Polarimeter for an Accelerated Spheromak

Transcription

1 Polarimeter for an Accelerated Spheromak by Patrick Jean-François Carle A thesis submitted to the Department of Physics, Engineering Physics & Astronomy in conformity with the requirements for the degree of Doctor of Philosophy Queen s University Kingston, Ontario, Canada April 2014 Copyright c Patrick Jean-François Carle, 2014

2 Abstract A three-beam heterodyne polarimeter has been designed and constructed to measure line-integrated density and Faraday rotation of accelerated spheromak plasmas in the Plasma Injector 1 and 2 devices (PI-1, PI-2) at General Fusion Inc. Faraday rotation is a function of the local magnetic field and electron density. Therefore, the polarimeter has the potential to provide information on the internal magnetic field of the plasma. A typical spheromak is about 1m in length and is accelerated to speeds on the order of 100km/s. At a bandwidth of 1MHz, the polarimeter can axially resolve the spheromak down to about 10cm. The polarimeter uses a CO 2 laser that produces a Faraday rotation signal of about 0.5 for a typical plasma with density and magnetic field on the order of m 3 and 1T. The Faraday rotation measurement noise floor for a null signal is about 0.1. Two important sources of Faraday rotation error are the ellipticity and collinearity of the plasma-immersed beams. These error sources are examined by sending the plasma beams through a rotating optic to mimic the path through a dense, magnetized plasma. The error due to the ellipticity effect has been reduced to below the noise floor by careful alignment and use of zero phase reflectors that minimize elliptical polarization of the beams. i

3 Collinearity error has been greatly improved by aligning the beams with a rotating ZnSe wedge. Measurements after the alignment match well with a model Faraday rotation signal generated from magnetic probe measurements. However, beam collinearity continues to be a significant source of error. For regions with strong density gradients, the size of this error can be on the order of the signal magnitude. For future work, steps should be taken to improve the alignment of the two plasmaimmersed beams, and to shorten the length of the beam path to further reduce the beam collinearity error. ii

4 Acknowledgments I came in with little experience in plasma physics and polarimetry, and have learned a lot. Many thanks to my tireless supervisors Stephen Howard at General Fusion and Jordan Morelli at Queen s. Also, thanks to David Brower and Roger Smith for helpful discussions on polarimetry. Thanks to Doug, Michel and Stephen for giving me a chance to be part of their amazing project at General Fusion. Thanks to Blake for photocopying my AOM and laser manuals and staying up late with me one night to help with a calibration experiment that was doomed to fail. Thanks to everyone else at General Fusion for their technical help and friendship along the way. My funding was mostly provided by an NSERC Industrial Postgraduate Scholarship scholarship. It is important that we keep programs like these available for future students. Thank you Canada! Your tax dollars went to good use. Last but not least thanks to my amazing family and friends for their overwhelming support. Special thanks to Mom and Dad (aka Colleen and François) for making life way easier than it should have been. Thanks to my brother Alex for travelling thousands of kilometres for Super Nintendo sessions. Honourable mention to Aunt Marg for hitting me with a snowball in the ear at a time when I was much shorter and knew way less physics. To Sophie, growl woof woof! Love you all! iii

5 Statement of Originality The work in this thesis is the original and independent work of the author. Chapter 1 reviews the state of the art in fusion and plasma magnetic field diagnostics, and contains little original material. Chapters 2-4 present established theory on plasma waves, polarimetry and spheromaks. Chapter 4 on spheromak theory greatly benefited from discussions with Stephen Howard. Original contributions from the author begin in Chapter 5, which presents the design of the polarimeter used to diagnose accelerated spheromaks. A novel use of a phase-retarding reflector in place of a waveplate is incorporated into the design, which works adequately and provides significant cost-savings. Another original concept is the placement of the polarimeter near the measurement chord, which is helpful to reduce the collinearity requirements of the beams passing through the plasma but also presents challenges in terms of electrical noise due to proximity to the pulse power banks. Chapters 6-7 analyse simulated and experimental measurements for a magnetized target fusion spheromak in a coaxial geometry, which is original work. Chapter 8 outlines original ideas that could be implemented to improve the polarimeter s performance including a two-translation stage alignment mechanism and a proposal for multi-chord polarimetry on an accelerated spheromak. Chapter 9 summarizes and concludes this thesis. iv

6 Contents Abstract Acknowledgments Statement of Originality Contents List of Tables List of Figures Acronyms i iii iv v ix x xxiii Chapter 1: Introduction Fusion Power The General Fusion Inc. Project Problem: Accelerated Spheromak Magnetic Field Measurement Immersed Probes Particle Spectroscopy Non-Perturbing Polarimetry Objectives Contributions Organization of Thesis Chapter 2: Plasma Waves Fundamental Plasma Parameters Appleton-Hartree Dispersion Relation Non-magnetized Plasma Magnetic Field Perpendicular to Wave Propagation Direction and the Cotton-Mouton Effect v

7 2.5 Magnetic Field Parallel to Wave Propagation Direction and Faraday Rotation Arbitrary Magnetic Field Direction Chapter 3: Polarimetry Amplitude-Based Polarimeters Phase-Based Polarimeters Two-Beam Heterodyne Polarimeter Three-Beam Heterodyne Polarimeter Signal Demodulation Beam Ellipticity Beam Collinearity Collinearity Error Theory Rotating Wedge Calibration Device Iris for Overlapping Beams Gaussian Beam Propagation Beam Refraction Interference Phase Contrast Beam Alignment Beam Coherence Length Acousto-Optic Modulator Chapter 4: Spheromaks and Plasma Injectors Helicity and Minimum Energy States Spheromak Magnetic Geometry Spheromak Formation and Acceleration λ Profile β Limit Safety Factor Related Spheromak Experiments PI-1 Geometry Chapter 5: Polarimeter Design Plasma Injector and Chord Geometry Complementary Injector Diagnostics Interferometers Magnetic Probes Thomson Scattering Polarimeter Laser Laser stability Laser Alignment vi

8 5.4 Acousto-Optic Modulators (AOMs) Optics Lens Waveplate Thin Film Polarizer Reflectors and Beamsplitters Polarimeter Component Layout Bottom Level Top Level Signal Measurement Hardware and Software Reference Signals Raw Signal Magnitude and Noise Bandwidth Calibration Ellipticity Calibration Collinearity Calibration Triggering Electrical and Vibrational Noise Related Work Chapter 6: Polarimeter Simulations Chord Profiles and Simulated Polarimetry Characteristic Plasma Frequencies Cotton-Mouton Effect Refraction WKBJ Approximation Beam Collinearity Optically Active Windows Plasma Light Chapter 7: Measurements of an Accelerated Spheromak Time Delays Noise Floor for a Null Signal Density Measurements Missed Fringes on the Polarimeter and Interferometer Density Signals Polarimeter Dual Densities Correlation between Interferometer and Polarimeter Densities Faraday Rotation Measurements Axial Resolution vii

