Investigation of possible diagnostic methods for plasma wakefield accelerator (PWFA) driver degradation using synthetic diagnostics in PIConGPU

Bereich Mathematik und Naturwissenschaften Fakultät Physik Institut für Strahlenphysik Investigation of possible diagnostic methods for plasma wakefield accelerator (PWFA) driver degradation using synthetic diagnostics in PIConGPU Nico Wrobel Geboren am: 10. März 2001 in Hoyerswerda Master-Arbeit zur Erlangung des akademischen Grades Master of Science (M. Sc.) Gutachter Prof. Dr. Ulrich Schramm Prof. Dr. Thomas E. Cowan Betreuer Dr. Richard Pausch Eingereicht am: 28. Oktober 2025 Abstract Plasma-based accelerators have the potential to revolutionize the field of particle acceleration. Compared to traditional radio-frequency accelerators, they offer an ultra-compact form factor while reaching three orders of magnitude higher field strengths. New diagnostic methods must be considered, in order to resolve the micrometer-scale plasma and driver structures within femtoseconds. This thesis investigates common electromagnetic radiation-based diagnostic methods in their ability to diagnose characteristic driver dynamics in plasma wakefield acceleration (PWFA). Focus was put onto diagnosing the oscillation and also the breakup of the driving electron bunch, due to the resulting degradation of PWFA-performance. Measurements of the electron self-emission spectrum and the coherent transition radiation (CTR) spectrum were implemented as synthetic diagnostic tools for drivers in particle-in-cell (PIC)-simulations. Additionally, a toymodel of the driver was developed to gain a better understanding of the correlations between driver behavior and radiation signal. One result of this thesis showed that the self-emission of the electron driver is unsuitable as a diagnostic tool for the dynamic processes, as the radiation intensity is overshadowed by the emission from the plasma background and therefore not able to be distinguished. Usage of the transition radiation spectrum provided more promising results, with clear evidence for both the oscillation and the bunch breakup being detectable. A novel approach to resolve the time-dependent dynamics of an evolving PWFA-driver in the CTR spectrum was also explored as part of this research. Kurzfassung Plasmabasierte Beschleuniger haben das Potenzial, das Gebiet der Teilchenbeschleunigung zu revolutionieren. Im Vergleich zu traditionellen Beschleunigern, werden hier ultrakompakte Formfaktoren geboten und gleichzeitig dreimal höhere Feldstärken erzielt. Neuartige diagnostische Methoden müssen in Betracht gezogen werden, damit Plasmaund Treiber-Strukturen im Mikrometer-Bereich in der kurzen Zeitspanne von Femtosekunden noch aufgelöst werden können. In dieser Arbeit werden gängige Diagnostiken, basierend auf elektromagnetischer Strahlung, in ihrem Potenzial untersucht, die charakteristischen Dynamiken des Treibers eines Plasma Wakefield Beschleunigers (PWFA) zu messen. Hauptaugenmerk sind die Diagnose der Oszillation und des Zerfalls des treibenden Elektronenstrahls, da diese potenziell zu einer Abschwächung der PWFA-Beschleunigung führen können. Die Messung von Eigenstrahlung und der kohärenten Übergangsstrahlung (CTR) von Elektronen wurden als synthetische diagnostische Methoden für Treiber in Particle-in-Cell (PIC) Simulationen implementiert. Zudem wurde ein vereinfachtes Modell des Treibers entwickelt, um ein besseres Verständnis über die Beziehung zwischen Treiber-Verhalten und den resultierenden Strahlungssignalen zu erhalten. Als ein Kernresultat dieser Arbeit ergab sich, dass die Nutzung der Eigenstrahlung des Elektronen-Treibers, als diagnostisches Tool für die dynamischen Prozesse, sich als ungeeignet erweist, da die Strahlungsintensität von der Strahlung des Plasmahintergrundes überschattet wird und somit nicht zu unterscheiden wäre. Vielversprechendere Resultate wurden durch die Übergangsstrahlung erzielt: Sowohl die Oszillation als auch der Zerfall des Treibers konnten mithilfe des Spektrums nachgewiesen werden. Im Rahmen dieser Arbeit wurde außerdem ein neuer Ansatz untersucht, um die zeitabhängigen Dynamiken des Treibers in einem CTR-Spektrum aufzulösen. Table of Contents Abstract III Kurzfassung III 1 Introduction 1 2 Theoretical background 3 2.1 Plasma wakefield acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Particle-in-Cellcodes................................... 5 3 Setup of the PIC-simulation 9 3.1 Parameters for a PIConGPU simulation . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Driver initialization from Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Definitionoftheplasma................................. 11 4 Analyzing the self-emission of a driving electron bunch in PWFA 13 4.1 Theoreticalprerequisites ................................ 13 4.1.1 The Liénard-Wichert potentials . . . . . . . . . . . . . . . . . . . . . . . . 14 4.1.2 Spectrally resolved emission from relativistic charges . . . . . . . . . . . 15 4.1.3 Direction shift in the relativistic regime . . . . . . . . . . . . . . . . . . . . 16 4.2 Creating a simplified toymodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.1 Numerical calculation of the radiation . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Recreating the relativistic particle dynamics . . . . . . . . . . . . . . . . . 18 4.2.3 Methodology at the example of the toymodel . . . . . . . . . . . . . . . . 23 4.3 Results from the particle-in-cell simulation . . . . . . . . . . . . . . . . . . . . . . 26 4.3.1 Comparison of different observation angles of the radiation . . . . . . . 27 4.3.2 Dependency of the polarization . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3.3 Exploring the effect of different input parameters on the radiation signal 29 4.3.4 Measurements with plasma background . . . . . . . . . . . . . . . . . . . 31 4.4 Conclusion of the self-emission diagnostics . . . . . . . . . . . . . . . . . . . . . 32 5 Analyzing the transition radiation of the driving electron bunch 35 5.1 Theoreticalprerequisites ................................ 35 5.1.1 Theoretical description of the transition radiation of an electron bunch 36 5.2 Simplified introductory example of the implementation . . . . . . . . . . . . . . 38 V Table of Contents 5.3 Diagnostics of the bunch breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4 Resolving the oscillation signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.5 Conclusion of the transition radiation diagnostics . . . . . . . . . . . . . . . . . . 45 6 Conclusion and Outlook 47 Bibliography 49 VI 1 Introduction The observation of our surrounding environment through the use of electromagnetic (e.m.) radiation began with the formation of the first photoreceptor cells [1]. A few hundred million years later, our eyes evolved to become our most critical sensory organ. Still, to surpass its shortcomings, even more advanced radiation detectors were developed that can capture frequencies far outside our visible spectrum and even capable of detecting the signal from single photons. Therefore, it should come as no surprise that radiation measurements have always been a fundamental diagnostic tool in all disciplines of science. One important application of radiation detection are the diagnostics of plasma-based accelerators. Such particle accelerators were first invented with the laser wakefield acceleration (LWFA) in 1979 by Tajima and Dawson [2]. They proposed the acceleration of electrons through the wakefield that is created by a laser beam when it moves through a plasma. Six years later, Chen and Dawson [3] proposed with the plasma wakefield acceleration (PWFA) a similar new acceleration-scheme that uses electron beams instead of lasers to drive the wake in a plasma. Since then, active research has been conducted on both accelerator types in the expectation to surpass the limits of traditional radio-frequency (RF)-accelerators. Through their design, RF-accelerators are limited to acceleration field strength of approximately 100MVm−1 [4]. This stands in contrast to more than 100GVm−1 reached in current plasma-accelerators [5]. Thus, energies beyond 200MeV can be achieved in plasma accelerators by an accelerated witness bunch over an acceleration length of only a few millimeters [6]. Further improvements were achieved recently by a new design, using the resulting beam from an LWFA to drive a PWFA (LPWFA) [7]. This hybrid accelerator concept has already been demonstrated in practice [8]. These overall capabilities of plasma-accelerators provide many potential use cases, ranging from particle physics to medical science. However, resolving the micrometer extent of the electron bunches in only a few femtoseconds provides some challenges in the diagnostics of LWFA and PWFA. In order to investigate the driving and accelerated bunch in and after the plasma, new methods had to be