以磁性波電漿尾隨場加速為極高能宇宙射線之產生機制

國

立

交

通

大

學

物理研究所

博

士

論

文

以磁性波電漿尾隨場加速為極高能宇宙射線之

產生機制

Magnetowave Induced Plasma Wakefield Acceleration as

a Production Mechanism for Ultrahigh Energy Cosmic Rays

研 究 生：張鳳吟

指導教授：林貴林 教授

陳丕燊 教授

以磁性波電漿尾隨場加速為極高能宇宙射線之產生機制

Magnetowave Induced Plasma Wakefield Acceleration as a

Production Mechanism for Ultrahigh Energy Cosmic Rays

研 究 生：張鳳吟 Student：Feng-Yin Chang

指導教授：林貴林 Advisor：Guey-Lin Lin

陳丕燊 Pisin Chen

國 立 交 通 大 學

物 理 研 究 所

博 士 論 文

A Thesis

Submitted to Institute of Physics

College of Science

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

in

Physics

July 2009

Hsinchu, Taiwan, Republic of China

以磁性波電漿尾隨場加速為極高能宇宙射線之產生機制

學生：張鳳吟

指導教授

：

林貴林

陳丕燊

國立交通大學物理研究所

摘

要

近幾十年來極高能宇宙射線的來源一直是個未解的謎團。目前一些現

有的機制，如震波擴散加速與其他，在解釋這些粒子上仍存在問題。

所以根據電漿加速器的概念，我們提出了新的機制—磁性波電漿尾隨

場加速—來解釋極高能宇宙射線的產生。這篇論文中，我們以哨波作

為電漿尾隨場的驅動脈衝來建立磁性波電漿尾隨場加速的理論及模

擬，電漿模擬的結果證實磁性波電漿尾隨場的存在，同時它的場強符

合我們新推導的相對論性理論預測；在適當的條件下，我們也證明磁

性波電漿尾隨場經過幾百個電漿肌膚深度後仍可以維持高度同調性及

高度加速梯度，這樣的特性使磁性波電漿尾隨場可以應用在加速器實

驗。在天文環境中，粒子與尾隨場的隨機交互作用讓加速粒子能譜遵

守指數律，我們最後說明活躍星系核加速粒子到 10

電子伏特的可能

性。

Magneowave Induced Plasma Wakefield Acceleration

as a Production Mechanism for Ultrahigh Energy Cosmic Rays

student：Feng-Yin Chang

Advisors：Dr. Guey-Lin Lin

Dr. Pisin Chen

Institute of Physics

National Chiao Tung University

ABSTRACT

The origin of ultrahigh energy cosmic rays has been puzzled over several

decades. So far, the existing mechanisms, such as diffusive shock

acceleration (DSA) and others, still present problems in explaining these

particles. Based on the concept of plasma wakefield accelerator, we

proposed a novel mechanism, the so-called magnetowave induced plasma

wakefield acceleration (MPWA) to elucidate the production of ultrahigh

energy cosmic rays. In this thesis we establish the general MPWA theory

and perform a particle-in-cell simulation that provides the evidence of the

generation of magnetowave induced plasma wakefield. Here we invoke

the high frequency and high speed whistler mode for the driving pulse.

The plasma wakefield obtained in the simulation compares favorably with

our newly developed relativistic theory of MPWA. We show that under

appropriate conditions, the plasma wakefield maintains very high

coherence and sustains high-gradient acceleration over hundreds of

plasma skin depths. In astrophysical setting, the power-law spectrum and

accelerating gradient are given in the theory. Invoking AGNs as the

acceleration site, we will show that the particle accelerated to 10

eV is

possible.

誌

謝

在交大 12 年的光陰碰過數不清的人事物，一切因緣成就我的慧命與學

識經歷，為此，我有無盡的感恩。林貴林老師及陳丕燊老師是我求學過程

中最重要也最感恩的兩位貴人，透過兩位老師之間的合作，我才有機會能

到史丹福線性加速器中心訪問、學習，浸潤當地的學術研究氣息。兩位老

師對缺乏自信的我，充滿無私的包容與鼓勵，讓我可以一步步踏實做好自

己的研究，我從他們身上也學習到很多學問與人生經驗，沒有他們就沒有

現在的我。

另外，我要感謝我的好夥伴建文，多難得有這樣一個人可以和我從博

一工作到現在，不管在美國或台灣都能有你的幫忙和陪伴，還有我人生中

最重要的伴侶: 邦昱，謝謝你七年來給我的照顧，你讓我對未來有了夢

想，我在等你為我戴上戒指喔。心中還有好多好多感謝，感謝 Alberto,

Kevin, Bob, Johnny 和 Rick 在研究上給我的幫助和指導;感謝待我如家

人一般的房東夫婦 Jesse, Pat 和他們的女兒 Chris; 感恩上人的悲心，

慈濟人的用心，讓我的生命有了深度和廣度、增長慧命;感謝台大交大的

學弟妹們，志清、宗哲、佳均、貝禎、家瑜、尚佑…，謝謝你們的熱情，

讓我的生活充滿樂趣; 感謝我的主治醫師呂聆音守護我的健康，讓我在課

業上無後顧之憂。

最後僅以這本論文，獻給我最愛的父母，謝謝您們對我長期的支持，

我是這麼讓您們操心，終於我可以很驕傲地說 ”爸、媽，您們的辛勞沒有

白費了。”

Contents

中文摘要

英文摘要

致謝

Contents

List of Figures

1 Introduction 1

1.1 The Origin of Ultrahigh Energy Cosmic Rays

1

1.2 Conventional Model

3

1.2.1 Diffusive Shock Acceleration

3

1.2.2 Limitations

5

1.3 Plasma Wakefield Acceleration as a Possible Mechanism

6

2 Basic Concept of Plasma 8

2.1 What is Plasma

8

2.2 Definition of Plasma

9

2.3 Dynamics of Plasma

11

2.3.1 Fluid Description

11

2.3.2 Kinetic Description

12

2.4 Waves in the Plasma

14

2.4.1 Plasma Oscillation

14

2.4.2 Electromagnetic Wave in Plasma

16

2.5 Plasma Wakefield Acceleration

18

3 Plasma Wakefield in Magnetized Plamsa I 20

3.1 General Formulation

22

3.2 Laser Wakefield Acceleration

23

3.2.1 Linear Regime

23

4 Plasma Wakefield Acceleration in Magnetized Plasma II 31

4.1 MPWA Condition

31

4.2 Linear Theory

32

4.2.1 Ponderomotive Force

32

4.2.2 Linear Formulation

34

4.3 Nonlinear Theory

36

4.3.1 MPWA Condition in Relativistic Regime

36

4.3.2 Nonlinear Formulation

36

4.3.3 Numerical Results

38

4.4 Limitation of MPWA

39

4.4.1 Three Cases

40

4.4.2 Maximum of MPWA

44

5 Particle in Cell Simulation 47

5.1 Introduction

47

5.2 The ”em1da” Code

48

5.2.1 Simulation Unit

48

5.2.2 Charge and Current Densities

50

5.2.3 Field Update

51

5.2.4 Particle Update

53

5.2.5 Computation Cycle

54

5.3 The MPWA Simulation

55

5.3.1 Initialization

55

5.3.2 Results

57

5.4 Summary

64

6 Applications to UHECR 65

6.1 Power-Law Spectrum

65

6.2 Possible Sources for UHECRs

68

6.3 Application to AGNs

70

7 Conclusions

75

List of Figures

1.1 The cosmic ray spectrum. . . 2 3.1 The full solutions of Eq. (3.1) with ωc/ωp = 2. The two curves

(R and L waves) above the light curve (dashed line) would have

vph > c and the two curves (whistler wave and ion cyclotron wave) below the light curve would havevph< c. . . . 21 3.2 Density variationδn = n−n₀(dashed curve) and the axial electric

