基於相對論性雷射電漿飛鏡之模擬黑洞研究

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國立臺灣大學理學院物理學系碩士論文

Department of Physics College of Science

National Taiwan University Master Thesis

基於相對論性雷射電漿飛鏡之模擬黑洞研究

Analog Black Hole Based on Relativistic Laser-Plasma Flying Mirror

劉詠鯤 Yung-Kun Liu

指導教授：陳丕燊博士 Advisor: Pisin Chen, Ph.D.

中華民國 109 年 6 月 June, 2020

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誌謝

三年的碩士生涯，即將到了尾聲。對於未來要繼續在學術這方向上前進的決定，除了期待還有點擔心，不太確定自己的能力是否能在這條路上立足。回想在碩士班研究期間，也是遇到許多迷茫與挫折，但也受到了許多人的幫助與鼓勵。希望藉此機會回顧，期許自己不會忘記這些時光以及各種收穫，也作為將來繼續邁進的養分。

首先，很感謝我的指導教授陳丕燊老師。老師深具啟發性的指導，讓我了解到物理學不同領域之間，並非是獨立無關的。透過深入研究不同領域，

往往可以找到互相借鑑、啟發的想法，甚至能夠做出重大的突破。例如我碩士期間所參與的類比黑洞計畫，即為結合雷射電漿以及宇宙學所提出的跨領域構想。這也提醒我在未來的學術生涯中，要時時保持對各領域的好奇以及勇於發想創意的熱情。感謝口試委員裴思達老師以及汪治平老師在組會以及口試時對於論文以及研究方向提出的意見。尤其是汪治平老師，總是會以實驗學家的嚴謹精神，督促我仔細考慮清楚每一個細節。使我不會得過且過，能夠盡量將每個環節想清楚，這過程往往可以成為更進一步研究的穩固基石。陳老師的敏捷熱情與汪老師的謹慎細緻，都是我未來努力的目標。此外，也很感謝幫我撰寫博士班推薦信的趙挺偉老師、薛熙于老師，在兩位老師的數值物理相關課程中，學習到了非常多的數值計算知識，對於進入模擬領域，有非常大的幫助。

此外，特別感謝研究室的博士後方遠學長。感謝學長不厭其煩的和我討論未來的研究方向、如何改進研究成果、如何撰寫期刊論文等等。如果沒有學長的幫忙，我碩士研究期間應該會走非常多的彎路、面對迷茫而不知所措。感謝在捷克的合作對象，Petr Valenta 對於我在模擬上提供了非常多的

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建議，期待我們能繼續合作。感謝中央大學的劉耀澧學長，引導我走進了 PIC 模擬的大門。

此外，感謝 LeCosPA 的所有成員們，一起 meeting、茶會、期刊讀書會等等活動，從大家身上學到許多東西。感謝 AnaBHEL 組內的學弟妹們，至恩、冠男教了我許多場論、理論相關內容。和為寧、冠宏討論模擬、理論相關問題時，激盪出了不少的想法。特別感謝秘書們，幫我們安排會議時間表、處理各種行政事務，以及最重要的: 帶我們出去聚餐!

感謝我的朋友們，家恩、育翔、中鳴、健庭、能賢、宇霆、芸瑄等等。

研究碰到阻礙時，很幸運有你們可以一起聊天、走走，不同的生活、背景總能使我走出自己的小圈圈，使我重新以不同角度思考問題。也謝謝系羽的大家，孟哲學長、邦漢、聖義，努力地找我去打球，為我的生活注入更多活力。也感謝在口試前一段時間，進入我生活圈的靜文，一起聊天、散步，為我分散了非常多準備口試、趕論文的焦慮。

特別感謝我的家人們，爸爸媽媽對於我不走尋常路、追求理想的無條件支持，使我能夠十分放心的做自己想做的事。爸爸持續讀法律書充實自己、

媽媽工作之餘參與編撰教科書以及哥哥同時準備司法官考試以及碩士的強大毅力，家人們身體力行的精進，給予我很大的鼓勵與鞭策。住在台北的秋味姑姑時常美食、水果、點心支援，是我生活中堅強的後勤部隊。十分感謝我的家人對我的全方位支持與幫助，沒有你們，就沒有今天的我。

最後，有太多太多的人需要感謝，感謝你們的鼓勵、提醒、刺激。謹以此論文獻給諸位!

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摘要

自 1974 年霍金提出黑洞蒸發理論後，關於霍金輻射是否導致資訊遺失的爭議，持續吸引理論物理界之興趣。但由於宇宙中自然存在的黑洞放出之霍金輻射過於微弱，超出了目前所具備的觀測能力。為了更深入研究此問題，以及和理論預測互相驗證，在實驗室中產生「類比黑洞」的設計被陸續提出。其中一種模擬黑洞模型：飛鏡模型（Flying Mirror Model）描述一在閔考斯基空間具有特定移動軌跡的邊界，可以用來類比彎曲空間附近的物理。基於此飛鏡模型以及強場雷射在電漿中產生相對論性電子飛鏡的現象，

Chen and Mourou 於 2017 年提出了「桌上型類比黑洞實驗」(Analog Black Hole via Lasers, AnaBHEL) 之構想。

本論文研究內容，主要集中於此雷射電漿飛鏡之性質，例如：此飛鏡之反射率、入射雷射及飛鏡交互作用後之反射頻譜以及如何透過改變背景電漿密度控制飛鏡軌跡等等問題。透過數值模擬及電漿理論模型，我們對於雷射電漿飛鏡進行深入的分析研究，能提供 AnaBHEL 實驗更多必要資訊。