developed. Therefore, multiple radiation-based methods were considered and are used in practice as diagnostic tools [9]. The acceleration of the electrons in the witness beam, and for PWFA also the driving beam, results in the emission of high-frequency radiation. This radiation can be measured to detect electron bunch properties in plasma-accelerators [10]. Additionally through this property, the electrons can also be used to power a free-electron laser (FEL) [11][12]. Not only the emission of the participating particles need to be considered. For example, in transition radiation the radiation of a metal foil is detected, when the particle bunch moves through it [13]. It provides information about the spatial charge distribution of the particle 1 1 Introduction bunch at the time of the transition. Lastly, shadowgraphy should be mentioned. In this technique, a probe laser pulse is shot perpendicular to the motion of the particle bunches into the plasma, and then detected on the other side. Therefore, the shadow of the cavities surrounding the particles can be observed by a detector, as the cavities diffract the laser passing through them. This technique can be applied to measure the extent of the cavities [14]. In this thesis, the particle self-emission and the transition radiation are investigated for an electron driver in a PWFA with the usage of particle-in-cell (PIC)-simulations. Under the influence of plasma fields, the driver undergoes its own dynamics. This results in a feedback loop between driver and plasma. Energy gets depleted in parts of the driver due to decelerating electric forces, resulting in a degradation of PWFA performance when the driver finally breaks apart. Additionally, focusing forces push particles of the driver toward the propagation axis, resulting in a collective oscillation of the particles. This study focuses on whether these dynamics of the driver can be measured through the radiation and if diagnostic techniques can be developed to reconstruct parts of the initial distribution. Especially, determining if the bunch degradation occurred for a given PWFA-shot was of interest. Chapter 2 will provide a brief theoretical introduction of the PWFA scheme and of the PICalgorithm used to run the simulations. The used simulation setup is described in chapter 3. Results of the investigations of self-emission and transition radiation are presented in chapter 4 and 5, respectively. These include the development of a toymodel of the PWFA-driver, as well as the investigation into a new approach for measurement of time-dependent driver dynamics with transition radiation. Both chapters provide their own theoretical introductions, followed by the explanation of the methodology. They also feature their own discussion, but an overall conclusion and outlook is given in chapter 6. 2 2 Theoretical background 2.1 Plasma wakefield acceleration Plasma wakefield acceleration (PWFA) is a modern particle acceleration scheme, notable for the high achievable energies over short acceleration lengths, while promising higher-quality beams than in LWFA. While still under active research, current PWFA schemes reach acceleration gradients of tens to hundreds of GVm−1 [8]. In comparison to the 100MVm−1 of classic radio-frequency (RF) accelerators, only millimeters of accelerating plasma are needed in PWFA to accelerate an electron bunch to hundreds of MeV . This allows for a very compact accelerator type, usable in applications like material research or free-electron lasers (FEL) [4][12]. A typical PWFA setup may consist of an ultra-relativistic charged particle bunch, the so-called driver, and a plasma [15]. As an example, a preceding ring-accelerator might be used to accelerate a bunch of electrons or protons close to the speed of light. Shooting the electron driver into the plasma will result in the repulsion of the plasma electrons in front of the electron beam. Due to their comparatively high mass, the repulsion of the positive ions can be neglected, and it can be assumed that they stay fixed at their position. The resulting charge separation behind the driver creates a strong electric field gradient, as the plasma electrons are attracted to the positively charged cavity. The electrons get pulled back in, closing the positively charged bubble but will overshoot and oscillate around the propagation axis of the driver. This process will create a train of cavities behind the driver, co-moving with it as illustrated in Figure 2.1. In the so-called blowout regime, most of the plasma electrons are repelled, and strong electric fields emerge inside the cavities. These fields result in strong Lorentz-forces acting on charged particles inside these cavities, including the driver electrons. If another electron bunch, a so-called witness bunch, is injected into the backside of a cavity, it will get accelerated by these forces. Through this process, energy is transferred from the driver bunch to the witness bunch in order to accelerate it. The impact of the fields at the front of the cavities will cause the driver itself to decelerate, until parts of it may fall back due to the energy loss. Additionally, focusing forces result in an oscillatory motion of the electrons in transverse direction [17]. Due to the position-dependent oscillation frequencies, the collective oscillation is visible as wing structures [18], as shown in Figure 2.2. Another possible source for the relativistic electrons to consider is a laser wakefield accel3 3 Setup of the PIC-simulation px,pz py p ψ Figure 3.1: Visualization of momentum components of a particle and influence of the divergence angle ψ. The artificial driver, if not noted otherwise, will be based on a 3D Gaussian distribution. The initialization is in most parts identical to the method from my bachelor thesis [18]. Only a brief overview over the properties of the initial bunch will therefore be given, while more details about the PIConGPU initialization and field generation can be found in chapter 3 of the bachelor thesis. For the positions of the particles in the bunch, a 3D normal distribution was used. It was assumed that the bunch, described in the following procedure, might have resulted from a preceding LWFA-stage (see LPWFA in section 2.1). The parameters of the distribution used are listed below and are based on results from LPWFA experiments [14]. Parameter Value radius rrms 10 µm duration τFWHM (width) 20 fs (≈6µm) charge Q400 pC Energy E250 MeV to 350 MeV Energy spread ∆E10 MeV divergence spread σψ1.6 mrad Table 3.1: Default parameter values used for Gaussian PWFA bunch. Radius rrms and bunch-duration τFWHM are used to determine the spatial extent of the 3D-Gaussian distribution. The position-components x , y and z of the particles are sampled independent of each other from the distribution to achieve an uncorrelated bunch from an LWFA exit. The kinetic energy is distributed as a Gaussian distribution as well, and used to determine the total momentum of the particles. The direction of the momentum is set by the divergence spread σψ . This parameter controls the spread of the portion of the momentum that diverges from the forward direction (y, or ζin the co-moving frame), see Figure 3.1. To proceed with the initialization, the script simulates a propagation through the vacuum, similar to an LPWFA-setup. The new positions of the particles after the propagation are calculated analytically from the previous positions and momenta. This propagation introduces a correlation into the phase-space of the particles, as expected from experiments. Afterward follows the transition through a metal foil (laser blocker in LPWFA, see Figure 2.3). The effects of the blocker foil on a bunch [32] are simulated in the script by increasing the divergence of 10 3.3 Definition of the plasma −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 y[mm] 0 1 2 3 4 ρplasma [cm−3] ×1018 vacuum Super Gauss plateau Figure 3.2: Plasma distribution function for the PIC-simulations. It consists of multiple pieces for a smoother transition between vacuum and plateau. the electron bunch to approximately σψ=4.2 mrad. The resulting particle distribution is placed into a PIConGPU simulation, where it interacts with a predefined plasma. 3.3 Definition of the plasma As the plasma medium in the PIC-simulation, a pre-ionized hydrogen plasma was chosen. After an initial vacuum, the plasma density function is described by a plateau with a constant particle density of ρ0=4×1024 m−3 [14], if not stated otherwise. To smooth out the transition between vacuum and plasma, a super-Gaussian transition function is used. This function connects smoothly to the plateau. The complete distribution function is therefore described piecewise: ρ=         0y≤0 ρ0exp −y−6σ σ60 < y≤6σ ρ0y> 6σ (3.1) The parameter σ in the super-Gaussian can be used to set length of the transition and is set to 20 µm in all simulations in this thesis. A visualization of the function can be found in Figure 3.2. The plateau persists usually until the end of a simulation, otherwise, the length of the plateau will be noted. 