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國立交通大學

物理研究所

碩 士 論 文

磁化電漿中的尾波場

Wakefield In Magnetized Plasma

研 究 生 : 張研俞 (Yen –Yu , Chang)

指導教授 : 林貴林 (Prof. Lin – Guey , Lin)

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題目:磁化電漿中的尾波場

學生:張研俞

指導教授:林貴林教授

國立交通大學物理碩士班

摘要

高能宇宙射線(UHECR)的來源,一直是天文物理所要探討的

題目之一。其中利用太空中的電漿所產生之電場(尾波場),讓帶電

粒子加速至極高能量,是可能的來源之一。

本篇論文主要呈現利用數值分析的方法,探討在均勻磁場下之電

漿,受到強場電磁波激發後,所產生之尾波場的行為。

關鍵字:電漿,尾波場,天文粒子,電漿加速

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Topic : Wakefield in Magnetized Plasma

Student : Yen – Yu, Chang Advisor : Prof. Lin – Guey, Lin

Institute of Physics

National Chiao Tung University

Abstract

The source of ultra high energy cosmic ray (UHECR) has been a

mystery in astrophysics for years. It has been proposed that the plasma

wakefield acceleration could be a possible acceleration mechanism for

UHECR.

In this thesis, we present a numerical calculation of wakefield in the

magnetized plasma, taking into account the relativistic effects.

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誌謝

首先,我要感謝一直在身後默默支持我的家人,尤其是

爸爸、媽媽;你們的關心與鼓勵,是讓我前進的最大動力。

其次,感謝林教授不厭其煩的諄諄教誨,除了學術上的

指導外,更讓我學習到作科學研究應有的觀念與態度。

最後,還要謝謝所有指導過我的師長、學長姐們、以及

身邊的同學、朋友們。在碩士班的這兩年裡,我從大家身上

學習到很多。

研俞 07.23.2008

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Contents

1 Introduction 1

2 Wakefield in Non-Magnetized Plasma - Weak Field Case 4

2.1 Introduction . . . 4

2.2 Analytical Solution . . . 4

2.3 Numerical Method . . . 9

2.4 Comparison . . . 11

3 Wakefield in Non-Magnetized Plasma - Strong Field Case 14 3.1 Introduction . . . 14

3.2 Numerical Solution . . . 14

3.3 Verifying Numerical Methods . . . 16

4 Wakefield in Magnetized Plasma 19 4.1 Introduction . . . 19

4.2 The Analytical Solutions in Weak Field Case . . . 21

4.3 Numerical Solution of Wakefield Induced by Whistler Wave . . 23

4.4 Numerical Solution of Wakefield Induced by R Wave . . . 26

5 Conclusion 31

A Derivation of Equation (3.11) 33 B Derivation of Equation (4.4) 35

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List of Figures

1.1 hello. . . 2 2.1 The plot of the pulse. We set the pulse moving toward positive

z direction with speed vg, the group velocity. Since we send

the pulse into the system at z = 0 and t = 0, the position of the front point of the pulse is always zero in the co-moving frame [12]. . . 8 2.2 This is the plot of wakefield, where Ewb = mcωp/e is the

nor-malization factor and a = eE/mcω. The x-axis represents the co-moving frame where the unit length is c/ωp. The

ampli-tude of the pulse is a0 = 0.05, the wave number is ˜k = 20 and

σ = 3c/ωp. . . 12

2.3 The comparison of wakefield obtained numerically and analyt-ically. . . 13 3.1 The plot of wakefield and normalized density in the co-moving

frame, where we set a0 = 7.5, ˜k = 40 and σ = 3c/ωp. . . 17

3.2 The plot of maximum wakefield versus pulse amplitude. The settings of the driving pulse are ˜k = 40 and σ = 3√2c/ωp. . . 18

4.1 The plot of dispersion relation of right-handed circularly polar-ized EM wave in the magnetpolar-ized plasma without considering relativistic effects. The upper branch is called R wave while the lower branch is called whistler wave. . . 20 4.2 This is the comparison of numerical solution with analytical

approximation (equation (4.20)) and PIC simulation. The x-axis is the amplitude of the pulse. Other settings of the pulse are k = πωp/c, ω ' 3ωp, σ = 3√8c

p. The setting of the

external magnetic field is ˜B0 = 12. . . 24

4.3 The plot of EzM ax for wakefield induced by whistler wave. We fix ω/ωp = 3 and ˜B0/a0 = 9, and increase both a0 and ˜B0.

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4.4 The plot of EzM ax for wakefield induced by whistler wave. We fix ω/ωp = 3 and ˜B0/a0 = 12, and increase both a0 and ˜B0.

The Gaussian width of the pulse is σ = 3c/ωp. . . 26

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 =

q

v2

x+ 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. . 27

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 Fig. 4.5. . . 28 4.7 The plot of the maximum value of γ versus a0. . . 28

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. . . 29

4.9 The plot of maximum value of wakefield versus a0 in

magne-tized plasma. The parameter settings are ω/ωp = 20, ˜B0 =

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Chapter 1

Introduction

The source of ultra high energy cosmic ray (UHECR) is still a mystery. So far, the model for the source of UHECR can be divided into two cate-gories, one is “top down”[1] scenario and the other is “bottom up”[2] scenario. From results of several recent observations like HiRes [3] and Auger [4], the “ankle”in energy spectrum of cosmic rays exhibit the Greisen-Zatepin-Kuzmin suppression [5] [6] (Fig. 1.1). Hence the “bottom up”scenario seems more favorable than the “top down”scenario. Therefore, it is desirable to construct a theory for UHECR acceleration.