fieldEznormalized byE0(solid curve). The Gray shaded region

is the Gaussian pulse, a = a₀exp[−(ζ + 5)2/22] with a₀= 1.5. . . 29 3.3 Density variationδn = n−n₀(dashed curve) and the axial electric

field Ez normalized by E0 (solid curve) for Gaussian pulse a =

a0exp[−(ζ + 5)2/22] witha0= 0.1. . . . 30

4.1 (a) Frequency and (b) phase velocity versus wavenumber for dif-ferent magnetic field strengths. Whenωc/ωp 1, the dispersion relation is approximately linear over a wider range of wavenum-bers with phase velocity approaching the speed of light. . . 33 4.2 Density variation δn/n and axial field Ez for whistler gaussian

pulse located at ζ = −5(c/ωp) anda0= 1. . . 39

4.3 Density variation δn/n and axial field Ez for whistler gaussian pulse located at ζ = −5(c/ωp) anda0= 4. . . 40

4.4 The plots of (a) |Ez| and (b) |1/Ez| versus ϕ with b = 0 and

a0= 3, whereϕ ≡ 1 + φ and α2≡ 1 + a20. . . 41

4.5 The plots of (a) |E_z| and (b) |1/E_z| versus ϕ with b = 5 and

4.6 The plots of (a) |E_z| and (b) |1/E_z| versus ϕ with b = 5 and

a0= 3 (>√b − 1(√b − 1) = 2.47). . . . 45

5.1 The general flow chart for the PIC scheme. . . 49 5.2 The smooth function with different parameters. The solid, dashed

and dotted curves represent the function sm(k) = exp(−k3), exp(−(2k)3), and exp(−(4k)2) respectively. . . 52 5.3 The masking function with ncdl = 29 and ncdr=Ng− ncdl. . . . 53 5.4 The sketch of the geometry in simulation, with an external

mag-netic field B₀ imposed along the z direction. The whistler pulse is set to propagate parallel toB₀. . . 56 5.5 The intensity plot of the driving pulses in k space (in arbitrary

unit) imposed with their associated phase velocities in case a and b. . . 58 5.6 The snapshot of the whistler pulse (gray dashed) and the excited

plasma wakefield (solid) in case a and b at Δt = 100ω−1p after pulse released. . . 59 5.7 The snapshot of the whistler pulse (gray dashed) and the excited

plasma wakefield (solid) in case a and b at Δt = 300ω−1_p after pulse released. . . 61 5.8 The intensity contours of the driving pulse as a function of (ω, k)

from PIC simulation. The light curve and the theoretical disper-sion curves for the whistler wave with ωc/ωp = 1, 6 and 12 are superimposed. . . 62 5.9 The total energy (in arbitrary unit) versus simulation time in

case b. . . 63 5.10 The plot of accelerating gradient G versus a₀. The simulation

data points agree well with the solid curve obtained by solving Eq. (4.21). The dashed curve is the extrapolation of the non-relativistic theoretical result, Eq. (4.10). . . 64 6.1 The famous Hillas plot, showing the astrophysical objects with

their magnetic field strength and sizes. The solid lines represent-ingEmax∼ ZBL and Emax=ZBLΓ are also shown. . . . 69

6.2 The simplified e−− p jet geometry with ignoring the divergence angle. The plasma density and background magnetic field strength are considered as constant. . . 71

Chapter 1

Introduction

1.1 The Origin of Ultrahigh Energy Cosmic Rays

The origin of ultrahigh energy cosmic rays (UHECR) has been a long-standing mystery in astrophysics. According to the detection of the giant air showers, the arrival of UHECR with energy up to 1020eV was confirmed[1, 2, 3] and the most energetic cosmic particle recorded was about∼ 3 × 1020 eV by the Fly’s Eye Observatory[2]. It is amazing that a subatomic particle can carry macroscopic kinetic energy equal to that of a baseball (142 g) traveling at 96 km/h. Having such high energy, UHECRs pose a serious challenge on the theoretical models.

Figure 1.1 shows the overall cosmic ray spectrum which simply follows a power law with index roughly−3. There are two kinks at energy 1015 eV(the knee) and 1018 (the ankle) eV denoting the changes of the power-law indices. We believe that the ankle is due to the transition of galactic source to extra-galactic sources and the change of composition. In addition, beyond energy 5× 1019 eV, the flux is expected to drop significantly due to the GZK effect taking place. The GZK effect was proposed in 1967 soon after the discovery of the cosmic microwave background (CMB) by Greisen, Zatsepin, and Kuzmin (GZK)[4, 5]. A cosmic protons with energy above the threshold (the GZK cutoff energy) would lose its energy through interaction with the CMB photons. As a result its spectrum would be subject to a cutoff. In the observation aspect, HiRes which uses the fluorescence method clearly exhibits a GZK suppression

signature, while AGASA instead shows a continue spectrum. This discrepancy deeps the puzzle of UHECR. Fortunately precision measurements[6, 7] on the yield of air-shower induced fluorescence lend support to the energy calibration of the HiRes observations[8]. Together with the recent data from the Pierre Auger Observatory[9] which exhibits a similar location of an ”ankle” and the GZK suppression as those observed in HiRes, we confirm the validity of the GZK mechanism. Nevertheless both AGASA and HiRes presented the exist-ing of super-GZK events, which are not observed by Auger. It implies that the super-GZK particles should have original energies even higher and be re-strictively located within 50∼ 75Mpc (the GZK attenuation length). However so far there is no source within this range identified response for the UHECR production.

There has been many mechanisms caming up with to solve the UHECR production issues. Thus far, the existing theories can be broadly categorized into two scenarios, top-down and bottom-up. The bottom-up model relies on an efficient acceleration mechanism for an ordinary particle, such as a proton, at some astrophysical site to ultra high energies. While the top-down scenario is an alterative model to the bottom-up scenario proposed in order to explain the super-GZK events. It resorts to the decay of some relics of Grand Unified scale (∼ 1024eV) from the early universe. The main challenges for the scenarios are their difficulties of complying with the observed event rates and the energy spectrum[10], and the fine-tuning of particle lifetimes. Meanwhile the top-down theories would predict high fluxes of photon and neutrino as the side prod-ucts. The lack of observation of photons or neutrinos strongly disfavors these models[11, 12]. Therefore finding a viable bottom-up mechanism to accelerate ordinary particles beyond 1020 eV becomes more acute.

1.2 Conventional Model

1.2.1 Diffusive Shock Acceleration

The first idea of the cosmic ray acceleration mechanism yielding a power law spectrum is proposed by Fermi in 1949[13]. He considered that cosmic particles in interstellar space can diffuse by scattering off the randomly moving

mag-netic clouds, resulting in an average energy gain per encounter proportional to the mean velocity square of the magnetic clouds. It is often referred to as the second-order Fermi acceleration. Particles can be accelerated to high energy by many turns of the accelerating cycles. However the mechanism is not efficient because the process is non-relativistic and the energy gain proportionally toβ2 is accordingly small. As a variant of Fermi mechanism in strong non-relativistic shocks, the so-called diffusive shock acceleration(DSA) mechanism, was inde-pendently proposed by several authors [14, 15, 16, 17] in the late 1970s. It is referred to as the first order Fermi mechanism. This mechanism was conven-tionally accepted as the origin of the high energy cosmic ray. According to the simple picture from Bell[14], the upstream particles injected crossing the shock front could be turned back by scattering off the magnetic turbulence generated in downstream and vice versa, resulting in the diffusion of particles on the both sides of shock front. Different from the Fermi mechanism, each crossing can gain energy proportional to the first power of shock velocity. It is because the particles at shock always encounter head-on collisions.