在第一章中簡單回顧了霍金輻射及資訊遺失悖論的議題，以及類比黑洞的概念。我們主要介紹了類比黑洞的飛鏡模型，不同的飛鏡軌跡，會釋放出不同的能量通量 (Energy Flux) 及頻譜。此外介紹了在研究雷射電漿交互作用使用的模擬工具、理論。模擬部分，我們介紹了粒子網格模擬 (Particle In Cell Simulation) 中用到的概念及重要的演算法。理論部分，回顧了在雷射電漿交互作用領域中用到的基本概念。

第二章我們討論了背景電漿密度是如何影響相對論性飛鏡之速度。飛鏡速度除了在類比黑洞實驗中扮演重要的角色外，也在雷射尾場粒子加速器中有著重要的影響。我們首先介紹了雷射在電漿中產生非線性尾隨場的理論。

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之後，我們利用此理論，對於飛鏡和驅動光的距離進行理論討論。再來介紹了在梯度電漿密度背景下，計算飛鏡速度之兩種方法。在以往文獻中，第一面相對論性飛鏡被認為和驅動雷射相距一個電漿波長，但我們透過理論研究發現，此距離和電漿波長實際相差一個係數。若無考慮此係數的修正，以上兩種估算飛鏡速度的方法，皆會高估速度的改變量。

第三章我們討論了相對論性飛鏡的反射率問題。相對論性飛鏡為一層密度極高的電子組成，其反射率可以透過估計此電子層密度分布以及解一入射電磁波在此電子層上的邊界條件、波動方程式來進行計算。我們首先回顧了過往對於此問題的研究，在將過往研究結果和我們執行的一維模擬比較時，

我們發現在特定的情況下，以往之電子密度模型對反射率有高估的現象。因此我們根據模擬中的電子分布，提出不同的擬合模型，並獲得和模擬數據吻合的結果。此外，以往研究集中討論於一相對論飛鏡及平面波的交互作用。

本章後半，我們將此理論延伸至具有有限脈寬的入射波 (以高斯分布為例)，

發現有限脈寬入射波的反射頻譜，其峰值會和平面波結果具有一定偏移。

關鍵字：類比黑洞、粒子網格模擬、雷射電漿交互作用、相對論性飛鏡、

雷射電漿尾隨場

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Abstract

Since Hawking proposed the theory of black hold evaporation in 1974, the debate that whether Hawking radiation causes information loss attracts theoretical physicists. One way to set down the debate is through direct ob- servation. However, the Hawking radiation emitted by astrophysical black holes is too weak to be observed due to the large mass of black hole. To dig into this issue and verify the theoretical predictions, several schemes of

“Analog Black Hole” had been proposed to observe the black hole radiation in the Lab. One of these Analog Black Hole models, the flying mirror model, describes that a boundary with specific trajectory in Minkowski space can mimic the physics around curved-spacetime. On the basis of this model and the phenomenon that an intense laser can generate a relativistic flying mirror in plasma, Chen and Mourou proposed the experiment “Analog Black Hole via Lasers, AnaBHEL”.

This thesis mainly focuses on properties of laser-driven flying plasma mirror, such as the reflectivity, the reflected spectrum as an incident laser pulse interacts with the mirror and the relation between the trajectory of the flying mirror and the background plasma density. These studies are based on numerical simulations and cold collision-less plasma theory. These studies can provide essential information for the AnaBHEL experiment.

In chapter 1, we briefly review the issue about Hawking radiation, information loss paradox and proposals about analog black hole. In the flying mirror model, different trajectories of the flying mirror emit different energy

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flux and frequency spectrum. Besides, we introduce the simulation tool and the theory to study laser plasma interaction. In simulation part, we explain the concept and algorithm of Particle In Cell simulation. In theory part, we review the basic plasma theory and the interaction between laser and plasmas.

In chapter 2, we describes how the background plasma density affects the velocity of the flying mirror. The velocity of mirror plays an important role in not only analog black hole experiment but also the Laser Wake Field Ac- celerator (LWFA). We first introduce the one-dimensional nonlinear theory of the laser-driven wakefield and utilize this theory to investigate the distance between driver laser pulse and the flying mirror. Then, we review two methods to calculate the velocity of flying mirror in an inhomogeneous plasma background. In previous literature, the distance between first plasma mirror and the driver is thought to be a plasma wavelength. However, we find the distance differs from plasma wavelength by a coefficient. With this corrected term, the velocity of flying mirror can be calculated more accurately.

In chapter 3, we study the reflectivity of the flying mirror. The relativistic flying plasma mirror is composed with a dense shell of electrons. The reflectivity can be estimated by the density distribution of electrons and solving the wave equations with proper boundary condition of an incident wave. First, we review previous studies on this problem. We found previous model of the electron distribution seems to overestimate the reflectivity compared to 1D simulation results. Therefore, we proposed a density distribution fitting model and get results which agree well with simulation data. Besides, previous study mainly discussed the interaction between the flying mirror and a plane incident wave. In the second half of this chapter, we extend the study to a finite bandwidth incident wave (the Gaussian profile is considered). We find a deviation of the peak frequency of reflected spectrum exists compared to the result of a plane wave.

Keywords: Analog Black Hole, Particle In Cell Simulation, Laser Plasma

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Interaction, Relativistic Flying Mirror, Laser Driven Wakefield

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List of Figures

1.1 Progress of peak laser intensity since 1960 [1]. . . 8

1.2 Schematic of PIC routine. . . 11

2.1 Wake wave potential with an optimal-length driver in linear limit. The bubble width is three quarters of linear plasma wavelength. . . 25

2.2 Solution domain of the wakefield with a non-optima-length driver pulse.

The domain is separated into three regions. . . 26

2.3 Wake wave potential with an ultra-short driver. The bubble width is shown to be three quarters of the plasma wavelength. . . 30

2.4 Dependence of α on the normalized pulse length. . . . 32

2.5 Comparison between PIC data and theoretical results of velocity of the flying mirror. Curves are calculated numerically from Eq.(2.55), (2.66).