From the experience of terrestrial particle acceleration, one obtains im-portant conditions for possible acceleration mechanism [8]: First, the trajec-tories of the accelerated charged particles should have no bending otherwise the effect of synchrotron radiation would reduce the energy of the particle. Second, the system should be collision free or else the energy of the acceler-ated particle would be transferred and spread out. To fulfill these conditions, plasma wakefield acceleration has been proposed as a possible mechanism for UHECR acceleration [7] [8].

When an EM wave packet injects into the plasma, the non-uniform elec-tric field of the pulse imposes a longitudinal force (ponderomotive force) to electrons. Thus an electrostatic wave with phase velocity close to the group velocity of the driving pulse is excited behind this packet. We call this ex-cited electric field as wakefield. When an electron has longitudinal velocity close to the phase velocity of this wave, it can be accelerated by this excited electric field [7] [10].

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Figure 1.1: The energy spectrum of cosmic ray.

them are not available in the astrophysical settings. In astrophysical set-tings, the “magnetowave”induced wakefield seems more possible.

The simulation of magnetowave induced wakefield has been carried out [9]. However, the behavior of wakefield induced by a strong driven pulse is still unclear. Although we can derive theoretical approximation of wakefield in the weak field case (Section 4.2), it is difficult to extend this result to the strong field limit because relativistic dynamics makes the problem much more complicated. Therefore, we directly solve differential equations govern-ing the wakefield by the numerical method.

Before jumping to the main theme of the thesis, we verify our numerical methods with well-studied cases.

In Chapter 2, we discuss the wakefield induced in non-magnetized plasma in the weak field case. We present the analytical derivation in Sec-tion 2.2 [11]. Our numerical method is presented in SecSec-tion 2.3. The comparison of analytical and numerical solution is given in Section 2.4.

In Chapter 3, we present the numerical method in Section 3.2. We then compare our result with previous works in Section 3.3 [12].

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in the weak field case, as presented in Section 4.2. We then verify this result by our numerical approach.

The main result of this thesis is presented in Section 4.3 and 4.4, which is the calculation of wakefield induced by right-handed circularly polarized pulse in arbitrary strength.

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Chapter 2

Wakefield in Non-Magnetized

Plasma - Weak Field Case

2.1

Introduction

To make sure that our numerical method is reliable, we apply this method to several well-studied cases.

In this chapter, we present the analytical expression of plasma wakefield [11] in Section 2.2 and demonstrate our numerical method in Section 2.3. The comparison of both approaches are presented in Section 2.4.

2.2

Analytical Solution

First of all, we consider the simplest case: an EM pulse with small am-plitude sent into the plasma. In this limit, the electric field of the pulse is so small that velocities of the expelled electrons are much less than the speed of light. Therefore,we do not need to consider relativistic effects here. Fur-thermore, the plasma is not subject to any external field.

Following the above conditions, we shall derive the longitudinal electric field (wakefield) analytically in this section [11], and show the numerical re-sult in the next section.

Lorentz force equation

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dP

dt = −e(E + (v × B)/c), (2.1) where P is the momentum of the electron, and E, B are electric and magnetic field respectively.

Relating EM field to the scalar and vector potentials,

   E = − 1c∂A ∂t − ∇Φ, B = ∇ × A (2.2)

we can rewrite the equation of motion as dP dt = −e(− 1 c ∂A ∂t − ∇Φ + (v × ∇ × A)/c). (2.3) If the given pulse is propagating in z direction, we may assume that the scalar and vector potentials (Φ, A) are only a function of z and t. Therefore, the right hand side of equation (2.3) is just

−e(− 1c∂Ax ∂t − vcz∂A∂zx)ˆex+ −e(− 1c∂Ay ∂t − vcz ∂Ay ∂z )ˆey+ −e(− 1c∂Az ∂t − ∂Φ∂z)ˆez, (2.4)

here we expend the vector form to three components. Focusing on x and y components, we have dp dt = −(− ∂a ∂t − vz ∂a ∂z), (2.5) where a = (ax, ay) = e(Ax, Ay)/mc2 is the normalized vector potential and

p = (px, py) = (Px, Py)/mc is the normalized momentum. We note that

d dt = ∂ ∂t+ v · ∇, (2.6) which means da dt = ∂a ∂t + v · ∇a. (2.7) Since a is function of z and t only, the above could be rewritten as

da dt = ∂a ∂t + vz ∂a ∂z. (2.8)

Therefore, equation (2.7) becomes dp

dt = da

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which implies

p = a + C0. (2.10)

At (z, t) = (0, 0), a is zero, hence p = C0. However, the momentum of the

electron should be zero since we are considering the cold plasma. Therefore, C0 = 0. Thus we have p = a.

Concerning the z-component of equation (2.3), we have dPz dt = −e(Ez + (vxBy− vyBx)/c) = −e(− ∂Φ ∂z + (vx(∂A∂zx) + vy( ∂Ay ∂z ))/c). (2.11) Since (vx, vy) = e(Ax, Ay)/mc by equation (2.10), we can rewrite the above

equation as dPz dt = −e(− ∂Φ∂z +mce2(Ax(∂A∂zx) + Ay( ∂Ay ∂z ))) = −e(− ∂Φ ∂z + mc 2 2e ( ∂|a|2 ∂z )) . (2.12) Since we are dealing with non-relativistic motion, we have pz = mvz. With

d/dt = ∂/∂t + vz∂/∂z we rewrite equation (2.12) as ∂βz ∂t + vz ∂βz ∂z = ( ∂φ ∂z − 1 2 ∂|a|2 ∂z ), (2.13) where φ = eΦ/mc2 and β

z = vz/c.