11 4 Analyzing the self-emission of a driving electron bunch in PWFA The emitted radiation of charged particles, such as electrons accelerated in an electromagnetic field, is a well-established diagnostic technique to obtain information about electron bunches in an LWFA or PWFA [10]. Typically, the betatron radiation from an injected witness beam is measured to quantify its beam quality [9][33]. However, the e.m. fields of the plasma cavities in a PWFA-setup also act on the bunch that is driving the wakefield. Therefore, the driver electrons are able to emit radiation as well, which can be detected in experimental setups [34]. This chapter concentrates on the usage of this self-emitted radiation to diagnose possible symptoms of the drivers’ degradation by analyzing the infrared signals it emits. The results presented in section 4.3 demonstrate, that it is indeed possible to reconstruct driver properties, with both the transverse dynamics (e.g. oscillation) and longitudinal dynamics (e.g. bunch breakup) leaving an imprint in the combined radiation of all bunch electrons. But it also highlights the remaining challenges in the experimental feasibility of this method in a realistic setup. This ultimately led to the conclusion that the detection of the signals might not be realizable due to the low signal-to-noise ratio. The current chapter therefore documents my research into this topic, up to the point where it became clear that an alternative diagnostic method, such as transition radiation, might suit the initial task better (see chapter 5 for that). Section 4.1 will present the necessary theoretical knowledge to follow this chapter, while section 4.2 uses these prerequisites to develop a simple theoretical model of the PWFA driver and detector. This model will become a helpful tool to understand the resulting signals of the emitted radiation in more realistic and complex PIC-simulations in section 4.3. 4.1 Theoretical prerequisites This section will establish the mathematical foundation, used for the calculation of particle radiation. The e.m. potentials from the motion of point-like charged particles is commonly described by the Liénard-Wichert potentials [35] [36]. 13 4 Analyzing the self-emission of a driving electron bunch in PWFA time space re rret observer at r |r−rret| c Figure 4.1: Time-space diagram, visualizing the relation between the observer (point at which potential is evaluated) at r and the position of a charged particle. While currently, the particle is at re , the observer receives information about its potential from the particle in the past at rret , as changes in the potential only propagate with the speed of light. 4.1.1 The Liénard-Wichert potentials Φ(r,t)=e 4πϵ0 1 1− β·ˆ n|r −rret| ret  A(r,t)=e 4πϵ0   β 1− β·ˆ n|r −rret| ret (4.1) This formula presents the scalar potential Φ(r,t) and the vector potential  A(r,t) as a function of position r and time t . Information about the updated potential of a particle spreads with the speed of light c . Therefore, the observer at r sees the position of the particle at the so-called retarded time tret =t−|r −rret| c in the past. While the particle currently is at position re(t) , for the observer it located still is at the retarded position rret =re(tret) and all the expressions in the brackets in Equation 4.1 must be evaluated at the retarded time tret , as is also marked by the sub-scripted ret . This relationship between the different positions and times is also visualized in Figure 4.1.  β in Equation 4.1 is the normalized velocity of the particle, while ˆ n is the normalized vector r−rret |r−rret | between observer and source. The dot-product  β·ˆ n in Equation 4.1 corresponds to the component of  βwhich is directed towards the observer. Equation 4.1 correctly describes the scalar and vector-potential of the particle following an arbitrary trajectory and can therefore be used to obtain the corresponding fields from the particle. The formulas remain accurate at relativistic velocities which makes them powerful tools in scenarios where quantum mechanical effects can be neglected. These formulas can 14 4.1 Theoretical prerequisites also be used to determine the emitted radiation of the particle. 4.1.2 Spectrally resolved emission from relativistic charges When the velocity of a charged particle changes, i.e. when the particle gets accelerated, a ripple effect will be induced into the potential it creates. This rippling is visible in the resulting electric and magnetic fields, which will propagate this change in the form of an e.m. wave. However, the fields cannot keep up with the motion of the particle and detach from it. Through this process, the particle emits energy in the form of radiation. The described emission effect can commonly be seen when a charged particle is under the influence of an e.m. field, for example, in a static electric field. Another important example is the field surrounding another particle which will induce a curvature into the trajectory of a particle that is passing by. In the special case of plasma acceleration, the e.m. fields emerge from the strong charge separation in the plasma cavities. This causes the particles in the driving bunch to radiate. But, also the electrons from the plasma itself are radiating due to their motion under the influence of all other electrons and ions. When discussing e.m. radiation in an experimental setup, one can imagine a potential detector. This detector has the form of a small camera with a finite size opening, observing the radiation under a given solid angle far away from the source. The experimentalist may want to change the position of the detector between runs because the emission varies in direction, and the detector will detect different signals for different observation directions. Since the detector is placed far away and the radiation source is microscopic, the exact positions of the sources in comparison to the detector become negligible. A far-field approximation can be made, which reduces the distance between different radiation sources to a simple phase difference between the emitted waves. Like with the shutter in a camera, the radiation can only be measured over a finite period of time. Thus, not the radiation energy emitted at a given point of time is measured, but instead, how much is detected over a set time span. The integration over these time spans allows one to determine the frequency-resolved radiation spectrum. For a mathematical description of the radiation, the emitted radiation energy of a particle source per given frequency and solid angle d2W dΩdω is wanted. The derivation of such a formula from the Liénard-Wichert potentials 4.1 will be skipped here and can be found in other works like [35]1. Instead, only the resulting formula will be presented: d2W dΩdω=q2 16π3ϵ0c N X j |{z} Sum over N particles Zt1 t0 ˆ n ×hˆ n − βj×˙  βji 1− βj·ˆ n2·eiω(t−ˆ n·rj/c)dt 2 (4.2) This formula describes the radiation energy for a given time-interval, frequency and N electron trajectories. ˙  β is the normalized acceleration and a shorthand notation for the time-derivative of  β: ˙  β:=∂ ∂t  β 1 Note the use of the symbol W for the emission energy instead of I as it may else be confused with an intensity. It is also done to distinguish it from electric fields, marked with  E 15 4 Analyzing the self-emission of a driving electron bunch in PWFA x y ˆ n θ Detector Particle Figure 4.2: Schematic of a radiation detector in a simplified setup. The detector is assumed to be infinitely far away from the radiation source. ˆ n points from the particle source in the direction of the detector. In this 2D image, the detector can be placed at a given angle θ , in reality there also is an angle ϕ to freely position it on a 3D sphere. The unit vector ˆ n has a different yet similar meaning here than the vector in Equation 4.1. It is the observation direction, stating where the detector is located on an (infinitely far away) unit sphere around the source, see Figure 4.2 for a visualization. The "aperture" and spectral resolution of the detector, described by this formula, is assumed to be infinitely small. So, one must integrate over a given frequency and solid angle to obtain results comparable to those of real detectors. The emission spectrum from multiple particles in the observed system is gained by summation over the complex integrals from the different particle trajectories. This is noted in Equation 4.2 as the sum over N particles. Due to the summation of complex amplitudes, interference effects between the particles need to be considered. Later, in section 4.2 and section 4.3, it is presented how many-particle interference effects result in a signal that cannot be explained by single-particle radiation alone. 