Circles are PIC simulation data. Driver with different a₀ are considered. . 33

2.6 Phase velocity of the flying mirror in a gaussian down-ramp plasma background. Solid lines are analytical predictions. Dots are 1D PIC data. It can be seen that α = 3/4 is a good approximation as long as the ultra-short pulse condition holds. . . 35

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3.1 Reflectivity of the relativistic flying mirror as a function of the source fre- quency ω_s. Solid lines are calculated from different reflectivity models (Eq.(3.40)-(3.42)). Distinct symbols are PIC simulation results with dif- ferent λ_s. SRLD model agrees well with PIC results and the cusp model approaches SRLD when a longer wavelength source is applied. Inset:

Comparison between three density distribution (Eq.(3.29)-(3.31)) models and the density of flying mirror from PIC simulations. Note that the PIC data (circles) is almost overlapped by SRLD (blue line). . . 49 3.2 Normalized reflected electric field amplitude calculated by Eq.(3.47) (blue

curve) and the naive estimation with ω = 4γ_m²ω₀ (red curve). The black curve shows the decaying term in Eq.(3.47) and is also normalized to the value calculated with ω = 4γ_m²ω₀. The deviation of both the frequency and amplitude at the peak of spectrum is demonstrated. . . 52 3.3 Dependence of δ on the pulse duration T and the Lorentz factor γ with

other parameters fixed. The deviation is evident for few cycle source pulse or flying mirror with higher lorentz factor. . . 53 3.4 Comparison among the estimated double-Doppler shift frequency (yellow

line), the theoretical prediction of ω_peakfrom Eq.(3.47) (blue line), and the PIC simulation result (red dots). γ_m = 4, n_m,0 = 3n_cand τ_s = 1.5T_s are used as the initial condition. The 1D PIC result agrees reasonably with theoretical prediction and the linear dependence of the deviation on mirror thickness is also illustrated. . . 54

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List of Tables

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Chapter 1 Introduction

1.1 Analog Black Hole and Moving Mirror Model

Black holes are sites with enormous gravity. In classical theory, the gravitation of black hole is so huge that nothing, not even photon can escape from this gravitational singular- ity. However, in 1974, within the framework of quantum field theory in curved spacetime, Stephen Hawking discovered that the quantum effect allows black hole to emit black body radiation [2], the so-called Hawking radiation. The Hawking radiation reduces the mass and angular momentum of the black hole, therefore leads to the “black hole evaporation”.

Such process may result in the loss of information [3]. Conservation of information, or probability, in a physical process is a fundamental basis of quantum mechanics and quantum field theory. The possibility that black hole evaporation may result in the loss of information therefore implies a conflict between general relativity and quantum theory, the two fundamental pillars of modern physics. There have been proposed solutions and endless debates about this paradox over the past 40 years, but are essentially all theoretical (see [4] for more details). The difficulties to observe black hole evaporation in our universe is due to the gentle evaporating rate. Without absorbing extra energy, a solar mass black hole will evaporate over 10⁶⁴ years which is apparently longer than the life of the universe. Accordingly, to conquer the information loss paradox, ideas of analog black hole are resorted. Unruh proposed the idea of acoustic black hole [5] to construct the horizon in the fluid system. Based on this scheme, analog black hole based on Bose-

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Einstein condensate [6, 7, 8] and Superconducting Quantum Interface Device (SQUID) [9]

are all demonstrated. On the other hand, it has been long recognized that a time-dependent Dirichlet boundary condition in 1+1D Minkowski spacetime is possible to generate particles out of the initial vacuum state [10, 11, 12]. The particle generation origins from the interaction between a moving boundary and the vacuum fluctuation of the quantized fields, therefore these phenomena are termed the names:“Dynamical Casimir Effect” or

“Moving Mirror Model”. The analogy between black hole evaporation and moving mirror model had been investigated in [13]. Based on this analogy, the idea of Analog Black Hole Evaporation via Lasers (AnaBHEL) [14] was proposed in 2017.

1.1.1 1+1D Moving Mirror Model

The moving mirror model in 1+1D can be described by a quantized massless scalar field ψ(t, x) in flat spacetime subjects to the Dirichlet boundary condition, ψ(x = z(t), t) = 0, where z(t) is the trajectory of the mirror. This boundary condition forces the field to disappear on the boundary therefore describes a perfectly reflecting mirror. The scalar field satisfies the Klein-Gordon equation,

□ψ = (−∂_t²+ ∂_x²)ψ = 0. (1.1)

The inner product of any two solutions of Eq.(1.1) is defined by,

(ϕ₁, ϕ₂) =−i Z

Σ

dΣ^µ[ϕ₁←→

∂_µϕ^∗₂] =−i Z

Σ

dΣ^µ[ϕ₁∂_µϕ^∗₂− ϕ^∗2∂_µϕ₁], (1.2)

where Σ is a Cauchy surface and dΣ^µis the unit-vector orthogonal to that surface. The orthogonal basis of the solutions can be constructed with Eq.(1.2), which obeys

(u_i, u_j) = δ_ij, (u^∗_i, u^∗_j) =−δij, (u_i, u^∗_j) = 0. (1.3)

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After expanding the field operator with a given basis, the creation and annihilation oper- ators a^†_i, a_iare obtained,

ϕ(x) =X

i

[a_iu_i(x) + a^†_iu^∗_i(x)]. (1.4)

The concept of particles is defined based on a^†_i, a_i, for example, the vacuum state is constructed by a_i|0⟩ = 0.

