2.1. General

In the main working equations for the dynamics of a single electron in chirped plane-wave laser fields, the electron is treated as a point particle of mass m and charge −e, the relativistic energy and momentum are given, respectively, by ε = γmc ² and $ \vec{p}= \gamma mc \vec{\rm \beta}$, where $\vec{\rm \beta}$ is the velocity of the particle scaled by c, the speed of light in vacuum, and γ = (1 − β²)^−1/2. The fields, on the other hand, are modeled by an infinite plane-wave and a finite-duration plane-wave pulse. In modeling those fields the combination η = ω₀t − k ₀z, in which ω₀ is the initial (unchirped) frequency and k ₀ = 2π/λ₀ is the wavenumber, is used as a variable. Thus, the chirped frequency is ω = ω₀(1 + bη), where b is the dimensionless chirp parameter. We work with the fields (SI units)

(1)$$\vec{E} \lpar {\rm \eta} \rpar = \hat iE_0 \sin\lpar {\rm \eta} + b{\rm \eta} ^2 \rpar \cos^2 \left[{\displaystyle{{\rm \pi} \over {{\rm \tau \omega}_0 }}\lpar {\rm \eta} - \bar {\rm \eta} \rpar } \right]\comma \;$$

(2)$$\vec{B} \lpar {\rm \eta} \rpar = \hat j\displaystyle{{E\lpar {\rm \eta} \rpar } \over c}.$$

These equations model the fields of a pulse with a cos² envelope, a temporal width τ = 50 fs, a wavelength λ₀ = 1 μm, provided the choice $\bar {\rm \eta} = 15{\rm \pi}$ is made. For an electron interacting with an infinite plane wave, the normalized field strength a = eE ₀/mcω₀ is related to the laser field intensity I by $a = e\sqrt {2I/c {\rm \varepsilon} _0 } /\lpar mc{\rm \omega} _0 \rpar $, where ε₀ is the permittivity of free space and λ₀ is the wavelength of the unchirped laser field. Inserting values of the universal constants, one gets

(3)$$I\left[{\displaystyle{{\rm W} \over {{\rm cm}^2 }}} \right]= 1.36817 \times 10^{18} \mathop {\left[{\displaystyle{a \over {{\rm \lambda}_0 [{\rm \mu m}] }}} \right]}\nolimits^2 \comma \;$$

thus making a ² a dimensionless intensity parameter.

In the next two subsections, two sets of conditions on the initial injection position and velocity of the electron will be considered. Section 2.2 will be devoted to the axial injection configuration, while the more general case of electron injection at an angle ζ₀ to the pulse propagation direction, will be taken up in Section 2.3. Various aspects of the electron dynamics will be discussed, including evolution of its kinetic energy and velocity components, as well as its trajectories. In order to be able to tackle the general injection situation, for which some of the analytic solutions become too cumbersome, we will resort here to numerical integration of the combined equations of motion

(4)$$\displaystyle{{d \vec{\rm \beta}} \over {dt}} = \displaystyle{e \over {{\rm \gamma} mc}}\left[{ \vec{\rm \beta} \lpar \vec{\rm \beta} \cdot \vec {E} \rpar - \lpar \vec{E} + c\vec{\rm \beta} \times \vec{B} \rpar } \right].$$

Instead of working directly with the time t as a variable, we will employ η. The integration limits will be denoted by η_i and η_f. Equation (4) is equivalent to three coupled differential equations. This system of differential equations has been solved numerically using Runge-Kutta-based Mathematica codes we have developed over the years. All of our results have independently been confirmed by codes written in Python, as well.

2.2. Axial Injection

In Salamin (Reference Salamin2012), the special case corresponding to ζ₀ = 0 (axial injection) was briefly discussed. Much of the discussion there was based on the analytic solution to the Newton-Lorentz equations. Issues pertaining to this particular case not covered in Salamin (Reference Salamin2012) will be discussed here for completeness and in order to demonstrate that the results obtained in the chirped pulse have their correct unchirped limits published elsewhere (Hartemann et al., Reference Hartemann, Fochs, Sage, Luhmann, Woodworth, Perry, Chen and Kerman1995).