Continuity equation

Knowing how electrons are affected by the EM pulse, we further study the collective effect on plasma.

In the following derivation, we consider only two species of particles in the plasma: electrons and ions. Since ions are much heavier than electrons, we assume ions are almost motionless in comparison with electrons. Thus we ignore the dynamics of ions.

Furthermore, we assume collisions between particles can be neglected, thus there are no abrupt change for the path of moving particles. Therefore, the total number of particles in unit volume should conserve. In other words, the number density of electrons obey the continuity equation given by

dn dt =

∂n

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where n is the number density of the electrons and v is the velocity of elec-trons.

Let us denote the electron density n as n = (n0 + δn), where n0 is the

density in the equilibrium state(the density when the electrons are not per-turbed) and δn is the perturbation. By replacing n with (n0 + δn) and

ignoring the quadratic terms in perturbations, the continuity equation be-comes

∂δn

∂t + n0∇ · v = 0. (2.15) Again we only consider the case where v is a function of (z, t) ( v ≡ v(z, t) ). Therefore, the continuity equation becomes

∂δn c∂t + n0 ∂βz ∂z = 0, (2.16) where βz = vz/c. Poisson equation

The electrostatic potential in the plasma is governed by the Poisson equa-tion

∇2Φ = 4πe(n − n0). (2.17)

Again we assume Φ = Φ(z, t) and take φ = eΦ/mec2. This leads to

∂2φ ∂z2 = wp2 c2n 0 (n − n0), (2.18) where wp = q 4πe2n

0/me is the plasma frequency.

Let us now return to equation (2.13). In the current limit, one may ignore the term vz∂βz/∂z. Taking one more derivative, ∂/∂z, on both sides of this

equation, we obtain ∂2βz c∂z∂t = ( ∂2φ ∂z2 − 1 2 ∂2|a|2 ∂z2 ). (2.19)

In this equation, we can replace ∂2φ/∂z2 by w2

p(n − n0)/c2n0 from equation

(2.18), and replace ∂2β

z/∂z∂t by −∂2δn/cn0∂t2, which can be derived from

equation (2.16). We then arrive at the following equations for the plasma oscillation

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Pulse g v t = 0 g z - v

Figure 2.1: The plot of the pulse. We set the pulse moving toward positive z direction with speed vg, the group velocity. Since we send the pulse into

the system at z = 0 and t = 0, the position of the front point of the pulse is always zero in the co-moving frame [12].

∂2δn ∂t2 + w 2 pδn = n0c2 2 ∂2|a|2 ∂z2 . (2.20)

For convenience, we change the coordinate to the co-moving frame (Fig. 2.1) with the pulse, ξ = ωc (z − vp gt), where vg is the group velocity of the pulse

in the plasma. Then the above equation can be recasted into ∂2δnf ∂ξ2 +δn =f c2 2v2 g ∂2|a|2 ∂ξ2 , (2.21) where δn = δn/nf 0 .

We note that equation (2.21) appears as an equation of oscillation, and the term a on the right hand side is determined by the pulse.

Analytical solution

Solving equation (2.21), we obtain

f δn(ξ) = c 2 2v2 g (|a(ξ)|2− Z ξ ∞ |a(ξ0)|2sin(ξ − ξ0)dξ0), (2.22)

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under the boundary condition that a = 0 for ξ → ∞. Using

∇ · E = −4πeδn, (2.23) and assuming that the electric field E is only a function of (z, t), we obtain

∂Ez

∂z = −4πeδn. (2.24) Changing the coordinate and normalizing all the quantities, we have

∂Efz

∂ξ = −δn,f (2.25) where Efz = eEz/mep . Finally, by combining equation (2.22) and (2.25),

we obtain the wakefield Efz: f Ez = − R f δndξ0 = − c2 2vg2 R (|a(ξ0)|2Rξ0 ∞|a(ξ00)|2sin(ξ0− ξ00)dξ00)dξ0. (2.26)

2.3

Numerical Method

Linearly polarized pulse

For convenience, we assume the pulse is linearly polarized in ˆx direction and propagating toward positive ˆz direction. First of all, the equations of motion for electrons are

           mdvx dt = −e(Ex+ 1c (vyBz − vzBy)), mdvy dt = −e(Ey+ 1c (vzBx− vxBz)), mdvz dt = −e(Ez+ 1c (vxBy − vyBx)). (2.27)

Since, Ey, Bx and Bz vanish here, the second equation is trivial and we need

only consider the first and third equations. Setting the pulse as E(z, t) = E0cos(kz − ωt)ˆex, we have mdvx dt = −eE0(1 − vz c )(cos(kz − ωt)), (2.28) mdvz dt = −e(Ez+ E0( vx c cos(kz − ωt)). (2.29)

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Again, we need to consider the collective effect of plasma as in the last section. Therefore, we apply the continuity equation as before (equation (2.16))

∂δn c∂t + n0

∂βz

∂z = 0. (2.30) Also, we apply the Gauss law for the electric field

∂Ez

∂z = −4πe(n − n0). (2.31) Applying partial derivative on t to equation (2.31) and combining with equa-tion (2.30), we obtain ∂2E z ∂t∂z = −4πen0 ∂vz ∂z . (2.32) Now we have three differential equations (2.28), (2.29), (2.32) and three un-known quantities vx, vz, Ez, where Ez is the wakefield to be determined.

