2. LONGITUDINAL MOTION

Assuming |P_⊥| << P_z (paraxial approximation), it is convenient to expand the Hamiltonian in a Taylor series around r_⊥ = 0, P_⊥ = 0. To leading order, this reduces

to a purely one-dimensional Hamiltonian (Esarey & Pilloff, 1995)

with (ζ,P_z) as canonical coordinates. In Eq. (5) the notations γ₀ = (1 + P_z²/m²c²)^1/2 and Ψ₀(ζ) = Ψ(r_⊥ = 0,ζ) are used. For convenience, the subscript-0 will be suppressed in the remainder of this section.

In the linear wakefield regime, the plasma wave equation reduces to a harmonic oscillator equation (Gorbunov & Kirsanov, 1987)

where k_p² = 4πne²/mc² defines the plasma wave-number k_p, and Φ_p is the dimensionless ponderomotive potential of the laser pulse. The solution in the region behind the driving laser pulse is

where E₀ is the dimensionless wakefield amplitude and without loss of generality, particular choice of the wakefield phase has been made.

The phase space diagram contains 3 types of orbits, as can be seen in Figure 1, which shows the (ζ,γ)-phase space diagram for E₀ = 1/10, γ_g = 50. As seen in Figure 1, there are closed orbits inside the separatrix and open orbits both above and below the separatrix. The orbits below the separatrix describe the motion of electrons that are too slow to be captured in the wave. The orbits above the separatrix correspond to the motion of electrons that are out-running the wave. The orbits inside the separatrix describe the synchrotron oscillation of electrons that are trapped inside the wave. The dynamics of high-energy electrons out-running the plasma wave (orbits above separatrix) have been studied by Cheshkov et al. (2000) and Chiu et al. (2000), in the context of a linear collider based on multi-stage laser wakefield acceleration. In this regime, the transverse motion is effectively decoupled and the analysis is considerably simplified. Instead, in this article, we will analyze the dynamics of electrons injected at relatively low energy from a compact conventional electron source (orbits inside separatrix), for which the coupling of longitudinal and transverse dynamics becomes important.

Phase diagram for with O-point (O), X-point (X), highest (H) and lowest (L) point of separatrix.

Stable equilibrium points (O-points) are found at ζ = nλ_p, γ = γ_g and unstable equilibrium points (X-points) at ζ = (n + 1/2)λ_p, γ = γ_g for all

. For orbits inside the separatrix, one defines the turning points by the condition

: at these points the backward phase slip of the electron changes to forward slip or vice versa. In Figure 1 these points are seen to be at γ = γ_g. Points of minimum and maximum energy, defined by

, are found at ζ = nλ_p /2 for all

. The minimum and maximum values of γ on the separatrix are denoted γ_max(min): these points are indicated as H and L in Figure 1. The values of γ_max(min) are

in the limit γ_g >> 1.

Orbits close to the O-point describe the motion of deeply trapped electrons. Using that P_z is close to v_gγ_g mc and ζ << λ_p one finds a harmonic oscillation with synchrotron frequency ω_s

Using the wakefield equation, one finds ω_s ≈ (E₀ /γ_g³)^1/2ω_p, where ω_p = k_p c denotes the plasma frequency. It is found that γ_g >> 1 implies ω_s /ω_p << 1, i.e., for under-dense plasma the motion of deeply trapped electrons in the comoving frame is much slower than the motion of plasma electrons. This justifies a posteriori the quasi-static approximation for the description of the plasma wave.

Once they are accelerated, the electrons eventually propagate faster than the wave, so the energy gain is limited by phase slippage. The acceleration distance is equal to v_gT, where T denotes the time during which the electron can remain in the accelerating region (i.e., half a synchrotron period). The maximum acceleration, distance is the dephasing length L_d. Since for a large part of the acceleration, the approximation γ >> γ_g is valid, the phase slippage can be taken as constant:

With this approximation, one finds ω_s ≈ ω_p /2γ_g² for the synchrotron frequency of orbits above the separatrix, which indicates that the motion of electrons out-running the wave in the comoving frame is even slower than the motion of deeply trapped electrons. The dephasing length corresponds to a phase slippage distance of half a plasma wavelength, so with Eq. (11) it is found that

To illustrate the dynamics further, results of numerical integration of the lowest-order equations of motion are shown in Figure 2. Two different initial conditions inside the separatrix have been chosen: (ζ,γ) = (−3λ_p /20,γ_g /5), (−3λ_p /10,γ_g /5) for E₀ = 1/10, γ_g = 50. The time variable is multiplied by v_g to get the acceleration distance L_a, which is expressed as a fraction of the dephasing length. From Figure 2 the approximation of constant phase slippage is seen to hold for a large part of the motion. It fails only during a short time, when the electron rapidly slips backward. This leads to typical sawtooth oscillations for ζ. Orbits near the O-point have a shorter synchrotron oscillation period than orbits close to the separatrix. The maximum energy scales about linearly with the synchrotron period.