For the initial conditions on particle injection, the assumption will be made that the front of the pulse catches up with the MeV electron at t = 0, precisely at the origin of coordinates (x ₀ = y ₀ = z ₀ = 0). Hence, η_i = 0 and η_f = 30π for the cos² pulse described above. Furthermore, the electron is assumed to be traveling axially (along the direction of propagation of the pulse, the z-axis) with a scaled speed β₀, derived from γ₀ = 10, which corresponds to the initial injection kinetic energy K ₀ ~ 4.6 MeV. For an unchirped laser pulse of field intensity I ~ 1.23 × 10¹⁹ W/cm² (a = 3), this configuration has been discussed thoroughly elsewhere (Hartemann et al., Reference Hartemann, Fochs, Sage, Luhmann, Woodworth, Perry, Chen and Kerman1995). What we will do below is investigate the same electron dynamics, albeit in a chirped laser pulse. The results will be displayed along with their unchirped counterparts, so that visual comparison may be facilitated.

To begin with, Figure 1a shows the unchirped scaled electric field of the pulse vs. η. Figure 1c shows how this field gets distorted when the frequency is chirped using b = −0.0103. The pulse develops an asymmetrical low-frequency, indeed quasi-static, part. In Figure 1b, evolution with η of the kinetic energy of the electron is shown for b = 0. As expected from the Lawson-Woodward theorem (Woodward, Reference Woodward1947; Lawson, Reference Lawson1979) the electron gains no net energy from interaction with the plane-wave unchirped pulse. Substantial gain, however, is shown in Figure 1d to result from synchronous interaction with the low-frequency part of the chirped pulse. As expected, too, interaction with the wings, whose frequency is minimally affected by the chirp, results in very little or no further energy gain at all.

Fig. 1. (a) Evolution of the normalized unchirped (b = 0) electric field of a cos² laser pulse of duration τ = 50 fs. (b) Evolution of the electron kinetic energy during interaction with the unchirped pulse. (c) and (d): Same as (a) and (b), respectively, but for a chirped laser pulse (b = −0.0103). The remaining parameters are: λ₀ = 1 μm, a = 3 (intensity I ~ 1.23 × 10¹⁹ W/cm²) and γ₀ = 10 (injection kinetic energy K ₀ ~ 4.6 MeV).

As has been shown elsewhere (Hartemann et al., Reference Hartemann, Fochs, Sage, Luhmann, Woodworth, Perry, Chen and Kerman1995; Salamin, Reference Salamin2012), in the case of axial injection, the trajectory of the electron is confined to the xz–plane, the polarization plane of the laser pulse. Instead of showing the xz trajectory, Figure 2 displays evolution of the x– and z– coordinates, separately, with η. By comparing the unchirped Figure 2a with the chirped Figure 2c one can read clearly the amount of net electron displacement, Δx as a result of interaction with the pulse (poderomotive scattering). For the parameter set used, Δx ~ 2 μm (unchirped) whereas Δx ~ 7.5 mm (chirped). Likewise, the corresponding axial excursions made by the electron during interaction with the pulse are: Δz ~ 8 mm (unchirped) and Δz ~ 3.8 m (chirped).

Fig. 2. (a) and (b): Evolution of the transverse and axial position coordinates, respectively, as functions of the variable η, while interacting with an unchirped (b = 0) cos² laser pulse of duration τ = 50 fs. (c) and (d):Same as (a) and (b), respectively, but for a chirped pulse (b = −0.0103). The remaining parameters are the same as in Figure 1.

Components of the electron scaled velocity vector are displayed in Fig. 3. Interaction with the unchirped pulse gives results identical to those obtained earlier (Hartemann et al., Reference Hartemann, Fochs, Sage, Luhmann, Woodworth, Perry, Chen and Kerman1995). Figures 3a and 3b show that the electron is left behind the pulse moving at exactly the same velocity with which it was injected initially, again as expected from the Lawson-Woodward theorem (Woodward, Reference Woodward1947; Lawson, Reference Lawson1979). By contrast, interaction with the chirped pulse results in clear particle acceleration, as shown in Figures 3c and 3d. The electron emerges from the interaction with a slight increase in its transverse velocity β_x, while β_z → 1.