4.1.3 Direction shift in the relativistic regime The total radiation power emitted per solid angle by one relativistic particle moving in a straight line is described by [35] dP(tret) dΩ =e2˙ β2 4πc sin2(θ) (1−βcos(θ))5(4.3) θ is the angle between the propagation direction and the observation direction. For slow particles ( β≪1 ), the power scales with sin2(θ) . Therefore, the emitted energy increases as θ approaches 90◦ . No radiation is emitted in forward direction and the maximum amount of power is emitted in the plane perpendicular to the propagation direction. For relativistic particles ( β≲1 ), the denominator in Equation 4.3 becomes significant and results in the radiation peak bending forward toward the propagation direction. It can be shown, that the direction of peak radiation scales like θmax ∼γ−1 . As a result, particles with a high Lorentz factor γ will radiate most intensely at θ close to (but still not at) 0 . The comparison 16 4.2 Creating a simplified toymodel a) θmax b) c) Figure 4.3: Angular-resolved emitted power radiated from an accelerated particle. a) Emission for particle (orange) that is accelerated in propagation direction at non-relativistic velocities ( β≪1 ). Power is emitted perpendicular to propagation axis. b) Emission for particle that is accelerated in propagation direction at relativistic velocities ( β≲1 ). Peak of power is emitted at angle θmax .c) Emission for particle at different points of oscillating trajectory. Most radiation gets emitted when the particle reaches the maximum of the trajectory, while none is radiated in the nodes ( ˙ β=0). of the radiation direction between the two cases is shown in Figure 4.3a and b. For particles, that also perform an oscillation motion in transverse direction, additional effects must be considered. This is the case for particles in the driver of a PWFA and will therefore be important for the interpretation in later sections. A "lighthouse" effect emerges as the radiation cone from the particle sweeps over an imagined screen in the propagation direction. The sweeping angle depends on the ratio of forwardand total velocity of the oscillating electrons [10]. Through this effect, the radiation detected on the screen is a superposition of the radiation, emitted from different positions of the particle during the oscillation cycle. Therefore, interference effects will emerge and be visible in the observed signal. Additionally, some radiation may appear directly in forward direction (θ=0◦), because the emission cone can sweep over this direction. The radiation for different points of the oscillation trajectory is visualized in Equation 4.3c. 4.2 Creating a simplified toymodel Instead of an immediate examination of the output of PIC-simulations, a look at a simpler example will be taken first. As part of this thesis, a toymodel of the PWFA-driver was built. Its purpose is to understand the complex dynamics of the driver and the resulting radiation. The model is based on simplified assumptions, with the goal to still reproduce the main findings of its big brother, the PIC-simulations. It will also be used as a pedagogical tool, to guide through the underlying physics. The goal is to demonstrate how the emission signal took the form observed later in section 4.3. As a first step, a numerical implementation of the multi-particle emission spectrum from the Liénard-Wichert potentials was created. 17 4 Analyzing the self-emission of a driving electron bunch in PWFA 4.2.1 Numerical calculation of the radiation In the radiation formula 4.2, the integral over the time period during which one radiation signal is detected has to be approximated by a transition to a sum: Zf(t)dt →X t f(t)∆t This sum goes over the complex amplitudes at given times t , each with a finite timestep width ∆t. This procedure results in the following approximation of Equation 4.2. d2W dΩdω=q2∆t2 16π3ϵ0c N X jX t hj,t(ω,n) ˆ n ×hˆ n − βj,t×˙  βj,ti 1− βj,t·ˆ n2·eiω(t−ˆ n·rj,t/c) 2 (4.4) Here, the timestep width ∆t is already factored out of the absolute square of the sum. An additional new function hj(ω,n)was also added, which acts as a frequency filter [27] hj,t(ω,n)= Θ   π1− βj,t·n ∆t−ω (4.5) with Θ(x) being the Heaviside function. According to the results of the Nyquist–Shannon sampling theorem [37], the bandwidth of the spectrum is limited to frequencies less than half of the sampling rate 2π ∆t . The filter is implemented to exclude high frequencies from the spectrum that exceed the maximum frequency that is resolvable with a step width ∆t . Equation 4.5 also accommodates for the retarded step width ∆tret from the local Nyquist-limit of relativistic particles [27]. Simple tests were conducted to verify the correct implementation of the radiation function 4.4 against the results of singleand multiparticle trajectories with analytical solutions. With the numerical implementation verified, arbitrary particle trajectories (e.g. information about position, velocity and acceleration) can be used and put into Equation 4.4 to create an emission spectrum, similar to the result of the PIConGPU radiation plug-in [27]. 4.2.2 Recreating the relativistic particle dynamics A simplified toymodel of a relativistic PWFA particle driver was created. It was implemented to fulfill two purposes: • Allow for quick testing of different driver parameters without the need for timeand resource-intensive PIC-simulations • Direct control of the acting forces on the driver instead of the complex feedback-loop between the charged electron driver and the e.m. fields from the cavities These goals were achieved by the usage of an ordinary differential equation (ODE)-solver in order to numerically calculate the trajectories of the relativistic particles at given timesteps. This required the definition of a system of ODEs that describes the particle dynamics and the implementation of a solver. As a start, the solutions to the former problem will be presented. 18 4.2 Creating a simplified toymodel −10 −5 0 5 ζ[µm] −10 0 10 Fζ[nN] a) −10 −5 0 5 ζ[µm] 0 1 2 3 4 5 6 Fr/r[mN/m] b) Figure 4.4: Distribution of forces on driver electrons over ζ after 1.8 mm of plasma propagation from a PICsimulation. Forces are sampled on particle positions and the median force in each bin is plotted. a) Longitudinal forces. b) Transverse forces, divided by distance from propagation axis. As a simplification of the dynamics of the PIC-simulations, interactions between the bunch electrons are not modeled. The interactions between bunch and plasma were considered as mean-field forces acting on the bunch electrons. These forces from the plasma were assumed to be quasi-static. This stands in contrast to the time-varying forces in a real plasma. Therefore, the transverse and longitudinal forces are only dependent on the relative position inside the simulated co-moving cavity. Furthermore, only 2D dynamics were calculated, motivated by the cylindrical symmetry of the problem, which further reduces computational costs. The functions implemented for the longitudinal and transverse forces were based on empirical data from PIC-simulations. A representative snapshot of these forces in a simulation is shown in Figure 4.4. In the chosen time-independent approach, the longitudinal force Fζ only depends on the co-moving position ζ=y−c·t. The specific function modeled is Fζ(ζ)=−16 nN ·cos ζ+ 2.3 µm 3.0 µm·1 + e ζ−2.0 µm 0.1 µm−1(4.6) The cosine leads to an alternating curve that simulates the sign-alternating forces between the frontand backside of the cavities. Meanwhile, the exponential damping function decreases toward the front of the driver, causing the function to converge to 0 , as there are no cavities in front of the bunch. For the transverse force Fx , a harmonic oscillator approach was taken. The force was defined as Fx=−kx·x with the force constant kx as a free parameter. In this setup, kx is again defined as a function of the co-moving position ζ . Similar to the longitudinal force, the force constant is assumed to be 0 in front of the driver ( ζ> 3 µm ). For simplification, it is also set to be a constant value of 6mN m−1 far behind the driver ( ζ<−8µm ). Between these two plateaus, the function is linear interpolated to recreate the real increase of the force over the length of the electron bunch. So, in total: kx(ζ)=     6mN m−1if ζ<−8µm 6mN m−1·ζ−3µm 3µm+8.0 µmif −8µm≤ζ≤3µm 0if ζ> 3 µm (4.7) 19 4 Analyzing the self-emission of a driving electron bunch in PWFA −20 −10 0 10 ζ[µm] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y[mm] a) 01234 ω[1e15 s−1] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y[mm] b) 102 103 104 −dQ/dζ[pC/µm] 10−1 100 101 102 |F(ω)| [a.u.] Figure 4.10: a) Charge distribution of the PIC-simulation from an area around the ζ -axis plotted over time. b) Fourier-transformation over ζof a) The question remains, if such signals also appear in a more realistic PIC-simulation and if the signal in the radiation-spectrum also has the same origin as for the toymodel. 