The phase velocity of the wake field plays an important role in wakefield accelerators. The roles include the accelerated length and the injection mechanism of electrons. Bulanov et al. proposed an self-injection mechanism [41] based on the fact that down-ramp plasma background can reduce the phase velocity of wake field. As Wave breaking happened, that is when the phase velocity equals to the quiver velocity of electrons, electrons will be trapped into the wakefield. The trapped electrons are from the background plasma rather than an external source in this scheme, therefore is termed as “self-injection”. In this scheme, to determine whether or when the wave breaking happened, a robust relation between background plasma density and the phase velocity is needed. In the following text, we briefly introduce two different ways to calculate the phase velocity and how the factor α between plasma wavelength and bubble width corrects the result.
The first method is using the dispersion relation of the plasma wave. In [46], the phase velocity under an inhomogeneous plasma is discussed. Consider the eikonal of the plasma
wave θ(x, t), the angular frequency ωp and wave number kp are, ωp = −∂tθ, kp = ∂xθ respectively. With the cross differentiation property, frequency and wave number have such relation,
The plasma frequency ωp depends on the local electron density, ωp(x) =
qne(x)e2
For a down ramp density profile, ∂x√
ne< 0, the phase velocity decrease with time.
The other method to calculate phase velocity is proposed in [21]. As we discussed in previous subsection, the flying mirror is a bubble width behind the driver. Therefore, the position of flying mirror can be described by
xM = xL− λB, (2.64)
where xM and xLare the position of flying mirror and the driver respectively. Let ˙x ≡ dx/dt, the velocity of mirror is then
˙
xM = ˙xL− ˙λB. (2.65)
This shows that the velocity of the flying mirror depends on the velocity of driver and the variation of the bubble width. In a homogeneous plasma, the bubble width can be treated as a constant provided that the depletion of driver can be neglected. This leads to the
velocity of flying mirror equals to the group velocity, which is the principle of wakefield.
However, in an inhomogeneous plasma, with d/dt = ∂/∂t + ˙xM∂/∂x, we can write the velocity of mirror as
˙xM = ˙xL
1 + λ′B. (2.66)
Recall that the bubble width can be described by λB = αλp(cf. Eq.(2.44)), the velocity can be written as
˙xM = ˙xL
1 + αλ′p+ α′λp. (2.67)
In general, α is a function of the position of flying mirror when graded background density is presented. In Fig.(2.4), we show the numerical result about the relation between α and the ratio between driver pulse length and the ambient plasma wavelength L/λp. As mentioned in the non-optimal-length case, the ratio L/λp varies for a fixed length driver in an inhomogeneous plasma background. This makes α not a constant.
Figure 2.4: Dependence of α on the normalized pulse length.
Therefore, for an arbitrary density profile and pulse length, numerically solving Eq.(2.55) and Eq.(2.66) is needed. In Fig.(2.5), a flat-top driver with fixed pulse duration L/c =
47.5f s is used. The background density profile is
n(x) = n0
1 + e−x2/2D2
2
, (2.68)
with n0 = 1024m−3 and the characteristic length D = 100µm. The numerical result agrees well with 1D PIC simulation data.
Figure 2.5: Comparison between PIC data and theoretical results of velocity of the fly-ing mirror. Curves are calculated numerically from Eq.(2.55), (2.66). Circles are PIC simulation data. Driver with different a0are considered.
As we find previously, α equals to a constant 3/4 in two limits. The first one is a non-relativistic optimal-length driver. However, this approximation is not valid in a graded plasma background because the optimal condition can not be maintained. The other one is the ultra-short pulse limit. Even for a graded plasma, the condition can be satisfied provided that L≪ λp.
Consider the Gaussian-like down-ramp density profile (Eq.(2.68)). To satisfy the ultra-short approximation, the FWHM pulse length L of the gaussian driver is chosen such that L≪ λp. The condition guarantees α = 3/4. With Eq.(2.67), Eq.(2.68) and applying the underdense approximation (ω2p/ω02 ≪ 1), the velocity of mirror can be described by [21]
˙ xM
c = 1
1 + (3b/2)(λp0x/D2)e−x2/2D2. (2.69)
1D PIC simulations are performed to verify the predictive ability of Eq.(2.69) with
pa-rameters: n(0) = 4n0 = 1.0× 1023m−3, D = 100µm. The simulations are separated into two groups. The first one is fixing L = 2.4µm but varying the intensity of driver with a0 = 1.5, 2.1, 2.19, 2.5. The second group is fixing the intensity of driver to a0 = 2.0 but varying the pulse duration L/c = 4, 6, 10, 14, 20, 25, 30f s. The comparison between analytical results (Eq.(2.69)) and 1D PIC simulation is shown in Fig.(2.6). The subplot (a) demonstrates the comparison among analytical prediction with α = 1 and α = 0.75 and 1D PIC data with driver pulse of different a0. The curve with α = 0.75 agrees well with PIC data for cases of different a0. The subplot (b) shows the agreement between analytical curve and PIC data with different length driver. This verifies the argument that as long as ultra-short pulse limit holds, λB= 0.75λpis a good approximation.
In the end of this section, we hope to discuss about the two different methods when applied to compute the velocity of the flying mirror. They are
˙
xM = x˙L
1 + λ′B, (2.70)
vph= vph,0
1− ∂ω∂xpkp,0t . (2.71)
As we demonstrated earlier, Eq.(2.70) can provide well prediction about the velocity of flying mirror. Here, we wonder whether Eq.(2.71) gives the same result. The vph,0 in the numerator is the phase velocity in the homogeneous region. Based on the wake field principle, we know the velocity of the mirror can be approximated with the velocity of driver pulse, that is ˙xL. For the denominator, first we need to clarify the meaning of t.
The time here means the time after the wake wave formed. In the scheme of laser-driven wakefield, the wake wave is induced by the driver pulse. Therefore, the time t corresponds to how long after the driver pulse passed. Here, we are discussing about the first flying mirror. According to the result in previous sections, the first mirror always trails behind the laser by a distance 3/4λpunder ultra-short pulse condition. This implies t = (3/4)(λp/c).
Other terms in the denominator can be rewritten as ∂ωp/∂x =−2πc/λ2pλ′p and 1/kp,0 =
(a) Driver with different a0
(b) Driver with different L
Figure 2.6: Phase velocity of the flying mirror in a gaussian down-ramp plasma back-ground. Solid lines are analytical predictions. Dots are 1D PIC data. It can be seen that α = 3/4 is a good approximation as long as the ultra-short pulse condition holds.
λp/2π. After putting these terms back into Eq.(2.71), we get
vph,M = x˙L
1 + 3/4λ′p (2.72)
which recovers Eq.(2.70) in the ultra-short pulse limit. A short conclusion is that these two methods are consistent with each other but focus on different face. Eq.(2.70) cares mainly about the flying mirror, it is more straightforward and easily-implemented if the problem is about the flying mirror like the study of trajectory of the mirror [21]. On the other hand, Eq.(2.71) can describe the phase velocity of all parts in the wake wave. This is useful in the discussion of electron self-injection scheme because we are only curious about “whether” but not “where” the electrons are injected [46].