9 Chapter 8: Future Work Translation Alignment Beam Profiler Shortened Beam Path Multiple Chords and Abel Inversion Chapter 9: Summary 166 Appendix A: 178 A.1 Immersed Probes Continued A.2 Particle Spectroscopy Continued A.3 Relationship between Linearly and Circularly Polarized Light A.4 Polarimeter Signal for Initially Elliptical Beams A.5 Combination of Two Sinusoids Oscillating at the Same Frequency viii

10 List of Tables 5.1 Temperatue-wavelength regimes of the polarimeter laser. The C range is chosen since it is wide and gives a wavelength of µm, which is close to 10.6µm Review of plasma polarimeters in other experiments. RFP stands for reversed-field pinch Frequency of a 10.6µm beam compared to characteristic frequencies of the plasma assuming n = m 3, B = 2T, and T = 10eV Ratios of the maximum internal probe array toroidal field at the magnetic axis to the maximum axial field from the wall probe. The mean ratio is 2 with a standard deviation of ix

11 List of Figures 1.1 ITER tokamak is a large fusion experiment currently under construction. Tokamaks confine plasma with large magnetic fields created by external coils and plasma current [4] Wendelstein 7-x stellarator [89] is an alternative MCF reactor design that uses precisely shaped magnetic fields to confine the plasma NIF experiment attempts to reach fusion conditions by focusing a short, high-energy laser pulse onto a fuel pellet. (a) NIF laser bays [38]. (b) NIF target [44] GF prototype MTF reactor compresses a magnetized plasma target, requiring a mid-range plasma confinement time that could avoid the challenges of MCF and ICF reactors. (a) Reactor assembly. (b) Injector with simulated spheromak plasma in between electrodes. (Pictures courtesy of General Fusion Inc.) Magnetic probe array with ceramic shielding in formation region. Plasma damage visible The magnetic field vector, B, is in the y-z plane and makes an angle, θ, with the vector wavenumber k, which points in the direction of the travelling electromagnetic wave x

12 2.2 Faraday rotation interpreted as the rotation of the polarization plane of a linearly polarized beam of light after passing through a magnetized plasma Amplitude-based polarimeter where a vertically polarized beam enters a magnetized plasma, which Faraday rotates the polarization plane. The amount of rotation can be determined by monitoring the change in detected intensity after the vertical polarizer (P). The reduction in the vertical component of the electric field is exaggerated in this example A left and right-circularly polarized beam of different frequencies are sent through a magnetized plasma. Each beam is phase shifted by a different amount due to the Faraday effect. After passing through a polarizer (P), the detector (D) records a signal from which a Faraday rotation measurement can be extracted from the signal s phase A third, linearly polarized beam that does not pass through the plasma can be used to obtain density information. The additional beam has frequency ω V, and is combined to the circularly polarized beams with mirror, M, and beamsplitter, BS An additional reference signal is needed to extract the time-dependent phase shifts from the plasma signal. The plasma signal is the usual signal from the three-beam system that interacts with the plasma. The reference signal uses the same initial beams, but they do not interact with the plasma. The plasma and reference signals are respectively recorded by detectors D P and D R xi

13 3.5 Demodulation example for the ω LR beat. Horizontal and vertical axes are respectively frequency and magnitude. (a) The Fourier transform of the initial plasma and reference signals. (b) A bandpass filter is applied to the plasma and reference signals to extract the ω LR beat. (c) The Hilbert transform is applied to the filtered reference signal. The negative frequencies are eliminated. (d) The plasma beat signal is multiplied by the modified reference signal, which shifts both frequency spikes up by ω LR. (e) The residual 2ω LR spike is filtered out Simulated phase shift response generated by a rotating λ/2-waveplate for three test ellipticities where ɛ R = 1 and ɛ L varies. When the ellipticity is greater or less than 1, the response is non-linear The collinearity of the left and right-circularly polarized polarimeter beams has an effect on the Faraday rotation measurement. (a) Near collinear beams give low error. (b) Non-collinear (exaggerated in this figure) beams can introduce a significant error Side and top views of wedge used to calibrate out beam collinearity error. The highest and lowest point on the wedge are respectively labelled P high and P low Iris can make beams purely overlap, but not collinear since beam intensity peaks are still offset Gaussian beam profiles. (a) Radial profile shows 99% of beam power is within circle of radius 1.5W 0. (b) Axial profile shows that beam radius expands to 2W 0 at an axial distance of z 0 from the beam waist, and diverges at an angle θ 0 when z z xii

14 3.11 Ray A and B each encounter a different refractive index due to the plasma s density gradient. Therefore, each ray incurs a different phase shift and the wavefront is refracted. The angle of refraction, θ R, is exaggerated for clarity Phase contrast diminished due to angle between two interfering beams. Rays at points P1 and P2 do not produce the same interference due to phase differences Simulation showing phase contrast effects of two beams interfering. Beams are assumed to have uniform power distributions. Ten equallyspaced rays are used. Results are presented for four values of φ(w ), the phase difference between the central and outer rays: (a) φ(w ) = 0, (b) φ(w ) = π/3, (c) φ(w ) = 2π/3, (d) φ(w ) = π. The overall ray signal for the case of φ(w ) = π is not exactly zero due to the small number of rays used in this simplified example Simulation showing relationship between inner-outer ray phase difference, φ(w ), for uniform and Gaussian beam intensity profiles. A hundred equally-spaced rays are used along the detector Acousto-optic modulator Doppler-shifts an incident laser beam with frequency ω B by the acoustic wave frequency ω A Wave fronts of light, initially in phase, enter an acousto-optic modulator and reflect off acoustic waves producing a diffraction pattern. The moving acoustic waves Doppler shift the frequency of the diffracted light. 65 xiii