4.3 Results from the particle-in-cell simulation Following the introduction of the self-emission signal for a simplified toymodel, results for the more sophisticated and realistic PIConGPU simulations can be presented now. Before exploring the influence of changes in the input parameters of a simulation later in subsection 4.3.3, the Fourier-images and radiation spectra of the example setup from chapter 3 are examined. To obtain the Fourier-image, the charge distribution in a small region around the co-moving ζ - axis was integrated over the transverse coordinates x and z and then plotted over time, similar to the toymodel. The resulting time series of the charge distribution is shown in Figure 4.10a. After the bunch entered the plasma (at 0mm ), one can observe the same forward-moving nodes forming as seen in the toymodel distribution in Figure 4.8. After 1.5 mm , these peaks from the oscillation seem to wash out. In the PIC-simulation, particles can leave the cavity in transverse direction and therefore escape the influence of the transverse force. This stands in contrast to the toymodel, where the focusing forces extend infinitely in this direction. Therefore, a reduction of the number of particles oscillating back to the center occurs over time in the PIC-simulation. An additional explanation stems from the fact that the toy model was only a 2D model. The third dimension in the PIC-simulation allows oscillating particles to orbit the propagation axis with sufficient distance to not get counted toward the charge distribution. After 3mm , the bunch breakup happens. Similar to the toymodel, the breakup is visible as the formation of a gap, here between 0µmand −4µm , and charge culminating at −12 µm . As already noted in subsection 4.2.3, due to inaccuracies in the toymodel, there is less charge falling back in this more realistic PIC-model than in the toymodel. 26 4.3 Results from the particle-in-cell simulation In Figure 4.10b, the Fourier-transformation of Figure 4.10a over the ζ -axis is depicted. Besides a strong peak at frequencies under 0.3 ×1015 s−1 , there is, like in the toymodel, the roughly linear increasing branch signal until 1.4 mm . In contrast to the clear signals in the toymodel, there is only a weak sub-branch sticking out above the main branch signal. This is likely due to the more washed out charge distribution in Figure 4.10a. Above 3mm , the bunch breakup is again visible as an interference pattern. Due to the weaker breakup, in comparison to half of the particles falling back in the toymodel, the interference pattern appears fainter. Like for the toymodel, observing the radiation spectrum of the PIC-driver makes the information from the Fourier-image visible to a hypothetical detector. The built-in radiation-plugin [27] from PIConGPU was used to record the radiation spectrum over a span of different frequencies and observation angles. As an input, the observation angles θ and ϕ can be set. Since the driver is cylindrically symmetric, only changes in the angle θ between propagation axis and detector direction are considered, as the spectra are qualitatively identical for different ϕ. 4.3.1 Comparison of different observation angles of the radiation The spectral evolution over time is shown and compared for multiple observation angles in Figure 4.11. The angle θmax where the maximum of total radiation is emitted scales with γ−1 . Therefore, most of the total radiation is measured close to the forward direction and decreases toward higher observation angles θ . Radiation is also measured at θ=0◦ due to the sweeping lighthouse effect from the particle oscillation (as shown in Figure 4.3c). Independent of the total radiation, the appearance of the different radiation signals seems to depend on the observation angle θ as well. The angle ranges above each image indicate the range in which the results qualitatively show similar signals. The frequency-increasing oscillation branch appears in the small region from 0.7◦to 2.1◦ around the propagation axis, while no signal appears at smaller angles. So, like for the toymodel, this feature from the Fourier-transformed image is projected onto the collective self-emission spectrum of the bunch. The branch disappears in the region from 2.1◦to 11.3◦ , after which it becomes visible again. A second signal appears above 5◦ with a peak constant in time below 0.4 ×1015 s−1 , and an additional signal spanning the entire observed spectrum after 3mm . Comparisons were done in subsection 4.3.3 between simulations with different initial parameters, resulting in varying times until the bunch breakup. They all show a temporal correlation between the spectrum-spanning signal and the timing of the bunch breakup, which justifies the assumption that the breakup causes the later signal. After 11◦ , all signals remain visible, making this the ideal region to observe these signals. Further considerations about the measurability are discussed in subsection 4.3.4. Efforts were made to research the cause of the signals vanishing and appearing in certain angular ranges. This has proven difficult, as the signal is the result of the complex superposition of the radiation from the individual electrons going through a relativistic and oscillatory motion. Due to a shift in the focus of this work, no explanation which can be presented with confidence was found. I will discuss this shift in focus further in the discussion section 4.4. 4.3.2 Dependency of the polarization Different vector components of the complex amplitude in Equation 4.2 belong to the different polarization directions of the emitted radiation. Comparisons were made between the different 27 4 Analyzing the self-emission of a driving electron bunch in PWFA 0 1 2 3 4 θ=0◦−0.4◦θ=0.7◦−2.1◦ 0 1 2 3 4 θ=2.1◦−5.0◦θ=5.3◦−11.0◦ 0 1 2 3 4 0 1 2 3 4 θ> 11.3◦ 0 1 2 3 4 θ=45◦ 10−9 10−8 10−7 10−6 10−9 10−8 10−7 10−6 10−11 10−10 10−9 10−8 10−7 10−6 10−14 10−13 10−12 10−11 10−10 d3W dωdΩdt[J] 10−14 10−13 10−12 10−11 10−10 10−18 10−17 10−16 10−15 10−14 10−13 10−12 ω[1e15 s−1] y[mm] Figure 4.11: Time evolution of the self-emission radiation spectrum of the driver in a PWFA simulation, taken for different observation angles. Angles above the images mark the regions with qualitatively similar spectra. Note the difference in the intensity scale between different spectra. 28 4.3 Results from the particle-in-cell simulation 0 1 2 3 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 a) 0 1 2 3 4 b) 10−16 10−15 10−14 10−13 10−12 10−11 d3W dωdΩdt[J] ω[1e15 s−1] y[mm] Figure 4.12: Comparison of a) In-plane and b) Out-of-plane polarization directions of driver radiation evolution, observed at θ=21.7◦. polarization directions in relation to the propagation direction of the radiation and the driver. Figure 4.12 compares the components of the emission spectrum that are polarized • in the plane, spanned by the propagation direction of the radiation and of the driver (in-plane) •in the direction that is perpendicular to this plane (out-of-plane) While the bunch breakup signal appears to be equally distributed across different polarization directions, there is a clearly visible difference for the other signals. Both, the nearly constant signal below 0.4 ×1015 s−1 and the oscillation branch, are strongly polarized and vanish in the out-of-plane polarization direction, while appearing fully in the in-plane polarization direction. The signal-to-noise ratio of these signals can therefore be increased in an experimental setup by the use of a polarization filter that blocks the background from the out-of-plane polarized radiation. 4.3.3 Exploring the effect of different input parameters on the radiation signal Basic parameter scans were conducted to test the behavior of the self-emission under varying initial parameters of the simulation. The original plan was to preliminarily test the behavior for three data points per parameter. This returns a trend showing how the parameter affects the oscillation signal and the time until the bunch breakup and its signal. More precise scans were planned afterward to obtain an accurate description of the radiation as a function of the input parameters. However, only the preliminary scans took place due to the shift away from the infrared self-emission as a diagnostic method. The results of these scans will still be presented. 