High-gradient gas-jet targets for laser wakefield acceleration

Transcription

1 University of Twente High-gradient gas-jet targets for laser wakefield acceleration T.A.W. Wolterink April 2011

2 High-gradient gas-jet targets for laser wakefield acceleration T.A.W. Wolterink Master of Science thesis April 2011 University of Twente Faculty of Science and Technology Laser Physics and Nonlinear Optics Graduation committee: Prof. dr. K.-J. Boller Dr. ing. H.M.J. Bastiaens Prof. dr. J.L. Herek Dr. ir. E.T.A. van der Weide

3 ABSTRACT I Abstract Laser wakefield acceleration is a novel concept for particle acceleration which can provide for a significant reduction of the accelerator length, compared to conventional accelerators. In laser wakefield acceleration, an ultra-short high-intensity laser pulse is focussed into a plasma. As the laser pulse propagates through the plasma it drives a plasma wave travelling in the wake of the laser pulse. In such a plasma wave extremely large longitudinal electric fields are generated, easily exceeding 100 GV m-1. When an electron bunch is properly injected into this wakefield it can be accelerated by these huge electric fields to ultra-relativistic energies over distances thousands of times shorter than in conventional particle accelerators. Currently, the main issue in laser wakefield acceleration is a lack of control over the injection of electrons into the wakefield, resulting in a large spread in energy of the accelerated electron bunches. A possible method to gain a larger degree of control over the injection of electrons is by making use of wave breaking of the plasma wave. This wave breaking can be induced in a downward gradient in the plasma density over a distance in the order of a few hundred micrometres. For producing this particular density profile with a sharp density downramp in a plasma supersonic gas jets are a promising approach which is under intense investigation. The edge of a supersonic gas jets provides a sharp downramp in gas density, which after ionization by the laser results in a gradient in plasma density. Furthermore, even sharper transitions in density on a length of a few micrometres are associated with shocks inside a supersonic gas flow, which may also allow for injection into the wakefield. We explored the feasibility of the integration of the injection stage and the first acceleration stage of a laser wakefield accelerator using a double gas jet. For the first time, we studied the usage of shocks in a double gas jet for injection of electrons into the wakefield. For this numerical simulations have been performed to determine the gas flow in the double gas jet. In this gas jet a structure of shocks is visible, arising from the interaction between the two gas flows in the double gas jet. The shocks are accompanied by density transitions with a width down to 5 µm. These sharp transitions make a double gas jet of interest for injection of electrons in laser wakefield acceleration. Several supersonic nozzles with different Mach number and geometry have been designed and constructed, from which one is selected for characterization. The supersonic gas jet emanating from an axisymmetric Mach 4.8 nozzle with an exit diameter of 0.75 mm has been characterized using a Mach-Zehnder interferometer. By a careful design aiming for high sensitivity and spatial resolution it is shown that in helium jets even gas densities of below cm 3 can be measured accurately. This is, to our knowledge, the first quantitative characterization of such helium gas jets. Operated with helium at a pressure of 70 bar, the nozzle produces a gas jet, having a flat-top density profile with a maximum density of cm 3 and a length of 0.60 mm at a distance of 0.5 mm from the nozzle exit. The width of the transition in density at the edge of the jet is 0.25 mm. By changing the inlet pressure to the nozzle the density in the gas jet can be varied without affecting the density profile. The gas density profiles produced by this nozzle are very suitable for laser wakefield acceleration. The nozzle is currently implemented in an experimental laser wakefield accelerator at the Helmholtz-Zentrum Dresden-Rossendorf, producing accelerated electrons there for the first time.

4 TABLE OF CONTENTS II Table of contents 1 Introduction Laser wakefield acceleration Breaking of plasma waves Injection at plasma density gradient Aim of this thesis Physics of supersonic gas jets Compressible gas flow Thermodynamic gas properties Flow regimes Dynamics of continuum gas flow Speed of sound Compressible flow Quasi one-dimensional supersonic flow Supersonic gas jet Normal shocks Influence of ambient pressure Supersonic underexpanded jet Cluster formation Flow simulations Nozzle design considerations Jet divergence Boundary layers Optimal nozzle contour Simulations on double gas jet Nozzle geometry Boundary conditions Simulation model Simulation results Gas jet characterization Measurement of supersonic gas jets Gas jet interferometry Interferometry set-up

5 TABLE OF CONTENTS III Fringe-pattern analysis Abel inversion Results Characterization of gas jet Measurements on helium jets First experiments on laser wakefield acceleration Discussion Conclusion Recommendations Double gas jet References 60 7 List of symbols 64 A Nozzle designs 67

6 1. INTRODUCTION 1 1 Introduction Particle accelerators are devices capable of accelerating charged particles to velocities approaching the speed of light. These highly energetic particles are of great interest for investigating the fundamental structure of matter and energy and are essential for advancing our understanding of the origin of the universe. Particle accelerators are employed in synchrotrons and free-electron lasers, which provide bright light sources at extremely short wavelengths. These light sources are used in material science, chemistry and biology for microscopy with atomic resolution and on ultra-short time scales. Next to these applications in fundamental research, particle accelerators have also many other applications, for example in medicine and industry. Examples are radiotherapeutic treatment of cancer, isotope production for nuclear medicine, ion implantation in semiconductor industry and treatment of industrial materials. However, despite their wide application, there are severe drawbacks and limitations. One major drawback is the large size and complexity of a conventional accelerator structure, and the high costs involved for building and operating the accelerator. Typically, a state-of-theart particle accelerator facility extends over multiple kilometres, costs billions of euros and can only be maintained by a large, worldwide collaboration of specialized engineers and scientists. The most prominent example of this is the Large Hadron Collider, located at the European Organization for Nuclear Research (CERN) near Geneva. Due to the large size, the enormous complexity and the high costs of such a conventional particle accelerator, the maximum particle energy that can be obtained is limited. Therefore it is of great importance to pursue novel concepts for particle acceleration which allow for a significant reduction of size and complexity. In order to be able to identify suitable novel acceleration concepts, it is instructive to discuss the origin of the large size of particle accelerators currently available. Conventional accelerating structures are based on radio-frequency (RF) cavities. In these cavities high electric fields are generated by microwaves to accelerate charged particles. To minimize the length required for acceleration of the particles to a specific kinetic energy, the electric field is set as high as possible. However, there is a fundamental limit on the maximum electric field strength that can be generated inside a RF cavity. At values of at most 100 MV m 1 the material at the walls of the cavity breaks down. To avoid this breakdown of the material, the electric field has to be set at much lower values in practice. This indicates that in order to accelerate electrons to an energy of several TeV the accelerator needs to be tens of kilometres in length. Yet, to achieve a significant reduction of the accelerator length and a substantial increase in particle energy, it is essential to explore novel acceleration concepts that provide electric fields orders of magnitude higher in strength than can be generated in RF cavities. One of such novel concepts is laser wakefield acceleration.

7 1. INTRODUCTION Laser wakefield acceleration A promising candidate for providing huge electric fields is found in laser wakefield acceleration (LWFA), which was proposed already in 1979 [1]. For laser wakefield acceleration, an ultra-short high-intensity laser pulse is sent into a plasma. As the laser pulse travels through the plasma, electrons are expelled from its path by the ponderomotive force, while the ions remain at an almost fixed position due to their higher mass. This leads to a separation of charge. After the laser pulse has passed the electrons are pulled back to their original position by the Coulomb force resulting from the charge separation. The electrons, however, overshoot their initial position due to their momentum, thereby creating an oscillating electron density modulation travelling in the wake of the laser pulse. This oscillation creates a longitudinal charge separation wave, called a plasma wave, trailing the laser pulse. Associated with the charge separations in such a plasma wave are extremely large longitudinal electric fields, forming the wakefield trailing the laser pulse, which can easily exceed 100 GV m 1 and can be used for acceleration of charged particles. The presence of these large longitudinal electric fields, thousands of times higher than encountered in conventional accelerators, is what makes laser wakefield acceleration a promising candidate for next-generation particle accelerators. When an electron bunch is properly injected into this wakefield it can be accelerated to ultra-relativistic energies over distances thousands of times shorter than in conventional accelerators. Next to this, LWFA is expected to enable generation of electron bunches with a short duration down to 100 as [2]. In figure 1.1(a) the relative electron density and longitudinal component of the electric field E z in the plasma wave trailing the laser pulse are shown. When an electron is injected in a region in which the electric field is negative, it can be accelerated in the forward direction by this field. In regions in which the electric field is positive the electron will be decelerated. The transverse component of the electric field, which alternately focusses and defocusses the charged particle bunches, is not shown. The plasma wave has a wavelength of λ p = 2πv p /ω p. The frequency ω p is called the plasma frequency and this is the frequency at which the electrons oscillate around their equilibrium position. The plasma frequency is given by equation (1.1) [3], with e the elementary charge, ε 0 the permittivity of free space and m e the electron mass. The phase velocity v p = ω p /k p of the plasma wave is equal to the group velocity of the laser pulse v g, which is approximately the speed of light c. ω p = n e e 2 ε 0 m e (1.1) For a typical plasma electron density of to cm 3 the plasma wavelength is 10 to 30 µm. To excite a laser wakefield of sufficiently large amplitude to accelerate electrons to ultra-relativistic energies, high-intensity ultra-short laser pulses with intensities of up to to W cm 2 are required. Such pulses can be provided by high-power Ti:sapphire lasers operating at a wavelength of 800 nm. The wakefield is most efficiently generated if the laser pulse length is equal to half of the wavelength of the plasma wave [4]. For typical plasma wavelength of 10 to 30 µm this means that ultra-short laser pulses with a duration of 15 to 50 fs are required. A first proof-of-principle experiment demonstrated laser wakefield acceleration for the first time in 1994 [5]. In the following years the LWFA concept was experimentally explored further. However, the accelerated electron bunches were of a poor quality. In all experiments only

8 1. INTRODUCTION a 2 / 2 E z n e/n e k p(z - v gt) (a) Linear wakefield excited by a laser pulse with a 0 = a 2 / 2 E z n e/n e k p(z - v gt) (b) Nonlinear wakefield excited by a laser pulse with a 0 = 2. Figure 1.1: Calculated one-dimensional laser wakefield for two different laser intensities. The intensity is expressed in terms of the maximum normalized amplitude a 0 = ee 0 /(m e ωc). Shown are the normalized profiles of the laser intensity a 2 (red), the normalized longitudinal component of the electric field E z = ee z /(m e ω p v g ) (blue) and the electron density relative to the equilibrium plasma density n e0 (green) [3]. 0

9 1. INTRODUCTION 4 a small tail of the energy spectrum extended to energies above 100 MeV, while the majority of the electrons had an energy below 10 MeV. This large spread in energy resulted from uncontrolled injection and trapping of electrons into all accelerating phases of the wakefield [3]. Yet, applications in for example particle physics and free-electron lasers require bunches with a rather small energy spread, of the order of 1 percent or less. Conventional accelerators in general surpass these requirements. For instance, the proton bunches accelerated in the Large Hadron Collider have an energy spread in the order of only 10 4 [6]. For future applications of laser wakefield acceleration, it is of great importance to reduce the energy spread in the bunches. To obtain monoenergetic electron bunches, all electrons should be injected into a volume with a length scale much smaller than the wavelength of the plasma wave and on a time scale much shorter than the plasma period. For a typical plasma wavelength of 30 µm, this requires the injection of an electron bunch having a length of the order of a few micrometre with a precision of about 1 µm and 1 fs [7]. Furthermore, for electrons to be trapped in the plasma wave they should travel with the proper velocity. Currently, the main issue in laser wakefield acceleration is a lack of control over the injection of electrons into the wakefield. A possible method to gain a larger degree of control is by making use of wave breaking of the plasma wave Breaking of plasma waves The wakefield that has been shown in figure 1.1(a) is a called linear wakefield. In this case the ponderomotive force of the laser pulse is small compared to the Coulomb forces in the plasma, resulting in sinusoidal wake profiles. When the laser intensity is increased not only the amplitude of the wake profiles increases, also the shape of the profiles changes. This nonlinear response is shown in figure 1.1(b). The profile of the electron density sharpens drastically and at these positions the longitudinal electric field steepens strongly. If the laser intensity is to be increased further, the electrons are displaced over larger distances in their oscillation. At some point the restoring Coulomb forces are not sufficient to let the electrons continue their oscillation. When this point is reached the plasma wave breaks, destroying the oscillation. In terms of velocities a plasma wave breaks when the velocity of the electrons in the plasma v e becomes equal to the phase velocity v p of the plasma wave. Wave breaking of plasma waves is of great interest for laser wakefield acceleration. This is because when a plasma wave breaks, a large fraction of the plasma electrons will be injected and trapped into the wakefield in a relatively small volume, on a short time scale and with the proper velocity. Subsequent acceleration of these electrons will result in monoenergetic electron bunches. In 2004 the first quasi-monoenergetic electron bunches were produced by laser wakefield acceleration [8 10]. By using a particular combination of high-intensity laser pulses and high plasma densities, researchers were able to produce electron bunches with energies in the order of 100 MeV and an energy spread of only a few percent. The parameter regime used is called the bubble regime [11]. More recently the energy of the electrons has been increased to 1 GeV. The bunches showed an relatively low energy spread of 2.5%, although the shot-toshot reproducability proved to be poor [12]. The low shot-to-shot reproducability originates because acceleration in the bubble regime relies on a nonlinear evolution of the laser pulse as it propagates some distance through the plasma before reaching a sufficient high intensity to induce breaking of the plasma wave.