Focusing on equations (2.28) and (2.29), we rewrite the time derivative d/dt = ∂/∂t + vz∂/∂z on the left hand side

mdvx dt = m( ∂vx ∂t + vz∂v∂zx), mdvz dt = m( ∂vz ∂t + vz∂v∂zz). (2.33) Changing the coordinate to the co-moving frame of the pulse ξ = ωc (z − vp gt)

as before, equations (2.28) and (2.29) are recasted into

m(−ωp∂v∂ξx + kpvz∂v∂ξx) = −eE0(1 − βz)(cos(kξ)),e

m(−ωp∂v∂ξz + kpvz∂v∂ξz) = −e(Ez+ E0βxcos(kξ)),e

(2.34) where βx, βz and k equal to ve x/c, vz/c and kc/ωp respectively. Taking E =e

eE/mecωp as the normalized electric field, we finally obtain

−vg∂βx

c∂ξ + βz∂β∂ξx = −Ee0(1 − βz) cos(ekξ),

−vg∂βz

c∂ξ + βz∂β∂ξZ = −Eez−Ee0βx(cos(kξ)).e

(2.35) With the new variables, equation (2.32) is converted into

∂2E z

∂ξ2 = −

vg∂βz

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or

∂Ez

∂ξ = −βgβz, (2.37) where βg = vg/c. Combining equation (2.35) and (2.37), we are now ready

to solve the plasma wakefield numerically. Boundary conditions

Since we assume the plasma is cold, the wakefield and the velocities of electrons are zero for ξ → ∞. Thus the boundary conditions for various quantities are: ~β = 0 and ˜Ez = 0 for ξ → ∞.

2.4

Comparison

In this section, we compare results of Sections (2.2) and (2.3). Gaussian pulse

Originally, the input pulse is E(z, t) = ˆxEM

Z ∞

−∞e

−(k−k0)2/2µ2ei(kz−ω(k)t)dk, (2.38)

where EM is the maximum amplitude of the pulse, and k0 is the average

wave number of the pulse and ω(k) is the frequency of the pulse given by the dispersion relation

ω2 = k2c2+ ω2p. (2.39) For k2c2 >> ω2

p, the phase velocity and group velocity are approximately

equal to c. That is,

ω k '

dk = vg ' c. (2.40) Therefore, we have kz − ω(k)t = k(z − vgt) = ˜kξ. Thus equation (2.38)

becomes

E(ξ) = ˆxEM

Z ∞

−∞e

−(k−k0)2/2µ2ei˜kξdk. (2.41)

In our calculation, we do not integrate the Fourier transform but simply set the Gaussian pulse as

E(ξ) = ˆxEMe−(ξ−ξ0)

2/2σ2

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t g z - v -30 -25 -20 -15 -10 -5 0 wb /Ez E -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -3 10 × 0 0.05 0.1 -0.05 -0.1 a Pulse Wakefield p ω c/

Figure 2.2: This is the plot of wakefield, where Ewb = mcωp/e is the

normal-ization factor and a = eE/mcω. The x-axis represents the co-moving frame where the unit length is c/ωp. The amplitude of the pulse is a0 = 0.05, the

wave number is ˜k = 20 and σ = 3c/ωp.

Result

Here we introduce two important parameters. One is called the “strength parameter”, defined as

a0 =

eEM

mcω (2.43)

is the normalized factor for the pulse. The other parameter is called the “cold wavebreaking field”[7], defined as

Ewb =

eE mcωp

(2.44) is the normalized factor for the wakefiled. The reason we use this normaliza-tion is that the maximum amplitude which the plasma wakefield can support is EM ax ' a0Ewb [8].

Setting a0 = 0.05, ˜k = ck/ωp = 20 and σ = 3c/ωp , we present the

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0 a 0.01 0.02 0.03 0.04 0.05 wb /E Max z E 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -3 10 × Blue Numerical Equation (2.26)

Figure 2.3: The comparison of wakefield obtained numerically and analyti-cally.

In Fig. 2.3, we varying the maximum amplitude of the pulse and taking the maximum value of wakefield. One can see the wakefield obtained numer-ically agrees very well with the analytical result. From equation (2.26), one can see that EM ax

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Chapter 3

Wakefield in Non-Magnetized

Plasma - Strong Field Case

3.1

Introduction

After checking how the numerical methods work in the weak field case, we consider strong field case. When the amplitude of the pulse becomes stronger, the speeds of driven electrons would be close to the speed of light. Thus the relativistic effect (P = γmv) shall be taken into consideration.

In Section 3.2, we demonstrate how we derive and solve the system of differential equations which are the equations of motion for electrons, the con-tinuity equation and the Poisson equation. Since wakefield in non-magnetized plasma with relativistic effects have been well studied [12], we use these re-sults to check the numerical method. One can find the comparison in Section 3.3.

3.2

Numerical Solution

By increasing the amplitude of the pulse, the speed of the driven electron also increase. When the speed of electron is close to c, the speed of light, one has to consider the relativistic dynamics for solving the equation of motion for the electron.

γ factor

To derive the Lorentz force equation, we start from dP/dt. Here the momentum P is no longer mv but γmv with γ = 1/q1 − |v|2/c2. Therefore,

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the Lorentz force equation should be modified as dP dt = mvdγ dt + γmdv dt = −e(E + (v × B)/c), (3.1) since γ is also a function of time.