Fig. 3. (a) and (b): Evolution of the electron transverse and axial scaled velocity components, respectively, as functions of the variable η, while interacting with an unchirped (b = 0) cos² laser pulse of duration τ = 50 fs. (c) and (d): Same as (a) and (b), respectively, but for a chirped pulse (b = −0.0103). The remaining parameters are the same as in Figure 1.

2.3. Sideways Injection

The axial injection configuration, discussed in Section 2.2, is probably difficult to realize experimentally. Assuming that the model itself is sound, a more practically-realizable configuration would be one in which the electron is injected in, for example, the yz–plane at an angle ζ₀ with the laser pulse propagation direction, the z–axis. Thus, the initial injection velocity of the electron, scaled by the speed of light c, may be written as

(5)$$\vec{\rm \beta}_0= {\rm \beta} _0 \lpar - \hat j\sin {\rm \zeta} _0 + \hat k \cos{\rm \zeta} _0 \rpar.$$

Here, too, the assumption will be made that the front of the laser pulse catches up with the electron at t = 0 at the origin of coordinates. Every result to be presented below has been produced employing numerical codes which return the corresponding result discussed in Section 2.2 in the limit ζ₀ → 0. More importantly, the following logical amendments to the working equations must be introduced (Salamin, Reference Salamin2012). Adopting Eq. (5) as the initial condition on the velocity requires that the constants of the motion be altered to read: c ₁ = γβ_y = −γ₀β₀ sinζ₀, and c ₂ = γ(1 − β_z) = γ₀(1 − β₀ cosζ₀). Taking all of this into consideration, the scaled energy expression takes the form

(6)$${\rm \gamma} = \displaystyle{{1 + \lpar {\rm \gamma\beta} _x \rpar ^2 + \lpar {\rm \gamma}_0 {\rm \beta} _0 \sin{\rm \zeta} _0 \rpar ^2 + {\gamma}_0^2 \lpar 1 - {\rm \beta} _0 \cos{\rm \zeta} _0 \rpar ^2 } \over {2{\rm \gamma}_0 \lpar 1 - {\rm \beta} _0 \cos{\rm \zeta} _0 \rpar }}.$$

On the other hand, the general expression of γβ_x(η) stays the same (Salamin, Reference Salamin2012)

(7)$${\rm \gamma\beta} _x \lpar {\rm \eta} \rpar = a \int_{{\rm \eta} _0 }^{\rm \eta} \, \sin\lpar {\rm \eta} ^{\prime} + b\!\mathop {{\rm \eta} ^{\prime}}\nolimits^2 \rpar \cos^2 \left[{\displaystyle{{\rm \pi} \over {{\rm \tau \omega} _0 }}\lpar {\rm \eta} ^{\prime} - \bar {\rm \eta} \rpar } \right]d{\rm \eta} ^{\prime}.$$

Before investigating the electron dynamics in this general scenario, values of the chirp parameter b, appropriate for obtaining substantial acceleration, will have to be obtained. To this end, exit electron kinetic energies will be calculated as b is varied from −0.04 to zero. The results are shown in Figure 4, for the cases of axial injection (ζ₀ = 0°) and sideways injection (at ζ₀ = 10°). All other parameters are the same as in Section 2.2. The peaks in the figures occur at b values which result in optimal synchronous motion between the electron and the pulse. Conversely, total lack of synchronization leads to the appearance of the minima, which express zero net energy gain. Note that the absolute maximum in Figure 4, which has already been used in the case of axial injection, occurs at b ~  − 0.0103.

Fig. 4. Electron exit kinetic energy, K _exit ≡K(η_f) = [γ(η_f) − 1]mc ², where η_f = 30π, vs. the chirp parameter, as a result of interaction with a chirped plane-wave laser pulse of a cos² envelope and duration τ = 50 fs. The laser wavelength is λ₀ = 1 μm, and a = 3 (I ~ 1.23 × 10¹⁹ W/cm²). Initial injection is at γ₀ = 10 (K ₀ ~ 4.6 MeV) axially (ζ₀ = 0) and sideways at ζ₀ = 10°. Onset of the interaction is assumed to have been at t = 0, the instant the electron passes through the origin of coordinates.