In 1+1D moving mirror model, it is convenient to move x− t coordinate to the u − v null coordinate defined with u = t− x and v = t + x because the massless modes are all null. Let us consider the condition without a moving mirror, the right-moving modes with positive and negative frequency are

ϕ_ωu= 1

√4πωe^−iωu, ϕ^∗_ωu = 1

√4πωe^iωu, (1.5)

respectively. The left-moving modes are,

ϕ_ωv = 1

√4πωe^−iωv, ϕ^∗_ωv = 1

√4πωe^iωv. (1.6)

These modes form a set of basis to represent the scalar field in the whole spacetime and the expansion is unique. However, if the moving mirror exists, two different sets of mode, ϕ_ω and χ_ω′ must be used to decompose the field due to the condition introduced by the mirror. In the literature, ω and ω^′ may be used to distinguish the different set of modes. This leads to different definitions of creation and annihilation operators and therefore different definitions of particle states. The transformation between different modes is the “Bogoliubov transformation”,

ϕ_ω = Z _∞

0

dω^′[α^∗_ω′ωχ_ω′ − βω^′ωχ^∗_ω′], (1.7) χω^′ =

Z _∞

0

dω[αω^′ωϕω+ βω^′ωϕ^∗_ω], (1.8)

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where αω^′ω and βω^′ωare the Bogoliubov coefficients that can be evaluated as

α_ω′ω = (χ_ω′, ϕ_ω), (1.9)

β_ω′ω =−(χω^′, ϕ^∗_ω). (1.10)

To see the physical meaning of the Bogoliubov coefficients, the expectation value of num- ber operator N_ω = a^†_ωa_ω in the vacuum of χ_ω′ is

⟨0χ_ω′|Nω|0χ_ω′⟩ = Z _∞

0

dω^′|βω^′ω|². (1.11)

Therefore, non-zero β_ω′ωimplies that the vacuum state defined by the two mode functions are different. The vacuum state for the first mode function is not vacuum for the second one but particles exist. This is an important result in quantum field theory in curved spacetime : vacuum state may not be universally unique.

Consider a perfectly reflecting moving mirror with timelike trajectory z(t), the solu- tions of Eq.(1.1) are,

ϕ_in,ω′ = 1

√4πω^′[e^−iω^′^v − e^−iω^′^p(u)], (1.12)

= 1

√4πω[e^−iωf(v)− e^−iωu], (1.13)

where p(u) and f (v) are called “ray tracing functions” which guarantee the mode functions to vanish on the mirror,

p(u) = 2t_u − u, u = t_u− z(tu) (1.14) f (v) = 2t_v− v, v = t_v+ z(t_v). (1.15)

The terms t_v and t_u can be understood as the time coordinate when the null rays and the mirror intersect. For a non-asymptotically null mirror, the general expression of β_ω′ωcan

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be derived [15],

β_ω′ω = 1 4π√

ω^′ω Z _∞

−∞

dv(ω^′− ωf^′(v))e^iω^′^{v+iωf (v)} (1.16)

= 1

4π√ ω^′ω

Z _∞

−∞

du(ω^′p^′(u)− ω)e^iωu+iω^′^p(u). (1.17)

Generally speaking, the particle spectrum of any trajectory of mirror can be calculated with the Bogoliubov coefficients. However, the analytical form of ray-tracing functions are difficult to obtain for arbitrary trajectories. Therefore, the analytical result only exists in quite limited case (see summary in [15]). Here, we briefly introduce two different famous trajectories which emit thermal radiation: the modified Davies-Fulling (DF) and Carlitz-Willey (CW) trajectories.

The most famous moving mirror trajectory is DF trajectory [16] because it was the first trajectory proposed to understand the appearance of a thermal spectrum. However, the original calculation utilizes some subtle approximations such that Fulling concluded that although the final results still hold, the approach may have an error [17]. In [15], the author suggested a “late time Davies-Fulling” trajectory which can prevent obscure approximations.

z(t) =









−t − Ae^−2κt+ B t → ∞,

0 t < 0,

(1.18)

where A, B are some constants and κ characterizes the acceleration of the mirror. The velocity of the mirror is 0 initially and approaches−1 in the future infinity (t → ∞). The Bogoliubov coefficient is,

|βω^′ω|² ≈ 1 2πκω^′

1

e^2πω/κ⁻¹ for ω^′ ≫ ω, t → ∞, (1.19)

which describes a late-time thermal emission in the high frequency limit, ω^′ ≫ ω.

On the other hand, the CW trajectory [18] gives all-time thermal spectrum and constant

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energy flux. The trajectory written in the form of z(t) is found in [15],

z(t) =−t − 1

κW (e^−2κt), (1.20)

where W is the Lambert W function (product logarithm) and κ > 0 is a characteristic parameter about the mirror acceleration. The Bogoliubov coefficient is

|βω^′ω|² = 1 2πκω^′

1

e^2πω/κ⁻¹, (1.21)

which is a thermal spectrum in 1+1D. The corresponded temperature of CW trajectory, k_BT = κ/2π, is constant for all time, therefore may be an analogy to an “Eternal Black Hole”.

The correspondence between trajectories and the emitted particle spectrum provides a way to investigate the evolution of black hole. Accordingly, different trajectories may mimic different candidates of the end stage of black hole [19]. In 2017, Pisin Chen and Gerard Mourou proposed a novel experimental concept using ultra-intense lasers to in- duce flying mirrors in plasmas with graded density [20]. With a tailored plasma density, different trajectories can be fulfilled [21] and provide a way to investigate different evolution of black hole. The relation between plasma density profile and the mirror trajectory will be discussed with more details in Sec.2.

1.2 Plasma

The term “Plasma” is first introduced by Langmuir in 1928 [22] to describe the ionized gas near the electrode and usually called the “fourth fundamental state of matter”. The plasma consists of a gas of ions and free electrons. Unlike usual gas that is an insulator, the conductivity of the plasma can be treated as infinity due to the free electrons. Typically, the plasma only exists in vacuum. Otherwise, the surrounding air will cool down the plasma such that free ions and electrons will recombine into neutral atoms. Therefore, on the earth, plasma state is rare near ground due to the atmosphere. Only when high energy

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source exists, the plasma state can maintain, such as the gas near lightning or in ionosphere where high energy cosmic rays intensely collide with the air molecular. On the other hand, plasma state is common in the universe, such as the stellar interiors, gaseous nebulas and the most part of galaxies.

