T = λ/c is the laser period.

Moving on to Figure 4, more interesting information may be extracted regarding the example just considered. Figure 4A is similar to Figure 3A, except here the kinetic energy evolution is shown as a function of the excursion distance along the direction of propagation of the pulse. First, note the obvious correlation between the two curves displayed in Figures 3A, 4A. Second, during the electron interaction with the pulse, its trajectory extends an axial distance Δz ~ 13.38 cm. Its exit kinetic energy, in this particular example, is equivalent to an energy gain of roughly ΔK ≡ K_exit − K₀ ~ 2.480 GeV. These numbers predict an energy gradient of ΔK/Δz ~ 18.533 GeV/m, or more than 185 times the natural limit (~ 100 MeV/m) on the performance of the corresponding conventional (linear) electron accelerator.

Figure 4B shows the actual trajectory of the electron in the example subject of the discussion above. It is a typical 2D trajectory one would expect for an electron interacting with a linearly-polarized laser pulse. Effect of the x−polarized electric field is obvious in accelerating the electron transversely, relative to the pulse propagation direction. At the same time the forward excursion made by the electron is quite substantial. Once the electron is left behind the pulse, it follows a straight line trajectory whose direction is the same as the direction of its final (exit) momentum vector. For the example at hand, the scattering angle is about θ~tan-1[βx(ηf)/βz(ηf)]~0.37°, with the direction of propagation of the pulse.

We acknowledge that, for the results presented above, the interaction lengths and times between a (semi)relativistic electron and a laser are very long compared to what can currently be achieved with Fourier optics in free-space, in order to achieve such relativistic intensity levels. One possible avenue of investigation to overcome this challenge is the use of spatially and spectrally shaped beams, tailored to abruptly focus and defocus from high on-axis intensity [30]. A single electron at rest would naturally exhibit much shorter interaction lengths that are more realistic under conventional Fourier optics. In Figure 5 results similar to those of Figure 4 are displayed, but for an electron accelerated from rest (β₀ = 0, γ₀ = 1). According to Figure 5A the electron gets accelerated from rest to a kinetic energy ~ 251 MeV, over an axial excursion distance of Δz ~ 1.36 mm, and 4.56 ps interaction time. These numbers yield an acceleration gradient of 183.8 GeV/m, ten times that of the case displayed in Figure 4. From Figure 5B one also reads that the maximum transverse excursion of the electron is about 26.7 μm. Finally, the scattering angle of the electron accelerated from rest is roughly 3.7°.

Attention in this work has been focused on a somewhat generalized model for chirping the frequency of a plane-wave laser pulse of finite axial extension. The un-chirped linearly-polarized electric field of the pulse is a simple sin oscillation under a sin⁴ pulse envelope. This representation is quite idealized. A more realistic model ought to take into account the finite extension of the pulse in both space and time. Despite the observation that the pulse has a finite spatial extension in the axial (propagation) direction, it is still plane-wave in character in the transverse plane. This only strengthens the case for the need to adopt a more realistic model for description of the fields of the laser pulse [31–33].

The model also ignores existence of the axial field components and the Gouy phase [34]. These are important pulse attributes that must be included in any proper description of the fields of a high-power, tightly-focused and ultra-short laser pulse [31–33].

Finally, the examples employed to demonstrate utility of our model for the chirped fields have been limited to free-space electron acceleration, employing laser field intensities of about 10²¹ W/cm². For these reasons, radiation-reaction effects have been ignored. Proper account of the radiation-reaction effects requires resort to quantum electrodynamical methods, which fall beyond the scope of the present work. Recent calculations have demonstrated that such effects can be significant in plasma-based laser acceleration of electrons, when the laser intensity employed exceeds 10²³ W/cm² [35].

YS is the conceptual author of the presented work. YS developed the methods. YS and SC contributed to the theoretical results and to the composition of the manuscript.