29 4 Analyzing the self-emission of a driving electron bunch in PWFA 0 1 2 3 4 200 pC 0 1 2 3 4 400 pC 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 800 pC 10−18 10−17 10−16 10−15 10−14 10−13 10−16 10−14 10−12 d3W dωdΩdt[J] 10−16 10−14 10−12 10−10 ω[1e15 s−1] y[mm] Figure 4.13: Comparison of pre-polarized radiation spectrum evolution for varying total charge of the driver. Observed at θ=22.7◦ 30 4.3 Results from the particle-in-cell simulation Figure 4.13 shows the radiation spectra of three simulations with varying initial total charge of the driver. All other parameters are kept constant as described in chapter 3. Only radiation that is polarized in the in-plane direction was considered, due to the increased signal-to-noise ratio, as discussed in the previous section. The shape and timing of the oscillation branch and the bunch breakup signal are changing dependent on the charge of the driver. For the 200 pC driver, the bunch breakup is not visible as it occurs after the set end of the simulation. For the other two drivers, a trend is visible where the time until the breakup decreases. For the 400 pC driver, it occurs after 2.8 mm , while already after 1.7 mm for the 800 pC driver. A similar trend is visible in the slope of the frequency over the distance of the oscillation bunch. The slope is increasing from 0.8 ×1015 s−1mm−1 up to 1.5 ×1015 s−1mm−1from the lowest to the highest charge driver. These trends can be explained by the effect of the charge on the PWFA performance. An electron driver with a lower charge also results in a lower current when moving through the plasma. This weakens its ability to repel electrons and decreases the field strength in the cavities, compared to a high-charge driver. With all other parameters held constant, the reduced force results in a longer time until the energy of the driver electrons is depleted, causing a delayed breakup. The same is true for the focusing force, which causes the transverse oscillation of the bunch electrons. Higher forces increase the oscillation frequency and therefore decrease the distance between multiple oscillation nodes (see e.g. Figure 4.8). A similar trend was also observed in multiple other parameter scans. For the reasons stated above, the slope of the oscillation branch and the time until breakup are always inversely correlated. An increase in the size of the driver, both in width and in length, results in a reduced charge density and current. Therefore, the time until breakup is reduced. If the density of the plasma gets increased, the time until the breakup decreases, as it results in a stronger charge separation and field strength. More precise parameter scans would be needed to get a complete image of the dependency of the spectrum on these parameters. As stated in the beginning, these scans were canceled due to a shift away from self-emission as a diagnostic tool. The limits of the observed trends would also be of interest, as further increases in the plasma charge density will hinder the driver in its ability to drive the wakefield into a blowout regime. This would result in neither strong oscillation nor a bunch breakup. 4.3.4 Measurements with plasma background Two tests were conducted to evaluate the viability of self-emission as a diagnostic method. First, the number of photons emitted per frequency range was calculated from the polarized emission spectrum. This was done to find out if sufficient photons are emitted from the featured signals to be detectable by common detectors. The emission spectrum was expressed as energy per angular frequency and solid angle. As a potential detector, a 1cm2 sensor is assumed in 1m distance at an angle θ of 21.3◦ . The number of photons that are observed by this detector per 8×1012 s−1 frequency range in 30 fs was calculated. These assumptions resulted in hundreds of photons counted for both oscillation branch and breakup signal, in comparison to fewer than one photon in the background noise. This is a sufficient number of photons, so that the signals should be observable by common detector types, which would have even longer integration windows. The second test was the comparison of driver emission with the emission from the plasma 31 4 Analyzing the self-emission of a driving electron bunch in PWFA 0 2 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 a) 0 2 4 b) 0 2 4 c) 10−16 10−14 10−12 10−10 10−8 10−6 d3W dωdΩdt[J] ω[1e15 s−1] y[mm] Figure 4.14: In-plane polarized radiation spectra evolutions, observed at an angle θ=22.7◦ .a) Radiation emitted by the driver electrons. b) Radiation emitted by the plasma electrons. c) Radiation emitted from both the plasma and the driver electrons. electrons. Due to thermal noise and the interaction with the driver, high intensity radiation is emitted by the plasma, which cannot be separated from the driver radiation in real measurements. Outside the plasma, no radiation is emitted by the driver, as no forces are acting on it. The resulting spectrum is shown in Figure 4.14. Radiation from the plasma background and radiation from background + driver are nearly identical. Peak emission of the driver is nearly 6 orders of magnitude weaker than the emission of the plasma in the same infrared range, as the observed frequencies are close to the plasma frequency of ωp=0.36 ×1015 s−1 . Therefore, neither the signal from the oscillation branch nor the bunch breakup can be distinguished from the plasma background in this frequency regime and used to determine bunch properties. This finding caused me to drop further investigations of self-emission as a diagnostic tool and shift the focus onto transition radiation (chapter 5) 4.4 Conclusion of the self-emission diagnostics The viability of self-emission as a diagnostic tool for electrons of a PWFA-driver was tested. A qualitative description about the correlation between features in the spectrum and dynamics in the bunch was researched. This included the development of a toydriver as a simplified model of the PWFA driver and an implementation of the radiation formula from the Liénard-Wichert potentials. This toymodel has proven useful to gain insight into the electron dynamics of the driver and how the electron’s collective motion translates into a radiation spectrum. The reduced noise, more direct control of the bunch, and faster experimentation all contribute to the success of the toymodel as a research tool. However, further improvements are still necessary. Although the model reproduces the relevant oscillation and bunch breakup dynamics, the timing of 32 4.4 Conclusion of the self-emission diagnostics these dynamics, and especially the back driving force, still strongly deviate from the PIC-result. The cause of the violent breakup could not be determined at the time of writing. Improvements to the governing force functions might suffice to fix this problem, but more investigation is still needed to identify the cause of these issues and to further improve the model. With further enhancements, the toymodel can be used as an investigative tool for PWFAdrivers in the future, as it can work independently of the radiation diagnostic. Further analytical descriptions that could not be fully fleshed out to be part of this thesis could also help to improve understanding of the radiation in the future. The results from the more realistic PIC-simulations provided further insights into the selfemission and presented challenges such as noise or constraints on the detector placement. Preliminary tests of the dependency of the spectrum on driver input parameters were conducted and explanations for these dependencies given. Time resolving the measured radiation signals in an experiment would be a considerable challenge, because the dynamics take place in a femtosecond-time frame. But, this problem could potentially be circumvented by an off-axis measurement of the radiation. A screen could be placed parallel to the propagation axis in such a way, that it captures the radiation emitted from the driver into a 45◦ angle. When the driver moves along the screen, it would project the evolution of the emission spectrum over the length of the screen in a single measurement without the involvement of femtosecond-cameras. So, this restriction could potentially be solved. However, the simulation results also led to the final conclusion that the self-emission in the infrared regime is not viable as a diagnostic tool, as it is overshadowed by the radiation of the plasma background. While measurements of the self-emission in the X-ray regime are able to determine other properties of the bunch [34], the usage of the emission in the regime of the plasma frequency is not viable, as no measurement can be taken without inclusion of the spectrum from the plasma background. This remains true despite found improvements to the driver signal, like the inclusion of a polarization filter. Therefore, as it stands, the self-emission can be ruled out to diagnose these features, and the focus of research was shifted to other methods (chapter 5). 