10 1. INTRODUCTION 5 n e1 n e2 Δz g z Figure 1.2: Schematic plasma density profile for injection and trapping of electrons at a downward plasma density gradient. The plasma density decreases from n e1 to n e2 over a distance z g Injection at plasma density gradient To improve the shot-to-shot reproducibility and gain a greater degree of control over the wave breaking process, and thus the injection and trapping of electrons, different schemes to induce breaking of plasma waves have been proposed. Recall that wave breaking occurs when the velocity of the plasma electrons becomes equal to the phase velocity of the plasma wave. In the experiments in the bubble regime the intensity of the laser pulse was increased, thereby increasing the electron velocity to match the phase velocity of the plasma wave. A different approach, aiming at exactly the opposite, is to decrease the phase velocity of the plasma wave to match the electron velocity. This can be achieved by sending the laser pulse through an inhomogeneous plasma, in the direction of a downward gradient in plasma density [13]. The plasma density profile considered is schematically shown in figure 1.2. The profile consists of two regions with homogeneous plasma densities, the first part with a higher electron density n e1 and the second part with a lower, density n e2. In between these two regions the plasma density decreases from n e1 to n e2 over a distance z g. This distance z g is called the width of the density gradient. The process which leads to the breaking of plasma waves at a downramp in plasma density can be explained as follows. In the downramp the plasma density decreases, leading to a decrease of the plasma frequency according to equation (1.1). For plasma waves in an inhomogeneous plasma, the plasma frequency and wave number k p are related by k p / t = ω p / z [13]. Thus, a decrement of the plasma frequency in space results in an increment in wave number over time. This increasing wave number yields a decreasing phase velocity, as v p = ω p /k p. Eventually, after some time, the phase velocity becomes equal to the velocity of the electrons, triggering breaking of the plasma wave. In this way, a downward gradient in plasma density induces breaking of the plasma wave, enabling injection and trapping of electrons into the wakefield. For the optimal trapping of electrons, gradients with a width of 10 plasma wavelengths or more are required [13], corresponding to widths of 100 to 300 µm for typical plasma densities. Recently this principle of injection of electrons into the wakefield by a downward plasma density gradient was first demonstrated [14], obtaining electron bunches with a 10 to 100 fold lower spread in momentum, compared to previous experiments. Yet, the average energy of the electrons in the bunch was less than 1 MeV. This is attributed to the fact that the plasmavacuum interface at the edge of a plasma had been used. At this edge the plasma density

11 1. INTRODUCTION 6 decreased from n e1 to n e2 0, so that, immediately after injection and trapping into the wakefield, the electrons leave the plasma. Acceleration to higher energies can be performed by injecting this monoenergetic electron bunch into following acceleration stages, preserving the narrow spread in energy. A different method for injecting electrons using the downward plasma density gradient shown in figure 1.2 has also been identified [15, 16]. This method employs a sharp density transition, with a width in the same order or smaller than the plasma wavelength, corresponding to 1 to 10 µm [17]. In this downward density gradient the plasma wavelength λ p abruptly increases from its high-density value to a low-density value, causing a rephasing of a large fraction of the electrons, injecting them into the accelerating phase of the wakefield. This method for injecting electrons by a sharp density transition has only been demonstrated very recently [18]. Using a gradient in plasma density with a width of only 5 µm quasi-monoenergetic electron bunches with an energy of 24 MeV have been generated. 1.2 Aim of this thesis As has been discussed in the previous section, a downward density gradient as shown in figure 1.2 can be used to induce wave breaking of a plasma wave, resulting in injection and trapping of monoenergetic electron bunches. By changing the location and width of the density downramp one may directly influence this wave breaking process, allowing for a large degree of control over the injection and trapping of electrons into the wakefield. It is expected that these monoenergetic electron bunches can be accelerated to higher energies in a following acceleration stage, while preserving the low energy spread [19, 20]. The focus of this thesis is on designing and characterizing suitable plasma targets for density gradient injection in laser wakefield acceleration. These targets need to provide plasma densities of to cm 3 with sharp density gradients, with widths of either 100 to 300 µm for injection via wave breaking, or 1 to 10 µm for injection by rephasing. The length of the plasma target should match to the dephasing distance of the generated electron bunches, the distance over which the electron bunches outrun the plasma wave. This requires plasma targets with a length in the order of 1 mm. The envisioned laser wakefield acceleration set-up is schematically depicted in figure 1.3. A high-intensity ultra-short laser pulse is focussed into a helium gas jet. The leading part of this drive laser pulse already fully ionizes the gas, creating a plasma channel in the gas jet. The major part of the drive laser pulse then drives the wakefield in this plasma. To achieve the desired plasma density profile for density gradient injection, the neutral gas density in the gas jet will be shaped after the profile in figure 1.2. After ionization this profile will be imposed upon the plasma density in the jet. For the laser wakefield acceleration a 150 TW Ti:sapphire laser will be used, which is located at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR). It provides pulses with an energy of 4 J in 25 fs and the beam can be focussed to a spot size of 3 µm, yielding intensities of over W cm 2. For producing the required gas density profile supersonic gas jets are employed. A supersonic gas jet is a flow of gas, travelling at a velocity greater than the speed of sound, emanating from an orifice, called the nozzle. The nozzle is shaped such that one is able to control the properties of the gas jet, most importantly the density. Supersonic nozzles can be designed to produce a collimated gas jet, in other words, a gas jet which does not diverge strongly after exiting the nozzle. Specifically the edge of such a gas jet, that is the transition region from the

12 1. INTRODUCTION 7 Supersonic gas jet Laser pulse Electron bunch Plasma channel Nozzle Helium gas flow Figure 1.3: Schematic set-up for laser wakefield acceleration gas jet to the vacuum, provides a density downramp with a narrow width. Moreover, even much sharper transitions in density can be produced by shocks inside a supersonic gas flow. Because supersonic gas jets offer a large degree of control over the gas density and dthe density profile they are very suitable for use as targets in laser wakefield acceleration experiments [21 23]. By using gas jets which provide a density profile as shown in figure 1.2, it is possible to accelerate the electron bunches that are injected and trapped at the density gradient in the second part of the gas jet. This allows for integration of the injection stage with a first acceleration stage in a so-called double gas jet [24]. The possibility of using a double gas jet is also explored in this thesis. In order to illustrate to what degree one can control the density distribution in a supersonic gas jet it is important to understand the nature of the flow inside a supersonic nozzle. Therefore, in this thesis, first of all an analytical description of the gas flow inside a supersonic nozzle is presented in chapter 2. To get quantitative information about the three-dimensional structure of a supersonic gas jet, eventually resulting in the gas density profiles, a more extended analysis of the gas flow is required. This is provided for by using computational fluid dynamics to perform numerical simulations on the flow inside the supersonic jet. To study the behaviour of a double gas jet simulations have been performed by E.T.A. van der Weide [25]. The results of these simulations are presented in chapter 3. To be able to start with experiments on laser wakefield acceleration, several supersonic nozzles have been designed and constructed. Measuring the structure of the supersonic gas jets emanating from these nozzles is not a trivial task. Chapter 4 covers the characterization of the gas jets and discusses the results. Finally, the conclusions of this work and recommendations for future research are presented in chapter 5.

13 2. PHYSICS OF SUPERSONIC GAS JETS 8 2 Physics of supersonic gas jets As mentioned in the previous section, supersonic gas jets are employed to provide the particular gas density profiles for the laser-plasma interaction experiments. A supersonic gas jet is a flow of gas, travelling at a velocity greater than the speed of sound, emanating from an orifice, called the nozzle. The nozzle is shaped such that one is able to control the properties of the gas jet, most importantly the density. Because supersonic gas jets offer a large degree of control over the gas density they are very suitable for use as targets in laser-plasma interaction experiments. In order to illustrate to what degree one can control the density distribution in a supersonic gas jet it is important to understand the nature of the flow inside a supersonic nozzle. The first step, in providing a clear picture of such a gas flow, is determining the thermodynamic properties and relations relevant to the flow of an ideal gas, after which a continuum approach is followed to construct a framework for determining the kinetic behaviour of the gas flow, neglecting viscous effects and conductive heat transfer. Because the gas is flowing at velocities equal to and above the speed of sound is it essential to take into account effects of compressibility of the gas. The development of this framework is covered in section 2.1. Having introduced the various concepts regarding compressible gas flows an analysis of the flow in a supersonic nozzle is given. Analytical solutions are available to quasi one-dimensional flows, which are discussed in section 2.2. Next, a connection is made between the flow inside the supersonic nozzle and the gas jet plume emanating from this nozzle. For the gas and vacuum pressures commonly encountered in laser-plasma experiments, the gas jet emanating from a supersonic nozzle will be in the underexpanded regime, resulting in the appearance of a structure of shocks in the jet plume. The formation of shocks is described in section 2.3, resulting in a description of the structure of the gas jet emanating from a supersonic nozzle, including the influence of the background pressure in the vacuum chamber. Finally, due to the high pressures and low temperatures encountered in supersonic expansions, gas particles can aggregate and form clusters of atoms or molecules. Because the presence of large clusters in a gas jet may influence the behaviour of the flow, the process of condensation of gas particles is considered in section Compressible gas flow In order to obtain a suitable description of the density distribution in a supersonic gas jet it is important to understand the nature of the flow inside a supersonic nozzle. The first step, in providing a clear picture of such a gas flow, is determining the thermodynamic properties and relations relevant to the flow. In the following analysis all thermodynamic properties needed

14 2. PHYSICS OF SUPERSONIC GAS JETS 9 to describe a supersonic flow of an ideal gas are presented Thermodynamic gas properties In local thermodynamic equilibrium, all thermodynamic properties of a homogeneous and isotropic one-component medium are completely determined by the variables pressure p, temperature, T and volumetric mass density ρ. The relation between the pressure, temperature and density of a medium is given by the equation of state. Using the equation of state, two variables are always sufficient to determine the thermodynamic state of the system. For an ideal gas the equation of state is formed by the ideal gas law (2.1), with R the ideal gas constant and M the molar mass of the medium [26]. p = ρrt M (2.1) In the description of an ideal gas two main assumptions are made. Firstly, the size of the gas atoms or molecules is insignificant with respect to the average distance between different gas particles. This means that each gas particle is treated as a point mass. Secondly, the work exerted by forces between different gas particles can be neglected when compared with the kinetic energy of the particles. This corresponds to the approximation that the gas particles do not experience repulsive or attractive forces and all collisions between particles are elastic. Likewise, phenomena such as nucleation and condensation of gas particles are not included. Consulting the phase diagrams of helium and argon [27, 28], one finds that the pressures and temperatures used for operation of the gas jets discussed in this work are located at a distance from the vapour-liquid and vapour-solid lines. Therefore the approximation of an ideal gas is justified. However, in the expansion of the gas in the nozzle the temperature can decrease dramatically, possibly giving rise to clustering of the gas atoms, depending on the initial conditions of the gas. The possible presence of condensation in gas jets is covered in section 2.4, while, for now, the medium is treated as an ideal gas, consisting only of single gas particles. The density of the medium can also be expressed as the number of particles in a volume. The resulting particle density n is related to the volumetric mass density by n = ρ N A M (2.2) where N A is the Avogadro constant. The constant reflects that an ideal gas at standard temperature and pressure of 298 K, 1 atm has a particle density of cm 3. For laser-plasma interaction experiments expressing the density in terms of number of particles per volume is more convenient than using the amount of mass enclosed in a volume. This is because, after ionization of the gas, the resulting electron density in the plasma is directly related to the particle density of the neutral gas. In fluid mechanics however, it is common practice to use the volumetric mass density. This practice is followed in the analysis in this section. The amount of energy that is contained in a medium is given by the internal energy. Using kinetic gas theory, the specific internal energy u for an ideal gas can be derived. In thermodynamics it is convenient to express a quantity concerning energy in terms of energy per unit mass, as a specific energy. This practice is followed throughout this work, thereby making the quantity independent of the amount of material considered. The specific internal energy is

15 2. PHYSICS OF SUPERSONIC GAS JETS 10 found to depend only on the temperature of the gas through u = 1 2 f RT M (2.3) where f is the number of degrees of freedom of the gas particles. For a monoatomic gas only the three degrees of freedom corresponding to translation of the atom contribute to the internal energy. Due to the non-zero moment of inertia, for a diatomic gas one has also to take into account the two degrees of freedom associated with rotation around the centre of mass of the molecule. Vibrations of the molecule only become important at very high temperatures, above 5000 K. At the much lower temperatures considered here these levels do not give any contribution to the internal energy. They are then called to be frozen out. The internal energy of a system can be changed by adding heat to or performing work on the system. The resulting change in internal energy after adding a small amount of heat δq or performing work δw is given by the first law of thermodynamics, in which the change in work is defined as ( ) 1 δw = p δ ρ The change in entropy δs corresponding to heat added is given by δu = δq+δw (2.4) δs = δq T When considering the total energy contained in a gaseous medium which is at rest and in equilibrium with its environment one has to take into account the energy that is associated with the non-zero volume of the medium. This expresses that, to bring the medium to its present state from absolute zero, energy must be supplied equal to its internal energy u plus p/ρ, where p/ρ is the work performed in pushing against the ambient pressure. The total energy contained in the amount of gas is thus the sum of the internal energy and the work performed in the expansion. It is called enthalpy and is defined as (2.5) (2.6) h u+ p ρ (2.7) The amount of heat needed to raise the temperature of a medium by 1 K is called the specific heat capacity. The amount of heat required depends on the conditions under which the heat is transferred to the medium. Two realizable methods are maintaining a constant pressure or maintaining a constant volume. The corresponding specific heat capacities c p and c V are defined as ( ) h c p (2.8) c V T ) ( u T By substituting equation (2.3) in these expressions one is able to relate the heat capacities to the system properties of an ideal gas, giving p V (2.9) c p c V = R M (2.10)

16 2. PHYSICS OF SUPERSONIC GAS JETS 11 And by defining the adiabatic index γ as the ratio between the specific heat capacities explicit relations for c p and c v are found. γ c p c V (2.11) c p = γ R γ 1 M (2.12) c V = 1 R γ 1 M (2.13) By inserting equation (2.3) into the expressions for for c p and c v, one finds that the adiabatic index γ only depends on the number of degrees of freedom of the gas particles through γ = f + 2 f (2.14) From this expression it is found that the adiabatic index for monoatomic gases is γ = 5/3 and for diatomic gases it is γ = 7/5. Since the number of degrees of freedom that are contributing to the internal energy is constant, except at high temperatures, the medium can be considered to behave as what is called a caloric perfect gas, for which c p and c V can be considered constant. Using the definitions of the specific heat capacities, (2.8) and (2.9), the internal energy and enthalpy with constant c p and c V can be expressed as u = c V T (2.15) h = c p T (2.16) The change in entropy resulting from a process in which a gas is brought from state 1 to state 2 is then given by s 2 s 1 = c p ln T 2 T 1 R M ln p 2 p 1 (2.17) From this equation one can derive the well-known relation between the pressure and density of a gas in an isentropic process, an process in which entropy is conserved. p = constant (2.18) ργ Any process which is adiabatic as well as reversible conserves entropy. In an adiabatic process no heat is transferred to the environment. A process is reversible when no energy is dissipated as heat. With this, all relevant thermodynamic properties and relations needed in order to describe the kinetic behaviour of an ideal gas are available. The next step for describing the flow in a supersonic nozzle is selecting a suitable description of the kinetic behaviour of the gas particles.