15 4.1 Spheromak magnetic structure. (a) Poloidal, B θ, and toroidal, B φ, field components. [81]. (b) Spheromak magnetic field is mainly toroidal near centre and poloidal at edges [49] PI forms and accelerates spheromaks. As it is accelerated down the injector, the spheromak is compressed and increases in density and temperature Spheromak formation by CHI [56]. (a) Gas puffed into chamber. (b) Gas breakdown and toroidal field formed. (c) Lorentz force pushes plasma out. Stuffing field frozen into plasma. (d) Stuffing field reconnects at back of spheromak, forming poloidal field Flux conserver placed in the target chamber at the end of the injector Select CHI experiments drawn to scale [42] Approximate dimensions of PI-1 [43] PI-1 injector and chord geometry. (a) The most often used polarimeter chord is positioned 352cm from the backflange of the injector. Sketches of the toroidal field B φ and poloidal field B θ are provided. (b) Cross section of the injector at the PI axial position looking from the end of the injector towards the backflange. The labelling convention for the injector angular positions is shown. The interferometer chord is omitted for clarity xiv

16 5.2 Interferometer schematic. Laser split into reference and samples beams by fibre optic coupler. Reference beam is reflected back through the fibre. Sample beam is phase shifted by the plasma and reflected back. Both sample and reference beams are split by the coupler on their way to detectors D1 and D2. The signal is phase shifted by 120 at one of the detectors. Reference and sample beams interfere, producing a signal from which the plasma line-average density can be determined Amplitude-based interferometer diagnostic on GF plasma injector. (a) Two out of phase interference signals are produced and can be plotted against one another to track density changes. (b) Computed densities from interfereometers at three axial locations Sample of magnetic field measurements for probes at the 118, 352 and 493 axial positions and several angular positions. The axial field signals show the location of the spheromak in the injector. The toroidal field signals increase once the spheromak has passed, which corresponds to the toroidal pushing field Probe array magnetic field measurements. The array is inserted into the injector at the 352 axial, 180 angular position from the bottom. The radial position of the coils, ρ, is given. For reference, the inner and outer electrode radii are 32.8cm and 51.0cm respectively. Two of the toroidal field signals appear to have saturated at about 1T. Noise from the plasma formation and acceleration discharges can be seen at about 270µs and 310µs Thomson scattering geometry [86] xv

17 5.7 Polarimeter CO 2 laser with vertically polarized beam path overlaid. Actuator that controls the laser shutter is shown Post-It notes change colour when exposed to heat and can be used to locate the invisible beam. For very low power beams, the more sensitive Macken probe is used, which is seen on the far right AOMs are used to frequency shift an incident beam to allow for heterodyning. (a) Original Brimrose AOM in the appropriate orientation to frequency-shift vertically polarized light. (b) New IntraAction AOM that replaced the broken 25MHz Brimrose AOM A λ/4 waveplate can be used to convert between linearly and circularly polarized light Thin film polarizer reflects s-polarized light and transmits p-polarized light when the angle of incidence is at the Brewster angle A phase retarding reflector used to convert between linearly and circularly polarized light. The s and p components of the initially linearly polarized light are shown for beam ω L Sketch of the bottom level of the polarimeter as viewed from above. The initial laser beam is split into three beams, two of which are frequency shifted by the AOMs and combined. The z-direction is out of the page xvi

18 5.14 Sketch of the top level of the polarimeter as viewed from above. Beams ω L and ω R are converted to circular polarization and their divergence is reduced. All beams are split into multiple parts, giving the possibility to have multiple polarimeter chords. Beams ω LR and ω V are interfered at the detector. The z-direction is out of the page Combined ω L and ω R beam conversion to circular polarization with a phase retarding reflector Simulated radius of the combined ω LR beam along its path through the plasma and into the detector. The radii of the of the lenses and injector windows are also drawn for reference Comparison of Faraday rotation measurements using software and hardware reference signals. The Faraday rotation measurement bandwidth is 0.5MHz. (a) Software reference with f LR = 15MHz. (b) Software reference uses the corrected frequency 15MHz + δf LR. The Faraday rotation signal for the hardware reference is offset for convenience Apparatus used for calibration of beam ellipticity and collinearity Elliptically polarized beams. (a) Nonlinear and linear responses from calibration with spinning λ/2 waveplate [15]. (b) Original configuration of polarimeter that produced elliptical beams. Beams ω L and ω R first converted to circular polarization and then combined Apparent Faraday rotation during beam alignment by placing a rotating, 14 arcmin (0.23 ) wedge of ZnSe in the path of the plasma beams. Both signals have bandwidths of 500Hz. Signal sampled at 10.4kHz xvii

19 5.21 Optical to electrical trigger circuit. The resistor typically has a value of about 500kΩ to provide a large gain to the small photodiode signal Simulated density and magnetic field radial profiles at the 352 axial position for n 0 = m 3 and B 0 = 2T. Plasma has an axisymmetric profile modelled after a coaxial Taylor state Simulated density and magnetic field chord profiles at the 352 axial position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state. (a) Toroidal magnetic field components along the beam chord. (b) Chord profiles for n 0 = m 3 and B 0 = 2T. The poloidal field B θ is taken to be purely in the axial direction. The components of the magnetic field parallel and perpendicular to the chord are respectively B and B Weighting of Faraday rotation contributions from points along the polarimeter chord at the 352 position. Also shown is the toroidal field magnitude at points along the chord normalized to the peak toroidal field in the plasma. Plasma has an axisymmetric profile modelled after a coaxial Taylor state Geometry used in refraction simulation to approximate the local transverse density gradient Simulated refraction of the 10.6µm polarimeter beam passing through the plasma at the 352 axial position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state. Simulation results are given for several peak densities n xviii

20 6.6 WKBJ approximation simulation results for 10.6µm beam for the polarimeter chord at the 352 position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state with n 0 = m Simulation of the collinearity parameter, C, at various positions along the polarimeter chord at the 352 position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state with B 0 = 2T and n 0 = m Polarimeter is typically out of sync with interferometer and magnetic probes. (a) Polarimeter and magnetic probes are synchronized by matching electrical noise spikes from the accelerator discharge. (b) Interferometer is already in sync with magnetic probes, and so can be shifted by the same delay as the probes Noise floor data from null signals for polarimeter measurements of (a) density and (b) Faraday rotation. The density and Faraday rotation null signals respectively have a bandwidth of 5MHz and 1MHz. The amplitude product variable is normalized so the highest measured value is xix