33 5 Analyzing the transition radiation of the driving electron bunch The measurement of coherent transition radiation (CTR) is another common diagnostic method for particle beams in plasma-based accelerators [9][13]. This type of radiation is produced when an electron bunch propagates through, e.g., a metallic foil after the plasma. By usage of this method, the longitudinal and transverse distribution of the beam can potentially be reconstructed from the detected transition radiation spectrum. For this thesis, tests were conducted using PIC-simulations to investigate how the effects of the driver degradation and transverse oscillation features in PWFA could be measured from the transition radiation spectrum. Due to the time-varying nature of the transverse and longitudinal features, new measurement approaches had to be explored. CTR has the inherent restriction, that it can be measured only once per PWFA shot, which makes these new approaches necessary. In section 5.1 and 5.2, the required theoretical knowledge about transition radiation will be conveyed and applied to a small example. The focus lies on a mathematical description of the resulting spectrum and the implementation of this description for this thesis. In sections 5.3 and 5.4, concepts of the mentioned novel measurement approach are presented, which could be employed to diagnose the bunch breakup and driver oscillation in experiments. Initial simulations showed first promising results, with signals corresponding to many prominent bunch features being detectable. These results and ongoing challenges are discussed in section 5.5. 5.1 Theoretical prerequisites Transition radiation is the e.m. radiation emitted when electrons pass through an inhomogeneous medium. In contrast to the self-emitted radiation discussed in the previous chapter, the emitter here is the medium and not the bunch itself. When the electrons move through such a medium, for example a metal foil, they will repel foil-electrons in their path. This repulsion induces radial currents into the foil around the propagating electron distribution. Inside the foil itself, the fields from the time-varying currents will cancel out through interference. However, at the transition interface where the electrons leave the foil and enter a medium with a different electric permittivity (e.g., a vacuum), these currents are not canceled out anymore. The currents excite the foil-electrons on the surface, causing them to emit e.m. radiation away from the surface. This transition radiation carries 35 5 Analyzing the transition radiation of the driving electron bunch 100 101 λ[µm] 1014 1015 ω[s−1] 10−21 10−19 10−17 d2W dωdΩ [Js] Dip Peak from Breakup Figure 5.5: Transition radiation spectrum of PWFA-driver taken at y=4.2 mm. corresponds to a slice in the evolution shown in Figure 5.4a. Therefore, all the same features of the bunch breakup described before are visible here. Such a spectrum can theoretically be measured through experimental means. Therefore, not only the splitting of the bunch, but also information about the extent and distance of individual constituents becomes visible in this spectrum and could be determined. However, if measurement is taken far after the plasma, the free expansion of the bunch in the vacuum must be considered. As visible in Figure 5.4a, transverse expansion of the bunch still influences the measured spectrum. Due to velocity differences between the low-energy particles in the back and the high-energy particles in the middle to front, the longitudinal distance between the sub-bunches increases over time. Here, these expansion effects become visible as a shift of the dip, between the two peaks, to lower frequencies. These effects must be taken into account, as they result in different spectra, dependent on the position of the metal foil that creates the transition radiation. Tests were also conducted for a more realistic bunch resulting from an LWFA-simulation operated in the self-truncated ionization-injection (STII) regime [42]. These tests highlighted additional challenges for drivers that already feature multiple charge peaks in the longitudinal direction when entering the plasma. The differentiation of these charge separations from those of the bunch breakup is non-trivial. Ideas to solve this problem are discussed further in section 5.5. 5.4 Resolving the oscillation signal As mentioned in the previous section, in addition to the bunch-breakup signal, there is a second prominent feature in Figure 5.4. The branch after 0.5mm is present both in the transition - radiation spectrum and in the Fourier-transformation of the driver’s charge distribution. It originates from the transverse oscillations of the bunch electrons, which generate wing - like structures (shown in Figure 2.2) and produce a series of localized peaks along the propagation axis of the charge distribution. The coherent signal in the transition radiation is an artifact of these peaks, because the branch in the Fourier-transformation and spectrum is 42 5.4 Resolving the oscillation signal 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 √ρplasma [m−3/2]∼ωplasma ×1012 0 5 10 15 20 −Fr,peak [nN] Figure 5.6: Comparison of the peak radial force Fracting on the driver electrons for different plasma densities. a projection of the spatial frequency of these peaks. Since these frequencies depend on the forces inside the cavity, conclusions about the driverand cavity dynamics can be drawn by taking measurements on this branch. In comparison to the bunch-breakup signal, this branch is strongly time-varying, which makes it impossible to measure with a simple transition radiation setup. To circumvent this problem, multiple PWFA-runs can be conducted and the transition radiation measured after each shot. This could be achieved by varying the plasma length between shots and measuring directly after the plasma, resulting in the same time evolution. However, this would be complicated to implement in practice, as it would require swapping gas-nozzles between each shot, with each nozzle having a different width. This becomes very impractical. Therefore, another measurement scheme was conceived. Instead of varying the length of the plasma, the plasma density can be varied comparatively easily to manipulate the fields and forces in the cavities, as is shown in Figure 5.6. A reduction in the plasma density decreases the amount of charge separation in the cavities and, consequently, the strength of the created fields. As a consequence, the forces acting on the driver are decreasing, which results in smaller oscillation frequencies. This effect was also demonstrated in subsection 4.3.3. Smaller frequencies of the oscillation reduce the distance between the density spike nodes at a given time. This distance is directly responsible for the measured emission peak, and changes in this distance over time are the cause of the slope of the emission branch. Reducing the plasma density at a given timestep, therefore, has a comparable effect on the frequency of the branch signal as observing the signal for a given plasma density at an earlier timestep. The dynamics of the driver are advancing slower at lower plasma densities and forces. Multiple PWFA shots with varying plasma densities can be used to represent different points in the driver evolution. It would be required for this method, that the shot-to-shot variance due to other contributions is minimal [43]. If given, the driver only needs to transition through the metal foil at the same position for all shots. Due to the variation of the plasma density, additional effects that are not present in the time evolution of the driver are introduced. For instance, an increase in the transverse force at higher densities not only increases the oscillation frequency of the electrons, but also reduces their amplitude. This results in a higher current of the driver, which may further affect the 43 5 Analyzing the transition radiation of the driving electron bunch 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ω[1e15 s−1] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 √ρplasma [m−3/2] ×1012 10−21 10−20 d2W dωdΩ [Js] Figure 5.7: Transition radiation spectrum of multiple PWFA-simulations with varying plasma density. Note the branch signal on the left increasing in frequency with increasing plasma density. plasma-driver feedback-loop. These differences must be considered and studied further. Parameter scans with PIConGPU simulations were set up to test the transition radiation response for different plasma density values. Between each run, the peak density ρplasma of the plasma was adjusted. A downramp was added again, this time after 2mm, and the transition radiation of the driver was calculated 0.2 mm after this downramp. The energy of the driver was increased to 350 MeV , compared to the setup in chapter 3. This increase was necessary to delay the time until the driver breaks up for higher densities. The same initial driver was used for all simulations. Results of the parameter scan are shown in Figure 5.7. The