17 2. PHYSICS OF SUPERSONIC GAS JETS Flow regimes Different approaches can be taken to describe the macroscopic kinetic behaviour of a gas, depending on the flow regime of the gas. For instance, when the gas particles can move freely throughout a volume and travel large distances before colliding with other particles, it is appropriate to use statistical methods to describe the flow of the gas. However, when the particles only travel minor distances between collisions and one is interested in the gross behaviour of the gas, it can be treated as a continuous medium. A dimensionless number to characterize the flow regime of a fluid flow is the Knudsen number, Kn. It compares the mean free path of the fluid particles l with a characteristic length scale L of the system. The mean free path of a particle is the average distance the particle has travelled between two collisions. Kn l L (2.19) For a Maxwellian distribution of the particle velocities, as it results from kinetic gas theory, the mean free path of of the particles is given by equation (2.20), with σ sc the scattering crosssection of the particles. By using the Van der Waals radius r W to estimate the scattering cross-section of the gas particles [29], the Knudsen number is given by equation (2.22). l = 1 2σsc n (2.20) σ sc = 4πr W 2 (2.21) RT Kn = 4 2πr 2 W pn A L (2.22) For Knudsen numbers below 10 3, the mean free path of the particles is much smaller than the length scale of the system and the fluid flow can therefore be regarded as a continuum flow, a flow of a continuous medium. For a description of the dynamics of a continuum flow the Navier-Stokes equations are used. Larger Knudsen numbers up to 0.1 correspond to what is called a slip flow. In this flow regime, the gas velocity and temperature at the wall of a channel differ significantly from the velocity and temperature of the wall itself. This can be accounted for when solving the Navier-Stokes equations. When the mean free path of the particles is larger or comparable to the length scale of the system (Kn > 10) one has to describe the flow as molecular flow using statistical methods. For supersonic nozzles, the Knudsen number reaches its maximum at the exit of the nozzle. This is due to the large pressure drop across the nozzle. The smallest nozzle considered for our experiment is a Mach 5 helium nozzle with a 0.75 mm exit diameter. When operated with helium gas at 1 bar, 298 K, the mean free path of the gas atoms is in the order of 10 µm at the nozzle exit, corresponding to a Knudsen number of This shows that for typical reservoir pressures of 10 bar or higher the approximation of continuum flow throughout the entire nozzle is well justified Dynamics of continuum gas flow For describing the dynamics of the continuum flow of a gas through a system, in other words, for describing its spatio-temporal development, one has to consider conservation of mass,

18 2. PHYSICS OF SUPERSONIC GAS JETS 13 momentum and energy. For convenience and completeness we recall here standard theory on fluid mechanics [30].Conservation of mass states that if any change in the amount of mass in a region occurs, this amount has to equal the amount of mass passing through the boundaries of the region. The conservation of mass is described by the continuity equation. Dρ Dt + ρ v = 0 (2.23) with v the fluid velocity and time t. This equation includes a temporal derivative called the material derivative. For any fluid property X, the material derivative is defined by equation (2.24). DX Dt = X t + v X (2.24) The material derivative forms a bridge between the Eulerian and Lagrangian description of a fluid flow. In the Lagrangian approach one follows a single gas particle moving through the fluid in order to describe the fluid flow. The gross behaviour of a fluid flow is the result of forces acting upon each particle. However, usually, one is interested in the properties of the flow at fixed points in space and time, instead of the properties of a fixed fluid particle. This is called an Eulerian description of the fluid flow. The Navier-Stokes equation governs the conservation of momentum for a continuum flow. Any change in velocity will be the result of a force exerted on the fluid particles. Assuming that the stress in the fluid can be separated into a pressure term and a viscous term, the conservation of momentum is implied by the Navier-Stokes equation ρ D v Dt = p+ρ g+ σ (2.25) with standard gravity g and σ the deviatoric stress tensor, which represents effects of viscosity. When the density of a compressible fluid changes, a change in temperature results. Therefore in addition to the continuity equation and Navier-Stokes equation the heat equation is used for maintaining energy conservation, in which ϕ represents an external heat flux. ρ Du Dt = ϕ p v+ σ : v (2.26) It is convenient to write the heat equation in a different form, in terms of enthalpy instead of internal energy. Calculating the material derivative of the enthalpy (2.7), making use of continuity (2.23), and inserting this into the heat equation one obtains a heat equation in terms of enthalpy. ρ Dh Dt = ϕ+ Dp Dt + σ : v (2.27) Using the Navier-Stokes equation (2.25) multiplied by the fluid velocity v and adding this to the heat equation in terms of enthalpy a more useful form of the heat equation is obtained. After noting that v 2 /2 corresponds to specific kinetic energy it is clear that the left-hand side of equation (2.28) gives a measure of the time rate of change in the total energy of the flow. D ρ (h+ 1 2 v 2) Dt = ϕ+ p t + ρ v g+ v σ (2.28)

19 2. PHYSICS OF SUPERSONIC GAS JETS 14 To simplify the relations needed to describe the compressible fluid flow, given by equations (2.23), (2.25) and (2.28), a few assumptions are made. When the velocity of the fluid flow is high, viscous effects are only present in the close neighbourhood of objects such as the walls. For the general behaviour of the flow, viscosity can be neglected. Neglecting viscous effects, implicating that no heat will be generated in the fluid makes the flow reversible. Because of the high flow velocity, also conductive heat transport is only of importance near an object or a wall. By not taking into account conductive heat transfer the fluid flow can be considered adiabatic. Therefore, when viscous effects and conductive heat transfer are of minor importance due to the high flow velocity, the compressible fluid flow will be isentropic. Furthermore, when describing motion on a small length scale it is not necessary to take into account gravity. Under these assumptions, the Navier-Stokes equation (2.25) transforms into the Euler equation, given by. ρ D v Dt = p (2.29) For the heat equation (2.28) D (h+ 1 2 v 2) ρ = p (2.30) Dt t is obtained. Together with the continuity equation (2.23), the equation of state for an ideal gas (2.1) and the relation for a caloric perfect gas (2.16), these expressions (2.29) and (2.30) provide the framework to describe the flow of a compressible fluid. This framework will now be used to study the behaviour of a supersonic gas flow, starting with taking a look at the speed of sound and the definition of the Mach number, which is useful in the context of supersonic flows Speed of sound The nozzles that are constructed and used in the laser-plasma interaction experiments are capable of producing flows with velocities far greater than the speed of sound. The speed of sound is the speed at which an acoustic wave travels through a medium. In a gaseous medium acoustic waves propagate by small disturbances in pressure. To see the importance of the speed of sound, note that any information about the system of the fluid flow, such as geometry or ambient pressure, can only travel through the medium at the speed of sound. When dealing with velocities above the speed of sound this will have important consequences for the behaviour of the fluid flow, which will be pointed out in the following sections. It is therefore necessary to take a closer look into the definition of the speed of sound and its connection to supersonic flows. In fluids, a small disturbance propagates in the form of an acoustic wave that is governed by the speed of sound. To calculate this speed of sound, imagine an infinitesimal perturbation in pressure δp propagating through a fluid. In figure 2.1 a pressure disturbance travelling to the right, in the positive x-direction, with velocity v s is shown. On the right side of the wavefront, ahead of the wave, the fluid is stationary. We assume that the flow has zero velocity, v = 0, and the pressure, density and temperature have values of p = p 0, ρ = ρ 0, T = T 0, called the stagnation properties. Behind the wave, the fluid is flowing at a velocity δv to the right and has properties p = p 0 + δp, ρ = ρ 0 + δρ, T = T 0 + δt. Using conservation of mass (2.23) and neglecting all terms of second order in δv, δp, δρ one arrives at δρ t + ρ 0 δv x = 0 (2.31)

20 2. PHYSICS OF SUPERSONIC GAS JETS 15 Wavefront p0 + δp p 0 T 0 T0 + δt ρ0 + δρ ρ 0 v s δv v = 0 x Figure 2.1: A disturbance propagates to the right into a still fluid [31]. Performing a similar substitution in the Euler equation (2.29), again neglecting all higher order terms, gives δv ρ 0 = δp (2.32) t x Finally, combining the derivative with respect to time of equation (2.31) and the spatial derivative of equation (2.32), and subsequently extracting the pressure from the spatial derivative, results in with 2 δρ t 2 = 2 δp x ( 2 ) p = ρ = v 2 s v 2 s = S 2 δρ x 2 2 δρ x 2 (2.33) ( ) p ρ s (2.34) Equation (2.33) is a one-dimensional wave equation for the propagation of a density wave travelling at the speed of sound of velocity v s. The speed of sound is defined in terms of the pressure change resulting from a change in density, while conserving entropy. The amount of irreversible entropy production in a passing acoustic wave is proportional to the square of the gradient of the temperature and velocity [31]. Therefore for infinitesimal changes of δv and δt each particle undergoes an almost isentropic process as the wave passes by. This justifies that the partial derivative in (2.34) is taken at constant entropy. Using equations (2.1) and (2.18) the following explicit expression for the speed of sound in an ideal gas is obtained. γrt v s = (2.35) M

21 2. PHYSICS OF SUPERSONIC GAS JETS 16 The equation shows that, within the named approximations, the speed of sound is only dependent on the temperature and species of the gas. As an example, at 298 K the speed of sound of helium is 1016 m s 1. Due to its higher molar mass, the speed of sound in argon is lower, reaching only 321 m s 1. The origin of the macroscopic kinetic behaviour of a gas flow is found in collisions of moving gas particles. This then also applies to the propagation of acoustic waves, which propagate via collisions of the moving gas particles. Therefore, one could expect the speed of sound to be in the same order of magnitude as, but limited to, the velocity at which the gas particles travel between collisions. This can be verified using kinetic gas theory as follows. From kinetic gas theory, the velocities of the gas particles in a continuum fluid flow can be described by a Maxwell distribution, with an average velocity of v = 8RT πm (2.36) A comparison of the speed of sound with the average velocity of a Maxwell velocity distribution given by (2.36) shows that, for a monoatomic gas, the speed of sound is a fraction of γπ/8 0.8 lower than the average velocity of the particles v, but in the same order of magnitude. When dealing with flow velocities with a magnitude in the same order as the speed of sound or higher it is standard to introduce an dimensionless number for the velocity of the fluid flow. This number is the Mach number and is defined as the ratio between the speed of the fluid and the speed of sound. Ma v v s (2.37) When a fluid is flowing with Mach 1, it is travelling at sonic speed, in other words, with the speed of sound. A flow with Ma < 1 is called subsonic and a fluid flow with Ma > 1 is called supersonic Compressible flow When dealing with fluid flows at relatively low velocities, the density of the fluid is often assumed to be constant in each infinitesimal volume element in the flow. This is possible if any changes in density resulting from pressure changes in the medium are very small, such that they have a negligible influence on the behaviour of the fluid flow. The fluid is then treated as incompressible, simplifying the description of the fluid flow. However, if large changes in density arise with respect to pressure, these will effect the behaviour of the flow. The category of fluid flows in which changes in density are so large that they give a large contribution to the dynamics of the flow is called compressible flow. For a description of a fluid flow at high velocities it is important to take into account these compressibility effects. In this section, an estimate is given of the flow speed at which it is no longer justified to neglect the compressibility of the fluid. For this, consider the continuity equation (2.23), for a steady one-dimensional flow. v ρ x + ρ v x = 0 (2.38) The continuity equation (2.38) indicates that a fluid flow can be considered incompressible when relative changes in density δρ are much smaller than changes in velocity δv. This gives

22 2. PHYSICS OF SUPERSONIC GAS JETS 17 Throat p 0 T 0 v = 0 A * p T v A x Figure 2.2: Quasi one-dimensional flow through a nozzle of varying cross-sectional area A. the criterion δρ ρ δv v (2.39) Using the definition of the speed of sound (2.34) it is possible to estimate the change in pressure δp corresponding to a change in density. δp v s 2 δρ (2.40) The relation between the small change in pressure and the change in velocity is given by the Euler equation (2.29). ρvδv δp (2.41) Combining this with expression (2.39) results in a criterion that has to be fulfilled for incompressible flow. δρ ( ) v 2 ρ δv v s v = δv Ma2 v δv (2.42) v Ma 2 1 (2.43) When tolerating a difference of one order of magnitude, such that Ma 2 < 0.1 this criterion states that incompressible flow can be assumed for flow speeds below Ma < 0.3. However, when dealing with fluid flows at velocities above Mach 0.3, compressibility effects have to be taken into account. Since supersonic flows, already by definition, lie beyond this limit, taking into account compressibility effects is therefore inherent to a suitable description of supersonic gas flows. For instance, the regime of interest for laser-plasma interaction experiments concerns supersonic flows from nozzles with Ma 4.8, of which the characteristics will be described in the next section. 2.2 Quasi one-dimensional supersonic flow As stated in the section before, including compressibility effects is essential in obtaining a proper description of the gas flow through a supersonic nozzle. For this description, consider

23 2. PHYSICS OF SUPERSONIC GAS JETS 18 a steady, isentropic, quasi one-dimensional flow through a nozzle with cross-sectional area A, as shown in figure 2.2. In a one-dimensional flow all flow variables vary only with the position x in the direction of the flow. When the cross-section of the nozzle is also allowed to change with the position, having A = A(x), this is called quasi one-dimensional flow. To describe a quasi one-dimensional flow, start from the continuity equation (2.23) and the heat equation (2.30). Combined they read for steady flow (ρ v) = 0 (2.44) ( ρ v ρ v h+ 2 1 v 2) = 0 (2.45) Integrating the continuity equation (2.44) over an infinitesimal fluid volume Adx states that the total mass flow is constant. ρva = constant (2.46) Performing a similar integration of the heat equation gives the following expression. ( ρv h+ 2 1 v2) A = constant (2.47) Combining the two expressions, one obtains h+ 1 2 v2 = constant (2.48) Equation (2.48) is a formulation of conservation of energy. It states that enthalpy h can be transformed into kinetic energy v 2 /2 and vice versa. The total energy inside the flow will remain constant. This can be seen more clearly by expressing equation (2.48) in the stagnation properties of the fluid h 0, T 0, v = 0, at which the fluid is at rest. Using the properties of a caloric perfect gas, described by equation (2.16), a relation between the speed and temperature of the flow inside the duct can be obtained (2.51). h+ 1 2 v2 = h 0 (2.49) c p T+ 1 2 v2 = c p T 0 (2.50) T 0 T = 1+ v2 2c p T (2.51) It is convenient to express the temperature of the gas in terms of the Mach number by eliminating the heat capacity from (2.51). In terms of the Mach number the temperature of the fluid reads T 0 T γ 1 = 1+ Ma 2 (2.52) 2 Next, for an isentropic flow equation (2.18) can be used to calculate the behaviour of the pressure and density inside the nozzle. ( p 0 p = 1+ γ 1 ) γ Ma 2 2 γ 1 ( ρ 0 ρ = 1+ γ 1 ) 1 Ma 2 2 γ 1 (2.53) (2.54)