21 7.3 Polarimeter has a higher chance of missing fringes since its wavelength is 10 times longer than the interferometer s. (a) Polarimeter density measurement does not return to zero after the plasma has passed, clearly indicating a fringe has been missed. (b) Polarimeter density measurement returns to zero after plasma passes, but there is a large difference in maximum densities, which is caused by the polarimeter missing fringes and does not represent an actual asymmetry in the plasma Interferometer also occasionally missed fringes. Note the large lowfrequency disturbance beginning at 500µs Polarimeter gives two density measurements since two beams pass through the plasma. The density signals are typically very similar Large differences between polarimeter density measurements. (a) Occasionally, one density measurement misses a fringe while the other does not. This is due to differing signal strengths between the signals beats. The beat with the weaker signal strength misses the fringe as can be seen from the Fourier transforms of the raw (b) reference and (c) plasma signals Correlation between the interferometer and polarimeter density measurements. (a) Distribution of Pearson correlation coefficients for all analysed shots. (b) Density measurements for shot with worst correlation, r = Polarimeter misses large density gradients. (c) Measurements for shot with the best correlation, r = xx

22 7.8 Model toroidal field at the magnetic axis is generated from the sum of the wall probe s modified poloidal and toroidal fields. The poloidal field measurement at the wall is multiplied by 2 to approximate the spheromak toroidal field. The pushing field contribution at the magnetic axis is obtained from the wall probe toroidal field multiplied by a factor of 1.2 to account for its 1/ρ dependence Comparison of modelled toroidal field at the magnetic axis to the field measured by internal probes for the (a) worst and (b) best match for maximum toroidal field Poor agreement between modelled and measured Faraday rotation before improved alignment with the rotating wedge. This suggest large collinearity noise Polarimeter measurements when the ω L and ω R beams are sent through a polarizer before passing through the plasma. Non-zero Faraday rotation indicates beam collinearity is not negligible. Faraday rotation and density signal bandwidths are respectively 100Hz and 1MHz Improved agreement between modelled and measured Faraday rotation after the rotating wedge was used to refine the alignment Faraday rotation filtered down to 300kHz makes the signal clearer at the expense of detail Inner toroidal field estimated from polarimeter Faraday rotation and density measurements. Faraday rotation measurement bandwidth is 300kHz. Chord length is 60cm xxi

23 7.15 Spline-fit to surface magnetic probe axial field measurements at time of peak polarimeter Faraday rotation (488µs) Proposed beam combination design with two translation stages for improved beam collinearity New lens arrangement with shortened path could reduce undesirable beam collinearity and refraction effects Multiple probing chords give spatial resolution with Abel inversion A.1 Magnetic coil probes measure the change in magnetic field with time. They are located primarily at the surface to avoid perturbing the plasma.179 A.2 Hall effect can be used to measure magnetic field without having to integrate the signal over time A.3 The Faraday rotation probe [47] observes the rotation of linearly polarized light through a magneto-optic material to directly measure local magnetic field. The probe is optically based and so is immune to electrical noise, but must be inserted into the plasma A.4 Observation of Zeeman splitting from a high energy lithium beam in a plasma allows for measurement of magnetic field [83]. (a) Lithium Zeeman splitting. (b) High energy lithium beam and observation optics mounted on a tokamak A.5 Deflected alpha particles can be used to reconstruct magnetic field. [93] 184 xxii

24 Acronyms AC AOM CHI CTF CTIX CTX DC DT FWHM GF ICF Alternating current. Acousto-optic modulator. Coaxial helicity injection. Compact Toroid Fueler. Compact Toroid Injection Experiment. Compact Toroid Experiment. Direct current. Deuterium-tritium. Full-width half maximum. General Fusion Inc.. Inertial confinement fusion. ITER International Thermonuclear Experimental Reactor. JET MARAUDER Joint European Torus. Magnetically Accelerated Ring to Achieve Ultra-high Directed Energy and Radiation. MCF MTF Magnetic confinement fusion. Magnetized target fusion. xxiii

25 NEP NIF PI PRR RACE RF SI SNR SSPX TEM TFP UPS USB WKBJ Noise equivalent power. National Ignition Facility. Plasma Injector. Phase retarding reflector. Ring Accelerator Experiment. Radio frequency. International System of Units. Signal to noise ratio. Sustained Spheromak Physics Experiment. Transverse electromagnetic. Thin film polarizer. Uninterruptible power supply. Universal Serial Bus. Wentzel Kramers Brillouin Jeffreys. xxiv

26 1 Chapter 1 Introduction 1.1 Fusion Power Fusion reactions power the Sun, the dominant source of energy on Earth. Even fossil fuels have their roots in photosynthesized sunlight. Fusion occurs when two light nuclei are brought within a close enough distance for the strong force to bind them together. The mass of the product particles is less than the mass of the reactants. This missing mass has been converted to energy, which is released in the form of kinetic energy to the product particles. For example, the deuterium-tritium (DT) reaction is important in fusion engineering due to its relatively high reaction cross section (ie. reaction probability). The DT reaction produces 17.59MeV [25] in kinetic energy, which is distributed between the product helium atom and neutron according to conservation of momentum: 2 1H + 3 1H 4 2He (3.54MeV) + 1 0n (14.05MeV). (1.1) Experiments have attempted to directly harness fusion energy since the mid 1900s [28]. The advantages of fusion power are numerous [16, 88, 84, 54, 62]:

27 1.1. FUSION POWER 2 1. The fuel is abundant. Deuterium accounts for 0.015% of hydrogen isotopes on Earth [25] and can be harvested from ocean water. Tritium is rare but can be bred from lithium, which is abundant. It is estimated current reserves could supply the Earth with DT fuel for millions of years. 2. Fusion reactors are inherently safe. No weapons-grade material is produced. Only small amounts of fuel need to be used at a given time. Meltdowns, which are a more significant threat in fission reactors, are not possible in fusion reactors since contamination from melting walls would quickly extinguish the reaction. 3. Fusion is clean. No greenhouse gases [84] or other atmospheric pollutants are emitted from reactions. Over time, a DT reactor s walls would need to be replaced due to neutron activation. However, this waste is low-level [78] compared to spent fission fuel. As reactor designs mature, aneutronic reactions might be feasible in which no neutrons are generated. 4. Fusion reactors can supply baseload power (ie. the minimum energy demand throughout the day). This is not the case for other clean-energy alternatives like wind and solar, which are intermittent. To initiate a fusion reaction, atoms in the fuel must be sufficiently energetic to overcome repulsive Coulomb forces between colliding protons. Fusion reactor schemes typically involve creating a plasma, which is a superheated gas where a majority of the atoms have been ionized. The three main parameters important to fusion energy gain, Q = P fusion /P heating, are plasma density, n, temperature, T, and confinement time, τ E. All other things being equal, a higher density plasma increases the chance of two reactants colliding. A higher temperature, up to a point, increases the chance