Phase (left) and energy (right) as functions of acceleration distance for initial conditions (ζ,γ) = (−3λp /20,γg /5) (solid lines) and (ζ,γ) = (−3λp /10,γg /5) (dashed lines).

3. TRANSVERSE MOTION

In three-dimensional geometry, the wave equation is (Gorbunov & Kirsanov, 1987)

Assuming that the ponderomotive potential can be written as a product Φ_p = Φ_z(ζ)Φ_⊥(r_⊥), the solution for Ψ is simply Ψ = Ψ_z(ζ)Φ_⊥(r_⊥), where Ψ_z is equal to Ψ in Eq. (7). In this subsection, it is assumed that the laser transverse profile is an axisymmetric Gaussian function Φ_⊥(r) = exp(−r²/r₀²).

The transverse electron motion follows from the second-order expansion

of the Hamiltonian (Eq. 3) with Reitsma (2002)

where the function Ψ₂ denotes the curvature of the potential Ψ in the vicinity of the propagation axis. The function Ψ₂ is given by

The transverse forces are focusing in regions with Ψ₂ < 0 → cos(k_pζ) > 0 and defocusing in regions with Ψ₂ > 0 → cos(k_pζ) < 0. Only one-fourth of the plasma wavelength is both focusing and accelerating, i.e., when cos(k_pζ) > 0 and sin(k_pζ) < 0. Therefore, the maximum attainable energy γ_max and minimum energy γ_min for electron trapping are not on the separatrix of Figure 1, but on the orbit through ζ = −λ_p /4, γ₀ = γ_g. Their values are given by

in the limit γ_g >> 1. In focusing regions,

is the Hamiltonian of a harmonic oscillator with time-dependent mass γ₀ m and focusing strength −Ψ₂. The time dependence enters through the dependence of γ₀ and Ψ₂ on P_z and ζ, respectively. The transverse oscillations are called betatron oscillations. From

one finds a condition for the laser pulse width

in order to satisfy ω_s << ω_β << ω_p, where γ_min < γ₀ < γ_max has been used. In this case, the betatron motion is much slower than the motion of plasma electrons, so that the wakefield is correctly described in the quasi-static approximation. Also, the betatron oscillation is much faster than the synchrotron oscillation, so that the (ζ,P_z)-dependence is adiabatically slow and the area α_x in (x,P_x) phase space

is an adiabatic invariant of the motion (similar for α_y). Note that the requirement that the longitudinal timescale is much longer than the transverse timescale may fail during the rapid backward slip of the electron (see Fig. 2). In this case, there is no adiabatic invariant.

4. COUPLING OF LONGITUDINAL AND TRANSVERSE DYNAMICS

In this section, it is assumed that the time-scales of longitudinal and transverse motion are sufficiently separated, so that the adiabatic invariants α_x and α_y can be defined. Defining x₀, P_x0 to be the betatron amplitudes for, respectively, x and P_x, one finds that the adiabatic invariant is α_x = πx₀ P_x0 (similar for α_y). The variation of x₀ and P_x0 due to the evolution of ζ and P_z on the slow time-scale is given by

which describes the coupling of longitudinal to transverse motion, i.e., adiabatic focusing due to acceleration. To estimate the magnitude of the focusing effect, consider injection with energy γ_i < γ_g and extraction at energy γ_f > γ_g. The injection phase and the extraction phase are taken as identical, so that the value of Ψ₂ is the same. In this case, the adiabatic focusing factor, defined as the ratio of initial to final x₀, equal to the ratio of final to initial P_x0, is found to be (γ_f /γ_i)^1/4. With Eqs. (16)–(17) it is found that

For E₀ = 1/10, γ_g = 50, this results in an upper limit of about 3.16 for the focusing factor. Note that acceleration leads to a decrease of opening angle P_x0 /γ, given by (γ_f /γ_i)^−3/4. A plot of the adiabatic focusing factor for γ_g = 50 is given in Figure 3, where the value of γ_f has been found by taking the zero-order Hamiltonian (Eq. 5) as a constant of the motion.