These two mechanisms both produce power law spectrums. Assuming the energy gain per encounter is Δ/ = ξ, the energy after n encounters is,

n =0(1 +ξ)n (1.1)

where ₀ is the energy at injection into the accelerator. If the probability of escape from the acceleration region isPescper encounter, then the probability of remaining in the acceleration region after n encounters is (1 − Pesc)n. The number of encounters needed to reach energyE is

n = ln   0  / ln(1 + ξ). (1.2) Thus, the proportion of particles with energy greater thanE is

N(> ) ∝ ∞

m=n

(1− Pesc)m=(1− Pesc) n

Pesc . (1.3)

Substituting the expression of n into Eq. (1.3), we arrive at the power-law spectrum N(> ) ∝ _P1 esc   0 −γ , (1.4) with γ = ln  1 1− Pesc  / ln(1 + ξ) ≈ Pesc_ξ (1.5)

From Fermi’s picture, the probability per encounter of escape from the ac-celeration regionP_esc is the ratio of the characteristic time for the acceleration cycle and the escape from the acceleration region. This resulting spectral index is not universal, but depends on the properties of the magnetic clouds. For the strong shock case, it can be shown that at a shock,

γ = P_ξesc = 3 uu/ud− 1 (1.6) with uu ud = (cp/cv+ 1)M2 (cp/cv− 1)M2+ 2 (1.7) given by Rankine-Hugoniot jump conditions at the shock front[18]. Here uu andud are the velocities of gas flows in upstream and downstream respectively and the Mach number M is defined as the ratio of uu to the sound speed in upstream gas. For an monatomic gas the ratio of specific heats cp/cv = 5/3,

γ ≈ 1 + 4

M2 ∼ 1 for the strong shock with M  1, which is independent of

the shock properties and is universal. The differential spectrum provided by diffusive shock acceleration mechanism is given by dN/dE ∝ E−2 at strong non-relativistic shock.

The above discussions are for non-relativistic shocks. Since the most pow-erful astrophysics objects often involves ultra relativistic flows, the application of DSA to ultra relativistic flows has also been massively studied over years (see [19]). When considering the relativistic shock, the distribution of scat-tered particles is no longer isotropic but has orientations on angle. As a con-sequence, the application of DSA becomes more difficult. The average energy gain ΔE =E_f − E_i in the rest frame of shock front is shown as order of E_i itself, hence in the first shock crossing cycle, a large initial boost in energy can be achieved, Ei/Ef ∼ Γ2 where Γ is the gamma factor of the bulk velocity of relativistic flows[20]. The power-index is fitted about−2.23 ± 0.01 [21, 22].

1.2.2 Limitations

With the successful application to supernova remnants for cosmic ray around the knee (1015) eV, which has been confirmed via the x-ray observation[23], DSA is conventionally considered as the possible solution of high energy cosmic ray.

However DAS relies on the random collisions of the high energy particle against magnetic field domains or the shock media. They restrict the accelerating time from the shock lateral size and the strength of background magnetic field, and therefore the maximum energy gain. At very high energy, the collision process in magnetic turbulence necessarily induces severe synchrotron radiation loss, which is proportional to the fourth power ofγ. Compiling above limitations, DAS has difficulties to explain UHECRs. Evidently, novel acceleration mechanisms that can avoid some of the difficulties faced by these conventional models should not be overlooked.

1.3 Plasma Wakefield Acceleration as a Possible

Mechanism

Plasma wakefield accelerators[24, 25] are known to possess two salient features: (i) The plasma can support an extremely high ”acceleration gradient,” i.e., energy gain per unit distance, which does not depend (inversely) on the parti-clesinstantaneous energy or momentum. This is essential to avoid the gradual decrease of efficiency in reaching ultrahigh energies. (ii) The acceleration field is collinear to the particle momentum. Therefore, bending of the trajectory is not necessary in this mechanism. This helps to minimize inherent energy losses that would be severe at ultrahigh energies.

So motivated by these considerations, it was proposed that UHECR can be produced from the plasma wakefield excited in astrophysical setting[26]. In-stead of using laser or charged beam which does not exist in astrophysical en-vironments, Chen et al invoked Alfven shocks as the driving sources to excite plasma wakefields. This idea of using shocks to excite plasma wakefield has at-tracted several astrophysical plasma physicists [27, 28]. Chen et al showed that the power-law spectrum is accounted for the stochastic encounters between the particles and the randomly generated wakefields. Using the short gamma ray burst(GRB) as the working source, Chen et al obtained the maximum accel-erating gradient and predicted the event rate. However, in that paper, their estimation of accelerating gradient was based upon the theory of laser wake-field acceleration without taking the background magnetic effect into account.

Furthermore, this concept has never been validated through computer simula-tion. Thus, we develop a new mechanism of plasma wakefield (the magnetowave induced plasma wakefield acceleration (MPWA) invoking the high frequency and high speed whistler mode as driving pulse, and confirm this concept via computer simulations[29]. The magnetowave with phase velocity vph < c has component|B| > |E| in nature. On the other hand the laser and charged beam have|E| > |B|.

In this thesis, we will discuss the complete theory of MPWA and its appli-cation to UHECR. The content is the following: in Chap.2, the basic plasma physics is viewed to give the way for subsequent discussions. The last section of Chap. 2 introduces the different types of the plasma wakeifled accelerator. In Chap. 3, we start looking at the plasma wakefield acceleration in magnetized plasma. With ω  ωc, the magnetic field effect can be ignored. In Chap.4, we focus on the driving pulse withω < ωc (MPWA) and introduce a MPWA condition. The theory of MPWA in linear and nonlinear regimes under the MPWA condition are presented and the limitation of MPWA is also discussed. In Chap. 5, the particle in cell(PIC) code to produce MPWA is introduced. We show that the simulation results are in good agreements with the theoretical prediction. Finally with the theoretical model established, we apply this mech-anism to explain the UHECR acceleration. In Chap. 6 we obtain the power-law spectrum from the stochastic process of the wakefield acceleration and estimate the accelerating gradient provided by AGN jets. The summary and conclusion are presented in Chap. 7.

Chapter 2

Basic Concept of Plasma

It is known that 99 percent of visible matter in the universe is in plasma state. All the astrophysical objects, such as stars, relativistic jets, accretion disks, etc.., are made of plasma. Therefore a though understanding of plasma physics could lead to an understanding of 99 percent of the visible universe. Based on that, the idea of plasma wakefield acceleration for UHECR is therefore possible. In this chapter I will briefly introduce the basic concept of plasma. The cgs unit system is used in the following treatment.