The plasma can be defined as follows [23]:

“A plasma is a quasineutral gas of charged and neutral particles which exhibits col- lective behavior.”

The “quasineutral” property describes the characteristic length scale of the plasma system. A fundamental property of the plasma is the ability to shield the electric potential applied on it. The shielding length is described by the Debye length λ_D =p

k_BT /4πn₀e², where k_Bis the Boltzmann constant, T , e and n₀are temperature, charge and the number density of electrons, respectively. Therefore, for the system with scale L much larger than λ_D, the plasma can be considered as “neutral”. The second property “collective”

implies that the plasma oscillation frequency ω_p =p

n₀e²/m_eϵ₀ is much larger than the collision frequency between electrons and neutral particles. This means the electrostatic effect dominates over the gas kinetics of neutral gas.

1.3 Laser Plasma Interaction

After Einstein investigated the relation between the stimulated and the spontaneous emission, people were considering a new way to amplify the electromagnetic field using this phenomenon. Tens of years later, the first working optical laser was finally invented by Maiman in 1960 [24] using the Ruby crystal as the gain medium. After that, this intense and coherent light source got success in quite diverse field, such as military and industry, not to mention the scientific research. In the past tens of years, new techniques to deliver high power and short pulse or extend available wavelength had been developed. Among these progress, we mainly focus on the blooming of the ultra-short pulse laser, which usually refers to the laser with pulse duration from pico-second (ps, 10⁻¹²s) to femto-second (fs, 10⁻¹⁵). In Fig.(1.1), the progress of peak intensity of the laser since 1960 is shown.

The increase of the intensity reaches a plateau in around 1970. At that time people can not

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amplify the light pulse further without damaging the laser gain medium. The invention of Chirped Pulse Amplification (CPA) [25] conquered this dilemma and opened a new era of ultra-short high intensity laser. Thanks to this technique, the laser peak intensity is still growing nowadays. Due to the short interaction time between the laser and the material, the ultra-short-pulse laser is widely used in material processing, cornea surgeries, molecular interaction, and so on. Among these applications, the one relevant to this thesis is the Laser Wakefield Accelerator [26]. When an intense laser propagates in the plasma, a wake field (longitudinal electric field) will be generated. The wake field can realize an accelerating gradient (∼ 100 GV /m) [27] that is much larger than the conventional radio-frequency accelerator (∼ 100 MV /m). This provides a promising way to construct next-generation accelerator for pursuing higher particle energies or more compact facili- ties.

Figure 1.1: Progress of peak laser intensity since 1960 [1].

The intensity of laser can be linked to a Lorentz invariant dimensionless “laser strength parameter”, a₀ = eE₀/m_eωc, where E₀ and ω are the electric field and angular frequency of the laser, with the relation

a₀ = 0.85λ[µm]p

I[10¹⁸W cm⁻²], (1.22)

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where λ, I are the wavelength and intensity of the laser. For a0 ≥ 1, the laser is called

“relativistic laser” or referred to as in the “nonlinear regime”. Roughly speaking, this terminology can be understood as follows. eE₀/m_e is the acceleration of electrons in electric field and 1/ω is roughly the order of a laser period. Therefore, eE₀/m_eω stands for the velocity that electrons can be accelerated in a laser period. Accordingly, a₀ ≥ 1 implies the electric field can accelerate electrons to near speed of light during one laser cycle, therefore relativistic effect should be taken into account. The intensity of the state- of-the-art 800nm Ti:Sapphire laser can achieve 10²³W cm⁻² [28], which corresponds to a₀ ∼ 70. This highly intense laser provides extreme light pressure within a very short time scale and acquires wide applications in the frontier scientific research (see review [29] for more discussions).

The interaction of ultra-short-pulse laser and matter can be studied from a simple case:

the interaction between a single electron and planar electromagnetic field. The motion of electrons can be described by the Lorentz equation,

dp dt =−e

E + v× B c

, (1.23)

where p = γm_ec²is the momentum and γ = p

1 + p²/m²_ec²is the Lorentz factor associ- ated with the electron. The evolution of electron energy follows

d

dt(γmec²) = −e(v · E). (1.24)

Note that the magnetic force v×B is always perpendicular to the trajectory of the electron, therefore does not contribute to the change of the electron energy. Besides, the laser pulse with frequency ω₀ can propagate in the plasma provided the plasma density is less than the “critical density” n_cthat is defined by

ω₀² = e²n_c meϵ0

, (1.25)

which corresponds to a plasma density such that the nature oscillatory frequency of the

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plasma is equal to the applied EM field frequency. As the nature frequency ωpbeing higher than ω₀, the electrons can react to the external field and damp it away. This makes the EM wave can only propagate into the plasma with a skin depth L_s= c/ω_p. On the other hand, if the plasma frequency is much smaller than the EM field frequency (ω_p ≪ ω0), referred to as “underdense plasma”, the EM field can propagate inside the plasma.

1.4 Particle In Cell Simulation

Traditionally, the investigations of phenomena in nature are carried out by experimental and theoretical techniques. Thanks to the rapid advance in computational power, computer simulation gradually becomes the third choice and benefits from the low cost compared to doing actual experiments and the ability to deal with complex physical systems. To numerically study the plasma behavior, there are two widely adopted methods. The first one is Magneto-hydrodynamics / Hydrodynamics (MHD/HD), in which the plasma is treated as fluid. The other one is particle-in-cell (PIC) simulation [30], where plasma is statistically sampled as macro-charged particles and the equation of motion is calculated kinetically.