Furthermore, since the kinetic energy of the electrons is given by

Ek = (γ − 1)mc2, (3.2) we have dEk dt = dγ dtmc 2. (3.3)

The left hand side of this equation is just the increasing rate of the electron kinetic energy, which is obviously −ev · E. Therefore, we have

dγ dt =

−ev · E

mc2 . (3.4)

Rewriting dγ/dt in equation (3.1), we arrive at −e(v · E)mv

mc2 +

γmdv

dt = −e(E + (v × B)/c). (3.5) Changing to the co-moving frame variable ξ and normalizing all quantities as before, we have              −(E · ~e β)βx+ γ(−βg∂βx ∂ξ + βz∂β∂ξx) = −(Efx+ βyBfz − βzBfy), −(E · ~e β)βy + γ(−βg ∂βy ∂ξ + βz ∂βy ∂ξ ) = −(Efy+ βzBfx− βxBfz), −(E · ~e β)βz + γ(−βg∂βz ∂ξ + βz∂β∂ξz) = −(Efz+ βxBfy− βyBfx), (3.6) where ~β = v/c and βg = vg/c.

The continuity equation (2.14) ∂n ∂t + ∇ · (nv) = 0 (3.7) can be written as −βg ∂ne ∂ξ + ∂(nβe z) ∂ξ = 0, (3.8)

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where n = n/ne 0.

Finally the Poisson equation can be written as −∂Efz ∂z = ωp c (n − 1),e (3.9) or −∂Efz ∂ξ = (n − 1).e (3.10)

Combining equations (3.6), (3.8) and (3.10), we have five equations and five unknown quantities βx, βy, βz,n ande Efz. We can solve the system of

differ-ential equations to find Ez.

Boundary conditions

Again we take the wakefield and velocities of electrons as zero for ξ → ∞. Furthermore, we have ˜n = 1 for ξ → ∞.

3.3

Verifying Numerical Methods

The solution of wakefield including relativistic effect is well-studied. Here we refer to the result by Sprangle, Esarey and Ting [12] and com-pare our calculation with their results . For simplicity, we just show their equations and present the solution. One can find the derivation of equation in Appendix A. The equation they derived (see Appendix A) is

∂2φ ∂ξ2 = 1 2( 1 + |a|2 (1 + φ)2 − 1), (3.11)

where |a|2 and φ is defined in equations (2.5) and (2.13) respectively.

The term a is determined by the given pulse, and the wakefield to be determined is Efz = −∂φ/∂ξ. Let Efz have a maximum value at ξ0, then

∂Efz

∂ξ |ξ=ξ0 = −

∂2φ

∂ξ2|ξ=ξ0 = 0, (3.12)

which implies that

1 + |a|2

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t g z - v -50 -40 -30 -20 -10 0 wb /Ez E -3 -2 -1 0 1 2 3 z E 0 10 -10 a p ω c/ Pulse Wakefield t g z - v -50 -40 -30 -20 -10 0 0 n/n 5 10 15 20 25 30 35 40 n p ω c/

Figure 3.1: The plot of wakefield and normalized density in the co-moving frame, where we set a0 = 7.5, ˜k = 40 and σ = 3c/ωp.

Hence [15] f Ez M ax ∝ a 2 0 q a2 0+ 1 (3.14) which implies Efz M ax is proportional to a0 for a0 >> 1. Result

Setting a0 = 7.5, ˜k = 40 and σ = 3c/ωp, one can see the wakefield in

Fig. 3.1. When the pulse amplitude increases, the ponderomotive force acts on the driven electrons becomes larger. Thus the longitudinal momentum of the driven electrons become larger, too. Such large longitudinal momen-tum allows electrons to “squeeze”in small region while oscillating (Fig. 3.1). Therefore, the electric field would goes down sharply during the crowds of electrons. Hence the behavior of the wakefield is no longer sinusoidal but saw-tooth-like.

In Fig. 3.2, we compare the results from two different methods. One can see that the numerical solution agrees well with the solution of equation

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0 a 1 2 3 4 5 6 7 wb /E Max z E 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Blue Numerical Equation (3.11)

Figure 3.2: The plot of maximum wakefield versus pulse amplitude. The settings of the driving pulse are ˜k = 40 and σ = 3√2c/ωp.

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Chapter 4

Wakefield in Magnetized

Plasma

4.1

Introduction

In this chapter, we will present our main result : the wakefield induced by right-handed circularly polarized pulse in the magnetized plasma. The strength of the pulse will be taken as arbitrary.

In the magnetized plasma, there are four modes of EM waves. Here we focus on the right-handed circularly polarized wave, which is described by the dispersion relation

k2c2− ω2+ ωω 2 p

ω − ωc

= 0 (4.1)

in the non-relativistic limit where ωc = eB/mc is the cyclotron frequency.

There are two solutions to this dispersion equation (Fig. 4.1), one is called R wave (the upper branch) and the other is called whistler wave ( the lower branch).

Since the ponderomotive force in the magnetized plasma in the non-relativistic limit has been well studied [13], we can use this result to de-rive analytical approximation of wakefield in this case. Furthermore, we can use this approximation to check the numerical method in the non-relativistic limit. The derivation of the analytical approximation is given in Section 4.2. In Section 4.3, we analyze the wakefield induced by the whistler pulse. We compare numerical solutions with analytical approximation (Fig. 4.2).

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p ω kc/ 0 5 10 15 20 p ω / ω 0 2 4 6 8 10 12 14 16 18 20 = 12 p ω / c ω

R wave

Whistler wave

Figure 4.1: The plot of dispersion relation of right-handed circularly polarized EM wave in the magnetized plasma without considering relativistic effects. The upper branch is called R wave while the lower branch is called whistler wave.