2.1 What is Plasma

Plasma is a partially ionized gas consisting of free negative electrons, positive ions and neutral atoms. When we heat a liquid, we can see more and more vapors created as the temperature rises till reaching the boiled point. After that, all liquid molecules are turned into gas molecules. If we continue to heat the gas, some atoms or molecules will eventually get ionized. Thus in addition to the three thermodynamic states, plasma is sometimes referred to as the fourth state of matter. Because the ionized energy of atoms is of the order 10 eV, plasma is usually created in a very high temperature. But in fact, the atoms still have chances to be ionized at the room temperature due to the tail of thermal distribution. The amount of ionization is very rare so that we can not feel the plasma around us. To estimate the portion of ionization in thermal equilibrium, we can use Saha equation[30]

ni

nn ≈ 2.4 × 10

15T3/2

ni e

−Ui/kT, _(2.1)

where ni and nn are the densities of ionized atoms and of neutral atoms re-spectively, T is the gas temperature in unit of K, and Ui is the ionization energy of the gas. If we take the room temperature T=300K, gas density

nn≈ 3 × 1019cm−3, andUi≈ 14.5 eV for nitrogen, we can predict the fraction of ionization ni nn+ni ≈ ni nn ≈ 10 −122

which is extremely low.

2.2 Definition of Plasma

Not any ionized gas can be called a plasma. As mentioned above, there is al-ways some small fraction of ionization in any gas. So the plasma is defined from its most important properties, collectiveness and the quasi-neutrality. Since plasma contains charged particles, the moves of these charges can generate local concentrations of positive or negative charges which give rise to electric fields. The motions of charges also generate currents and then the magnetic fields. These fields are long-range and could affect the motions of other charged par-ticles far away. To see the effect, let us imagine two small charged regions of plasma separated by a distancer. Even if the Coulomb interaction between the two individual charged particles diminishes as 1/r2, for a given solid angle, one region can feel a total force from the other region with volume increasing as

r3_{. Therefore, elements of plasma can experience a force on one another even}

at large distances. By ”collectiveness” we mean that plasma motions depend not only on local conditions but on the state of the plasma in remote regions as well.

The quasi-neutrality comes from a fundamental characteristic of the plasma, which is the capability to shield out electric potentials that are applied to it. Suppose we set up an electric field by inserting a ball charged with posi-tive charges, the ball would naturally attract an electron cloud with the same amount of positive charges surrounded. If we assume an electron distribution

which follows the Boltzman’s equation so that

ne(Φ) =n0eeΦ/kBT (2.2)

where Φ, is the potential associated with the slight separation of electrons and ions, n₀ is the plasma density at Φ = 0 and T is the electron temperature. Since an ion is 1800 times heavier than an electron, the ion background can be regarded as motionless. Therefore the ion densityni, where the subscripti denotes the ion background, is approximately equal to the plasma density n₀. Considering only the one-dimensional case, the Poisson equation turns into with Eq. (2.2)

∂2_Φ

∂z2 = 4πen0



eeΦ/kBT _{− 1}_{. (2.3)}

In the region where|eΦ/kBT |  1, we can expand the exponential to the first order, ∂2_Φ ∂z2 = 4πen0  eΦ kBT +· · ·  (2.4) that gives the solution of Φ

Φ = Φ₀e−|z|/λD _(2.5)

with the characteristic lengthλD defined as

λD= 

kT

4πne2 =vth/ωp (2.6)

whereωp≡4πe2n0/m is the plasma neutral frequency and vth =kBT/m is the velocity of electron thermal motion. λD is called Debye length, named after the Dutch physicist Peter Debye. If the plasma is cold, T = 0, then λD = 0 and the shielding is perfect. It allows no electric field being presented outside the electron cloud. However if T = 0 , λD is accordingly finite. The potential will be no longer perfectly shielded but decay exponentially with the distance. Because of the shielding, the distant particles will not feel the existence of the charged ball in the plasma. Therefore for remaining the quasi-neutrality, the condition for a plasma isλD L, where L is the plasma size. In addition, the Debye shielding itself is actually a statistic concept. Thus for the validity of Debye shielding, we should compute the number of particles in a Debye sphere g and require

g ≡ n4

3πλ

3

wheren = ne=ni is the plasma density andg is called the plasma parameter. Combining the two conditions, we can make the criteria for plasma

1) λ  L. 2) g ≫ 1. 3) ωτ > 1 .

where ω is the frequency of typical plasma collision and τ is the mean time between collisions. Finally the item 3 requests a low collision rate for plasma.

2.3 Dynamics of Plasma

2.3.1 Fluid Description

Since a typical plasma density might be a huge number of ion-electron pairs per cm−3, it is impossible to deal with each plasma particle. Fortunately, the majority of plasma presents a macroscopic behavior. So we are able to treat plasma as fluids, composed of electrons , ions and neutral atoms. As a result, the motion of individual particle is neglected and only the averaging motion is taken into account. The plasma fluid containing an additional electromagnetic effect is different from an ordinary fluid. Such effect leads to the complexity of plasma, and the varieties of phenomena could occur in a plasma.

In plasma, Maxwell’s equations can tell us how E and B are associated with a given state of the plasma. To maintain the self-consistency, we include equations that describe the plasma response to the E and B field such that

∇ · E = 4πe(ni− ne) = 4πρ ∇ × E = −∂B_∂t ∇ · B = 0 c∇ × B = 4πe(niui− neue) +∂E ∂t = 4πJ + ∂E ∂t

whereρ and J are the charge density and charge current given by the plasma, and u is the fluid velocity of from averaging the total velocity in the fluid unit.

These E and B fields above also act back on the plasma species, therefore the equation of motion regarding the electromagnetic force is described,

mjnj ∂uj ∂t + (uj· ∇)uj = qj(E + u_cj × B) − ∇p

wherej = i, e stands for fluid of ions and electrons respectively. The mj is the mass of the fluid element. The equation above is in Eulerian representation, dealing with the time and space derivatives separately. Sometimes we describe the fluid in either Eulerian (the coordinate scheme) or Lagragian(the co-moving scheme) representations. The relation between the two representations is

d dtLagragin=  ∂ ∂t+ v· ∇  Eulerian .

The second term on the right hand side is called the convective term. Finally combining the above equations and the continuity equation, we obtain the com-plete set of fluid equations

∇ · E = 4π j qjnj, (2.8a) ∇ × E = −∂B ∂t , (2.8b) ∇ · B = 0, (2.8c) c∇ × B = 4π j qjnjuj+∂E_∂t, (2.8d) ∂nj ∂t +∇ · (njuj) = 0, (2.8e) mj ∂uj ∂t + (uj· ∇)uj = qj(E +uj c × B) − ∇pj nj , (2.8f) pj = C(mjnj)γj. (2.8g)

with 11 unknowns (E, B, u, n, p) for each species. Here Eq. (2.8e) is the conti-nuity equation and the last equation is the equation of state, withC a constant andγ = Cp/Cv the ratio of specific heats.

2.3.2 Kinetic Description

Beside the fluid theory, the alternative way to describe plasma is the kinetic theory. In most cases, the fluid equations can solve the plasma problems with acceptable good accuracy. But for some special cases, such as the instabilities, the fluid treatment will be inadequate. Thus, we directly look at the distribution

function fj(r, v, t) for each plasma species (here v is the individual velocity). By knowing the distribution function, we are able to derive the macroscopic physical variables from integrating the function over all velocity spaces. This treatment is called kinetic theory.