In general, MHD/HD methods are mostly used for investigating phenomenon of time scale larger than nanometer. On the other hand, PIC is for shorter time scale interaction, such as pico-second or femto-second , where the thermal equilibrium state is not arrived. In the scope of this thesis, the driver pulse is in fs scale and the interaction period is sub-ps.

Therefore, we choose PIC as our numerical tool to study the laser plasma interaction.

PIC method combines the kinetic theory of plasma with Electromagnetic theory. Pro- vided that the collision frequency between plasma is much smaller than the nature oscillation frequency of plasma, the system can be described by the collision-less Vlasov equation. In relativistic regime, the equation takes the form,

h∂

∂t+ p

msγ ·q_s(E + v× B) ms · ∇p

i

f_s(t, x, p) = 0, (1.26)

where subscript s denotes species. In the implement of simulation code, the procedures

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can be divided into two main parts: particle pusher and field solver. As the name suggests, particle pusher is responsible for moving particles with known electromagnetic force and field solver accounts for solving electromagnetic field from charge density and current.

The routine of PIC simulation is summarized in Fig.(1.2).

In the following subsections, we briefly introduce the concept and algorithm used in PIC simulation.

Figure 1.2: Schematic of PIC routine.

1.4.1 Finite Sized Particles

In principle, the particle simulation code should calculate the position and momentum of all particles in the system. However, even with the state-of-art supercomputer, it is impractical to consider interaction between 10²⁰particles, which is a typical number en- countered in the plasma experiment. On the basis that the phenomena we usually concern about plasma are “collective”or “macroscopic”effect, huge amount of particles can be represented by a quasi-/pseudo-/macro- particle. This strategy drastically reduces the simulation particle numbers and makes computer simulation of plasma possible. Instead of treating the macro-particle as a point charge, finite sized particle is introduced. The reason comes from the fact that macro-particles overestimate Coulomb force, which is proportion to the multiplication of charge, among themselves and have divergent force in short range.

Finite sized particles can eliminate the overestimation, more detail about this problem can be found in [30].

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1.4.2 Field Solver

The electromagnetic fields (E, B) generated by charge density ρ and current J are de- scribed by Maxwell equations:

∇ · E = ρ

ϵ₀ (1.27)

∇ · B = 0 (1.28)

∇ × E = ∂B

∂t (1.29)

∇ × B = µ0J + 1 c²

∂E

∂t. (1.30)

To solve the field numerically, the problem domain is discretized into grids/cells. Sev- eral techniques for solving the electromagnetic field on the grids are available, e.g. finite- difference time-domain (FDTD), finite element method (FEM) and fast Fourier transform (FFT). The last two methods transform the partial differential equation problems into a global Eigen-value problem. Here, “global”means the solution of a specific point may depend on all the points in the problem domain. This property makes global field solver hard to be implemented efficiently in parallel computation. In modern PIC code, the widely used parallization technique, domain decomposition, demands the minimization of exchanging data between subdomains and implement of local equation solvers. There- fore, the FDTD method is in common use. The procedure begins with obtaining the density of plasma, which is extrapolated from the macro-particles onto the grid. After acquiring the density, the current can be evaluated with the help of continuity equation,

∇ · J +∂ρ

∂t = 0. (1.31)

Combining with Maxwell equations, electromagnetic field, which is discretized on the so-called Yee-grid [31], can be obtained. The detail of this algorithm can be found in the book [32].

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1.4.3 Particle Pusher

The force experienced by a (relativistic) charged particle in electromagnetic field is described by the Lorentz force (cf. Eq.(1.23)). It should be noted that the fields are defined on the grid but particles are not. To calculate the field experienced by the macro-particles, the field should be interpolated to the position of particles first. After acquiring fields on each particle, the force can then be calculated. When updating the position of particles, the “leap-frog”scheme [33] is implemented :

x_k+1− xk

∆t = v_k+1/2 (1.32)

v_k+1/2− vk−1/2

∆t = q

m

Ek+ v_k+1/2+ v_k_−1/2

2 × Bk

(1.33)

The subscript k denotes the time steps. Velocity is defined at half-integer time steps, on the other hand, position and fields are defined at integer time steps. Therefore, the pro- cedure of the scheme is: update half-step quantity (v) with full-step one (E, B), and then update full-step value (x) with half-step one (v). This method can prevent the numerical instability in the naive method, that is updating all quantities in integer time steps. The discussion about the stability with different updating strategy can be found in [34].

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Chapter 2 Velocity of the Flying Mirror in Inhomogeneous Plasma

2.1 Introduction

The property of nonlinear plasma oscillation had been investigated in 1956 by Akhiezer and Polovin [35]. For the plasma wakefields (driven electron plasma waves) excited by an intense laser, a nonlinear one-dimensional theory was developed by Bulanov et al. [36];

Sprangle et al. [37, 38]; Berezhiani and Murusidze [39]. A set of coupled equations is derived to describe the vector potential of the laser field and the electrostatic potential of the plasma (wake potential). This model provides a self-consistent description of the interaction of intense laser with plasmas.

The wakefields can be a promising way to accelerate electrons due to its ability to sustain extremely large acceleration gradients. The scheme was first proposed by Tajima and Dawson [26], where the plasma wakefield is induced by the laser that traverses the plasma. Later, electron-bunch-driven accelerator was also proposed [40]. Among various laser plasma accelerator (LPA) configurations, the laser wake field accelerator (LWFA) is the most adopted scheme in modern LPA experiments. The phase velocity of the wake wave is a critical factor for determining the maximum energy gain, minimum injection energy, and the dephasing length of electrons. In an uniform plasma background, if the

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evolution of driver is neglected as it propagates, the phase velocity of the plasma wave is equal to the group velocity of the driver, which is called the principle of wakefield. On the other hand, by introducing density gradient in the plasma background, the phase velocity is no longer equal to the group velocity and can be controlled artificially. This technique had been applied to fulfill the “self-injection” of electrons in LWFA [41, 42].

