In this section, we analyze the wakefield induced by the R wave, which is seen to be the upper branch shown in Fig. 4.1.
In the last section, we need to increase both the amplitude of the pulse and the external magnetic field to keep the driving pulse in the whistler branch. In this section, we fix the external magnetic field and increase the amplitude of the pulse. The frequency of the pulse remains larger than the cyclotron frequency in this process. Hence the pulse stays as a R wave in the magnetized plasma when we increase the amplitude of the pulse.
g t
Figure 4.5: This is the plot of velocities of electrons at different positions.
The upper graph is vr and the lower graph is vz, where vr = qvx2+ vy2. We set ˜B0 = 12. The settings of the driven pulse are a0 = 0.6 and ω/ωp = 20, the Gaussian width is σ = 3cωp.
g t
Figure 4.6: The plot of γ factor of electrons at different positions, where γ = 1/q1 − (|v|2/c). All settings of parameters in this plot are the same as
Figure 4.7: The plot of the maximum value of γ versus a0.
g t
Figure 4.8: The plot of wakefield in the co-moving frame. The settings of the driven pulse are a0 = 4, ω/ωp = 20 and σ = 2c/ωp. The external magnetic field is ˜B0 = 12.
γ factor
For the wakefield induced by R wave, it is desirable to know when we need to consider relativistic motion of the driven electrons. Thus we focus on the value of γ factors of driven electrons, where γ = 1/q1 − |v|2/c2 .
Fig. 4.5 are plots of electron velocities in different positions. We present vr =qv2x+ v2y rather than vx, vy because electrons are in cyclotron motions around the uniform magnetic field. Therefore, vx and vy oscillate in time and vr is more suitable for presenting the transverse motions of the elec-trons. From the velocities of electrons, we calculate γ factors of the electrons in different positions (Fig. 4.6). Taking the maximum value of γ factors of electrons driven by the pulse, we show the relation between γM ax and a0. In Fig 4.7, one can see the maximum value of γ (γM ax) increases linearly with a0.
Wakefield
We present the wakefield in Fig. 4.8. One can see the saw-tooth-like shape again in the plot which is similar to Fig. 3.1. For larger values of a0,
a0
50 100 150 200 250 300 350 400
wb/EMax zE
10 20 30 40 50 60
Figure 4.9: The plot of maximum value of wakefield versus a0 in magnetized plasma. The parameter settings are ω/ωp = 20, ˜B0 = 12, σ = 2c/ωp.
we find that the relation between EzM ax and a0 is different from that in the non-magnetized plasma case (Fig. 4.9).
In Fig. 4.9, we find that the growing rate of EzM ax reduces with a0 when a0 < 50 but remains constant when a0 becomes larger.
Chapter 5 Conclusion
First of all, we have found that the maximum value of γ factors of the driven electrons increase linearly with a0 in the magnetized plasma (Fig. 4.5).
According to this result, we expect there exists a simple relation between γ and a0 in the magnetized case.
Secondly, the plot of wakefield in the magnetized plasma (Fig. 4.8) is similar to that in the non-magnetized plasma (Fig. 3.1). For wakefield in-duced by the strong field pulse, we see the saw-tooth-like shape of wakefiled in both non-magnetized case (Fig. 3.1) and magnetized case (Fig. 4.8). This is because the uniform magnetic background does not affect the longitudinal motions of the electrons. Therefore, the longitudinal waves in two different cases have similar behavior.
Another interesting result is the relation between EzM ax and a0. If we keep the driving pulse as the whistler pulse and increase both amplitudes of the pulse and the external magnetic field, we find EzM ax approaches to a certain value for a0 >> 1. This asymptotic value is smaller when the ratio of the external magnetic field to the amplitude of the pulse is larger.
Finally, for wakefield driven by the R wave, we find that EzM ax grows linearly with a0 for a sufficiently larger a0 < 50.
Although numerical solutions work well as seen from many comparisons, there are still some limitations. First of all, we did not consider the dispersion effect of the pulse. Since the pulse is not made of single wave length in real-ity, it shall disperse in the plasma according to dispersion relation. However, in our numerical analysis, we simply assume the pulse is solid. Secondly, we did not consider the feed back effect of electrons to the pulse. This can only
be taken into account in a self-consistent plasma simulations.
Appendix A
Derivation of Equation (3.11)
First of all, we focus on the z component of the Lorentz force, dPz
dt = −e(Ez+1
c(vxBy − vyBx)). (A.1) From Maxwell equations, we have
Ez = − ∂Φ∂z − ∂Ac∂tz,
B = ∇ × A. (A.2)
Let us rewrite equation (A.1) dPz
where βg = vg/c and βz = vz/c.
Besides the electron equation of motion, we have Poisson equation and continuity equation
∇2Φ = −4πe(n − n0), (A.8)
∂n
∂t + ∇(nv) = 0. (A.9)
With new coordinates (ξ, τ ), we have
∂2φ
Integrate ξ on both side of equation (A.11), we have
Z ∞ equation (A.12) can be written as
Z 0 ξ
∂n
kp∂τdξ − (n(βg− βz))|∞ξ = 0. (A.13) Furthermore, the first term on the left hand side is very small if ω >> ωp. That is, if the frequency of the pulse is very large, the growth rate of density is very small. Therefore, we could drop out the first term of equation (A.12).
Hence
−(n(βg− βz))|∞ξ = 0
⇒ −n0+ n(ξ)βg− n(ξ)βz = 0. (A.14) Similarly, equation (A.6) could also be written as
γ(1 − βgβz) − φ = 1. (A.15) Combining equation (A.10), (A.14), (A.15), we finally have
∂2φ
∂ξ2 = 1
2(1 + |a|2
(1 + φ)2 − 1). (A.16)
Appendix B
Derivation of Equation (4.4)
Let us begin with the electron equation of motion mdv
dt = −e(E + (v × B)/c). (B.1)
In the x and y component form:
Recall that Bz is strong uniform magnetic field and Bx, By are just induced by the E field. Besides, we assume vzis much smaller than vx, vy here. Hence we may ignore the terms vzBy and vzBx on the right hand side of equation (B.2). Then we have
where ωc= eBz/mc is the cyclotron frequency.
Defining ¯v = vx + ivy, ¯E = Ex + iEy, we can combine the above two
For right-handed circularly polarized pulse,
E = E¯ x+ iEy = E0e−i(kz−ωt), (B.5) where E0 is the amplitude of the pulse, which is a function of z and t.
Therefore, ¯v can be solved as Using integration by part, we arrive at
¯ E0 vanishes as t goes to infinity. For the second term, we perform integration by part again because the former represent the slow variation of the amplitude while the later represent the fast oscillation. Hence we can ignore the third term on the right hand side.
Assuming ˙E0 is zero as t goes to infinity we conclude
¯
From the Maxwell equation
∇ × E = −1 In equation (4.3), we have defined
fk = −e
c (vxBy − vyBx). (B.12)
Hence, by combining equation (B.9) and (B.11), we obtain
fk = e2
mω(ω − ωc)(−E0∂E0
∂z − k
ωE0E˙0+ k ω − ωc
E0E˙0− 1 ω(ω − ωc)
E˙0∂ ˙E0
∂z ).
(B.13) Again, the fourth term on the right hand side is very small since the second derivative term ∂ ˙E0/∂z is negligible compare to (ω − ωc). Therefore, we finally have
fk = −1 2 ( ∂
∂z − kωc ω(ω − ωc)
∂
∂t) e2E02
mω(ω − ωc). (B.14)
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