28 1.1. FUSION POWER 3 a collision produces a fusion reaction. A longer confinement time increases the total number of collisions between the reactants. An important reactor goal is ignition, which occurs when the heating power from fusion reactions balances power losses from the system. For example, the fusion heating power could be the product helium ions from a DT reaction. About 20% of the energy generated from the fusing of DT is given to the helium particle. If these particles are confined, they can continue to heat the plasma through collisions. Considering this additional heating mechanism, it can be shown [88] that ignition is achieved when nt τ E = m 3 kevs. (1.2) where n is in m 3, T is in kev, and τ E is in seconds. The plasma temperature is typically optimal around 10keV, near a peak in the cross section curve where it is most likely for a collision to result in a fusion reaction. The plasma tends to become violently unstable when exceeding a certain density limit. The Greenwald density [32] is an approximate limit to the maximum density obtainable in a magnetically confined plasma: n G = I p πa 2 (1.3) where I p is the plasma current in MA and a is the plasma minor radius in metres. A main distinction among reactor designs is the trade-off between density, n, and confinement time, τ E. If a reactor is limited to a low density limit, it must compensate with a long confinement time in order to meet the ignition condition and have a greater chance of breaking even (ie. Q=1).

29 1.1. FUSION POWER 4 Figure 1.1: ITER tokamak is a large fusion experiment currently under construction. Tokamaks confine plasma with large magnetic fields created by external coils and plasma current [4]. There are several fusion reactor projects that are quickly converging on ignition and break-even gain. The largest of these (see Figure 1.1) is the International Thermonuclear Experimental Reactor (ITER) currently under construction. ITER [4] is a tokamak reactor, which aims to achieve fusion by confining a plasma inside a toroidal vacuum vessel with large, externally applied magnetic fields (> 1T). An induced plasma current also generates additional magnetic field to stabilize the plasma for relatively long periods of time. ITER plasma discharges are limited to densities on the order of m 3. The goal for ITER is to maintain Q=10 for several seconds with a density of about m 3 [36]. The energy gain record for a tokamak is currently held by the Joint European Torus (JET), which obtained Q=0.7 for 0.15s [52]. Tokamaks belong to a fusion scheme known as magnetic confinement fusion (MCF) where a long confinement time compensates for low plasma density. Stellarators are

30 1.1. FUSION POWER 5 Figure 1.2: Wendelstein 7-x stellarator [89] is an alternative MCF reactor design that uses precisely shaped magnetic fields to confine the plasma. another class of MCF reactor in which perfectly shaped magnetic field coils virtually eliminate the need for a stabilizing plasma current. The Wendelstein 7-X stellarator (see Figure 1.2) currently under construction in Germany is expected to run at about m 3 for more than a second [89]. However, net energy gain is not expected until future generation stellarators are developed. In contrast, inertial confinement fusion (ICF) typically uses a powerful laser to ignite the fuel at a high density for a short time. The National Ignition Facility (NIF) [38] concentrates a 1.8MJ, 500TW laser onto a fuel pellet to heat it to fusion conditions at a density of m 3 for a matter of nanoseconds [60]. NIF was expected to achieve ignition by It has fallen short of this milestone, but has nonetheless made significant progress having attained 70% of its target ignition density [70]. An encouraging result on a recent shot shows that the fusion power has exceeded the

31 1.1. FUSION POWER 6 (a) (b) Figure 1.3: NIF experiment attempts to reach fusion conditions by focusing a short, high-energy laser pulse onto a fuel pellet. (a) NIF laser bays [38]. (b) NIF target [44]. incident laser beam power for the first time [44]. MCF and ICF are on opposite ends of the confinement time spectrum and consequently face difficult challenges to achieve an economically viable solution to controlled fusion energy. For example, MCF seeks very long confinement times, which requires large, expensive superconducting coils to generate the magnetic field precisely. ICF occurs over a very short time and therefore requires powerful lasers that are highly inefficient and expensive. A third approach, magnetized target fusion (MTF), works at mid-range confinement times and densities in the hopes of obtaining significant cost savings [54]. In MTF, a magnetized plasma is created and compressed to fusion conditions. The plasma magnetic field insulates it from the wall, allowing for a longer confinement time than ICF. In addition, the fast plasma compression enables the quick extraction of fusion energy without having to keep the plasma stable indefinitely as in MCF. In

32 1.2. THE GENERAL FUSION INC. PROJECT 7 this way, it is believed that MTF can circumvent some of the more difficult technological challenges of other fusion schemes. 1.2 The General Fusion Inc. Project General Fusion Inc. (GF) 1 [58][42] is working to develop a prototype net-gain scale MTF reactor. The exact path to fusion is an uncertain one that still requires a great deal of exploration. For this reason, GF is investigating several paths towards an MTF reactor. The essence of the original concept centres around the Plasma Injector (PI) that forms and accelerates a magnetically self-confined plasma into a compression chamber (see Figure 1.4(a)). This special plasma is known as a spheromak [7][49], which is a type of compact toroid plasma that confines particles with magnetic fields generated from the plasma s own internal currents. PI spheromaks are in the shape of a torus as shown in Figure 1.4(b). The spheromak is pushed down the injector into an ever-decreasing volume, which compresses the spheromak to a higher density and temperature. On the inside of the compression chamber, a liner consisting of a layer of liquid lithium-lead is spun to form a vortex. Once the accelerated spheromak reaches the compression chamber, an array of pistons strikes the outer walls causing the liquid liner to close in on the spheromak and acoustically compress it to fusion conditions. The liquid liner absorbs fusion neutrons, which protects the walls and allows heat from the neutrons to be extracted. The liner also breeds tritium from reactions between lithium and neutrons, which can be extracted and used as fuel in subsequent reactor pulses. An alternate version of this core design, involves accelerating two spheromaks 1

33 1.2. THE GENERAL FUSION INC. PROJECT 8 (a) (b) Figure 1.4: GF prototype MTF reactor compresses a magnetized plasma target, requiring a mid-range plasma confinement time that could avoid the challenges of MCF and ICF reactors. (a) Reactor assembly. (b) Injector with simulated spheromak plasma in between electrodes. (Pictures courtesy of General Fusion Inc.) towards one another. The two spheromaks then merge in the compression chamber via a process known as magnetic reconnection [13] and form a hotter, stationary magnetized plasma that could offer a more convenient target for acoustic compression. The PI-1 and PI-2 injectors were built with this goal in mind. Although they have produced valuable data separately, they have not yet been combined for a merging experiment. In order to obtain the temperature, density and lifetime needed for net energy gain during compression, the pre-acoustic compression plasma parameters must meet certain criteria. PI initially forms a plasma with a density of about m 3 and temperature of about 10eV. It is then accelerated down a narrowing conical tube, which compresses it to higher magnetic field, density, and temperature. At the end of the