On the pure theoretical side, in 1993, Wilczek [13] suggested that a flying mirror can serve to investigate the information loss paradox [3] using suitable mirror trajectories.

Based on this analogy, an experimental scheme was proposed by Chen and Mourou [20]

with the intent to investigate the information loss paradox using a laser-induced plasma flying mirror in a tailored plasma target. By carefully designing the plasma density, different trajectories can be realized to mimic different candidate resolutions to the information loss paradox [19].

In this section, we start from reviewing the procedure to calculate the solution of wake potential induced by a nonlinear laser field. The solution exists when an optimal-length flat-top laser is considered. After that, the term “bubble width” is defined as the distance from the energy average position of driver and the first density cusp, that is the first flying mirror. This distance has been long considered as equal to the plasma wavelength. How- ever, it is shown the ratio between bubble width and plasma wavelength is generally not equal to one. In the linear limit (a₀ → 0), the ratio is found to be three quarters.

For an inhomogeneous plasma background, the optimal-length condition of the driver can not be maintained due to the change of local plasma density. We therefore extend previous studies on optimal-length to a driver with non-optimal length in Section 2.4.

Unfortunately, a generally analytical solution can not be found in this case due to the inverse function appears in the solution is hard to be solved. However, as the driver is ultra-short, the approximated solution of wake potential can be found [37, 38]. Therefore, the ratio can be calculated and is found to be three quarters.

The bubble width ratio is important in calculation about the phase velocity of the flying mirror. In previous literature, based on the principle of wakefield, studies about the phase velocity of wake wave focuses on the nonlinear correction to the group velocity [43, 44,

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45]. For an inhomogeneous plasma, the phase velocity of the wake wave was studied both analytically and numerically using the dispersion relation in [46]. This method focuses on the local phase velocity. On the other hand, another method which focuses on the velocity of the flying mirror is proposed in [21] recently. With the fact that the flying mirror is a bubble width behind the driver, if the rate of bubble size change can be known, the velocity of flying mirror can be obtained. Therefore, the bubble width should be carefully treated to acquire accurate result. In Section 2.5, we use this method and the derived formulae to calculate the velocity of flying mirror and compare with PIC data.

2.2 Wave Excitation by an Electromagnetic Pulse

To study the wakefield excited by an EM pulse, one can consider a 1D model based on cold relativistic hydrodynamics and Maxwell’s equations. The plasma is assumed to be unmagnetised and ions are immobile. Consider the fluid is moving in z direction, the equation for electron momentum is

∂p

∂t + vz

∂p

∂z =−e(E +1

cv× B), (2.1)

where p = m₀γv, γ = p

1 + p²/m²₀c², m₀ and v are the electron rest mass and velocity, respectively.

The electromagnetic field (from here referred as “driver”) which propagates along z direction can be described by

E =−1 c

∂A_⊥

∂t − ˆz∂ϕ

∂z; (2.2)

B =∇ × A⊥, (2.3)

where A_⊥ = ˆxA_x+ ˆyA_yis the vector potential and ϕ is the potential for charge separation in the plasma (also called wake potential). Note here that the Coulomb gauge is considered (∇ · A = 0).

With Eq.(2.1)- (2.3), the perpendicular component of electron momentum can be found

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to be

p_⊥

m₀c = e

m₀c²A_⊥ ≡ a(z, t). (2.4)

This shows the conservation of momentum in the transverse direction. The Lorentz factor can be separated into transverse and longitudinal direction:

γ =

"

1 +

p_⊥ m₀c

2

+

p_z m₀c

2#1/2

≡ γaγ_∥, (2.5)

where γ_a = (1 + a²)^1/2 and γ_∥ = (1 − v²z/c²)^1/2 are the transverse and longitudinal gamma factor. To complete the description of the plasma fluid, we need the longitudinal component of Eq.(2.1), continuity equation, Poisson equation and the wave equation for the driver. The longitudinal component of Eq.(2.1) is,

1 c

∂

∂t(γ_ap

γ_∥− 1) + ∂

∂z(γ_aγ_∥) = ∂φ

∂z, (2.6)

where φ≡ |e|ϕ/m0c²is the normalized scalar potential. The continuity equation is

1 c

∂n

∂t + ∂

∂z(nβ_∥) = 0, (2.7)

where β_∥ = v_z/c. The Poisson’s equation is,

∂²ϕ

∂z² =−4πρ, (2.8)

where ρ is the charge density. In the system, the ion is assumed to be immobile due to the large mass compared to electrons (n_ion = n₀ everywhere). Therefore, the charge density is ρ = n_e− nion = n− n0. With the definition of ambient plasma wave number k²_p ≡ 4π|e|²n0/m0c², the Poisson equation can be written into

∂²φ

∂z² = k²_p

n n₀ − 1

. (2.9)

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The wave equation is

∂²A

∂z² − 1 c²

∂²A

∂t² = −4π

c J, (2.10)

where the current J = n0ev_⊥. With the definition of normalized vector potential a and the conservation of transverse momentum (Eq.(2.4)), the equation can be reformed into,

c²∂²a

∂z² −∂²a

∂t² = k_p² n

n₀β_⊥ = k_p² n n₀

a

γ. (2.11)

It is convenient to transform from the lab frame coordinate (x, t) to a co-moving co- ordinate with the speed of the driver (ξ, τ ), where ξ = x− vgt and τ = t. The derivative in the co-moving coordinate is ∂/∂x = ∂/∂ξ and ∂/∂t = ∂/∂τ − vg∂/∂ξ. With this transformation, Eq.(2.6),(2.7),(2.9) and (2.11) become

