2.6. Acceleration of Electrons in Laser Wakefields
2.6.3. Limits
Depletion Dephasing Diffraction Beam loading
There are several processes that limit the maximum energy gain of the accelerated electrons. In the following, each process is discussed including an estimate on the max-imum acceleration distance that it allows. With this, the maxmax-imum possible electron energies can be roughly estimated by assuming that the electrons experience the maxi-mum possible accelerating field Emax, equation (2.33), given for a certain plasma density and laser intensity over the respective acceleration distance Wmax =eEmaxLacc
Electron dephasing. Since the laser beam (and therefore the plasma wave) propagates in a plasma with a velocityv < c, electrons can get accelerated to higher velocities than the group velocity of the laser pulse. They can outrun the accelerating fields of the plasma wave and get decelerated which is called dephasing. Since the longitudinal (accelerating) and the transverse focusing fields of the plasma wave are out of phase by a factor of π/2 with respect to each other [Akhiezer and Polovin, 1956], both linear accelerating and transverse focusing fields only exist in a quarter of the plasma wave period. Therefore, the dephasing length is reached after the electron beam outruns the plasma wave by a distance of 'λp/2. The length can be estimated by assuming an electron moving with c and the plasma wave moving with the group velocity of the laser: Ld/vp = (Ld+λNp/2)/c. Using equation 2.35, this leads to
Ld = λ3Np λ2
a20+ 2 2
. (2.44)
This equation can be simplified for a20 1 and in the case of a20 1, a lengthy calculation including relativistic effects leads to [Esarey et al., 2009]:
Ld=
λ3p
λ2 fora20 1
√2 π
λ3p λ2
a0
Np fora20 1
(2.45)
and a corresponding maximum energy gain of [Esarey et al., 2009]
∆Wd[MeV] =
1260·I[W/cm2]
n0[cm−3] for a20 1 1260· 2
πNp
I[W/cm2]
n0[cm−3] for a20 1,
(2.46)
whereNp is the number of plasma periods behind the driver laser.
In the linear regime (a0 1), the maximum acceleration distance is mainly limited by the dephasing length. This can be overcome by using a nonuniform axial (along the acceleration distance) plasma density. Especially for an axially increasing plasma density, the phase velocity of the plasma wave also increases [Sprangle et al., 2001]. This means that the dephasing length (and thus the maximum energy gain) can be extended. For appropriate tapering of the plasma density, the acceleration limit is given by the pump depletion (see below).
In the nonlinear regime (a0 1), the dephasing and the pump depletion lenghts become comparable which means that an increase of the dephasing length does not lead to higher energy gains. Therefore a simpler setup with no density tapering can be employed, leading to energy gains comparable to the linear gain, but using higher accelerating gradients and therefore shorter channel lengths.
For the parameters of the experiment described below (laser powerP = 20 TW at a wavelength of 800 nm, laser intensity ofI '2·1018W/cm2 and correspondingly a0 '1, a plasma density ofn0 '5·1018cm−3and correspondinglyλp '15µm) the dephasing length is calculated to beLd= 5.2 mm and a correspondingly maximum energy gain of ∆Wd = 500 MeV. However, this has to be considered only as a coarse approximation since the laser intensity a0 ' 1 which is in between the limits of the estimations given above (the formula in the limit a0 1 has been used for the calculation).
Laser diffraction. Without any forms of optical guiding, the laser pulse undergoes Rayleigh diffraction which increases the beam size and correspondingly decreases the laser intensity. Since a plasma wave can only by driven by a sufficiently intense laser, the acceleration distance is limited to a few Rayleigh lengths (ZR)
Ldiff 'πZR, (2.47)
where ZR =πw20/λ, with λ being the laser wavelength and w0 the rms laser spot size. In this case, the energy gain is limited to
∆Wdiff[MeV]'740 λ λp
p 1
1 +a20/2 ·P[TW]. (2.48) To overcome this limitation, the laser pulse can be kept focused beyond the Rayleigh length by a medium with an index of refraction that is higher on-axis
2.6. Acceleration of Electrons in Laser Wakefields than off-axis (∂η(r)/∂r <0) [Sprangle et al., 1992]. The index of refraction for a laser pulse in a plasma depends on both the radial plasma density and implicitly on the radial laser intensity through the relativistic mass increase of the electron in the laser field:
η(r) = r
1−ωp
ω 2
'1− 4πe2 2ω2
n(r)
γ(r)me,0. (2.49) This suggests that a radially increasing appropriate plasma density profile n(r) as well as radially different electron energies γ(r) can form such a channel. The first possibility can be achieved by a second laser pulse or an electrical discharge in a gas that ionizes and heats electrons. The hot electrons expand and form a channel that has a radial plasma density distribution with a minimum (and thus a maximum index of refraction) on axis. This can be used as a plasma waveguide to guide the laser beam over several Rayleigh lengths [Butler et al., 2002; Geddes et al., 2004].