34 1.3. PROBLEM: ACCELERATED SPHEROMAK MAGNETIC FIELD MEASUREMENT 9 tube, the 40cm diameter pre-acoustic compression plasma must have a temperature of 100eV, density of m 3. To reach these conditions and produce energy, a plasma lifetime of about 100µs is needed as the liner compresses the plasma. To meet these pre-acoustic compression parameters, it is critical to have an understanding of the spheromak s inner magnetic field. The research for this PhD thesis, in partnership with GF, investigates this problem. 1.3 Problem: Accelerated Spheromak Magnetic Field Measurement Plasmas generated from the PI machines offer a difficult challenge for magnetic field diagnostics. Diagnostics extending into the plasma must be avoided wherever possible. Spheromaks rely on internal currents to generate their insulating magnetic fields. Therefore, probes interfering with these currents could significantly perturb the spheromak. Currently, PI spheromaks have been created with a density of m 3 and temperature of 100eV (10 6 K). Considering this in addition to MA currents and occasional electrical arcing, the PI machines present a hostile environment, which can damage immersed devices. Although there is some degree of axisymmetry in the generated spheromak, they evolve in a fast and often unpredictable manner. A typical PI spheromak has an axial length of about 1m and can be accelerated to an axial speed of about 100km/s. Therefore, measurements on the MHz timescale are required to resolve the 1T magnetic fields generated. Plasma magnetic field diagnostics are separated into three main categories: immersed probes, particle spectroscopy, and non-perturbing polarimetry. The sections to follow give brief examples of some of these diagnostics, and their respective

35 1.3. PROBLEM: ACCELERATED SPHEROMAK MAGNETIC FIELD MEASUREMENT 10 strengths and weaknesses. More detailed information can be found in Appendices A.1 and A Immersed Probes Immersed probes require a device to be inserted into the plasma to measure local magnetic field. The most common example is the magnetic coil probe, which is a multi-turn loop of wire. A changing magnetic field through the coil generates a changing voltage between the wire s terminals. The voltage can then be measured to produce an estimate for the local magnetic field passing through the coil. Magnetic coil probes are used almost universally in plasma experiments due to their effectiveness and simplicity. This type of device must be immersed, which can perturb the plasma and damage the probe causing contamination of the plasma and vacuum. As such, probes are not ideal to measure the inner magnetic field of hot, dense plasmas. On the PI machines, probes are mainly placed on the outer electrode to measure the magnetic field at the outer surface of the plasma. An array of probes shielded in a ceramic tube can also be inserted into the plasma to measure the inner magnetic field (see Figure 1.5). However, this is usually only in the plasma-formation region of the injector where the plasma still has low density and low temperature Particle Spectroscopy There are a number of low-perturbation techniques to measure magnetic field that rely on spectroscopically observing interactions between the magnetic field and injected particle beams. Electrons transitioning from a higher to a lower energy level emit

36 1.3. PROBLEM: ACCELERATED SPHEROMAK MAGNETIC FIELD MEASUREMENT 11 Figure 1.5: Magnetic probe array with ceramic shielding in formation region. Plasma damage visible. light at a specific frequency. This is known as a line emission. The interaction between an electron and a magnetic field splits the line slightly. This splitting is known as the Zeeman effect and can be used to estimate the magnetic field [83]. An analogous phenomenon for electric fields is called the Stark effect [90]. These effects are typically observed from neutral particle beams injected into the plasma to facilitate the measurement of the magnetic field spatially. While these particle-based, spectroscopic magnetic field diagnostics are largely non-perturbing, it is challenging to extract a magnetic field measurement. For example, for the Zeeman and motional Stark diagnostics, the line emission splitting must be distinguished from other potentially strong light emissions originating from plasma dynamics. The consequence is a large increase in the complexity and cost of the system compared to magnetic probes. In addition, due to the large, immobile equipment involved (eg. high-energy particle beam generators), these devices are likely confined to a certain part of the machine. This may not be an issue for highly symmetrical machines like tokamaks. However, for the PI machines, it is of interest to track the evolution of the plasma as it is accelerated.

37 1.3. PROBLEM: ACCELERATED SPHEROMAK MAGNETIC FIELD MEASUREMENT Non-Perturbing Polarimetry The Faraday effect is the rotation of the polarization plane of a linearly polarized beam of light in a magneto-optic medium, such as a magnetized plasma. The amount of Faraday rotation is given by [17] φ F [rad] = λ 2 0 nb dl (1.4) where λ 0 is the beam wavelength in metres, n the plasma electron density in m 3, B the magnetic field in tesla, and dl is an infinitesimal vector in the direction of the beam s path through the plasma. The International System of Units (SI) is used throughout the thesis unless otherwise noted. As can be seen, the Faraday rotation depends on the component of magnetic field parallel to the beam direction, B. A non-perturbing, plasma polarimeter can be devised where a properly conditioned laser beam is sent through a magnetized plasma and analysed to measure the resulting Faraday rotation [15, 71, 74]. The Faraday rotation measurement depends on the line-averaged product of plasma density and magnetic field. This means the local magnetic field cannot be determined unless the density is known, and a profile shape for the density and magnetic field is either known or assumed. For this reason, polarimeters can be configured to simultaneously measure line-averaged density. In addition, if multiple polarimeter chords are used at a particular axial position and axisymmetry can be assumed, profiles can be estimated using Abel inversion [45]. An advantage of the polarimeter is that the beam can easily be split and analysed at different locations on the plasma device, giving an efficient way to estimate magnetic field and density at multiple locations in the machine. The device is also

38 1.4. OBJECTIVES 13 non-perturbing and does not require any material to be placed inside the plasma, nor does it require particle beams to be fired into the plasma. 1.4 Objectives The objectives of this research are to 1. Design and build a non-perturbing plasma polarimeter to estimate the density and internal magnetic field on the PI machines and future MTF experiments at GF, 2. Build the prototype polarimeter to allow for multi-chord measurements, 3. Test and improve the prototype version iteratively during plasma experiment operations, 4. Identify noise sources and fundamental diagnostic performance limitations, 5. Compare polarimeter data to the existing set of diagnostics on the PI machines, 6. Use polarimeter data to better understand the magnetic structure of an accelerated spheromak. 1.5 Contributions A number of novel contributions have been made. A polarimeter was constructed to estimate the internal magnetic field of an accelerated spheromak [15]. Having reviewed the literature, this is the only polarimeter to be designed and tested for an accelerated spheromak. The polarimeter is capable of having measurement bandwidths in the MHz, which is among the fastest plasma polarimeters found in the