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Since there exist simulation results (Particle In Cell simulation) of wake-field in the magnetized plasma [9], we also compare our numerical solutions with the simulation results. Furthermore, we will consider the case of strong whistler pulse and calculate the wakefield.

In Section 4.4, we take the driving pulse as the R wave with arbitrary amplitude and analyze γ factors of the driven electrons and the wakefield .

4.2

The Analytical Solutions in Weak Field

Case

In chapters 2 and 3, we dealt with unmagnetized plasma. However, in many physical systems, the external magnetic field should be taken into consideration. Here we focus on the particular situation that a circularly po-larized EM pulse is propagating along the external magnetic field direction. For convenience, this direction is taken to be the z-axis.

Derivation of wakefield

To derive the wakefield, let us first rewrite the z component of Lorentz force,

dPz

dt = −e(Ez+ 1

c(vxBy − vyBx)). (4.2) In the non-relativistic limit, we can approximate the last term on the right hand side by fk (see Appendix B). That is, we have

−e c (vxBy− vyBx) = fk, (4.3) where fk = −1 2 ( ∂ ∂z − kωc ω(ω − ωc) ∂ ∂t) e2E02 mω(ω − ωc) , (4.4) where k and ω are the wave number and angular frequency of the pulse respectively, ωc = eB0/mec is the cyclotron frequency of the electron, and

E0 is the electric field amplitude of the pulse. Thus equation (4.4) can be

written as dPz dt = −eEz+ fk. (4.5) Noting that dPz dt = ∂Pz ∂t + vz ∂Pz ∂z , (4.6)

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and assuming that the quadratic term (the last term in the right hand side) is negligible in the non-relativistic limit, equation (4.5) can be written as

m∂vz

∂t = −eEz+ fk, (4.7) where we just replace Pz by mvz.

Equations (2.16) and (2.18) are still valid in this case. ∂n ∂t + n0 ∂vz ∂z = 0, (4.8) ∂2Φ ∂z2 = 4πe(n − n0). (4.9)

Since ∂Φ/∂z = −Ez, equation (4.9) becomes

−∂Ez

∂z = 4πe(n − n0). (4.10) Taking ∂/∂t on both sides of equation (4.10), we have

∂2n

∂t2 + n0

∂2v z

∂t∂z = 0. (4.11) Also, by taking ∂/∂z on both sides of equation (4.9), we arrive at

m∂ 2v z ∂z∂t = −e ∂Ez ∂z + ∂fk ∂z . (4.12) Combining equations (4.13) and (4.14), we acquire

−m n0 ∂2n ∂t2 = −e ∂Ez ∂z + ∂fk ∂z . (4.13) Furthermore, we can operate ∂2/∂t2 on both sides of equation (4.10), which

gives −∂ 3E z ∂t2∂z = 4πe ∂2n ∂t2. (4.14)

Finally, combining equations (4.13) and (4.14), we obtain m 4πen0 ∂3Ez ∂t2∂z = −e ∂Ez ∂z + ∂fk ∂z , (4.15) which is the differential equation for Ez.

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Solution

Changing the variables (z, t) to ξ = ωp

c(z − vgt), equation (4.15) becomes ( ∂2 ∂ξ2 + 1) ∂Ez ∂ξ = 1e ∂fk ∂ξ = − 12 ∂ ∂ξ( ωp c ∂ξ∂ + kωcωp ω(ω − ωc) ∂ ∂ξ) eE02 mω(ω − ωc) = − 12c +p kωcωp ω(ω − ωc) ) e mω(ω − ωc) ∂2E02 ∂ξ2 , (4.16)

where fk is given by equation (4.6). Finally, Ez is solved as

e Ez(ξ) = −1 2 ( c vg + ω˜c ˜ ω(˜ω − ˜ωc) ) 1 ˜ ω(˜ω − ˜ωc) Z ∞ ξ dξ0Ee2 0(ξ 0 ) cos(ξ − ξ0), (4.17) where E = eE/mcωe p, ˜ω = ω/ωp and ˜ωc = ωcp. We can further simplify

the solution to e Ez(ξ) = χ(ξ)( c vg + ω˜c ˜ ω(˜ω − ˜ωc) ) 1 ˜ ω(˜ω − ˜ωc) e EM2 , (4.18) where EM is the maximum amplitude of the pulse, and

χ(ξ) = −1 2 1 e E2 M Z ∞ ξ dξ0Ee2(ξ0) cos(ξ − ξ0). (4.19)

4.3

Numerical Solution of Wakefield Induced

by Whistler Wave

In the following sections, we do not restrict ourselves in the small field limit. It is obvious that as the amplitude of the pulse increases, the motions of the affected electrons might be relativistic. Hence the solution in the last section might not be correct if the amplitude of the pulse is large.

We will show the numerical results by solving differential equations de-rived in chapter 3 (equation (3.6), (3.8), (3.10)).

               −(E · ~e β)~β + γ(−βg∂ ~β ∂ξ + βz∂ ~∂ξβ) = −(E + ~e β ×B),e −βg∂∂ξne + ∂(∂ξnβe z) = 0, − ∂Efz ∂ξ = βg2(n − 1).e (4.20)

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0 a 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 wb /E Max z 30 E 0 0.5 1 1.5 2 2.5 Analytical Approximation Numerical PIC Simulation

Figure 4.2: This is the comparison of numerical solution with analytical approximation (equation (4.20)) and PIC simulation. The x-axis is the am-plitude of the pulse. Other settings of the pulse are k = πωp/c, ω ' 3ωp,

σ = 3√8c

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0 a 10 Log 0 0.5 1 1.5 2 2.5 3 3.5 4 wb /E Max z E 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 = 9 0 /a 0 B~ = 3 p ω / ω

Figure 4.3: The plot of EzM ax for wakefield induced by whistler wave. We fix ω/ωp = 3 and ˜B0/a0 = 9, and increase both a0 and ˜B0. The Gaussian width

of the pulse is σ = 3c/ωp.