Laser pulses with a sufficiently high powers (P >Pcrit[GW]'17.4(ω/ωp)2) undergo relativistic self-focusingand can also be guided in a plasma (for relativistic self-focussing, see section 2.6.4).
A plasma waveguide has several advantages over the relativistic self-guiding regime.
Since the channel does not require a high laser power P >Pcrit, it can be operated at lower intensities with the advantage of not having to rely heavily on nonlinear effects which may result in a more stable regime.
In the case of the experimental parameters given above, the limit of the energy gain given by the laser diffraction is ∆Wdiff = 660 MeV. It can be seen that in this case the dephasing is limiting the acceleration rather than the diffraction. However, for a slightly lower plasma density, the dephasing length can be increased and the maximum energy gain is limited by laser diffraction.
Pump depletion. As the laser excites a plasma wave, it transfers energy to it and starts to deplete [Horton and Tajima, 1986],[Ting et al., 1990]. The depletion length can be estimated by assuming that the laser pulse energy is completely transferred to the plasma wave. In the 1D case, assume a plasma wave with an electric field of Emax (energy density of Emax2 ) over a length of Lpd and a laser pulse with a longitudinal square profile over a pulse length of L=λNp/2 (forλNp, see equation (2.31)). With a laser electric field EL, this can be computed by: Emax2 Lpd 'EL2L which results in a depletion length of [Esarey et al., 2009]
Lpd=
2 a20
λ3p
λ2 for a20 .1
√2a0
π λ3p
λ2 for a20 1
(2.50)
and a corresponding maximum energy gain of [Esarey et al., 2009]
∆Wpd[MeV] =
3.4×1021· 1
λ[µm]n0[cm−3] for a20 1 400·I[W/cm2]
n0[cm−3] for a20 1.
(2.51)
Since both the dephasing as well as the diffraction lengths can be extended by either target engineering or certain physical effects, the pump depletion sets the upper limit of the single-stage energy gain. In order to run a laser-plasma accelerator beyond pump depletion, it has to be operated over several stages, each driven by a “fresh” laser pulse.
For the experimental parameters given above, the depletion length is given by Lpd'12 mm which leads to a maximum energy gain of ∆Wpd'850 MeV.
Beam loading. The electric fields of the plasma wave can be significantly modified by the fields of the highly dense injected electron bunches. This is referred to as beam loading and can set severe limitations on the number of accelerated electrons, the quality of the accelerated beam and the efficiency of the process. The maximum number of electrons that can be loaded into a wave bucket can be estimated by calculating the number of electrons in a small axial region (λp) which produce an electric field that cancels out the accelerating field of the plasma wave. For a linear wakefield far from wavebreaking (Emax<E0), this number is calculated to be [Katsouleas et al., 1987]
Nmax'5×105Emax
E0
A[cm2]p
n0[cm−3], (2.52) where A is the cross-sectional area of the bunch (i.e. the transverse area over which the fields of the bunch and the plasma wave interact. The equation assumes Aπ/kp2; 1/kp is the skin depth).
The energy spread of an infinitesimally short electron beam can be estimated by assuming that the front of the bunch is accelerated by the whole electric field of the plasma wave and gains an energy ∆γmax, whereas the back of the bunch experiences only the accelerating field shielded by the front and gains an energy of
∆γmin. Since the reduction in accelerating field for the electrons at the back of the bunch is linear in the number of electrons N contained in the bunch, the relative energy spread can be estimated as [Katsouleas et al., 1987]
∆γmax−∆γmin
∆γmax = N
Nmax. (2.53)
The efficiency of transferring wake energy into accelerated electron energy can be estimated by the maximum decrease of the electric field of the plasma wave due to
2.6. Acceleration of Electrons in Laser Wakefields the loaded electron bunch (which is at the back of the bunch) and can be written as [Katsouleas et al., 1987]
ηb = N Nmax
2− N Nmax
, (2.54)
which means that for N → Nmax, the efficiency approaches 100% but also the energy spread approaches 100%. Therefore the electron beam quality sets an upper limit on the number of accelerated electrons in a bunch. However, these are only 1D estimations and laser pulses with higher energies over a larger focal spot (same intensities) can lead to a decrease of the beam loading effect.