The time evolution of distribution functionfj(r, v, t) is govern by the Boltzmann equation, dfj dt = ∂fj ∂t + vj· ∇fj+ qj mj(E + vj c × B) · ∇vfj = ( δfj δt )c, (2.9)

where (δfj/δt)c is the collision term. The plasma density can be obtained from

nj(r, t) = 

vfj

(r, v, t)d3v,

and the average velocity uj is given by

uj(r, t) = 

vvfj(r, v, t)d3v, 

vfj(r, v, t)d3v

If the plasma is collisionless, the collision term vanishes and Eq. (2.9) takes the form ∂f ∂t + v· ∇f + q m(E + v c × B) · ∇vf = 0, (2.10)

where we drop the subscriptj. This is called the Vlasov equation, most com-monly studied in the kinetic theory. Regardless of the collision term, the zero moment of Eq. (2.9) is obtained by integrating over the velocity space,

 ∂f ∂tdv +  v· ∇fdv + q m  (E +v c × B) · ∇vfdv = 0 (2.11)

and the next moment is obtained by multiplyingmv to the equation and inte-grating over v such that

m  v∂f ∂tdv + m  v(v· ∇)fdv + q  v(E +v c × B) · ∇vfdv = 0. (2.12)

Taking the above two moments of the Boltzmann’s equation leads to the con-tinuity equation Eq. (2.8e) and the equation of motion of fluid(2.8f). We show that the fluid theory can be derived from the kinetic theory; therefore the fluid and kinetic representations of plasma are equivalent.

2.4 Waves in the Plasma

We have already established the complete set of fluid equations for plasma. To solve these equations, we introduce the perturbation theory describing a small deviation of physical quantities to their equilibrium state. These quantities can be decomposed into the equilibrium solution plus a small perturbation. After taking the Fourier expansion, the perturbations are transformed into a super-position of sinusoidal oscillations in different frequencies. As a result the fluid equations Eq. (2.8a) to (2.8g) can be linearized in (ω, k) space and the result-ing equations are easier to solve. In this section, I review the physics of plasma oscillation and electromagnetic wave in plasma for demonstrating the technique for solving the fluid equations.

2.4.1 Plasma Oscillation

When the electrons in plasma are displaced by some perturbations from a uni-form background of ions, electric fields are built up in such a direction as to restore the neutrality of the plasma by pulling the electrons back to their origi-nal positions. Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions. The process is known as the plasma oscilla-tion, also known as Langmuir wave, with a characteristic frequencyωp[31]. To derive the expression for the plasma frequencyωp, we solve the fluid equation in the simplest case that (1)B = 0, there is no magnetic field; (2) T = 0, hence

∇p = 0, there are no thermal motions; (3) background ions are fixed; (4) the

electron motions only take place along to z direction. The fluid equations in this simplification are obtained,

mne ∂ve ∂t + (ve· ∇)ve = −eneE (2.13a) ∂n ∂t +∇ · (neve) = 0 (2.13b) ∇ · E = 4πe(ni− ne), (2.13c)

with electron fluid velocity uereplaced by vefor convenience. Since the plasma is assumed to be slightly perturbed, the variables in the equations can be sepa-rated into two parts: the equilibrium part, subscripted by 0, and the

perturba-tion part by 1 that

ne = n0+n1,

ve = v0+ v1,

E = E₀+ E₁.

If the plasma is initially stationary, we have the equilibrium state∇n₀= v₀=

E₀= 0. The fluid equations are then given by

m∂v1

∂t = −eE1, (2.14a)

∂n1

∂t +n0∇ · v1 = 0, (2.14b) ∇ · E1 = −4πen1, (2.14c)

where the convective term

(v· ∇)v = (v₁· ∇)v₁

vanishes due to the higher order perturbation. For plasma oscillation, it is assumed that the quantities oscillate sinusoidally and,

n1 = n1ei(kz−ωt)

v₁ = v₁ei(kz−ωt)ˆz E₁ = E₁ei(kz−ωt)zˆ.

So the time derivative ∂/∂t can be replaced by −iω and the space derivative

∇ can be replaced by ik in the equations. Then the differential equations

Eq. (2.14a) to (2.14c) are linearized such that,

−imωv1=−eE1 (2.15a)

−iωn1+n0ikv1= 0 (2.15b)

ik · E1=−4πen1. (2.15c)

where E₁, which is associated with the plasma oscillation, is an electrostatic field along thek direction. We can rewrite Eq. (2.15a) by applying Eq. (2.15b) and (2.15c) as  ω2₋4πn0e2 m  v1= 0. (2.16)

Eq. (2.16) is the dispersion relation for the plasma oscillation. Because there is nok dependence in this expression, the plasma oscillation does not depend on the wavelength. Hence the phase velocity defined asω/k and the group velocity defined as ∂ω/∂k are both zero. When v₁ is finite, a non trivial solution for Eq. (2.16) requires terms in the parentheses to be 0. Therefore the frequency

ω = ωp≡  4πn₀e2 m 1/2 . (2.17)

is defined as the plasma frequency. Numerically, with the known physical para-meter numbers, one can make the approximate formula

fp≈ 9000√n0

which only depends on the plasma density. So far the treatments are all done in cold plasma case (T = 0). For warn plasma (T = 0), the pressure term

∇p should be taken into account in Eq. (2.13a). The dispersion relation then

becomes ω2_=ω2 p+  γBkBT m  k2_{, (2.18)}

where (γBkBT/m)1/2 is the electron sound speed and the γB, usually taken to be 1 in the isothermal sate, is the adiabatic constant for the pressure term Eq. (2.8g). So that the plasma oscillation starts to propagate asymptotically with electron sound speed. Such wave is called the electron acoustic wave.

2.4.2 Electromagnetic Wave in Plasma

Next we study the case of electromagnetic waves in plasma. When an elec-tromagnetic wave travels through a plasma, its associated elecelec-tromagnetic field shall push the charged particles from their original states and the resulting plasma motions will induce the currents that contribute back to the fields them-selves. As a consequence, the dispersion relation of the electromagnetic wave in the plasma contains the plasma effect. If there is no magnetic background field,

B0 = E0 = 0, the electric and magnetic fields in plasma, denoted by E1 and

B₁, are related to each other according to the Maxwell equations

∇ × E1=−1_c∂B_∂t1 (2.19)

where the term (4π/c)J₁ ≡ −(4π/c)en₀v₁ is the plasma current. Taking the time derivative on Eq. (2.20), we combine the above two equations and obtain

c2_{∇ × (∇ × E}

1) = 4πen0∂v_∂t1 +∂E_∂t1. (2.21)

Here v₁ is directly related to the oscillating electric field because the second force term v₁/c × B₁ in Eq. (2.8f) is neglected since it is of the second order. Assuming a plane wave varying as exp[i(kz − ωt)], the electron velocity v₁ is given by

−iωv1=−_meE1. (2.22)

We then rewrite Eq. (2.21) as

−c2_{k× (k × E}

1) =−i4πen0ωv1− iωE1. (2.23)

Substituting Eq. (2.22) to Eq. (2.23), we obtain

−k(k · E1) +k2E1=−4πe 2_n0

mc2 E1+

ω2

c2E1. (2.24)

We note that k· E₁= 0 because the wave is transverse. Then the equation can be rearranged as k2₊ωp2 c2 − ω2 c2  E₁= 0. (2.25)

, which leads to the dispersion relation for electromagnetic wave in unmagnetized plasma.

ω2_=ω2

p+k2c2. (2.26)

We can calculate the phase velocity vph by using this dispersion relation and obtain vph= ω k = c  1−ωp ω > c. (2.27)

The phase velocity is real only whenω > ωp. Therefore a threshold of frequency exists for the electromagnetic wave to penetrate into the plasma. If the wave has frequency ω < ωp, it will be reflected by the plasma surface and decays exponentially in the plasma within a skin depth defined asc/ωp. This concept

is the working principle for radio station to transmit the signals. Similarly, The group velocity can be calculated as

vg=∂ω

∂k = c2

vph. (2.28)

Clearlyvg is always smaller that the speed of lightc.