We note that these differential equations are derived without any approxi-mation. They are also applicable to the current case.

Comparison with analytical approximation and PIC simulation We set the pulse as E = E0e−i(kz−ωt), where E0(z, t) is taken to be a

Gaussian shape. Here the driving pulse is whistler pulse of which the fre-quency is smaller than the cyclotron frefre-quency. Other settings of the pulse are kc/ωp = π, and the Gaussian width of the pulse σ = 3√8c

p. We denote the

external magnetic field as B0, which can be normalized as ˜B0 = eB0/mcωp.

In Fig. 4.2, we show the comparison of wakefield obtained by different methods. One can see that the numerical solution agrees well with theoreti-cal approximation (equation (4.20)) and “Particle In Cell”(PIC) simulation in the weak field limit. As the amplitude gradually increases, the analytical approximation fails. In other words, the relativistic effect becomes non-negligible.

Having checked the numerical result in the weak field limit, we like to study the behavior of wakefield in the strong field limit. It is to be noted that, if we simply fix the frequency of the pulse and the external magnetic field while verifying the amplitude a0 of the pulse, the driving pulse would

not always stay as the whistler wave. The wave could become R wave when a0 >> 1.

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0 a 10 Log 0 0.5 1 1.5 2 2.5 3 3.5 4 wb /E Max z E 0.02 0.025 0.03 0.035 0.04 0.045 0.05 = 12 0 /a 0 B~ = 3 p ω / ω

Figure 4.4: The plot of EzM ax for wakefield induced by whistler wave. We fix ω/ωp = 3 and ˜B0/a0 = 12, and increase both a0 and ˜B0. The Gaussian

width of the pulse is σ = 3c/ωp.

To ensure that the driving pulse is a whistler wave, we fix the ratio of the external magnetic field to the amplitude of the pulse while allowing both of them to grow. In Fig. 4.3, we fix ω = 3ωp and ˜B0/a0 = 9 while in Fig.

4.4 we fix ω = 3ωp and ˜B0/a0 = 12. Under this settings, we find that the

maximum value of the wakefield approaches to certain value as a0 >> 1.

We note that the asymptotic value becomes smaller when the ratio of the external magnetic field to the amplitude of the pulse becomes larger.

4.4

Numerical Solution of Wakefield Induced

by R Wave

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.

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t g z - v -30 -25 -20 -15 -10 -5 0 /cr v 0.1 0.2 0.3 0.4 0.5 0.6 0.7 /c r v 0 0.2 0.4 0.6 -0.2 -0.4 a p ω c/ Pulse t g z - v -30 -25 -20 -15 -10 -5 0 /cz v -0.4 -0.2 0 0.2 0.4 /c z v p ω c/

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 =

q

v2

x+ vy2. We

set ˜B0 = 12. The settings of the driven pulse are a0 = 0.6 and ω/ωp = 20,

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t g z - v -30 -25 -20 -15 -10 -5 0 γ 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 0 0.2 0.4 0.6 -0.2 -0.4 -0.6 a p ω c/ Pulse Max γ

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

Fig. 4.5. 0 a 0.5 1 1.5 2 2.5 3 3.5 4 Max γ 1 1.5 2 2.5 3 3.5 4 4.5 5

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t g z - v -50 -40 -30 -20 -10 0 wb /Ez E -3 -2 -1 0 1 2 3 z E 0 2 4 -2 -4 a p ω c/ Pulse Wakefield t g z - v -50 -40 -30 -20 -10 0 0 n/n 5 10 15 20 25 30 35 n p ω c/

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 =

q

v2

x+ 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,

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0 a 50 100 150 200 250 300 350 400 wb /E Max z E 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 EM ax

z 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 EM ax

z reduces with a0 when

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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 EM ax

z 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 EM ax

z 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 EM ax

z 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

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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 dt = −e(− ∂Φ ∂z + − ∂Az c∂t 1 c(vx ∂Ax ∂z + vy ∂Ay ∂z )). (A.3) With d/dt = ∂/∂t + vz∂/∂z, we change the left hand side of equation (A.2)

such that ∂Pz ∂t + vz ∂Pz ∂z = −e(− ∂Φ ∂z + − ∂Az c∂t 1 c(vx ∂Ax ∂z + vy ∂Ay ∂z )). (A.4) Using the relation between (Ax, Ay) and (Px, Py) from equation (2.10),

we have ∂Pz ∂t + vz ∂Pz ∂z = −e(− ∂Φ ∂z + e∂ 2γmc2∂z|A| 2), (A.5)

where γ = 1/q1 − |v|2/c2. Changing coordinates

ξ = z − vgt,

τ = t. (A.6)

Equation (A.5) becomes ∂ ∂ξ[γ(1 − βgβz) − φ] = − 1 c ∂ ∂τ(γβz), (A.7)

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where βg = vg/c and βz = vz/c.