24 2. PHYSICS OF SUPERSONIC GAS JETS 19 Now, by assuming a steady, isentropic, quasi one-dimensional gas flow through a nozzle, one has arrived at three important equations for dealing with supersonic nozzle flows. They state that once the Mach number throughout the flow is known, the relative temperature (2.52), pressure (2.53) and density (2.54) of the fluid throughout the flow are fully determined. The absolute gas properties are then only dependent on the stagnation conditions T 0 and p 0 in the reservoir connected to the entrance of the nozzle. To explain this, note that any information about the exit of the nozzle, such as ambient pressure, can only travel upstream as a small disturbance with the speed of sound. Because the velocity of the flow is larger than the speed of sound the net velocity is directed towards the nozzle exit and this information cannot propagate upstream. Therefore the flow properties inside the nozzle are not influenced by anything that occurs downstream. According to the three equations above, (2.52), (2.53), (2.54), the temperature, pressure and density throughout the gas flow in the nozzle are governed by the Mach number. To determine the Mach number of the gas flow one first relates the gas density in the flow to the gas density ρ at the point the flow reaches the speed of sound, at Ma = 1. Starting from conservation of energy (2.48) this is performed in a similar way as is done to derive equation (2.54), resulting in an expression for ρ /ρ. The continuity equation (2.46) then relates this to the cross-sectional area of the nozzle. In this way equation (2.55) is obtained, in which A is the area of the nozzle at which the flow reaches the speed of sound. A A = 1 ( ( 2 Ma γ+1 1+ γ 1 2 )) 1 γ+1 2 Ma 2 γ 1 (2.55) Equation (2.55) shows that the Mach number of the gas flow only depends on the relative area of the nozzle. Note that this area-mach number relation is double-valued concerning the Mach number, resulting from the Ma 2 term. For each area of the nozzle there exist exactly two isentropic solutions for the gas flow, corresponding to a subsonic and a supersonic flow. The area A at which the flow reaches the speed of sound can be found by differentiating the continuity equation (2.46) with respect to x. Rewriting the result in relative changes of density, velocity and area gives δρ ρ + δv v + δa A = 0 (2.56) After eliminating the density term using equation (2.42) it relates a change in flow velocity δv to a change in nozzle cross-section δa. δv v = 1 δa 1 Ma 2 A (2.57) Inspecting equation (2.57) reveals that any resulting change in velocity is proportional to the change in area. For a subsonic flow, Ma < 1, δv is proportional to δa. This shows that if the area of the nozzle decreases, the flow speeds up. Thus, compressing a subsonic flow results in an acceleration of the flow. In contrast, the change in velocity δv for a supersonic flow, having with Ma > 1, is proportional to +δa. A supersonic flow thus speeds up with an increase in area, that means, with an expansion of the flow. This behaviour of accelerating supersonic flow is opposite to the way a subsonic flow behaves. Combining these implies that if a gas flow reaches exactly the speed of sound at Ma = 1, it will do so wherever the nozzle area is at its minimum, δa = 0.

25 2. PHYSICS OF SUPERSONIC GAS JETS 20 The difference in behaviour of subsonic and supersonic flows to changes in area is commonly used to accelerate a flow to supersonic speeds as follows. First the flow is compressed to accelerate it to the speed of sound. Exactly at this point the cross-section is increased again and the flow is allowed to expand in a diverging section such that it accelerates further to supersonic speeds. The minimal cross-section of the nozzle, at which the gas flow becomes sonic, is called the throat. Nozzles with a convergent-divergent shape used for generating supersonic gas flows are called De Laval nozzles. Applying the quasi one-dimensional flow analysis discussed above to a supersonic De Laval nozzle one can now calculate the properties of the flow inside such a nozzle. Figure 2.3 shows an example of a calculation for an axisymmetric convergent-divergent nozzle with a throat diameter of 0.25 mm and an exit diameter of 0.75 mm. The nozzle is operated by helium gas with a temperature of 298 K and a pressure of 10 bar at the inlet. In figure 2.3(a) the nozzle geometry is shown, next to the Mach number and the absolute velocity along the direction of the flow in figure 2.3(b), and in figure 2.3(c) the relative gas temperature, pressure and density with respect to the inlet conditions. Figure 2.3(b) shows that, as the flow accelerates (from left to right) in the converging part of the nozzle, the Mach number increases until it reaches unity at the throat. At this point the flow is travelling with the speed of sound. In the diverging section the Mach number increases further reaching Ma = 4.8 at the exit. The absolute flow velocity approaches a finite value of 1.7 km s 1 in the diverging part of the nozzle. This is results from conversion of enthalpy into kinetic energy and will be discussed in the following section. The temperature of the gas drops significantly along the direction of the flow in the diverging part of the nozzle, as is shown in figure 2.3(c), reaching 34 K at the exit. Because the speed of sound decreases with the temperature, as described by expression (2.35), the Mach number keeps increasing while the absolute flow velocity approaches a finite value. Similar to the temperature, also the pressure and density exhibit a large decrease, as depicted in figure 2.3(c). The pressure at the nozzle exit has dropped to 44 mbar, which results in a particle density of cm 3. Looking back at equation (2.50), it is easy to see that the fluid flow in the nozzle cannot obtain an arbitrarily high velocity. As the flow accelerates from rest inside the nozzle to a velocity equal to and beyond the speed of sound, the temperature, pressure and density of the gas all drop significantly. If the gas flow reaches a maximum velocity the temperature, the pressure and the density approach zero. Starting at rest, at a temperature T 0, the fluid can only be accelerated up to a point at which the temperature T of the fluid has dropped to zero. All initial enthalpy has then been converted into kinetic energy and the flow has obtained a maximum velocity. Setting T = 0 in equation (2.50) to obtain an explicit expression for the maximum flow velocity and subsequent substituting c p from equation (2.12) yields 2Rγ v max = M(γ 1) T 0 (2.58) To give an example, starting at rest at 298 K, a flow of helium in a supersonic nozzle can reach a maximum velocity of 1760 m s 1. Under similar conditions an argon flow can only obtain a velocity of 557 m s 1. With this a suitable description of the gas flow internal to a supersonic nozzle has been obtained. To make a connection to the gas jet exiting from this supersonic nozzle, additional effects, such as the formation of shocks have to be taken into account. The formation of shocks and its influence on the structure of a gas jet exiting from a supersonic

26 2. PHYSICS OF SUPERSONIC GAS JETS Throat r (mm) x (mm) (a) Geometry of the supersonic nozzle Ma v (m s ) x (mm) (b) Mach number (black) and velocity (red) 1.0 p / p 0 ρ / ρ 0 T / T x (mm) (c) Temperature (blue), density (green) and pressure (red) Figure 2.3: Calculated properties of a gas flow inside an axisymmetric supersonic nozzle, reaching Mach 4.8 at the nozzle exit. The nozzle is operated with helium at 298 K.

27 2. PHYSICS OF SUPERSONIC GAS JETS 22 Shock Ma 1 Ma Figure 2.4: The properties of a flow change significantly and abruptly when it encounters a normal shock while going from region 1 to region 2. x nozzle are discussed in the next section. 2.3 Supersonic gas jet Having derived an analytical solution for the compressible gas flow in a quasi one-dimensional supersonic nozzle, it is possible to make a connection between the flow inside the supersonic nozzle and the gas jet plume emanating from this nozzle. For the gas and vacuum pressures commonly encountered in laser-plasma experiments, the gas jet emanating from a supersonic nozzle will show a structure of shocks in the jet plume. A shock is a large disturbance of the flow, characterized by a significant, abrupt change in velocity, pressure, temperature and density on a very small length scale. Shocks are commonly encountered in aerodynamics, and a propagating shock can for instance be produced by an explosion. Formally, shocks are discontinuous solutions to the equations which govern compressible flow. It is appropriate at this point to first introduce the concept of the occurrence of shocks in supersonic flows, before continuing with the characteristic properties and features of the plume of a supersonic gas jet in following sections Normal shocks Until now only continuous solutions for the gas flow have been discussed. In addition to these, the governing equations for compressible flow allow also for discontinuous solutions. These exist in the form of shocks, which play a major role in supersonic expansions. A shock is characterized by a large change in flow velocity, pressure, temperature and density over a very small length scale, in the order of a few mean free paths of the gas particles [32]. To calculate the behaviour of the gas flow at a shock consider the situation depicted in figure 2.4. Gas is flowing through a nozzle in which a shock occurs, oriented normal to the direction of the flow. The flow upstream of the normal shock, in region 1, has a relative velocity with Mach number Ma 1. The flow downstream of the shock, in domain 2, has a Mach number of Ma 2. The shock is assumed to be fixed in position, so that steady flow can be assumed. To describe the properties of a shock one has to evaluate the equations for conservation of

28 2. PHYSICS OF SUPERSONIC GAS JETS 23 mass, momentum and energy for a one-dimensional flow from region 1 into region 2. As the analysis is one-dimensional, it is only valid for a shock normal to the direction of the flow. For oblique shocks, with a shock front oriented under an angle β to the direction of the upstream flow velocity, the analysis holds as well, after substituting Ma 1 sin β for Ma 1. Because only the direct vicinity of the shock is considered, any change in area across the shock can be neglected. Rewriting the continuity (2.46), Euler (2.29) and heat (2.48) equations results in ρ 1 v 1 = ρ 2 v 2 (2.59) p 1 + ρ 1 v 1 2 = p 2 + ρ 2 v 2 2 (2.60) h v 1 2 = h v 2 2 (2.61) Assuming an ideal, caloric perfect gas to eliminate the pressure and enthalpy variables in favour of the speed of sound, the Euler (2.60) and heat (2.61) equations become ρ 1 v 2 s1 + ρ γ 1 v 2 1 = ρ 2 2v s2 2 + ρ 2 v 2 γ (2.62) v s1 2 2 γ v 1 2 = v s2 γ v 2 2 (2.63) The equations (2.59), (2.62) and (2.63) are solved by elimination of ρ 2. Rewritten in terms of a characteristic speed of sound v s, the speed of sound at the position where the flow reaches Ma = 1, the equations can be solved for (v s 2 v 1 v 2 ) (v 1 v 2 ) = 0 (2.64) The trivial, continuous solution of the set of equations is v 1 = v 2, which means that the flow does not change when going from region 1 into 2 in figure 2.4. The non-trivial solution is given by v s 2 = v 1 v 2 (2.65) which is known as the Prandtl relation. The fluid flow properties for this discontinuous solution are given by the Rankine-Hugoniot relations (2.66), (2.67). Ma 2 2 = (γ 1) Ma 1 2 γma (γ 1) (2.66) p 2 = 1+ 2γ ( ) Ma 2 p 1 γ (2.67) These relations can be used to determine the change in Mach number, pressure and other flow parameters when the flow goes through the transition from region 1 to region 2 in figure 2.4. These relations show that only shocks with Ma 1 > 1 > Ma 2, or vice versa are possible. To determine which of the solutions following from these relations is valid one has to consider the change in entropy across the shock. When calculating this entropy change using formula (2.17), one discovers that the entropy across a normal shock is not conserved. For a situation with Ma 1 > Ma 2, a compression shock, an increase in entropy is found. An expansion shock, with Ma 1 < Ma 2, would result in a decrease in entropy. However, a decrease in entropy would violate the second law of thermodynamics from which it has to be concluded that in

29 2. PHYSICS OF SUPERSONIC GAS JETS 24 subsonic flows no shocks can appear. Thus, shocks can only occur in supersonic flows in the form of compression shocks. The flow upstream of the shock abruptly changes to a subsonic flow downstream. Across the shock, the velocity and Mach number of the flow decrease, the temperature, pressure and density increase. Because expansion shocks would lead to an increase of entropy and thus cannot exist, a large expansion of a supersonic flow will occur isentropically. This is called a Prandtl-Meyer expansion [31]. Throughout such an expansion the flow parameters change gradually and entropy is conserved. In section the propagation of small disturbances, in the form of acoustic waves, through a fluid has been discussed. Large disturbances travel as moving shocks. To obtain the propagation speed of a moving shock wave, as produced by for instance an explosion, one has to subtract v 1 from the velocities in equations (2.59), (2.60) and (2.61), such that one enters the frame of reference in which the upstream flow is stationary. What results is that the shock wave propagates with a velocity v 1, which is larger than the speed of sound. The propagation of large disturbances is facilitated by shock waves and is not limited by the speed of sound, in contrast to small disturbances. Now that the concept of shocks in a supersonic flow has been clarified, it is possible to move forward to the implications that shocks have on expansions, both inside as well as outside a supersonic nozzle. A first appearance of shocks in the supersonic gas jet will become visible when looking at the influence of the pressure drop across the nozzle on the flow through the supersonic nozzle Influence of ambient pressure The parameter driving the flow through a supersonic nozzle is the pressure drop across the nozzle. The way the pressure drop influences the flow inside a De Laval nozzle can be explained using figure 2.5. In this figure the pressure distribution throughout the nozzle is displayed for different downstream ambient pressures p a. The upstream reservoir is kept at constant pressure p 0. Recall that given the nozzle geometry, the temperature and the pressure, only two isentropic solutions for the flow inside the nozzle exist. A subsonic and a supersonic solution. If now the ambient pressure is reduced, starting from p 0 the flow starts as an entirely subsonic flow and exits the nozzle at the ambient pressure, as is shown by curve a in figure 2.5. As the ambient pressure is reduced further, at some point the flow will reach exactly the speed of sound at the throat, but continues at subsonic speeds in the divergent section, as is visible from curve b. Both of these situations concern solutions on the subsonic branch of the nozzle flow. The flow inside the nozzle is fully controlled by the ambient pressure at the exit of the nozzle. Yet, a further reduction of the ambient pressure will not influence the flow upstream of the throat any more. The pressure at the throat cannot decrease further and from now on the mass flow through the nozzle remains constant. This situation in which the mass flow is limited to a certain maximum is called choked flow. At lower ambient pressures, pressure distributions as depicted by curves c and d arise. The flow continues along the supersonic branch downstream of the throat, but encounters a normal shock inside the nozzle to adapt to the ambient pressure. Downstream of the shock, the flow travels at subsonic speeds. Reduction of the ambient pressure moves the shock towards the exit of the nozzle. This is the situation depicted in curve d, where the flow in the entire divergent section is supersonic.