Adiabatic focusing ratio as a function of initial energy for γg = 50.

By rewriting the second-order Hamiltonian

in terms of the adiabatic invariants, one finds the one-dimensional Hamiltonian

that describes the coupling of transverse to longitudinal motion, e.g., the effect of the r-dependence of the accelerating field on the energy gain. The Hamiltonian

depends on the adiabatic constant α = α_x + α_y and is defined only in the focusing region, where Ψ₂ < 0.

The influence of the transverse motion on the longitudinal dynamics is illustrated in Figure 4, which shows phase diagrams of

for α/πr₀ mc = 0, 1/2, and 3/2. Also indicated are contours of

(points of maximum or minimum energy) and

(turning points). For α > 0, energy maxima and minima are found around ζ = ± λ_p /4, which are absent in the case α = 0. The turning points, which are always at γ₀ = γ_g for α = 0, are seen to occur at γ₀ ≈ γ_g(1 + σ)^1/2, where the approximation σ = (Ψ₀γ_g /2)^1/2α/πr₀ mc << γ_g² has been used. In Figure 4, X-points are seen to exist near ζ = ± λ_p /4, γ₀ = γ_g. The area inside the separatrix decreases with increasing α in such a way that γ_min increases with α and γ_max decreases with α.

Phase diagrams for with α/πr0 mc = 0 (left), 1/2 (middle), and 3/2 (right).

The influence of transverse motion on longitudinal dynamics is further illustrated in Figure 5. This figure shows one orbit for α = πr₀ mc and two orbits for α = 0, chosen such that one of them has the same maximum energy and the other one has the same minimum energy. The main difference between the α = 0-orbits and the α > 0-orbit is seen to be in the low energy (γ₀ < γ_g) part, where the electron with a finite value of α has a larger backward slip in the wakefield. As a consequence, for a collection of electrons (i.e., a bunch) a finite spread in α effectively leads to bunch length increase and possibly a growth of energy spread. It also means that in the low-energy regime the electron bunch cannot be described as a collection of “slices” labeled by the longitudinal coordinate. In the high energy (γ₀ > γ_g) part, the α > 0-orbit is barely different from the large α = 0-orbit, indicating that the radial variation of the accelerating field has only little effect on energy gain. This is because the electron moves close to the axis as a result of strong adiabatic focusing during the rapid backward slip—see also Eq. (21).

Selected orbits for α = 0 (dashed lines) and α = πr0 mc (solid line).

To check the validity of the paraxial approximation, it is instructive to look at some results of numerical integration of the full equations of motion Eqs. (1)–(2). These simulation results are given in Figure 6, which shows

as functions of the acceleration distance. The following initial conditions have been chosen: (ζ,γ₀) = (0,γ_g /5), (y,P_y) = (0,0). For (x,P_x), the cases (0,mc/2) (a) and (0,mc) (b) are compared. The wakefield parameters are E₀ = 1/10, r₀ = λ_p, γ_g = 50.

Simulation results for initial conditions (a) (x,Px) = (0,mc/2) (solid lines) and (b) (x,Px) = (0,mc) (dashed lines) showing influence of betatron oscillation on longitudinal dynamics.

The quantity x₀ P_x0 is seen to be nearly constant for electron a, while there are some fluctuations for electron b. This does not mean that there is not an adiabatic invariant α_x for electron b, it only reflects that α_x ≠ x₀ P_x0. The change of x₀ P_x0 is most pronounced during the rapid backward slip of electron b, when x₀ reaches its maximum. The behavior of

indicates that the second-order approximation breaks down for electron b. This implies that the parabolic approximation of the focusing potential Ψ is not satisfied, Since P_x0 << γ₀ during the whole acceleration.

There is a considerable difference in longitudinal dynamics for the two electrons: electron b is seen to slip closer to the defocusing regions ζ < −λ_p /4, ζ > λ_p /4 and reaches a higher energy than electron a. The influence of phase slippage and acceleration on the betatron motion is seen in the graphs of x₀ and P_x0 as functions of acceleration distance. The maximum of x₀ is at the point of minimum energy, the maximum of P_x0 is at the point of maximum energy, indicating that the influence of γ₀ on x₀, P_x0 dominates over the influence of Ψ₂.