39 1.6. ORGANIZATION OF THESIS 14 literature. In addition, a novel optical arrangement replacing waveplates with phase retarding reflectors is shown to be more cost effective while maintaining adequate polarimeter performance. 1.6 Organization of Thesis The document is organized as follows. A review of plasma wave, polarimeter, and spheromak theory is provided in Chapters 2, 3, and 4. Chapter 5 details the design of the PI polarimeter. Simulation results exploring the expected performance of the polarimeter are discussed in Chapter 6. Polarimeter measurements of density and Faraday rotation are presented in Chapter 7. Options for future work and a summary can be found in Chapters 8 and 9.

40 15 Chapter 2 Plasma Waves This chapter describes basic theory regarding the propagation of electromagnetic waves through a magnetized plasma [17, 45, 79, 88]. The results from this analysis are crucial in the development of a plasma polarimetry diagnostic. 2.1 Fundamental Plasma Parameters Several parameters fundamental to the collective behaviour of plasmas must be introduced [17]. A plasma is a highly ionized gas composed of a mixture of ions and electrons. An ion is over 1000 times more massive than an electron. Therefore, compared to electrons, ions are effectively stationary and can be ignored in this analysis. Consider a plasma at equilibrium. If a test charge is suddenly added to the plasma, the electrons quickly rearrange themselves in response to the test charge s electric field. During the transient response to this perturbation, the electrons oscillate on a time scale given by the electron plasma frequency ( ) ne 2 1/2 ω p = (2.1) ɛ 0 m

41 2.2. APPLETON-HARTREE DISPERSION RELATION 16 where n is the electron density, e the elementary charge, ɛ 0 the vacuum permittivity, and m the electron mass. Once a new equilibrium is reached, the rearrangement of nearby electrons shields out the test charge s electric field from the rest of the plasma. The boundary around the test charge where shielding begins is roughly given by a sphere with length-scale radius of one Debye length ( ) 1/2 ɛ0 T λ D = (2.2) ne 2 where T is the electron temperature. This equation has been expressed using the shorthand notation T in place of the implied k B T, where k B is the Boltzmann constant. This shorthand is used throughout the document unless otherwise noted. In order for Debye shielding to work effectively, there must be a large number of electrons within the Debye sphere. The number of particles within a Debye sphere is called the plasma parameter, which is given by Λ = 4 3 πλ3 Dn. (2.3) An ideal plasma reacts collectively to changes in its environment. This greatly simplifies the analysis since overall plasma dynamics can be examined in terms of largescale electrostatic interactions rather than individual particle collisions. A plasma is said to be ideal when the time scale of interest is much greater than 1/ω p, the length scale of interest is much greater than λ D, and the plasma parameter has Λ Appleton-Hartree Dispersion Relation Consider an electromagnetic plane wave propagating inside a homogeneous plasma. The goal is to determine the plasma index of refraction, N, and to understand how

42 2.2. APPLETON-HARTREE DISPERSION RELATION 17 z k θ B x y Figure 2.1: The magnetic field vector, B, is in the y-z plane and makes an angle, θ, with the vector wavenumber k, which points in the direction of the travelling electromagnetic wave. changes in N affect the travelling wave. The energy in the wave is weak, amounting to a small perturbation in the plasma. The incident electromagnetic plane wave has angular frequency ω, wavelength λ, and travels in the ẑ direction: k = kẑ (2.4) where k is the wavenumber, which is related to the wavelength by k = 2π/λ. The plasma contains a magnetic field, B, which is defined to be at an angle θ relative to k as shown in Figure 2.1. For simplicity, B is chosen to be in the y-z plane without loss of generality: B = ŷb 0 sin θ + ẑb 0 cos θ (2.5) where B 0 is the magnetic field magnitude.

43 2.2. APPLETON-HARTREE DISPERSION RELATION 18 Plasma parameters are separated into an equilibrium component and a perturbation component. For example, the density can be written as n = n 0 + n 1 (2.6) where n 0 is the equilibrium part and n 1 the perturbation. Perturbations are assumed to be sinusoidal and can be written as the real part of a complex exponential. For a wave travelling in the ẑ direction, this gives: n 1 = R {ˆn 1 e i(kz ωt)} (2.7) where ˆn 1 is the complex amplitude of the oscillation, z is position, and t is time. The symbol R is omitted for the rest of the derivation for clarity. The plasma equilibrium electric field is assumed to be zero, so the electric field consists only of the perturbation, E 1 : E = E 1 = E 1xˆx + E 1y ŷ + E 1z ẑ. (2.8) Compared to the wave propagating through the plasma at near the speed of light, the plasma is assumed to be initially at rest. Therefore, only the velocity perturbation v 1 must considered: v = v 1 = v 1xˆx + v 1y ŷ + v 1z ŷ. (2.9) Pressure gradient effects are assumed to be small, which is reasonable for a plasma with electron temperature T 1keV [73]. Collisions can be modelled as a frictional force between the perturbed electrons and stationary ions. However, the contribution

44 2.2. APPLETON-HARTREE DISPERSION RELATION 19 from collisions is small if ω ν (2.10) where ν is the electron-ion collision frequency and is defined using the Spitzer resistivity [17] ν = ne 4 ln Λ. (2.11) 16πɛ 2 0m 1/2 T 3/2 where ln Λ 10 is the Coulomb logarithm. The equation of motion for the electron fluid is then ( ) v mn + (v ) v = en (E + v B) (2.12) t Cancelling the density parameters, and writing in terms of perturbed variables gives ( ) v1 m t + (v 1 ) v 1 = e (E 1 + v 1 B). (2.13) The velocity perturbation can be written in complex exponential form as in Equation 2.7. Taking the partial derivative, it is seen that the convective term (v 1 ) v 1 is the product of two velocity perturbation terms. These are both assumed to be small, therefore the convective term can be dropped. The linearized result is v 1 = i e ωm (E 1 + v 1 B). (2.14)