30 2. PHYSICS OF SUPERSONIC GAS JETS 25 Upon a further reduction of the ambient pressure the pressure distribution in the nozzle does not react, as is shown by curves e, f and g. The flow inside the nozzle is completely determined by the conditions of the gas in the reservoir. Because information can only travel upstream at the speed of sound, the exit pressure of the nozzle p e does not have to be equal to the ambient pressure for supersonic flow. Any adaption to the ambient pressure then occurs outside the nozzle in the supersonic jet. For curve e the flow exiting the nozzle is at a lower pressure than the ambient. The jet is called overexpanded. To adapt to the ambient pressure the flow has to be compressed. This is done by an oblique shock. At some lower ambient pressure, the exit pressure of the nozzle and the ambient pressure will exactly match and no shock structures will appear in the jet. The jet is fully expanded, as is depicted by curve f. Curve g shows the flow inside the nozzle at very low ambient pressures. In this situation the exit pressure of the nozzle is higher than the ambient pressure and the flow is called underexpanded. In matching the ambient pressure the jet undergoes an isentropic expansion. In laser-plasma experiments, and for the nozzles that we construct for such experiments, the nozzles all exit into a vacuum and are driven by high pressures. Common configurations are reservoir pressures of 1 to 100 bar and a vacuum pressure below 10 4 mbar. Therefore, all supersonic jets that are discussed in this work operate in the highly underexpanded regime. The gas jets will show a particular structure, with the formation of shocks in the jet plumes, which will be discussed in the following section Supersonic underexpanded jet As mentioned in the section before, the flow of an underexpanded supersonic jet will show a particular structure due to the large difference in the exit pressure of the nozzle and the ambient pressure in the vacuum chamber. Figure 2.6 gives a schematic representation of the features encountered in the plume of a supersonic underexpanded jet. To start, the gas is exiting the nozzle at supersonic speeds. As the nozzle exits into a vacuum chamber the pressure of the gas at the nozzle exit is higher than the ambient pressure. Therefore the gas exiting the nozzle will expand in a Prandtl-Meyer expansion, turning the flow away from the centreline. Waves emanating from the nozzle exit as a result of the abrupt expansion reflect at the boundary of the jet and coalesce to form an intercepting shock at some distance downstream [32]. This is called the intercepting barrel shock and it surrounds the core of the gas jet. The additional expansion of the supersonic flow accelerates the flow to even higher velocities. At larger distances from the nozzle exit the jet boundary and barrel shock curve towards the centreline. At some distance the pressure of the core of the jet has dropped below the ambient pressure. There a normal shock can form, which decelerates the flow to subsonic velocities. This shock is called the Mach disc. The Mach disc and barrel shock intersect, giving rise to the formation of a reflected shock. The supersonic flow through the reflected shock is turned outwards again. This structure of shocks will repeat itself along the length of the jet, giving rise to a diamond-shaped shock pattern. For the underexpanded jets used in this work the formation of a Mach disc is not visible. It would be located at a distance in the order of 1 m from the nozzle exit, much greater than the length of the region in the vacuum chamber influenced by jet [34]. Therefore only the jet boundary, and at large distances from the nozzle exit possibly the barrel shock, is of importance when using an underexpanded supersonic jet for laser-plasma interaction experiments. Due to the relatively high pressures and low temperatures encountered in supersonic expansions, as depicted previously, gas particles can aggregate and form clusters of atoms or

31 2. PHYSICS OF SUPERSONIC GAS JETS 26 Exit Ambient Reservoir p 0 p e p a Shock Throat Shock Overexpanded p / p 0 1 Subsonic a b c Oblique shock Fully expanded d Supersonic e f g Underexpanded x Expansion fan Figure 2.5: Influence of the ambient pressure onto the pressure distribution throughout a De Laval nozzle [31, 33]. Shown are a schematic geometry of the nozzle exiting into the ambient and the pressure distribution along the direction of the flow for decreasing ambient pressure p a. For curves c to g a schematic view of the shock structures in the gas jet is shown.

32 2. PHYSICS OF SUPERSONIC GAS JETS 27 Intercepting barrel shock Jet boundary External flow streamline Expansion fan Ma > 1 Reflected shock Slip surface Ma > 1 Ma < 1 Internal flow streamline Mach disc Figure 2.6: Schematic representation of an underexpanded supersonic jet [35]. molecules. Because the presence of large clusters in a gas jet may influence the behaviour of the flow, the process of condensation of gas particles is considered next. 2.4 Cluster formation Supersonic jets have long been known as a useful tool for producing clusters of atoms or molecules [36]. Clusters are essentially formed due to collisions of particles. The aggregation of gas particles is promoted by high pressures and low temperatures. It are these circumstances that are encountered in supersonic expansions and make that supersonic jets are widely being used as sources for atomic or molecular cluster beams. The properties of a cluster jet deviate from those of a gas jet. First of all, due to the formation of clusters, locally the particle density in a cluster jet is higher than in a gas jet. In addition to this, because of their much larger mass, the transverse velocity at a given temperature is smaller for clusters than for single gas particles. This offers the possibility to achieve sharper gradients in density at large distances from the nozzle exit, for example by placing objects or skimmers in the jet [37]. Furthermore, the absorption coefficient for light in a cluster jet is much higher than in a gas jet. Thus, ionization of a cluster jet is easier, which can be employed in high-harmonic generation (HHG) of extreme ultraviolet (XUV) radiation. Because the presence of large clusters in a gas jet may influence the behaviour of the flow, and to prepare the grounds for experiments on quasi-phasematching of HHG in ionized cluster jets, where the sharpness of the achievable gradients determines the shortest XUV wavelength that can be phasematched, the process of condensation of gas particles is considered. The formation of clusters in supersonic expansions can be understood from a p,t phase diagram, as presented in figure 2.7. The starting point is the gas reservoir, in point A. The gas expands from the reservoir through the nozzle along an isentropic path. The path of this isentropic expansion crosses the vapour pressure line at point B, allowing the gas to become supersaturated. At some point, C, clusters of particles start to form, leading to a collapse of the supersaturated state and a return to the liquid-gas equilibrium curve. This onset of nucleationcondensation is determined by the thermodynamics and kinetics of the gas expansion and influenced strongly by the vapour pressure, the nozzle geometry and the reservoir conditions [38].

33 2. PHYSICS OF SUPERSONIC GAS JETS 28 log p (solid / liquid) A B Isentrope C (gas) Vapour pressure log T Figure 2.7: Phase diagram showing the path for cluster formation in gases [38]. Unfortunately a rigorous condensation theory is currently unavailable. Yet, with the development of semi-empirical scaling laws, the process of cluster formation in a supersonic expansion can be described approximately [39 41]. In these descriptions, it is assumed that the condensation starts at the nozzle throat and continues up to a certain distance at which there is no contribution to the clustering process any more. This is called the sudden-freeze model. For the jets considered in this work, the sudden-freeze distance can be assumed to be corresponding to several exit diameters away from the nozzle exit in the vacuum [40]. Evaluating the scaling laws at the sudden-freeze distance therefore provides an upper limit for the condensation in the supersonic jets. Following theoretical considerations as well as numerous experimental observations on condensation in different supersonic expansions from sonic nozzles, a dimensionless scaling parameter Γ, the so-called Hagena parameter, has been defined to describe the degree of clustering [42]. Γ = k cd 0.85 p 0 T (2.68) In this formula, d is the diameter of the sonic nozzle. In a sonic nozzle is the smallest crosssectional area located at the nozzle exit, such that the gas exits the nozzle at the speed of sound. The characteristic properties of each material are incorporated into a condensation parameter k. This parameter is dependent on the sublimation enthalpy at zero Kelvin and the Van der Waals bond length, which can be related to the volumetric density of the bulk solid. For various gases the condensation parameters are given in table 2.1 [43, 44]. To apply this expression (2.68) for the Hagena scaling parameter to supersonic nozzles one substitutes an equivalent diameter for sonic nozzle diameter d, depending on the adiabatic index of the gas and the geometry of the supersonic nozzle [38]. For axisymmetric nozzles

34 2. PHYSICS OF SUPERSONIC GAS JETS 29 k c (K 2.29 µm 0.85 mbar 1 ) He 3.85 Ne 185 Ar 1650 H N Table 2.1: Condensation parameter for various gases [43, 44]. used for monoatomic gases the substitution d = d tan β (2.69) is performed, with β the opening half-angle of the diverging section of the nozzle. Experiments reveal that no clustering is observed for Γ < 200. For Γ > 1000 massive condensation sets in, with formation of aggregates exceeding 100 atoms per cluster [42]. According to this classification, no significant clustering will be observed in the nozzles reported on in this work. However, in order to obtain, indeed, cluster formation for different experiments on high-harmonic generation, we have designed a slit nozzle. A technical drawing of this nozzle is found in appendix A, next to the drawings of the other, axisymmetric, nozzles. When using the slit nozzles, ionization of the emerging cluster jet will be performed with a 1 J, picosecond laser system which is currently under construction [45]. At this point all relevant properties of a supersonic gas jet have been described. The description was presented in terms of algebraic expressions. This, however, required severe approximations such as assuming a quasi one-dimensional geometry of the flow. In addition, effects of viscosity and conductive heat transfer have been neglected. In order to get quantitative information about the three-dimensional structure of a supersonic gas jet, eventually resulting in the gas density profiles, an extended analysis of the gas flow is required. This is provided for by using computational fluid dynamics to perform numerical simulations on the flow inside the supersonic jet.

35 3. FLOW SIMULATIONS 30 3 Flow simulations The properties of the gas flow inside a supersonic nozzle have been described in chapter 2, assuming a quasi one-dimensional flow geometry. This description can only be used to approximate the gas flow on the axis of the nozzle. For studying flow in the gas jet outside the nozzle, including shocks and density variations in the transverse direction of the flow, jet this is insufficient. To obtain a full description of the flow, including the structure of the supersonic jet, a three-dimensional analysis is necessary. In this analysis effects of viscosity and turbulence have to be included as well. For this we rely on numerical simulation methods used in computational fluid dynamics (CFD). Using computational fluid dynamics it is possible to study the properties of the supersonic jet. Recently an extensive study on the optimal nozzle shape for an underexpanded supersonic micro-jet has been carried out by a research group at the Max-Planck-Institut für Quantenoptik [46]. Simulations have been used by them to determine the required shape of the divergent section of the nozzle to obtain a flat-top density profile with sharp gradients in an axisymmetric gas jet. The influence of the expansion ratio, defined as the ratio between the exit and throat area of the nozzle, as well various characteristic parameters of the divergent section of the nozzle onto the gas density profile of the gas jet was studied. These parameters included the the length of the diverging section, the divergence angle, which is the half-angle of the divergent section of the nozzle, and the contour of the divergent nozzle section. The optimal nozzle shape is a result of a trade-off between the divergence, the gradient steepness and the flat-top profile of the jet. This is discussed in section 3.1. In chapter 1 it is suggested to apply a double gas jet for the integration of the injection stage and a first acceleration stage of the laser wakefield accelerator. The feasibility of this concept depends on the density profile that will generated in such a double gas jet. To study the possibility and potential advantages in forming sharp gradients of a double gas jet, CFD simulations have been performed by E.T.A. van der Weide [25] on the jet emanating from a coaxial double nozzle. In section 3.2 the results of these simulations are presented. 3.1 Nozzle design considerations The small divergence of a gas jet produced by supersonic nozzles is essential for laser wakefield acceleration experiments. A steep edge in the density profile of the jet is maintained over a distance of several nozzle diameters, allow plasma wave breaking and efficient injection of electrons into the wakefield for acceleration. To illustrate the origin of the small divergence of a gas jet emanating from a supersonic nozzle, it is instructive to first consider the expansion of a gas flow through an arbitrary nozzle into a vacuum chamber.