As it turns out, the plots of K _exit vs. b, corresponding to the different cases of particle injection into the laser field, have the same structure and exhibit minima and maxima at precisely the same values of b, with all other parameters the same. The heights of the corresponding maxima, however, are different. For the chosen parameters, the exit kinetic energies associated with the configuration of sideways injection at ζ₀ = 10° are roughly a factor of 4 smaller than those of the axial injection case.

The above conclusions should come as no surprise, in light of the following. To find the values of b that correspond to maximum exit kinetic energies, one sets equal to zero the derivative with respect to b of the expression of γ given in Eq. (6). This is tantamount to extremizing γβ_x with respect to b, Eq. (7). In fact, this process amounts to finding the zeros of the following function, obtained by differentiating with respect to b under the integral in Eq. (7)

(8)$$S\lpar b\rpar = \int_{{\rm \eta} _0 }^{\rm \eta} \, \mathop {{\rm \eta} ^{\prime}}\nolimits^2 \cos\lpar {\rm \eta} ^{\prime} + b\!\mathop {{\rm \eta} ^{\prime}}\nolimits^2 \rpar \cos^2 \left[{\displaystyle{{\rm \pi} \over {{\rm \tau \omega} _0 }}\lpar {\rm \eta} ^{\prime} - \bar {\rm \eta} \rpar } \right]d{\rm \eta} ^{\prime}.$$

Careful inspection of S(b) quickly reveals that it is independent of the initial velocity of the electron and the laser field intensity parameter a. Thus, those zeros as well as the maxima exhibited by the K _exit vs. b plots, will always have the same general structure and the same b values at which the zeros of S(b) and maxima of K _exit occur, provided the pulse-shape is the same. In other words, the positions of the maxima are model-dependent. The heights of the maxima, however, depend upon a ², γ₀ and ζ₀. This is demonstrated in Figure 5 where, in addition to S(b), the exit kinetic energy is shown over the same range of b values for the case of an electron initially at rest at the origin. Vertical lines are added to indicate the zeros of S(b) and the corresponding exit kinetic energy maxima.

Fig. 5. Same as Figure 4, but for electron initial conditions of rest at the origin of coordinates. Shown also is the function S(b) given by Eq. (8) whose zeros are the values of b corresponding to each of which the exit kinetic energy attained is a maximum. The vertical lines indicate the zeros of S and the exit kinetic energy maxima.

3.1. Acceleration from Rest

In laser-assisted atomic ionization, an electron may be produced in a state of near rest (Hu & Starace, Reference Hu and Starace2002; Maltsef & Ditmire, Reference Maltsev and Ditmire2003). When subsequently subjected to the fields of the chirped pulse, such an electron may be accelerated in vacuum. If the front of the pulse reaches the point at which the electron is born at the same time, its subsequent dynamics will be dominated by the fields of such a pulse. This is clearly an idealized state of affairs and a probably difficult scenario to realize experimentally. Nevertheless, several aspects of the electron motion will be discussed here based on the solutions to the relativistic equations of motion.

Note first the synchronized rise in the kinetic energy of the electron in Figure 7, during interaction with the part of the pulse that has been distorted by the chirp. That accelerating part of the pulse witnesses a substantial decrease in the frequency due to the negative chirp. The electron interacts with an essentially quasi-static electric field and gains about 4.53841 GeV from it, for the parameter set used and b = −0.01. Interaction with the higher frequency parts of the pulse results in no further energy gain or loss.

Fig. 7. Evolution in η of the normalized electric field strength, E _x/E ₀, of a chirped (b = −0.01) cos² laser pulse of 50 fs duration, and the kinetic energy, K, of a single electron interacting with it. The field intensity is I ~ 1.37 × 10²⁰ W/cm² (a = 10, λ₀ = 1 μm). Initial conditions are those of rest at the origin of coordinates.