calculated spectra for the square root of the density values are plotted here. This scaling is motivated by the expected nonlinear relationship between the density and the resulting oscillation frequency of the driver electrons. However, despite the measures taken, there is still a visible curvature in the oscillation branch signal. This stands in contrast to the mostly straight line in the plot over time in Figure 5.4a. One possible reason for this behavior could be the deviation of the cavity-force response at high plasma densities. The comparison in Figure 5.6 shows a nearly linear dependency between √ρplasma and Fζ for low to mid densities. It was expected that the force would scale with the oscillation frequency of the plasma itself: ωplasma ∼√ρplasma. But at high densities, a point is reached where the driver becomes too weak to drive the cavities into the blowout regime, reducing the resulting forces. Therefore, it is assumed that even higher driver charges are needed to maintain the expected scaling. Further research is needed here. It has been demonstrated that the transition radiation spectra of a PWFA driver under an increase of the plasma density in multiple shots can produce a branch signal. This signal is analogous to the frequency-increasing branch-signal produced when observing the spectrum at different times during the evolution of the driver bunch. Questions remain about the scaling and how the different densities are mapped to different times in the evolution. 44 5.5 Conclusion of the transition radiation diagnostics 5.5 Conclusion of the transition radiation diagnostics Two CTR-based approaches for diagnosing dynamic features of a PWFA driver through the measurement of the transition radiation spectrum emitted from a metal foil after the plasma were presented. The transition radiation spectrum of the bunch in a PIC-simulation was calculated by the use of a theoretical formula to approximate the results of a real metal foil. In this initial investigative study, coherent signals were observed from the breakup of the bunch due to the decelerating fields in the cavity, as well as signals from the transverse oscillation of driver electrons. For an ideal Gaussian bunch, the breakup resulted in a spectrum that is clearly distinguishable from that of an intact driver by the appearance of a secondary peak at higher frequencies. This peak is separated from the main peak by a dip in intensity due to destructive interference. If the spectrum is measurable in an experiment, it can reveal if a breakup of the driving bunch occurred. Further research still has to be conducted to reconstruct the longitudinal features of more complicated initial driver beams. One challenge for this method is to differentiate bunches that are fragmented due to the PWFA from bunches that were already fragmented when entering the plasma. A potential idea to approach this problem might be to include the knowledge about the plasma-cavity length that can be obtained through the combination with other plasmadiagnostic methods, such as shadowgraphy [14]. The fallen-back part will, with high certainty, land in the backside of the leading cavity, where it gets accelerated again. Therefore, the distance between the separated bunch parts could be estimated from the cavity length. This distance is imprinted as a phase-difference into the transition radiation spectrum and will result in dips in the signal, whose frequencies are characteristic for such a distance and allow to differentiate them from other charge-separation patterns. Regarding the research of the transverse bunch features, which originate from the oscillation of the electrons, a novel reconstruction technique was discussed. This technique might be able to approximate the temporal information of the bunch evolution through the transition radiation spectra from multiple PWFA shots by varying the plasma density. Initial results have shown a virtual "slow-down" of the transverse evolution, as a decrease in the plasma density also causes a decrease of the oscillation-frequencies of the driver. Through this slow-down, the spectra from different points in the evolution could be mapped between the different points in time and different plasma densities. A preliminary approach to this mapping was demonstrated. While the increase in the frequency for higher densities is evident, the exact scaling and mapping to the time domain remain as open questions and have to be further investigated. For higher plasma densities, this mapping approach might break down if the driver lacks sufficient current to reach the blowout regime in the plasma. This results in a slowdown of the oscillation-signal slope for higher densities, which does not appear in the time-evolution, where the branch frequency increases linearly. Therefore, the response in the spectrum to higher-energy drivers has to be tested, as these could solve the problem. Nevertheless, the initial results were promising and might lead to a new diagnostic method that is able to time-resolve PWFA-driver dynamics in CTR signals. This method is therefore worth further investigation. 45 6 Conclusion and Outlook In this thesis, two different radiation-based methods were investigated for their ability to diagnose the dynamics of a driving electron bunch in PWFA-simulations. The collective oscillation of the bunch electrons as a transverse driver dynamic and the bunch breakup due to energy depletion were examined with these methods. The objective was to determine if an approach could be developed that would help to investigate these dynamics in real experiments. The collective self-emission of the driver electrons was researched in chapter 4. This type of radiation is emitted when the driver is under the influence of the plasma forces and can therefore detect bunch properties during the PWFA. As part of this investigation, a toymodel of the PWFA-driver was developed that provided improved control over the driver dynamics and helped to understand the observed spectra. While the overall behavior of the bunch could be reproduced, there were still inaccuracies that must be addressed in the future development. Possible causes and solutions to these inaccuracies were discussed in sections 4.2.2 and 4.4. The applicability of the toydriver as a general-purpose investigation tool for PWFA-drivers makes this model worthy of further research and improvement. Both investigated bunch dynamics were observed in the radiation spectrum of the toymodel and the PIC-simulations. The radiation spectrum contains a projection of the form-factor of these dynamics as signals in the infrared range. The dependency of these signals on parameters of the observation device and the driver was investigated, with the goal to improve the measurability under experimental circumstances and draw conclusions about the bunch from the observed spectrum. Lastly, the combined spectrum from driver and plasma was examined because they cannot be separated by real detectors. Due to an overlap of the observed frequency range with the broad-band plasma radiation, there were strong plasma radiation signals from the driver-plasma interaction in the observed region that overshadowed the dynamic driver signals. Consequently, none of these features can be made distinguishable from the background and actually detected in experimental setups. This renders the self-emission of the driver unusable as a diagnostic tool for these features. The focus of this thesis therefore had to be shifted to the second diagnostic method, as further research on the self-emission was impractical under these circumstances. In contrast to the self-emission, transition radiation can only be measured once the driver left the PWFA plasma. Additionally, only one measurement can be taken per bunch, making time-resolved measurements impossible under normal circumstances. A novel approach was 47 6 Conclusion and Outlook developed in chapter 5, so time-varying processes of the driver were still able to be resolved in the emitted spectra. Similar to self-emission, the projection of the form-factor was visible in the transition radiation spectrum. Through variation of the plasma density, the time-dependent frequency of the driver oscillation could be captured. Challenges remain in the mapping of plasma density to the corresponding time in the driver evolution, and in cases with more complex bunch distributions. More parameter studies are also needed to investigate when the density variation causes qualitative differences in the driver dynamics that cannot be mapped to its evolution. If these challenges can be overcome, the measurement of the oscillation dynamics, and potentially also other PWFA dynamics, would be possible with transition radiation, and a fundamental limit of this diagnostic method would be circumvented. 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