36 3. FLOW SIMULATIONS Jet divergence Every high-pressure gas inlet into a vacuum chamber will generate a supersonic jet as in figure 2.6, because the gas expands to adapt to the background pressure. Consider for instance a nozzle only consisting of a converging section that exits into a vacuum chamber. Gas flowing through this nozzle is accelerated and reaches the speed of sound at the position with the smallest cross-section, which in this situation corresponds to the exit of the nozzle, as follows from equation (2.57). A nozzle in which the gas flow reaches the speed of sound at the exit of the nozzle is called a sonic nozzle. As the gas exits the nozzle and enters into the vacuum chamber, it expands such that the pressure matches the background pressure in the vacuum chamber, thereby accelerating the flow to supersonic speeds. Because the gas enters the vacuum chamber at a low flow velocity and a high pressure, the expansion takes place in all directions, obviously leading to a large divergence of the jet and thus a gas distribution with only a weak density gradient. In contrast, using instead of a sonic nozzle a supersonic nozzle, the first part of the expansion takes place in the diverging nozzle section. This expansion is guided due to the constrainment by the nozzle wall. The gas is accelerated in the forward direction, to supersonic speeds before it exits the supersonic nozzle with a high forward velocity and at a low pressure. Because the gas then enters the vacuum chamber at a low pressure, the expansion required to accommodate to the background pressure is much smaller than when a sonic nozzle would be used. In addition to this, due to the high forward velocity component of the gas flow the effect of the expansion on the divergence of the supersonic jet is limited and a collimated jet, with little divergence, is formed. The particles are following almost ballistic trajectories as they exit the supersonic nozzle and the gas may form a sharp gradient in gas density as desired. Therefore, the key solution for reducing the divergence of a supersonic jet is to guide the major part of the gas expansion by the diverging section of a supersonic nozzle. To obtain supersonic jets with little divergence, supersonic nozzles with large expansion ratios are favourable. For obtaining a steep edge in the density profile of the gas jets, it is important to consider next to the divergence of the jet also the formation of boundary layers inside the nozzles. Outside the nozzle these layers govern the minimal width of the density gradient at the edge of the gas jet Boundary layers A major issue for microscopic nozzles is the formation of boundary layers along the walls of the nozzle, which may strongly influence the profile of the exiting gas jet. Such boundary layers are thin layers which connect the free flow in the centre of the nozzle to the flow at the nozzle wall, where the velocity decreases to zero. Inside a boundary layer the behaviour of the gas flow is dominated by viscous effects. The flow in the centre of the nozzle can be regarded as inviscid. A common measure for the thickness of a boundary layer is the distance from the wall δ 99 over which the flow velocity changes from 0 to 99% of the velocity in the centre of the nozzle. Figure 3.1 shows a simulated profile of the forward velocity component in the diverging section of a Mach 4.8 supersonic nozzle, at a position of 2 µm before the nozzle exit, when operated with 10 bar helium. These data are obtained by numerical simulations [25], as presented in the next section. It can be seen that the velocity boundary layer has a thickness of 87 µm. In nozzles with radii approaching this thickness the profile of the flow is then solely determined by the boundary layer. In addition to the velocity boundary layer,

37 3. FLOW SIMULATIONS δ 99 Velocity boundary layer v z (m s ) p (mbar) n (10 cm ) T (K) r (mm) 0 0 Figure 3.1: Simulated profile of the forward velocity component (black), pressure (red), density (green) and temperature (blue) in the diverging section of a Mach 4.8 supersonic nozzle, at a position of 2 µm before the nozzle exit. The nozzle is operated with 10 bar helium at 298 K [25]. similar boundary layers can be defined in terms of pressure, density and temperature, as shown in figure 3.1. The density boundary layer is most important for applications to laser wakefield acceleration, as it governs the minimum width of the density gradient in the jet, at short distances after the nozzle exit. For small nozzles the presence of a boundary layer leads to a decrement of the effective nozzle diameter for the supersonic expansion, resulting in a drop of the Mach number of the flow at the exit of the nozzle. When the thickness of the boundary layer is known, this drop in Mach number can be avoided by incorporating this into the design of the nozzle. To obtain a crude estimate on which flow properties influence the boundary layers consider a flow over a semi-infinite flat plate. The boundary layer that forms is known as the Blasius boundary layer [31]. The thickness of this boundary layer scales according to equation (3.1), where µ is the dynamic viscosity of the fluid and x the direction of the flow. From equation (3.1) one can see that the boundary layer grows as the flow travels greater distances along the plate. Furthermore, the boundary layer shrinks as the density of the gas increases. µx δ ρv (3.1) In the context of supersonic nozzles, this indicates that the boundary layer can be reduced by using short nozzles and by operating at high pressures. Next to this, following previous studies of boundary layer formation in supersonic nozzles [46], the thickness of the boundary layer increases with the expansion ratio of the nozzle. To come to an optimal design for the shape of a supersonic nozzle, the formation of a boundary layer has to be taken into account.

38 3. FLOW SIMULATIONS Optimal nozzle contour In section 2.2, the shape of the diverging section of the supersonic nozzle is depicted as a cone. When operated shock-free, a conical nozzle with a small divergence half-angle produces an almost uniform gas jet [46], with a flat-top density profile. Next to this conical shape various other contours are commonly used. Widespread in use is the approximately parabolic contour determined by the Method of Characteristics [32]. Using such nozzles, a completely uniform gas jet can be obtained. However, manufacturing these exact parabolic contours on a micrometre scale length is extremely challenging. Nozzles for generating molecular beams are commonly shaped as a trumpet, increasing rapidly in diameter close to the nozzle exit. The downside of using trumpet-shaped nozzles is that these nozzles produce a non-uniform gas-jet [46]. Because a uniform gas jet exiting from the nozzles is desirable for obtaining a flat-top density profile, and considering challenges in the fabrication of the nozzles, usage of supersonic nozzles with a conical diverging section is the most suitable. To avoid the formation of shock structures in the supersonic flow, the nozzle contour should be smooth. Next to this, the contour should not be strongly diverging, such that the gas can follow the changes in cross-section. If not, shock structures, emanating from for instance the throat, will destroy the uniform flow downstream in the nozzle. In order to achieve a shockfree flow, we designed all of our axisymmetric nozzles with a throat with a length of 1/5 times the throat diameter. In addition, the edges of the the throat are rounded to a radius of 4 throat diameters on the supersonic side, following results of a study on supersonic micro-jets [46]. In chapter 2 it has been indicated that the supersonic flow in the nozzle is only weakly dependent on the subsonic flow upstream of the throat. Therefore the exact design of the converging part of the nozzle is not critical. All of our nozzles have been designed with a conical convergent section, with a half-angle of 30. For obtaining a uniform gas jet is it essential that the flow along the wall of the nozzle is able to follow the wall and to not separate from this wall, distorting the flow in the centre of the nozzle. According to simulations [46] this requires the divergent section of the nozzle to have a half-angle of less than 10. For these small divergence angles the divergence of the free jet has been shown to be only weakly dependent on the divergence of the nozzle contour [46]. In conclusion, to obtain a uniform gas jet, the divergence of the supersonic nozzle should not be too large. Recall that for obtaining small boundary layers, and thus steep density gradients, short nozzles and small expansion ratios are favoured. However, to produce a collimated gas jet with little divergence, the expansion ratio of the nozzle should be large. So, for an optimal design this results in a trade-off between the collimation, gradient steepness and flat-top profile of the jet. An adequate compromise seems to be a divergence half-angle of the nozzle of 7 and an expansion ratio of 3 [46]. These parameters are used for designing and simulating the gas jets in this work. We started with a design that is not too far off standard [46], to save time with fabricating a first nozzle. The simulations were then used to investigate nozzles for double gas jets, in search of density distributions with higher gradients. 3.2 Simulations on double gas jet The idea and potential advantages of integrating the injection stage and a first acceleration stage of the laser wakefield accelerator by means of a double gas jet has been described in section 1.2. To study the feasibility of a double coaxial gas jet, CFD simulations have been performed by E.T.A. van der Weide [25] on a configuration consisting of two supersonic nozzles,

39 3. FLOW SIMULATIONS 34 Ambient r Inlet 2 Nozzle Inlet Nozzle z Figure 3.2: Cross-section of the double coaxial nozzle used in the flow simulations. Indicated are the boundary conditions of the simulation problem at the two inlets of the nozzles and to the ambient in the vacuum chamber. All dimensions are in mm. spaced a short distance apart Nozzle geometry The double gas jet exhausts from two conical supersonic nozzles as shown in figure 3.2. The nozzles are axisymmetric with respect to the horizontal axis, with an outer annular shaped nozzle, nozzle 2, configured coaxially around the inner nozzle, nozzle 1. The inner nozzle as well as the annular nozzle have an exit of 0.75 mm in size, and the length of the diverging sections is taken 2.0 mm long. The throats of the nozzles are dimensioned such that both nozzles produce a Mach 4.8 flow at the exit, according to quasi-one dimensional inviscid analysis. As this theory neglects viscosity and boundary layers this Mach number can be overestimated. A Mach 4.8 nozzle operated at pressures of 1 to 100 bar produces in the jet, at the exit of the nozzle, a gas density of to cm 3. To maximize the interaction between the two supersonic jets, which can possibly result in a steep gradient with an adjacent plateau in density, at a short distance behind the nozzles, the nozzle exhausts are separated by only 0.10 mm. Decreasing this distance, to move the point of first interaction between the two jets closer towards the nozzle exits, would result in great challenges in fabricating the double nozzle. A large region of about 300 by 100 mm 2 of the vacuum chamber is included in the geometry for the flow simulation, allowing for the development of the supersonic jets in the vacuum. To resolve all boundary layers and shocks in the double gas jet mesh dimensions down to 0.5 µm are used Boundary conditions The driving parameter for the supersonic jets is the pressure difference between the reservoirs and the vacuum. The reservoir conditions are modelled by imposing a fixed pressure, temperature and direction of flow at the inlets of the nozzles, indicated with respectively inlet 1

40 3. FLOW SIMULATIONS 35 and 2 in figure 3.2. The gas enters both nozzles parallel to the axis of symmetry at a temperature of 298 K. The pressure at inlet 1 is set to 10 times the pressure at inlet 2 to obtain an equally large difference in gas density in the gas jets exhausting into the vacuum chamber. Typically plasma densities of to cm 3 are used for laser wakefield acceleration, so that the maximum drop in plasma density across any gradient would be a factor in the order of 10. This is the motivation to conduct the first simulations on a double gas jet for which the operating pressures of the two nozzles differ by a factor of 10. Conformation of the pressure in the flow to the ambient is forced by setting the pressure at great distances of the nozzles equal to the background pressure in the vacuum. To ensure that the supersonic jets are in the highly underexpanded regime for inlet pressures of 1 to 100 bar this background pressure is set at 1 mbar. The walls of the nozzles are modelled as viscous, isothermal walls. At each wall boundary the flow taken is stationary, corresponding to the no-slip condition, and fixed to a temperature of 298 K. Imposing a stationary flow at the walls results in the formation of a boundary layer along the nozzle walls. In the simulations helium is used, which is treated as an ideal gas. Sutherland s law [47] is used for describing the temperature dependence of the viscosity of helium [48] Simulation model The simulations are performed by using a finite volume method to obtain a steady-state solution of the Reynolds-averaged Navier-Stokes equations for compressible flow of an ideal gas. These equations are the Navier-Stokes equations (2.23), (2.25) and (2.26), modified to allow inclusion of turbulence effects. The modification consists of a separation of all flow variables in a time-averaged and a fluctuating part, which is known as the Reynolds decomposition. Effects of turbulence can be modelled using the fluctuating part of the flow variables. One can confirm that the gas flow inside the nozzles is turbulent by looking at the Reynolds number of the flow. The Reynolds number Re is a dimensionless number that gives a measure of the ratio of the inertial forces ρv 2 /L to the viscous forces µv/l 2. Thereby the Reynolds number quantifies the relative importance of these two types of forces for given flow conditions. It is defined according to equation (3.2), where L is the characteristic length of the gas flow. The origin of the Reynolds number is found in a nondimensionalization of the Navier-Stokes equation (2.25). Re ρ v L (3.2) µ At low Reynolds numbers a flow is laminar, while for large Reynolds numbers it is turbulent. The values of the Reynolds number characterizing the flow regime depend on the geometry of the flow. For instance, a fully developed flow in a smooth pipe is laminar for Re < 2100 and the flow is turbulent for Re > 4000 [49]. Calculating the Reynolds number for a helium flow at the exit of a typical supersonic nozzle, taking the 0.75 mm throat diameter as a characteristic length and setting the operation pressure to 10 bar, yields a value of This number indicates that the flow in all supersonic nozzles discussed in this work will be turbulent. It is therefore important to take into account possible influences of turbulence on the macroscopic behaviour of the flow. The Spalart-Allmaras model [50] was used by us for the modelling of turbulence in the simulations. The simulations have been performed using the SUmb flow solver [51].

41 3. FLOW SIMULATIONS Simulation results The result of the simulations on the gas flow of the double coaxial nozzle is shown in figure 3.3. The inner nozzle, nozzle 1, is operated at 100 bar and the annular nozzle, nozzle 2, at 10 bar. Figure 3.3(a) shows the Mach number throughout the flow. From this figure it can bee seen that the flow inside both nozzles reaches Mach 1 at the throat an is free from any shocks further downstream. At the exit of the inner nozzle, the flow reaches Mach 4.5 and at the exit of the annular nozzle it reaches Mach 4.3. This is slightly lower than the predicted Mach 4.8 by using quasi one-dimensional inviscid theory. This decrement of the achieved Mach number of the flow at the nozzle exit results from the formation of boundary layers along the nozzle walls, decreasing the effective expansion ratio of the nozzle. In nozzle 2 the development of a boundary layer is clearly apparent, while in nozzle 1 the boundary layer remains small, due to the 10 times higher operating pressure. After exiting the nozzles, the extra expansion in the vacuum chamber accelerates the flow further. At a distance of approximately 4.0 mm from the nozzle exits, it eventually reaches Mach Directly at short distances from the nozzle a layer of almost stationary flow remains in between the two nozzle exits. Also shown in figure 3.3(a) are streamlines of the flow in the supersonic jet, equally spaced at the inlets of the nozzle. A streamline is a line which is tangent to the velocity vector at each point throughout the flow. For steady flow, which has been assumed in the simulations, these streamlines coincide with the pathlines of the flow, which are the paths traced out by following the trajectories of individual fluid particles as they move through the flow. The expansion of the flow is clearly visible by the divergent streamlines, both inside and outside of the nozzles. The outer most streamline gives an indication of the boundary of the double gas jet. The complex structure of the double gas jet arises from the large pressure difference between the gas exiting from the two nozzles. At the nozzle exits, the pressure in the inner gas jet is 10 times higher than in the annular jet. As a result of the expansion of the inner jet, the surrounding annular gas jet is pushed outwards. Due to the abrupt change of direction of this gas flow, shocks are induced in both the inner and the annular jet. These two shocks are indicated in figure 3.3 as the black dashed lines, and are also recognizable by the abrupt bending of the streamlines in figure 3.3(a). The particle density in the gas jets is shown in figure 3.3(b). At the centres of the nozzle exits the density amounts to cm 3 for the inner nozzle and to cm 3 for the annular nozzle. This ratio of these numbers corresponds exactly with the 10 times difference in operating pressures of the nozzles and taking into account the small difference in Mach numbers. A more clear picture of the flow profile can be obtained using figure 3.4. In these graphs the Mach number and gas density are shown along a cross-section at the nozzle exits. In figure 3.4(a) the Mach number is shown for operation of the double gas jet, with 100 bar at the inner nozzle and 10 bar at the annular nozzle, corresponding to the situation in figure 3.3. Next to this, results of a simulation in which the inner nozzle is operated at 10 bar and the annular nozzle at 1 bar, are plotted. For the operation at a combination of 100 and 10 bar both nozzles produce a relatively uniform, flat-top, flow profile. For the combination of inlet pressures of 10 and 1 bar, however, only the profile of the inner nozzle exhibits a flat top. The profile of the Mach number for the annular nozzle is strongly non-uniform. Comparing the two curves for the inner nozzle, it is clearly apparent that the boundary layer increases when the pressure decreases. This is indicated by the difference in width of the gradients of the flow profiles, representing the thickness of the boundary layer that have formed along