Figure 8 shows the x– and z–components of the velocity of the electron, scaled by c, while it is interacting with the pulse. In the absence of a force y–component, the electron motion is confined to the xz–plane and, thus, a β_y is absent. Note that during the initial stage of interaction with the pulse, β_x only rises, due to the x–component of the force exerted by the laser's oscillatory electric field. Soon after that, the z–component of the v^→ × B^→ force of the laser begins to impart axial kicks to the electron. Thus, while β_x continues to exhibit essentially equal-amplitude oscillations for a while, β_z oscillates between zero and an increasing positive value which approaches unity quickly as the electron is left behind the pulse. Note that the exit value of β_x is nonzero, which shows that the electron will suffer net deflection from pure axial motion (ponderomotive scattering). A scattering angle, indicating the direction in which the electron will emerge after interaction with the pulse has ceased, may be defined by

Fig. 8. Evolution in η of the components, β_z and β_x, of the velocity vector, normalized by the speed of light in vacuum, of the electron whose kinetic energy is shown in Figure 7.

(9)$${\rm \theta} _{sc} = \arctan\left[{\displaystyle{{{\rm \beta} _{x\comma exit} } \over {{\rm \beta} _{z\comma exit} }}} \right]\comma \; $$

where β_x,exit = β_x(η_f) and β_z,exit = β_z(η_f). Using the exit values β_x(η_f) ~ 0.0150046 and β_z(η_f) ~ 0.999887 results in a scattering angle of θ_sc ~ 0.859732°.

As has just been pointed out, the electron trajectory, while interacting with the laser pulse, is two-dimensional (in the xz–plane). A sample trajectory is shown in Figure 9. Initial interaction with the linearly-polarized pulse results in the familiar structure shown in the inset. After these initial oscillations the electron follows an essentially straight-line trajectory while interacting with the quasi-static part of the electric field. For the parameter set employed, the calculation returns exit transverse and axial excursions Δx ~ 0.944792 mm, and Δz ~ 5.75264 cm, respectively.

Fig. 9. Two-dimensional trajectory of the single electron whose kinetic energy is shown in Figure 7. The inset is a zoom-in on the part of the trajectory which results from interaction with the first few laser cycles.

Performance of a conventional accelerator is often indicated by quoting its average acceleration gradient. An estimate of this quantity may be obtained by dividing the exit kinetic energy (see Fig. 7) by the exit axial excursion (see Fig. 9)

(10)$$\bar G \equiv \displaystyle{{\Delta K} \over {\Delta z}}\comma \;$$

where ΔK = K(η_f) − K(η₀) and Δz = z(η_f) − z(η₀). For example, a natural limit of ~ 100 MeV/m exists on the average gradient of a conventional electron linear accelerator. For the electron accelerated from rest by the present scheme, our results show that $\bar G \sim78.8926$ GeV/m, or close to three orders of magnitude times the conventional limit.

3.2. Acceleration from Axial Injection

An already fast electron, pre-accelerated by a table-top linear accelerator or an electron gun, may be subjected to a static magnetic field in order to bend its trajectory prior to shining the laser pulse on it. This can approximate the case of axial injection, to be discussed here. Again, the assumption is made that the front end of the pulse will catch up with the electron at t = 0, exactly at the origin of the coordinate system. Following the pattern of the previous subsection, the normalized electric field of the high-intensity chirped pulse is shown in Figure 10, together with the evolving kinetic energy of the electron during interaction. In addition to exhibiting the same general features as in Figure 7, Figure 10 shows an increase in the electron's kinetic energy from K ₀ ~ 2.044 MeV (γ₀ = 5) to K _exit ~ 44.9277 GeV, or more than four orders of magnitude.

Fig. 10. Same as Figure 7, but for an electron injected axially with an initial kinetic energy K ₀ ~ 2.044 MeV (γ₀ = 5).

Evolution of the electron's velocity during interaction with the chirped pulse is shown in Figure 11. Here, too, subsequent motion is two-dimensional (in the polarization plane of the laser). As for the case of acceleration from rest, β_x starts from zero due to the strong action of the oscillating E _x of the laser pulse on it and oscillates between roughly –0.1 and 0.1. Upon exit from the interaction region β_x ~ 0.00151586. On the other hand, β_z starts from a relativistic value, oscillates for some time between that value and an increasing maximum, due to the combined action of E _z and (ν^→ × B ^→)_z, and finally approaches unity as a result of interaction with the quasi-static electric field. It is eventually left behind the pulse moving at β_z ~ 0.999999.

Fig. 11. Same as Figure 8, but for axial injection with γ₀ = 5.