Besides the electron equation of motion, we have Poisson equation and continuity equation

∇2Φ = −4πe(n − n

0), (A.8)

∂n

∂t + ∇(nv) = 0. (A.9) With new coordinates (ξ, τ ), we have

∂2φ ∂ξ2 = −β 2 g( n n0 − 1), (A.10) ∂n kp∂τ − ∂ ∂ξ(n(βg− βz)) = 0, (A.11) where kp = ωp/vg. Quasistatic approximation

Integrate ξ on both side of equation (A.11), we have

Z ∞ ξ ∂n kp∂τ − ∂ ∂ξ0(n(βg− βz))dξ 0 = 0. (A.12)

However, when ξ > 0 (see Fig 2.1 ), βzis zero and n is equal to n0. Therefore,

equation (A.12) can be written as

Z 0

ξ

∂n kp∂τ

dξ − (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)

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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:

     mdvx dt = −e(Ex+ (vyBz− vzBy)/c), mdvy dt = −e(Ey+ (vzBx− vxBz)/c). (B.2) 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      dvx dt = − em Ex− vyωc, dvy dt = − em Ey+ vxωc. (B.3) where ωc= eBz/mc is the cyclotron frequency.

Defining ¯v = vx + ivy, ¯E = Ex + iEy, we can combine the above two

equations d¯v dt = − e m ¯ E + iωcv.¯ (B.4)

For right-handed circularly polarized pulse, ¯

E = Ex+ iEy = E0e−i(kz−ωt), (B.5)

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Therefore, ¯v can be solved as ¯ v(z, t0)e−iωct0 = − e m Z ∞ t0 dt ¯Ee−iωct= −e m Z ∞ t0 dtE0e−ikz+i(ω−ωc)t. (B.6)

Using integration by part, we arrive at ¯ v(z, t0)e−iωct0 = − e m 1 i(ω − ωc) [E0e−ikz+i(ω−ωc)t|∞t0 − Z ∞ t0 dt ˙E0e−ikz+i(ω−ωc)t], (B.7) where ˙E0 = dE0/dt . For the first term on the right hand side, the value of

E0 vanishes as t goes to infinity. For the second term, we perform integration

by part again ¯

v(z, t0)e−iωct0 = −me i(ω−ω1 c)[E0e−ikz+i(ω−ωc)t0 −i(ω−ω1 c)[ ˙E0e−ikz+i(ω−ωc)t|∞t0

−R∞

t0 dt ¨E0e

−ikz+i(ω−ωc)t]],

(B.8) where ¨E0 = d2E0/dt2. Since E0(z, t) represents the amplitude of the pulse,

the second derivative term ¨E0 must be much smaller than (ω − ωc). This is

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

¯ v(z, t) = −e m 1 i(ω − ωc) (E0e−ikz+iωt− 1 i(ω − ωc) ˙ E0e−ikz+iωt). (B.9)

From the Maxwell equation

∇ × E = −1 c ∂B ∂t, (B.10) we have ¯ B = Bx+iBy = − c ω( ∂E0 ∂z e −i(kz−ωt)−ikE 0e−i(kz−ωt)+i 1 ω ∂ ˙E0 ∂z e −i(kz−ωt) +k ωE˙0e −i(kz−ωt) ). (B.11)

In equation (4.3), we have defined fk =

−e

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Hence, by combining equation (B.9) and (B.11), we obtain fk = e2 mω(ω − ωc) (−E0 ∂E0 ∂z − k ωE0 ˙ E0+ k ω − ωc E0E˙0− 1 ω(ω − ωc) ˙ E0 ∂ ˙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) e2E2 0 mω(ω − ωc) . (B.14)

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Bibliography

[1] Pierre- Alain Duc, et al. arXiv: astro- ph/ 0408524v1. [2] E.Fermi, Phys. Rev. 75, 1169 (1949)

[3] S. C. Corbato et al., Nucl. Phys. B(Proc. Suppl.) 28B, 36 (1992) [4] R. Knapik et al. (Auger Coll.), arXiv: 0708.1924; presented at 30th Int.

Cosmic Ray Conf. (ICRC), 2007.

[5] K. Greisen, Phys. Rev. Lett. 16 (1966) 748

[6] G. T. Zatepin and V. A. Kuz’min, Sov. Phys. JETP. Lett. 4 (1966) 78 [7] T.Tajima, J.M.Dawson, Phys. Rev. Lett. 43 267 (1979)

[8] P.Chen, T.Tajima and Y.Takahashi, Phys. Rev. Lett. 89 161101 (2002) [9] F-Y.Chang et al.,arXiv: 0709.1177

[10] E.Esarey, et al.,IEEE Trans. Plasma Sci: 24 (1996)

[11] L.M.Gorbunov and V.I.Kirsanov, Zh Eksp. Teor. Fiz. 93, 509 (1987) [Sov. Phys. - JETP 66, 290 (1987)]

[12] P.Sprangle, E.Esarey, and A.Ting, ”Nonlinear theory of intense laser pulses in plasma”, Phys. Rev. A, vol. 41, pp. 4463-4469 ,1990.

[13] H.Washimi and V.I.Karpman, JETP 71, 1010 (1976)

[14] M. Marklund, P. K. Shukla, L. Stenflo, G. Brodin and M. Servin. Plasma Phys. Control. Fusion 47 (2005) L25-L29

[15] V. I. Berezhiani and I. G. Murusidze. Physica Scripta. Vol. 45, 87-90, 1992.

數據

Figure 1.1: The energy spectrum of cosmic ray.
Figure 2.1: The plot of the pulse. We set the pulse moving toward positive z direction with speed v g , the group velocity
Figure 2.2: This is the plot of wakefield, where E wb = mcω p /e is the normal- normal-ization factor and a = eE/mcω
Figure 2.3: The comparison of wakefield obtained numerically and analyti- analyti-cally.
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