42 r (mm) Nozzle 1 Nozzle 2-2 Nozzle 1 Nozzle z (mm) (b) Particle density z (mm) 1 3 (a) Mach number and streamlines (white) Figure 3.3: Simulations of the gas flow produced by the double coaxial nozzle. Nozzle 1 and 2 are operated with 100 and 10 bar helium respectively. Two shocks, formed due to the interaction between the two jets, are indicated (black dashed lines). r (mm) Ma n (cm-3) 3 3. FLOW SIMULATIONS 37

43 3. FLOW SIMULATIONS 38 the nozzle wall. The non-uniform flow profile in the annular nozzle, when operated at 1 bar, reflects that at 1 bar the boundary layers have increased to such an extent that they greatly influence the flow profile. The decrement of the Mach number of the flow at the nozzle exit for low pressures is the result from a decreasing effective expansion ratio due to the increasing boundary layers. Figure 3.4(b) shows the density profiles for the combination of inlet pressures of 100 and 10 bar, as well as for the combination of 10 and 1 bar. The density profile for the combination of 100 and 10 bar exhibits a flat top for both jets. In between these flat tops the density drops significantly. The width of the flat top in the inner jet at 90% of the maximum density is 0.61 mm, for the annular jet it is 0.44 mm. The density of the inner jet drops from 90 to 10% of the maximum on a width of 59 µm. For the outer density gradient of the annular jet this width is 77 µm. For the combination of inlet pressures of 10 and 1 bar the particle density is smaller by a factor of 10. The profile is, however, similar in shape, although only the density profile in for the inner jet has a flat top. The flat top in the outer jet has disappeared due to the large boundary layers in this low-pressure jet, resulting in a non-uniform profile. The width of the inner flat top is 0.55 mm and the width of the density gradient in the inner jet is 86 µm. To display the evolution of the gas density in the supersonic jet, in figure 3.5 the density profiles are plotted at various distances from the nozzle exit. The profiles result from operation of the inner nozzle at an inlet pressure of 100 bar and the annular nozzle at 10 bar. As one moves further away from the nozzle exit, in the first 0.5 mm the the flat-top profiles of the two jets are relatively well preserved, although the maximum density obtained in the central jet somewhat decreases. From 0.5 mm onwards the development of the two shocks becomes visible, as has already been shown in figure 3.3. One shock develops in the inner jet, recognizable by the sharp density gradient located at radial positions ranging from 0.4 to 0.6 mm. The length scale on which this density gradient is present is only 5 µm at 1 mm from the nozzle exit. The second shock occurs in the outer jet, arising from the change in flow direction due to the expansion of the central jet. This shock starts at a radial position of 0.7 mm and curves outwards at greater distances from the nozzle exit, resulting in a density gradient with a width of 42 µm. The profiles of figure 3.5 show the development of shocks, resulting in density gradients with widths down to 5 µm. Because of the ability to produce such sharp gradients with solely gas jets, a double gas jet appears promising for injection of electrons for laser wakefield acceleration. A quantitative comparison to previous work on shocks in the context of laser wakefield acceleration [18, 46] may be difficult because both jets can be operated over a broader range of parameters and measurements of the respective gas densities are not available so far. A qualitative difference is, however, that in [18] shock formation is achieved with a solid obstacle in the jet. We investigate shocks induced by a second gas jet which can be seen as an obstacle as well that moves, however, away from the nozzle exit plane. As a result, the doublejet approach enables shock generation, and thus sharp gradient features, further away from the nozzle exit plane or other objects, allowing for a minimum working distance of 0.5 mm or more. Thereby, a double gas jet may allow for usage of much higher laser intensities which would lead to destruction of solid obstacles, such as knife edges or nozzle exits too close to the focus of the TW drive laser beam.

44 3. FLOW SIMULATIONS Ma r (mm) (a) Mach number n (10 cm ) n (10 cm ) r (mm) (b) Density Figure 3.4: Flow profiles along the cross-section at the exits of the double coaxial nozzle. The double coaxial nozzle is operated at inlet pressures of 100 bar for the inner nozzle and 10 bar for the annular nozzle (black), and at a combination of 10 and 1 bar (red). 0

45 3. FLOW SIMULATIONS z (mm) 19-3 n (10 cm ) r (mm) Figure 3.5: Density line-outs of the double gas jet at different distances from the nozzle exit. Nozzle 1 and 2 are operated with respectively 100 and 10 bar helium.

46 4. GAS JET CHARACTERIZATION 41 4 Gas jet characterization To be able to start with laser wakefield acceleration experiments several supersonic nozzles have been designed and constructed. The first nozzle that has been manufactured and characterized is an axisymmetric conical Mach 4.8 nozzle with a length of 2 mm. The throat of this nozzle has a diameter of 0.25 mm and the nozzle exit is 0.75 mm in diameter. The contour of the nozzle is identical to that of the central nozzle used in the simulations of the double gas jet, shown in figure 3.2. When operated at pressures of 1 to 100 bar this nozzle will produce a gas density at the exit of the nozzle of to cm 3, confirmed by the simulations presented in section As a second type of nozzle, to provide for larger interaction lengths between laser pulse and gas jet, two slit-shaped nozzles have been manufactured, providing interaction lengths of 5.0 and 15.0 mm. The technical drawings of all nozzles that have been constructed are included in appendix A. Manufacturing these small nozzles is a great challenge. To obtain the specified shape with the required high accuracy the nozzles are produced by electrical discharge machining. 4.1 Measurement of supersonic gas jets Measuring the structure of supersonic gas jets is not a trivial task. The density of the gas in the supersonic jets is relatively low, even in the centre as low as cm 3, and due to the small dimensions of the jets only a short interaction length is available for any characterization method. Furthermore commonly used neutral gases, especially noble gases, are transparent up to very short optical wavelengths and possess an index of refraction extremely close to 1. Table 4.1 gives the index of refraction η for various gases at a temperature of 273 K and a pressure of 1 atm, for light with a wavelength of nm [52]. When recalling that in laserplasma experiments the nozzles will be operated with helium, with an index of refraction closest to unity of all gases, it is evident that measuring the structure of these supersonic gas jets is met with great challenges. However, with a sufficiently sensitive imaging technique, such as based on interferometry, where even tiniest differences in optical path lengths can be detected, a quantitative optical imaging of helium gas jets should be possible. Interferometric characterizations of gas jets using argon have been performed on gas jets with a diameter of 5 mm and densities above cm 3 [21, 22] and also at gas densities of cm 3 using 0.20 mm jets [46]. Using helium, measurements on 1 to 2 mm diameter gas jets have been reported for densities of to cm 3 [53, 54]. However, to our knowledge no quantitative measurements on helium gas jets have been presented yet for jets with a diameter of 0.75 mm at gas densities of cm 3 and lower, nor any measurements on gas jets with similarly small differences in optical path lengths.

47 4. GAS JET CHARACTERIZATION 42 η 1 (10 5 ) α (10 41 F m 2 ) Φ (rad) He Ne Ar H N Table 4.1: Index of refraction of various gases at 273 K, 1 atm for light with a probe wavelength of nm [52]. Also given are the polarizability of the gas and the accumulated phase difference after travelling through 1 mm of gas with a density of cm 3. The index of refraction η of a gas is proportional to the particle density n and depends on the used probe wavelength. This dependence is described by the Lorentz-Lorenz equation (4.1), with α the polarizability of the gas atoms and ε 0 the permittivity of free space [55, 56]. For η 1 this can be approximated by equation (4.2). η(λ) 2 1 η(λ) = nα(λ) 3ε 0 (4.1) n 2ε 0 (η(λ) 1) (4.2) α(λ) Using reference values of the refractive index [52] the atomic polarizability of a gas can be calculated. Table 4.1 gives the index of refraction and polarizabilities for various gases using light with a probe wavelength of nm. The index of refraction is given at a temperature of 273 K and a pressure of 1 atm. Also shown is the accumulated phase shift after traversing 1 mm gas with a density of cm 3. By using the Lorentz-Lorenz equation an interferometric measurement of the refractive index distribution throughout the supersonic gas jet allows determination of the density structure of this gas jet. 4.2 Gas jet interferometry For the characterization of the supersonic gas jet we used a Mach-Zehnder interferometer. One of the arms of the interferometer passes through the gas jet. Due to the difference in refractive index between the gas and vacuum light waves traversing the gas jet accumulate a phase shift Φ. This phase shift, between measurements with and without a gas flow, can be derived from the interference pattern that results after combining the two arms of the interferometer Interferometry set-up A schematic representation of the interferometry set-up used for characterizing the supersonic gas jets is shown in figure 4.1. The beam of a 30 mw continuous-wave helium-neon laser is expanded to a diameter of about 20 mm using a telescope and split into the two arms of a Mach-Zehnder interferometer. One arm of the interferometer remains in the ambient, while the other arm enters a vacuum chamber and traverses the gas jet. Both arms are recombined at a small angle to create an interference pattern which is imaged onto a CCD camera. The camera

48 4. GAS JET CHARACTERIZATION 43 HeNe laser 30mW CW Vacuum chamber Gas jet 60 ms CCD camera 67.5 µ s Figure 4.1: A schematic representation of the Mach-Zehnder interferometer used for imaging the supersonic gas jets. One arm of the interferometer enters the vacuum chamber and passes through the gas jet, the other arm traverses through the ambient. Next, both arms are combined and imaged at a CCD camera. we used is a PCO Sensicam QE, which limits the minimum resolution of the interferograms to 2.6 µm. To retrieve only the phase information due to the presence of the gas jet, a reference measurement, without gas jet, is subtracted from all measurements after extraction of the phase. The nozzle is mounted inside the vacuum chamber to obtain a supersonic outflow. To reduce the gas load on the vacuum pumps and maintain an average pressure below 10 4 mbar inside the vacuum chamber the nozzle is operated by a pulsed valve. Parker Hannifin Series 99 pulse valves are used by us for this purpose. The time it takes for the flow out the valve and nozzle to completely develop has been determined by positioning a fast pressure transducer in the vacuum chamber, facing the exit of nozzle. Measurements show that supersonic flow reaches a steady state after approximately 25 ms. To ensure a steady flow, the valve is opened for a much longer duration, of 60 ms. At 50 ms after opening the valve to the nozzle, the interferogram is recorded by the CCD camera. Initially, the shutter time was set to 67.5 µs, which provided sufficient contrast for evaluating the interferograms Fringe-pattern analysis From these interferograms extraction of the phase information is performed by making use of Fourier transformations [57]. The intensity I(y, z) of a general fringe pattern in the z-direction can be described using equation (4.3). I A (y, z) represents the background intensity and I B (y, z) the variations in the visibility of the fringes. These intensity fluctuations are assumed to vary slowly compared to the modulation introduced by the carrier frequency of the fringes ν 0, which is well satisfied in the experiments due to the low gas index and thickness of the jet. Φ(y, z) is the local fringe phase which is to be retrieved from the fringe pattern because it is related to the particle density of the gas. I(y, z) = I A (y, z)+ I B (y, z) cos(2πν 0 z+φ(y, z)) (4.3)

49 4. GAS JET CHARACTERIZATION 44 It is convenient to rewrite the expression for the intensity of the fringe pattern using Euler s formula, introducing I C (y, z) defined according to equation (4.5). I(y, z) = I A (y, z)+ I C (y, z) e i2πν 0z + I C (y, z) e i2πν 0z (4.4) I C (y, z) = 1 2 I B(y, z) e iφ(y,z) (4.5) Fourier transforming (4.4) with respect to z results in the spectrum of the fringe pattern intensity (4.6). Î(y, ν) = Î A (y, ν)+ Î C (y, ν ν 0 )+ Î C (y, ν+ν 0 ) (4.6) Next to the slowly-varying background Î A (y, ν) it consists of two spectra separated from the zero frequency spectrum by the carrier frequency ν 0. After selecting one of these spectra Î C (y, ν ν 0 ) and translating it by +ν 0 one can apply the inverse Fourier transform to obtain I C (y, z) (4.5). In this process the slowly-varying background I A (y, z) is essentially filtered out. Finally the phase information modulo 2π can then be extracted using equation (4.7) and subsequently be unwrapped. Intensity fluctuations of the fringes I B (y, z) are cancelled out in this stage. ( ) I IC (y, z) Φ(y, z) = arctan (4.7) R I C (y, z) The step of selecting the Î C (y, ν ν 0 ) spectrum means essentially filtering out all spatial frequency components of the interferogram that are not related to the phase shift resulting from the gas jet. In this filtering process one has to take care that all information of the gas jet is preserved. Selecting too narrow a spectrum results in the disappearance of valuable highfrequency information. Specifically, as we will see later, the information that yields the steepness of the edges of the gas jet. Preserving, however, a wide range of frequencies will result in a large amount of high-frequency noise. Filtering, extracting and unwrapping the phase we performed using the Interferometrical Data Evaluation Algorithms program [58]. For unwrapping of the phase we used the Minimum-Cost-Matching Branchcut method [59] Abel inversion To obtain the refractive index distribution and thus the density distribution inside our gas jets from the phase images, additional calculations are needed. Because the gas jet is illuminated from one side in the interferometric set-up the obtained phase image of the gas jet is a two-dimensional projection of the three-dimensional gas jet. Principally, by rotating the gas jet and obtaining multiple projections of the three-dimensional structure of the gas jet it would be possible to reconstruct the density distribution of arbitrarily shaped gas jets. For the much simpler case of an axisymmetric gas jet, as we use here, reconstructing the structure of the gas jet is already possible using only a single projection measurement. To deduct the relation between the measured phase-shift and the density of the gas jet consider the geometry depicted in figure 4.2, where a ray of light is propagating in the positive x-direction through an axisymmetric gas jet, positioned at the origin in the x, y-plane. The axis of symmetry of the jet is along the z-axis, perpendicular to the x, y-plane. The relative refractive index of the gas is assumed to drop to zero at large distances from the origin. The amount of gas the light encounters results in a projection of the refractive index η(r, z) 1 of the gas jet onto the y, z- plane. Calculating the accumulated phase-shift Φ of the light results in equation (4.8), which