Finally, the two-dimensional trajectory for the case of axial injection is shown in Figure 12. The trajectory looks similar in overall features to the corresponding one displayed in Figure 9 for the case of acceleration from rest. The dimensions in this case, however, are roughly 10-fold transversely and a hundred-fold axially, as a result of the initial forward momentum. Note that the electron undergoes the transverse and axial displacements Δx ~ 9.35248 mm and Δz ~ 5.63773 m, respectively. The latter, when combined with K _exit ~ 44.9277 GeV, yields $\bar G$ ~ 7.96911 GeV/m. Note further that, although the case of axial injection results in roughly ten times overall exit kinetic energy for the electron, compared with the case of initial rest at the origin, the acceleration gradient is almost ten times less in the latter case compared with the former. The initial forward momentum results in a much larger axial excursion, in this case, than in the case of acceleration from rest.

Fig. 12. Same as Figure 9, but for axial injection with γ₀ = 5.

3.3. Acceleration from Sideways Injection

It seems much easier, from an experimental perspective, for the electron to be injected sideways (at an angle ζ₀ with respect to the direction of pulse propagation). This configuration will be discussed next. The injected electron has initial transverse and axial momenta. Thus, its subsequent motion will not be confined to the xz–plane. In fact, the trajectory will be three-dimensional, in spite of the fact that the initial injection is made within the xz–plane. This is due to the influence of the magnetic field of the laser pulse. Discussions will now be presented of the case of injection at γ₀ = 5, but at the angle ζ₀ = 5°.

The normalized electric field of the pulse and evolution of the kinetic energy of the electron are shown in Figure 13. The fact that the electron, in this case, is injected with less initial axial momentum, than in the case of purely axial injection when the same parameters are used, will result in lower exit kinetic energy for the accelerated electron; see Eq. (6). The exit kinetic energy that may be read off of Figure 13 is about 38 GeV, compared to ~ 45 GeV for an axially injected electron.

Fig. 13. Same as Figure 7, but for an electron injected sideways at the angle ζ₀ = 5° with the direction of pulse propagation, and an initial kinetic energy K ₀ ~ 2.044 MeV (γ₀ = 5).

Components of the scaled velocity vector β are shown in Figure 14. As expected, β_x starts initially from a value of zero, oscillates during interaction and retains a small exit value, ~ 0.00179557 for the parameter set employed. On the other hand, the y– and z–components, β_y and β_z, have nonzero initial values, which help them accumulate successive increases after interaction with every laser cycle. Partly due to the fact that the injection angle is too small and, hence, the initial value of β_z is greater than that of β_y, the exit values of these two components are widely different (β_y →  − 5.75237 × 10⁻⁶ and β_z → 0.999998). Using the exit values of β_x and β_z in Eq. (9) gives θ_sc ~ 0.102879°. In conclusion, the electron emerges from interaction with the pulse essentially axially.

Fig. 14. Same as Figure 11, but for an electron injected sideways at the angle ζ₀ = 5° with the direction of pulse propagation, and an initial kinetic energy K ₀ ~ 2.044 MeV (γ₀ = 5).

Sideways injection of the electron leads to the three-dimensional trajectory shown in Figure 15. Details of the part of that trajectory which result from interaction with the first few laser cycles are displayed in Figure 15b. The existence of turning points in the x– and y–coordinates is quite evident. More clearly pronounced is the forward drift in z. The following exit coordinates have been calculated: x _exit ~ 7.89547 mm, y _exit ~  − 53.5223 μm, and z _exit ~ 4.01806 m. Therefore, using the calculated exit kinetic energy of K _exit ~ 37.9288 GeV, Eq. (10) yields $\bar G$ ~ 9.43905 GeV/m, or about two orders of magnitude better than the limit of a conventional linear accelerator.

Fig. 15. Three-dimensional trajectory of a single electron submitted to a chirped laser pulse (b =− 0.01). In (b) we zoom-in on part of the trajectory followed during roughly the first one-third of the interaction time with the pulse. Injection is at ζ₀ = 5° and γ₀ = 5 (K ₀ ~ 2.044 MeV). The laser parameters are: a = 10, λ₀ = 1 μm, and τ = 50 fs.