2.5 Plasma Wakefield Acceleration

It has been 30 years since T. Tajima and J. Dawson proposed plasma as an accelerator[24], to transfer electromagnetic energy from a laser pulse into the kinetic energy of the accelerated electron by letting the short laser pulse excites large-amplitude plasma waves. In fact the ”plasma wave” we call here is the plasma oscillation but having a phase velocity exactly equal to the pulse speed. An electrostatic field relate to this plasma wave is called plasma wakefield. If its phase velocity is closed to the speed of light, a test particle with similar velocity injected to its accelerating phase can surf on the wave and continually gain energy from it. The characteristic accelerating gradient for the plasma wakefield isG = eEwb =mcωp∼



n0[cm]−3 Since the mechanism provides a

great accelerating gradient which can accelerate charged particles to very high energy in a short distance, it is very attractive to accelerator physics, plasma physics and astrophysics.

Since then there has been several reviews discussing about the plasma based accelerators [32, 33, 34]. So far there have been three plasma wakefield accelera-tors utilizing laser pulses: laser wakefield accelerator(LWFA), plasma beat-wave accelerator(PBWA) and self-modulated LWFA (SM-LWFA)[32]. In the PBWA [24, 35], two long pulse laser beams with frequencies differed by ωp are used to resonantly excite the plasma wave. This method was first proposed as an alternative to the laser wakefield accelerator because of the lack of technol-ogy for generating ultra-intense picosecond laser pulses at that time. The last one SM-LWFA is somewhat similar to LWFA with a single short pulse but operated at higher density[36, 37, 38]. Therefore SM-LWFA involves a longer length that L > λp and slightly larger laser power P than the critical power

Pc = 17ω2/ω2p for relativistic optical guiding. In the high density regime, the pulse becomes self-modulated at the plasma period due to the self-modulation

instability [39, 40, 41]. Then the plasma wave is generated coinciding with the modulated regime. Instead of utilizing laser as driving pulses, Chen et al in 1985 proposed another way to excite the plasma wake by using the charged particle beams [25]. The charged particle which moves relativistically generates the quasi-perpendicular electric field in the lab frame and the magnetic field as well according to the relativistic dynamics. Therefore the charged particle beams behave similar to the laser pulse and the dynamics of plasma wakefield for the two schemes was also shown to be similar by Ruth and Chen[42].

However, either laser beam or charged particle beam is the external impulse and could not be found in the astrophysical environment. Motivated by the ultrahigh energy cosmic ray acceleration issue, Chen, Tajima and Takahashi in 2002 proposed the third type of plasma wakefield acceleration invoking Alfven-shocks as the driving pulses. Different from the laser and the particle beam, Alfven wave is a medium wave which only exist with the support from plasma. Therefore the wakefield driven by Alfven wave is more relevant to the astro-physical settings. F. Y. Chang et al. [29] extended the concept to the high frequency mode (whistler wave). According to Maxwell’s equations, these waves have theB component exceeding the E component since their phase velocities are less than the speed of light. We categorize such wave as ”magnetowave”. In the following chapters, I will discuss the plasma wakefield in magnetized plasma and introduce the theory of magnetowave induced plasma wakefield ac-celeration (MPWA) in both linear and nonlinear regimes. I will also present a self-consistent plasma simulation which is performed to validate this theory.

Chapter 3

Plasma Wakefield in

Magnetized Plamsa I

We have studied the dispersion relation of electromagnetic wave traveling in plasma (Eq. (2.26)). Once the plasma is imposed a background magnetic field, the electromagnetic wave presents various different modes at arbitrary angles to the external magnetic field. Among that, we concentrate on the modes parallel to the external magnetic field for our purpose to ensure the linear acceleration that minimizes the energy loss. With the parallel background magnetic field

B0, the electromagnetic wave becomes circularly polarized and its dispersion

relation is given by ω2_=k2_c2₊ ωip2 1± ωic/ω+ ω2 p 1∓ ωc/ω, (3.1)

where the upper (lower) signs denote the right-hand (left-hand) circularly polarized waves. ωc=eB0/mc is the electron cyclotron frequency and the

sub-scripti denotes the ion species. Each polarization has two real solutions with high and low frequency branches and both have a frequency cutoff which forms a forbidden gap for wave propagation. Figure 3.1 exhibits the solution of all possible modes and the light curve in vacuum (dashed line) is superimposed. Above the light curve, there are two curves labeled L and R waves to stand for the left-handed and right-handed circularly polarized electromagnetic waves respectively. Whereas the two solutions below the light curve are the whistler

0 1 2 3 4 5 0 1 2 3 4 5 ckΩp Ω Ωp L wave R wave whistler wave

ion cyclron wave

ΩcΩp2

Figure 3.1: The full solutions of Eq. (3.1) withωc/ωp= 2. The two curves (R and L waves) above the light curve (dashed line) would havevph > c and the two curves (whistler wave and ion cyclotron wave) below the light curve would

havevph< c.

wave and the ion cyclotron wave, having a lower phase velocity than the speed of light. We call such waves the ”magnetowaves” because of their exceedingB components in all reference frames. To explain the production of UHECR, Chen et al proposed Alfven shocks as the driving pulses for plasma wakefield. Since the Alfven wave is an ion wave, having very low frequency and low phase veloc-ity, it was the first idea of magnetowave induced plasma wakefield acceleration (MPWA).

In fact the non-relativistic plasma wakefield in magnetized plasma was first studied by P. K. Shukla[43] in 1994. Shukla introduced the ponderomotive force from a circularly polarized electromagnetic pulse that is applicable for all frequency range to excite the plasma wakefield. However he only addressed the upper branch issue (the laser case) in his calculation. For R and L waves which have frequenciesω  ωc, the dispersion relation Eq. (3.1) can be reduced to that in unmagnetized plasma. Therefore the background magnetic field doesn’t play a significant role to the wakefield excitation. Whereas the wakefield induced by the wave with ω < ωc will greatly determined by the ratio of ωc/ω. In the

following two chapters we will discuss the physics of wakefield induced in the two branchesω  ωc (the laser limit) andω < ωc(magnetowaves). The general theory of MPWA is established in the next chapter.

3.1 General Formulation

In the laser plasma interaction community, the related fields in one dimension along the z direction are often described by the normalized scalar potential

φ(z, t) ≡ eΦ(z, t)/mc2 _{for the plasma electrostatic field (plasma wakefield) and}

the normalized vector potential a(z, t) ≡ eA(z, t)/mc2 for the laser field. We have Az = 0 if choosing the Coulomb gauge ∇ · A = 0. The peak of the normalized vector potentiala₀, called laser strength parameter by the plasma community, is often used to determine the strength of the driving laser. Since the vector potential A is the spacial component of the 4-vector (ρ, A), the transverse components of A_⊥ (Ax, Ay) are Lorentz invariant in any reference frame boosting along z direction. As a result, the a₀ by definition is also a Lorentz invariant quantity. In MPWA study, we still follow the convention for laser case. The plasma fieldφ(z, t) is governed by the set of fluid equations

∂2_φ ∂z2 =kp2  n n0− 1  , (3.2a) ∂n ∂t +c ∂ ∂z(nβz) = 0, (3.2b) d dt(γβ⊥) = da_dt − iωcβ_⊥, (3.2c) d dt(γβz) =c ∂φ ∂z − e mc(βxBy− βyBx), (3.2d) dγ dt =− e mc(βxEx+βyEy+βzEz), (3.2e)

with ignoring the ion motions. The influence of background magnetic field only takes place in the transverse momentum equation (Eq. (3.2c)), see Appendix A. In these equations, we may neglect the thermal effect due to the assumptions: (i) the electron quiver motion is much greater than the electron thermal motion (ii) the plasma temperature is so low that the thermal energy spread is not sufficient for the plasma to be trapped by the plasma wave. The Lorentzγ of plasma here defines (1−β_⊥2−β_z2)−1/2and theωcin Eq. (3.2c) gives the influence from the external magnetic field. While the normalized vector potential a(z, t)

satisfies the wave equation ∂2 ∂z2 − 1 c2 ∂2 ∂t2 a =k2_p n n0β⊥. (3.3)

kp≡ ωp/c is the plasma wavenumber.