50 4. GAS JET CHARACTERIZATION 45 y y r 0 x Φ(y,z) 0 η(r,z) - 1 Figure 4.2: Schematic representation of the measurement of an axisymmetric distribution (blue) of relative refractive index η(r, z) 1. At large distances the distribution is bound to zero. The accumulated phase shift of a light ray traversing the distribution will be the integral of the relative refractive index along the path travelled through the distribution (red). Retrieving the refractive index distribution is possible using an Abel inversion. is known as the Abel transform. Φ(y, z) = 2π λ 2 y (η(λ, r, z) 1) r r 2 y 2 dr (4.8) Inverting the Abel transform allows one to retrieve the index of refraction of the gas jet from a measured phase-shift. η(λ, r, z) 1 = λ 1 dφ(y, z) 1 dy (4.9) 2π π r dy y 2 r2 The particle density profile of the gas jet is then obtained by making use of the Lorentz-Lorenz equation (4.2). n(r, z) = ε 0λ dφ(y, z) 1 π 2 dy (4.10) α(λ) r dy y 2 r2 The phase images obtained by interferometry often show a slight asymmetry. As the Abel transform assumes an axisymmetric profile in density, the phase images are symmetrized with respect to the centre of the jet. Determination of the centre position of the jet was performed by using the half-maximum points of the phase shift profile. To be able to perform the Abel inversion one has to take care that after symmetrization the edges of the phase image are still bound to zero. The numerical Abel inversion was performed using the Interferometrical Data Evaluation Algorithms program [58]. The result of an Abel inversion is highly sensitive to the type of numerical algorithm used for performing the integrations. To determine the most suitable inversion algorithm for usage in our measurements on supersonic gas jets, results obtained by multiple numerical inversion algorithms have been compared with analytical solutions to the Abel inversion, for various density profiles. The f-interpolation algorithm [60] has been shown to give the closest approximations to analytical inversions for density profiles with a flat top

51 4. GAS JET CHARACTERIZATION 46 and steep gradients. This algorithm employs a piecewise polynomial fitting to the measured phase data which is subsequently analytically inverted. 4.3 Results The supersonic gas flow emanating from the conical Mach 4.8 nozzle is characterized using the Mach-Zehnder interferometer of figure 4.1. Initially, we used argon gas because of its relatively high index and resulting phase shifts, indicated in table 4.1, which should facilitate an easier testing of our characterization method. Because argon and helium are both monoatomic gases the adiabatic index of both gases is equal, which theoretically results in identical supersonic expansions, as discussed in section 2.2. Characterizations of the supersonic gas flow using argon will therefore produce similar results as characterizations of helium flows. The difference in molar mass and in viscosity between argon and helium can only lead to minor changes in the gas flow. [27, 61]. If required, this can be accounted for using the concept of dynamic flow similitude [31] Characterization of gas jet In figure 4.3 measurements obtained using interferometry are presented, for operating the nozzle with argon at a pressure of 91 bar. The nozzle exit is 0.75 mm in diameter and is located at the bottom centre of the figure, at z = 0. The supersonic gas jet is exiting to the top. Figure 4.3(a) shows the interferogram obtained. Close to the nozzle exit bending of the fringes due to the gas jet is clearly visible. Here, the phase shift reaches a relatively high value, due to the high refractive index or argon. Extracting the phase from this interferogram results in figure 4.3(b). It can be seen that the jet is collimated, indicating the supersonic behaviour of the flow. The standard deviation in phase over 5 images is 8.1 mrad. To obtain a more clear view of the supersonic gas jet the density profile of the jet was reconstructed from the phase image, making use of the Abel inversion as described in section Figure 4.4 shows the density profile along cross-sections of the gas jet at various distances from the nozzle exit. Qualitatively it can be seen that, close to the nozzle exit, the density profile of the supersonic jet exhibits a more or less flat-top profile with steep gradients at the edges of the jet. In the measurements presented in figure 4.4 oscillations in density are present, with periods of about 0.1 to 0.2 mm. Since the integrand in the Abel inversion (4.9) is singular for r, y 0, the oscillations are more pronounced in the centre of the jet. It has been verified that these oscillations are resulting from secondary interference patterns in the interferometer set-up. These interference patterns arise from reflections at the windows of the vacuum chamber and the protective glass window of the CCD camera. Because the oscillations occur on a similar length scale as the gradients in the supersonic gas jet it is not possible to remove these oscillations by frequency filtering in the analysis of the fringe pattern without loss of details in the edges of the gas jet. Removal of the oscillations was possible through a realignment of the interferometer such that the secondary interference patterns do not end up onto the detector. This has been realized before performing the measurements using helium, presented in the next section. Despite the oscillations a quantitative analysis of the gas density profiles shown in figure 4.4 can be carried out by averaging the density in the flat-top part of the jet. At the nozzle exit

52 4. GAS JET CHARACTERIZATION z (mm) y (mm) (a) Interferogram of the supersonic gas jet z (mm) 1 3 Φ (rad) y (mm) (b) Phase image of the gas jet, retrieved from the interferogram Figure 4.3: Results of an interferometric measurement on the supersonic gas jet, operated with argon at a pressure of 91 bar. The gas is exiting from the nozzle to the top. 0

53 4. GAS JET CHARACTERIZATION z (mm) 19-3 n (10 cm ) r (mm) Figure 4.4: Density profile along cross-sections of the supersonic gas jet, operated by argon at a pressure of 91 bar. Profiles are shown for various distances from the nozzle exit. the density in the centre of the jet is about cm 3 and the density drops from 90 to 10% over a distance r g of 0.11 mm at the edge of the jet. The width of the 90-to-10% density drop is named the width of the density gradient hereafter. The width of the flat-top profile itself r f where the density remains at above 90% of the maximum density is about 0.45 mm. Based upon a quasi one-dimensional analysis discussed in section 2.2 the maximum density of the gas jet is cm 3. For distances up to 0.2 mm from the nozzle exit the shape of this profile is nearly preserved, only the maximum of the flat-top decreases somewhat. Going to larger distances from the nozzle exit the height of the flat-top further decreases and the width of the density gradient increases. At 0.5 mm from the nozzle exit the maximum density has dropped to cm 3. The width of the flat-top is 0.51 mm and the width of the density gradient is 0.34 mm. The only practical method for influencing the flow in the supersonic gas jet is changing the inlet pressure to the nozzle. According to equation (2.54) the density in the supersonic flow scales with the inlet density, in the reservoir, and thus inlet pressure at which the nozzle is operated. Results of measurements on the gas jet for various operating pressures are shown in figure 4.5. Plotted are the density profiles at 0.5 mm from the nozzle exit for inlet pressures of 30 to 91 bar. The distance of 0.5 mm corresponds to the minimum working distance that is kept for the laser wakefield acceleration experiment, in order to minimize the risk of destroying the nozzle by a slight misalignment of the high-power drive laser beam. It can be seen that each profile represents a flat top with gradients. Comparing these measurements at different operating pressures shows that the density profiles are scalable in a simple fashion. The scaling becomes more clear by plotting the density in the centre of the jet, and, the width of the flat-top profile together with the the density gradient width as a function of operating pressure. The results are shown in figure 4.6.

54 4. GAS JET CHARACTERIZATION p 0 (bar) 19-3 n (10 cm ) r (mm) Figure 4.5: Density profile of the gas jet at 0.5 mm behind the nozzle exit. operated by argon at various pressures. The nozzle is Figure 4.6(a) shows the dependence of the density in the centre part of the gas jet on the operating pressure. The gas density scales proportional to the operating pressure, which is confirmed by the experimental data matching well to the linear fit, drawn as a dashed line. This behaviour is predicted by quasi one-dimensional analysis in equation (2.54). In figure 4.6(b), the influence of the operating pressure on the dimensions of the most important features in the density profile is presented. Taking into account the average standard deviation of the phase images yields a precision of about 0.05 mm. The width of the flat-top section in the profile is approximately 0.47 mm, not much depending on the pressure used for operation. Also plotted in figure 4.6(b) is the width of the density gradient at various operating pressures. For operating pressures between 30 and 91 bar the width of the density gradient is approximately constant at about 0.32 mm. For argon, gas densities down to cm 3 have been measured in the flat-top sections of the gas jets as presented in figures 4.4 and 4.5.For these measurements specifically argon is used, because the difference in index of refraction of argon with respect to the vacuum is one order of magnitude larger than for helium, as indicated in table 4.1. Considering the phase shifts associated with these argon gas densities, it is found that helium jets with densities of cm 3, densities which have been achieved in the experiment, should result in similar shifts in phase. This indicates that the interferometer set-up should be suitable for a quantitative imaging of helium gas jets with a diameter of 0.75 mm at gas densities in the order of cm 3.

55 4. GAS JET CHARACTERIZATION n (10 cm ) p 0 (bar) (a) Gas density in the flat-top section of the jet profile. Measurements (red) and linear fit (black) Δr g (mm) Δr f (mm) p 0 (bar) (b) Width of the flat-top (black) and density gradient (red) Figure 4.6: Pressure dependence of the density profile of the argon gas jet at 0.5 mm from the nozzle exit.

56 4. GAS JET CHARACTERIZATION Measurements on helium jets In the first laser wakefield acceleration experiments in the Helmholtz-Zentrum Dresden-Rossendorf based on the described jet, as briefly reported on in section 4.4, helium is used to operate the gas jets. To determine the density profiles in the helium jets, interferometric measurements have been performed, similar to the measurements in section The Mach-Zehnder interferometer of figure 4.1 is used again, however, much care has been taken to remove the oscillations present in density profiles that were seen in the argon measurements in the previous section. This has been realized by a realignment of the interferometer, by slightly changing the angle at which the laser beam was send through the windows of the vacuum chamber, such that the secondary interference patterns, responsible for the oscillations, do not end up onto the detector. In figure 4.7 measurements of the supersonic helium jet are presented. Figure 4.7(a) shows the phase image of the helium jet, operated at a pressure of 70 bar. It can be seen that the shape of the supersonic jet using helium is similar to the jet of argon. The density profile at 0.5 mm from the nozzle exit is shown in figure 4.8. In the centre of the jet a density of cm 3 is obtained. The width of the flat top is 0.60 mm and the width of the density gradient at the edges of the jet is 0.25 mm. The high-frequency oscillations that were discussed in the previous section regarding the argon measurements, are now found to be successfully eliminated to a large extend by the realignment of the interferometer. To illustrate this statement a density profile obtained before realigning the set-up is included in figure 4.8 as the dashed curve. With the set-up described, we found that measuring helium jets is possible down to a operating pressure of 9.7 bar. The corresponding phase image is shown in figure 4.7(b). At this operating pressure the phase shift due to the gas jet is very low, such that fluctuations in the background level become more pronounced. The standard deviation over 49 images of the background phase has been determined to be 5.1 mrad, as an average over the image. Using the data of figure 4.7(b) the density profile at 0.5 mm behind the nozzle exit is reconstructed and plotted in figure 4.8. The profile reveals a flat-top density of cm 3 over 0.43 mm. The density gradient is present on a length scale of 0.22 mm. This is, to our knowledge, the first quantitative characterization of the gas density distribution in helium jets. Our measurements set the current state of the art in lowest gas density and highest resolutions with such jets. The current limits in our density measurements are formed by fluctuations in the detected background signal. The main source of the background fluctuations is a flow of air through the arms of the interferometer. The influence of any airflow on the background level can be seen by performing measurements while the flow box above the set-up has been turned on, increasing the airflow through the arms of the interferometer. This resulted in an increase of the average standard deviation of the background phase to 13.7 mrad. This relative large increase in fluctuations arises because the reference arm of the interferometer remained in the ambient, not passing through the vacuum chamber, the complete interferometry set-up had not been enclosed, nor were any beam paths shielded from the environment. The next step of improvement, to increase the lowest gas density that can be detected and thus also the spatial resolution, would be to install additional air flow shieldings.

57 4. GAS JET CHARACTERIZATION Φ (rad) z (mm) y (mm) (a) Operated at a pressure of 70 bar Φ (rad) z (mm) y (mm) (b) Operated at a pressure of 9.7 bar Figure 4.7: Phase images of the supersonic gas jet, operated with helium.

58 4. GAS JET CHARACTERIZATION n (10 cm ) r (mm) Figure 4.8: Density profile of the gas jet at 0.5 mm behind the nozzle exit. The nozzle is operated with helium at 70 bar (black) and 9.7 bar (red). For comparison a profile of the gas jet before the realignment of the set-up is included (dashed curve).

59 4. GAS JET CHARACTERIZATION 54 Figure 4.9: Beam of electrons accelerated in the wakefield of the multi-tw laser at the HZDR excite a phosphor screen, producing a bright fluorescence spot. The screen is positioned at a distance of 40 cm behind our gas-jet target [62]. 4.4 First experiments on laser wakefield acceleration Recently, the laser wakefield acceleration experiments have started at the Helmholtz-Zentrum Dresden-Rossendorf. In figure 4.9 a preliminary result on the very first generation of electron bunches by laser wakefield acceleration in a helium gas jet is shown. The image shows a bright spot generated by a fast beam of electrons striking a phosphor screen, positioned at a distance of 40 cm behind the gas-jet target. The gas jet is produced by the 0.75 mm Mach 4.8 nozzle, which has been designed, constructed and characterized in this work. The nozzle is operated at a pressure of 10 bar. The drive laser provided a pulse with an energy of approximately 2 J and has been focussed at a distance of 2 mm above the nozzle exit in the centre of the jet. With these parameters the bubble regime of laser wakefield acceleration is entered. A next step would be focussing the laser onto the back end of the gas jet, to make use of the downward density gradient for injection of electrons into the wakefield. As of this moment, no information about the energy or charge of the electron bunches is available, except preliminary data that still needs to be evaluated and checked. We expect more extensive results of the laser wakefield acceleration experiments from a next experimental run in summer Discussion A Mach-Zehnder interferometer has been built for measuring the density profile of supersonic gas jets. The interferometer is able to measure quantitatively the tiny differences in optical path lengths arising from helium jets with a diameter of 0.75 mm at gas densities of cm 3 and lower. Measurements of supersonic helium jets have been obtained at operating pressures down to below 10 bar. On average the standard deviation in the background level in phase