∂

∂ξ[γ(1− βgβ_∥)− φ] = −1 c

∂

∂τ(γ_aβ_∥), (2.12)

∂²φ

∂ξ² = k²_p

n n₀ − 1

, (2.13)

∂

∂ξ[n(β_g− β∥)] = 1 c

∂n

∂τ, (2.14)

1 γ_g²

∂²

∂ξ² + 2β_g c

∂

∂ξ∂τ − 1 c²

∂

∂τ²

a = k²_p n n0

a

γ, (2.15)

where β_g = v_g/c. Eq.(2.12)-(2.15) form a complete set of fully nonlinear, relativistic, cold fluid equations which describe the 1D laser-plasma interaction. The 1D model is valid provided that the spot size of driver is much larger than the plasma wavelength, i.e., r_s ≫ λp. The set of equations can be further simplified with the so-called quasistatic approximation [37]. This approximation implies that if the laser pulse is sufficiently short, there exist a quasistatic state for the macroscopic quantities, n, β_∥ and γ. More explicitly, this approximation means that the right-hand side of Eq.(2.12) and (2.14) can be neglected, that is ∂/∂τ ≪ ∂/∂ξ for the macroscopic quantities. In this case, the first integral of

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Eq.(2.12) and (2.14) can be evaluated:

γ(1− βgβ_∥)− φ = 1 (2.16)

n(β_g− β∥) = n₀β_g, (2.17)

with integration constant chosen such that for γ_a = 1, n = n₀, β_∥ = 0, φ = 0. The initial condition means that when there is no driver field initially, no perturbation of density and wake potential exists. With Eq.(2.16) and (2.17), Eq.(2.12)-(2.15) can be reduced to,

d²

dξ²γ(1− βgβ_∥) = k_p² β_∥

βg− β∥, (2.18)

2 ∂

∂τ

iω₀a0+ cβ₀∂a₀

∂ξ

+ c²ω_p0² ω₀²

∂²a₀

∂ξ² =−ωp0²

1− β_g γ(βg− β∥)

a0. (2.19)

In the ultra-relativistic limit (β_g ≈ 1), the equations can be simplified (using the potential φ from Eq.(2.16)),

d²φ dξ² = k_p0²

2

γ_a²

(1 + φ)² − 1

(2.20) 2 ∂

∂τ

iω₀a₀+ cβ₀∂a₀

∂ξ

+ c²ω_p0² ω₀²

∂²a₀

∂ξ² =−ωp0²

φ

1 + φa₀. (2.21)

These two equations together describe the evolution and coupling between the wake potential and the driving laser field.

2.3 Bubble Width with an Optimal-Length Pulse

The evolution of wake field which excited by a driver laser pulse can be described by the coupled equations discussed in previous subsection. Providing that the propagation distance is smaller than the depletion distance, the evolution of driver pulse can be neglected.

Under this condition, the evolution of wake field can be simply described by Eq.(2.20).

In the following derivation, we follow the notation and normalization using in [39]. The

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equation for scalar potential (Eq.(2.20)) can be written in the form

∂²y

∂x² = 1 2

γ₀²_⊥ y² − 1

, (2.22)

where y ≡ 1 + ϕ, ϕ is the scalar potential, γ0⊥ is the Lorentz factor of electron in per- pendicular direction and x = k_pξ is the co-moving normalized spatial coordinate. For a linearly polarized driver pulse, γ₀²_⊥= 1 + a²₀/2 by conservation of transverse momentum.

Here we consider a flat-top driver such that γ₀_⊥ = γ₀_⊥ for −L ≤ x ≤ 0 and γ0⊥ = 1 elsewhere. The potential inside the driver (−L ≤ x ≤ 0) can be analytically solved as follows.

By multiplying ^∂y_∂x on both sides of Eq.(2.22), we have

∂y

∂x

∂²y

∂x² = 1 2

γ₀²_⊥ y² − 1

∂y

∂x, (2.23)

which can be organized into

1

2[(y^′)²]^′ = −1 2

γ₀²_⊥ y + y

_′

(2.24)

Integrate both sides over x, we have:

(y^′)² =−

γ₀²_⊥ y + y

+ C (2.25)

where C = 1 + γ₀²_⊥is the integration constant which can be determined by the boundary condition : y(0) = 1, y^′(0) = 0. With the demand that real solution of y^′exists ((y^′)² ≥ 0), the value of y is bounded by

1≤ y ≤ γ0²⊥. (2.26)

Eq.(2.25) can be written into

dy dx =±

s

(y− 1)(γ₀²_⊥− y)

y (2.27)

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After integrating y from 1 to γ₀²_⊥and x from 0 to x on both sides, we obtain the solution of y

x(y) =−2γ0⊥E(ϕ_i, κ_i) + 2 s

(y− 1)(γ0²⊥− y)

y (2.28)

where E(ϕ, κ) is elliptical integral of the second kind with argument ϕ_i ≡ sin⁻¹q

γ_0⊥² (y−1) (γ₀²_⊥−1)y

and κ_i ≡q

γ₀²_⊥−1 γ_0⊥² .

By substituting ymax = γ₀²_⊥ into Eq.(2.28), the maximum scalar potential is found at x = −2γ0⊥E(κ_i). Here E(κ) is the complete elliptical integral of the second kind. This means the excited wake field becomes maximum when the flat-top pulse has an optimal length,

L = L_opt ≡ 2γ0⊥E(κ_i). (2.29)

The scalar potential behind the pulse (x ≤ −L) can be solved by noting that γ0⊥ = 1 due to the absence of driver pulse in this region. Therefore, Eq.(2.22) becomes

∂²y

∂x² = 1 2

1 y² − 1

(2.30)

with boundary condition y(x = −Lopt) = γ₀²_⊥ and y^′(x = −Lopt) = 0. With similar procedure above, the range and the solution of y in this region are