3.2 Laser Wakefield Acceleration

Since the technology of high field laser has been well developed in a laboratory, laser wakefield acceleration is widely studied because of the possibility to the next generation of high accelerating gradient accelerators. When the driving pulse with frequencyω  ωc (orωc → 0), the dispersion relation of the pulse approximates to that in unmagnetized plasma case. It is reasonable to study the LWFA mechanism under this limitation. Based on this consideration, the right hand side of Eq. (3.2c) can be ignored and Eq. (3.2c) is rewritten as

d

dt(γβ⊥− a) = 0. (3.4) Therefore the transverse canonical momentum γβ_⊥− a is conserved and the transverse velocity β_⊥ = a/γ is easily obtained. Substituting the transverse velocity expression into Eq. (3.2d), we rewrite the Lorentz force in terms of the normalized vector potential a

dγβz dt =c ∂φ ∂z − c 1 2γ ∂a2 ∂z (3.5)

where the second term on the right hand side is the ponderomotive force, the average of the second order Lorentz force. It is on the opposite direction to the gradient of laser intensity and is independent of the charge sign. Thus the electrons within the pulse are pushed away from the center and leave a positive region (ions are only barely moved by the same force), which generates the plasma wakefield.

3.2.1 Linear Regime

With the full set of fluid equations, we first study LWFA in the non-relativistic regime whereγ ∼ 1. In this regime, β_⊥ is equal toa  1 so that the condition

for non-relativistic isa  1. Hence the complete set of fluid equations can be rewritten as ∂2_φ ∂z2 =kp2N, (3.6a) ∂N ∂t +c ∂ ∂z(βz) = 0, (3.6b) ∂βz ∂t =c ∂φ ∂z − c 1 2 ∂a2 ∂z , (3.6c)

whereN ≡ (n − n₀)/n₀. We eliminate thecβz∂βz/∂z term in Eq. (3.6c) since

βz  1. It is convenient to write the equations in a co-moving coordinate system (ζ, τ) [44, 45], in which τ = t and ζ = z − ct. Then the derivatives

∂/∂z and ∂/∂t are replaced by ∂/∂ζ and ∂/∂τ − c∂/∂ζ respectively. If the laser

pulse is sufficiently short, the field a andφ are expected to change very little during the transit time of the plasma through the pulse and the changes can be ignored in plasma reaction. Assuming that the laser envelop changes on a characteristic time scaleτe∼ 2|n0/n|(ω/ωp)/ωp, the quasistatic approximation (QSA) is applicable. In the QSA,∂/∂τ which determines the plasma response to the laser pulse are neglected in the plasma fluid equations . However,∂/∂τ is retained in the wave equation because it describes the evolution of the laser pulse [45, 32]. Thus for a short laser pulse, we can write Eq. (3.6a) to (3.6c) as

∂2_φ ∂ζ2 =kp2N, (3.7a) ∂ ∂ζ(N − βz) = 0, (3.7b) ∂βz ∂ζ =− ∂φ ∂ζ + 1 2 ∂a2 ∂ζ , (3.7c)

Substituting Eq. (3.7a) and (3.7b) into Eq. (3.7c), we arrive at  ∂2 ∂ζ2 +k2p  φ = k2p 2 a 2 _(3.8)

The solutions to the equation are easily calculated with the Green’s function such that[46] φ ∼=kp 2  ζ 0 a 2_(ζ_{) sin[k} p(ζ− ζ)]dζ (3.9) and the related axial field is obtained

Ez Ewb =− 1 kp ∂φ ∂ζ =χa20

whereχ = kp/(2a2₀)₀ζa2(ζ) cos[kp(ζ− ζ)]dζis the form factor which depends on the pulse shape. HereEwb≡ mcωp/e is the cold wavebreaking limit, charac-terizing the accelerating gradientG (G = eEz) of the plasma accelerator. Since

kp=ωp/c, the solutions to Eq. (3.8) describe the plasma waves generated at the frequencyωp and are valid far from wavebreaking, Ez  Ewb. Meanwhile the wakefields are generated sinusoidally and are efficient when the envelope scale length is on the order of the plasma wavelengthλp= 2πc/ωp[46].

3.2.2 Nonlinear Regime

When the laser power is extremely high such thata₀ 1, the plasma particle quiver motions become highly relativistic and a variety of nonlinear phenomena happens in the laser plasma interaction. It includes [45] (a) relativistic optical guiding of the laser beam[47, 48, 49], (b) the excitation of coherent radiation at harmonics of fundamental laser frequency, (c) the generation of large plasma wakefield, (d) frequency shift induced in the laser pulse by plasma waves[50, 51], (e) frequency amplification using an ionized front, and (f) the snow-plow acceleration[52, 53]. A full set of the fluid equations is required to describe the nonlinear phenomena. For the study of the generation of large plasma wakefield, we have the equations from Eq. (3.2a) to (3.2e) in terms ofa and φ

∂2_φ ∂z2 =k2p  n n0 − 1  , (3.10a) ∂n ∂t +c ∂ ∂z(nβz) = 0, (3.10b) d dt(γβ⊥) = da dt, (3.10c) d dt(γβz) =c ∂φ ∂z − c 2γ ∂a2 ∂z , (3.10d) dγ dt =cβz ∂φ ∂z + 1 2γ ∂a2 ∂t , (3.10e)

with neglectingωc. From Eq. (3.10c), the conservation of transverse canonical momentum gives γβ_⊥ = a or γ = (1 + a2)1/2/(1 − β_z2)1/2. So the pulse is described by the wave equation of a

∂2 ∂z2 − 1 c2 ∂2 ∂t2 a =k2p_nn 0β⊥=k 2 p_nn 0 a γ, (3.11)

which leads to the dispersion relation in the relativistic regime

ω2_=k2_c2₊ωp2

γ (3.12)

by assuming a plane wavea ∝ exp[i(kz−ωt)] and n = n₀. We may combine this wave equation of a with the fluid equations to form a self-consistent equation set.

Insetting Eq. (3.10e) to Eq. (3.10d), we arrive at

d dt(γβz− γ) = c(1 − βz) ∂φ ∂z − 1 2γ  c_∂z∂ + ∂ ∂t  a2_{. (3.13)}

It is convenient to transform Eq. (3.13) into the new coordinate system (ζ, τ), then the second term on the right hand side vanishes. Together with the Poisson equation and the continuity equation, we obtain the complete equations forφ in (ζ, τ) coordinate with QSA applied

∂ ∂ζ[n(1 − βz)] = 0 (3.14a) ∂ ∂ζ(γβz− γ + φ) = 0 (3.14b) ∂2_φ ∂ζ2 =kp2 n n0 − 1 (3.14c) in which Eq. (3.14c) expresses the potential of plasma wakefield. Equation (3.14a) and (3.14b) can be solved from the integration overζ. Since the plasma keeps stationary until the driving pulse passes through, the boundary conditions for the two equationsγ = 1, n = n₀andβz= 0 are applied and give the solutions

γ(1 − βz)− φ = 1, (3.15a)

n(1 − βz) =n0. (3.15b)

So that the ratio ofn/n₀ is given,

n n0 = 1 1− βz = γ 1 +φ, (3.16)

and its quadratic form can be expressed as  n n0 2 =  1 1− βz 2 = γ2 (1 +φ)2 = 1 +a2 (1− βz)(1 +βz)(1 +φ)2

withγ2= (1 +a2)/(1 − βz2). Hence the differential equation ofφ is written as ∂2_φ ∂ζ2 = k2p n n0 − 1 = k2p βz 1− βz = k 2 p 2 (1 +a2) (1 +φ)2 − 1 (3.17) and the plasma wakefield normalized by E₀ is obtained as ∂φ/∂ζ. We then express the plasma quantities in terms of the fieldsa and φ as

n/n0= 1 +1₂[(1 +a2)/(1 + φ)2− 1] (3.18a)

γ = [1 + a2_{+ (1 +φ)}2_{]/[2(1 + φ)] (3.18b)}

βz= [1 +a2− (1 + φ)2]/[1 + a2+ (1 +φ)2]. (3.18c)

Considering the weak field limitφ  1, Eq. (3.17) becomes  ∂2 ∂ζ2 +k2p  φ = k2p 2 a 2 _(3.19)

after taking the Taylor expansion to the first power. This equation reduces to that in linear case (Eq. (3.8)).

Since Eq. (3.17) is fully nonlinear, its analytical solution only exists for a circularly polarized laser pulse with a square pulse profile as the laser envelop,

aL=a0 for−L < ζ < 0 and aL = 0 otherwise [54, 55, 56]. For simplification, we rewrite Eq. (3.17) ∂2_ϕ ∂ζ2 = 1 2 α2 ϕ2 − 1 (3.20) with ϕ ≡ 1 + φ, ζ ≡ kpζ and α2 ≡ 1 + a2. It can be integrated by first multiplying∂ϕ/∂ζ on both sides

 ∂2_ϕ ∂ζ2 ∂ϕ ∂ζdζ =  1 2 α2 ϕ2− 1 ∂ϕ ∂ζdζ. Then we get  ∂ϕ ∂ζ 2 =−α2 ϕ − ϕ + C1, (3.21)

where C₁ = α2+ 1 is the integration constant determined by the boundary condition that ∂ϕ/∂ζ = 0 and ϕ(ζ) = 1 at ζ = 0. Within the pulse that

−L < ζ < 0, the formal solution for ϕ(ζ_{) is [54, 55, 56]}

ζ_{=−2αE(θi, ki) + 2}(α2− ϕ)(ϕ − 1)

ϕ

1/2

with θi= arcsin  α2_{(ϕ − 1)} (α2− 1)ϕ 1/2 and ki=  α2_{− 1} α2 1/2 .

E(θ, k) here is the incomplete elliptic integration of the second kind and the

second term on the right hand side of Eq. (3.22) indicates that ϕ is allowed to lie in the range 1 ≤ ϕ ≤ α2. Thus the maximum ϕ = α2 occurs at ζ =

−2αE(ki)(E(ki) =E(π/2, ki)) which gives the optimal pulse length

Lo= 2

kpαE(ki)

λp 2 forα ∼ 1 (ki 1 , E(0) = π/2), and

Lo a0

πλ

forα  1 (ki ∼ 1 , E(1) = 1). We notice when ζ=−4αE(ki),ϕ = 1 implies

φ = 0. There is no wakefield excited behind the pulse.

Next with the laser pulse of lengthL = Lo, the equation of the plasma wakefield potential behind the pulse (ζ < −L and a = 0) is

 ∂ϕ ∂ζ 2 =−1 ϕ− ϕ + C2, (3.23)

where C₂ =α2+ 1/α2 is given from the boundary condition (∂ϕ/∂ζ)Lo = 0

andϕ(Lo) =α2. The solution ofϕ is therefore

ζ_{=−kpLo− 2αE(θe, ke) (3.24)} with θe= arcsin  α2_(α2_{− ϕ)} α4_{− 1} 1/2 and ke=  α4_{− 1} α4 1/2 .

Finally the axial electric fieldEz (wakefield) related toϕ is

Ez

Ewb ≡ ˜Ez=−

∂ϕ ∂ζ,

25 20 15 10 5 0 2 1 0 1 2 3 Ζ cΩp Ez Ew b Δn n0 Δnn0 Ez

Figure 3.2: Density variationδn = n − n₀ (dashed curve) and the axial electric fieldEznormalized byE0(solid curve). The Gray shaded region is the Gaussian

pulse,a = a₀exp[−(ζ + 5)2/22] witha₀= 1.5.

and the field is given by Eq. (3.23) such that ˜ E2 z = − 1 ϕ− ϕ + 1 α2 +α2 = a2₀− φ + 1 1 +a2₀ − 1 1 +φ.

Because ˜Ez is π/2 offset with φ, ˜Ez reaches the maximum when φ reaches 0. So that the maximum ˜Ez is

˜ Ezmax= a 2 0  1 +a2₀. (3.25)

We notice that in highly nonlinear regimea₀ 1, ˜E_zmax a₀, the acceler-ating gradient is linearly proportional toa₀, while in the linear regimea₀ 1,

˜

Ez a2₀, the accelerating gradient is proportional to the quadratic ofa0, which

is consistent with the result in the linear regime Eq. (3.10).

If in general cases with arbitrary laser pulse shapes, numerical calculations are essential to solve the equation Eq. (3.17). Assuming a circularly polarized gaussian pulse,a(ζ) = a₀exp[−(ζ −ζ₀2/22)], we plot the solution of Eq. (3.17) in

25 20 15 10 5 0 0.010 0.005 0.000 0.005 0.010 Ζ cΩp Ez Ew b Δn n0 Δnn0 Ez

Figure 3.3: Density variationδn = n − n₀ (dashed curve) and the axial electric field Ez normalized by E0 (solid curve) for Gaussian pulse a = a0exp[−(ζ +

5)2/22] witha₀= 0.1.

Fig. 3.2 and 3.3 fora₀= 1.5 and 0.1. Here the solid curve presents the plasma wakefield and the dashed curve presents the density perturbationδn = n − n₀. In the nonlinear case (a₀= 1.5), the plasma wakefield exhibits a sawtooth-like shape and the plasma density piles up as a delta function. These were caused by the totally expelled electrons from a strong laser ponderomotive field. The plasma density piling up forms parallel charged plates which result in a linearly-varying electrostatic field between every two plates. As fora₀ = 0.1, the plot as shown in Fig. 3.3 is purely sinusoidal, consistent with the result in the linear regime.

Chapter 4

Plasma Wakefield

Acceleration in Magnetized

Plasma II

In Chap. 3 we have discussed the plasma wakefield underω  ωc. In which the dispersion relation in Eq. (3.1) approaches a linear relation ofω to k, whose phase velocities are roughly equal to the speed of light. It is appropriately utilized as the driving pulse for plasma wakefield excitation. But whenω < ωc, the phase velocities of modes below the light curve (magnetowaves) are generally much less than c and vary with different k. Therefore a magnetowave pulse which is composed of different modes will quickly spread out during traveling. Nevertheless, we will show that, under a special condition (MPWA condition), the magnetowave will behaves like the light in vacuum and can be considered as a new type of driving pulses. In this chapter, we will focus on the magnetowave modes and establish the general theory of MPWA.

4.1 MPWA Condition

The idea of MPWA is first working on Alfven modes[26]. Alfven wave is a magnetic tension wave, only existing in magnetized plasma (medium wave) and having a very low phase velocity. However for an effective plasma accelerator,