45 2.2. APPLETON-HARTREE DISPERSION RELATION 20 Expanding the cross product, the spatial components of Equation 2.14 are ( e v 1x = i ωm E ω c 1x + v 1y ( e v 1y = i ωm E ω ) c 1y v 1x ω cos θ ( e v 1z = i ωm E ω ) c 1z + v 1x ω sin θ ω cos θ v 1z ω ) c ω sin θ,, (2.15) where the electron gyrofrequency is defined as ω c = eb m. (2.16) In order to eliminate the velocity perturbation terms from the component equations of motion, the electromagnetic wave equation must be introduced. The wave equation can be derived from Maxwell s equations. Faraday s law of induction is B = 1 ( 1 j + E ) c 2 ɛ 0 t (2.17) where c is the vacuum speed of light, and j is the current density. Taking the partial derivative with respect to time gives B t = 1 ( ) 1 j c 2 ɛ 0 t + 2 E. (2.18) t 2 Faraday s law of induction is B t = E. (2.19) Substituting Equation 2.19 into Equation 2.18 gives the wave equation for electric

46 2.2. APPLETON-HARTREE DISPERSION RELATION 21 fields in the plasma: E = 1 ( ) 1 j c 2 ɛ 0 t + 2 E. (2.20) t 2 The motion of the ions is assumed to be negligible, therefore the current density can be expressed as j = env. (2.21) Expanding into equilibrium and perturbation quantities gives j = e (n 0 v 1 + n 1 v 1 ). (2.22) The term containing the factor of two perturbations is small and can be dropped. This approximates the current density to j en 0 v 1. (2.23) Substituting Equation 2.23 into Equation 2.20 and expressing the electric field as a perturbation gives E 1 = 1 ( ) en0 v 1 c 2 ɛ 0 t 2 E 1. (2.24) t 2 Applying the partial derivatives on the right side gives E 1 = 1 ( i ωen ) 0 v c ω 2 E 1. (2.25) ɛ 0

47 2.2. APPLETON-HARTREE DISPERSION RELATION 22 The left side of Equation 2.25 operates on the spatial part of E 1 : E 1 = ( E 1 ) 2 E 1 = ( E1x x + E 1y y + E 1z z ) 2 E 1 z 2 = ( ike 1z ) + k 2 (E 1xˆx + E 1y ŷ + E 1z ẑ) = k 2 (E 1xˆx + E 1y ŷ). (2.26) Substituting this result into Equation 2.25, isolating the velocity terms, and separating into components gives v 1x = i ( N 2 1 ) eω E mωp 2 1x, v 1y = i ( N 2 1 ) eω E mωp 2 1y, (2.27) v 1z = i eω E mωp 2 1z where the definition for the plasma frequency, ω p, has been used (see Equation 2.1), and the index of refraction is defined as N = ck/ω. (2.28) A system of equations containing only the components of the electric field perturbation can now be obtained by substituting the wave equation results (Equations 2.27)

48 2.2. APPLETON-HARTREE DISPERSION RELATION 23 into the equation of motion (Equations 2.15). In matrix form, this is X + N 2 1 i(n 2 1)Y cos θ iy sin θ i(n 2 1)Y cos θ X + N i(n 2 1)Y sin θ 0 X 1 = 0 0 0, (2.29) where the cyclotron frequency ratio is Y = ω c /ω, (2.30) and the square of the plasma frequency ratio is X = ω 2 p/ω 2. (2.31) The electric field is a continuous quantity. Therefore, an infinite number of solutions exists for this system of equations. The determinant of the system must then be zero. Setting the determinant to zero leads to a quadratic equation in N 2, which can be solved to give the Appleton-Hartree dispersion relation: N 2 X(1 X) = 1 1 X 1Y 2 2 sin 2 θ ± ( 1 Y 4 4 sin 4 θ + (1 X) 2 Y 2 cos 2 θ ). (2.32) 1/2 A number of special cases of this solution can be investigated to obtain a better understanding of the propagation of an electromagnetic wave in a plasma.

49 2.3. NON-MAGNETIZED PLASMA Non-magnetized Plasma If the magnetic field is zero, then ω c = 0, and Y = 0. This leads to the special dispersion relation k = 1 c ω 2 ω 2 p. (2.33) If k becomes imaginary, then the wave is evanescent, exponentially decaying inside the plasma. Therefore, the wave propagates through the plasma only when ω > ω p. (2.34) The plasma frequency, ω p, is a proportional to the electron density, n 0. Therefore, a dense plasma with a high plasma frequency can block an incident beam. This is an important consideration when attempting to transmit a beam through a plasma. It is also observed that the wavenumber, and therefore phase of the wave, is a function of only one plasma parameter: the electron density. Interferometer diagnostics use this property to measure the density of the plasma. 2.4 Magnetic Field Perpendicular to Wave Propagation Direction and the Cotton-Mouton Effect For the case where the magnetic field is perpendicular to the wave propagation direction, θ = 90, the two solutions to Equation 2.32 reduce to N 2 = 1 X, (2.35) N 2 = 1 X(1 X) 1 X Y 2 sin 2 θ (2.36)

50 2.4. MAGNETIC FIELD PERPENDICULAR TO WAVE PROPAGATION DIRECTION AND THE COTTON-MOUTON EFFECT 25 The first solution results in the same dispersion relation as Equation Plugging this result into Equation 2.29 forces the electric field to be purely in the direction of the magnetic field, B: E 1x = E 1z = 0 (2.37) This is known as the ordinary wave. The ordinary wave can propagate when its frequency is greater than the plasma frequency (Equation 2.34). The wavenumber for the second solution is k = 1 ( ω c 2 ω2 p ω2 ωp) 2. (2.38) ω 2 ωp 2 ωc 2 The wave can propagate when its wavenumber is real. Solving for the roots of Equation 2.38, this occurs when ω > 1 2 ( ω 2c + 4ω 2p + ω c ) (2.39) or when ωp 2 + ωc 2 > ω > 1 ) ( ω 2 2c + 4ω 2p ω c. (2.40) Substituting Equation 2.38 into Equation 2.29, the electric field components have the relationship E 1x = ie 1z ω ( ω 2 p + ω 2 c ω 2) ω 2 pω c (2.41) E 1y = 0 (2.42) Therefore, the electric field is perpendicular to the magnetic field. This is known as

51 2.5. MAGNETIC FIELD PARALLEL TO WAVE PROPAGATION DIRECTION AND FARADAY ROTATION 26 the extraordinary wave. The difference between the wavenumbers of the ordinary and extraordinary waves (Equation 2.33 and Equation 2.38) indicates that the plasma can affect the phase difference between the two components. As a result of this phase difference, the combination of the ordinary and extraordinary waves gives an elliptically polarized state. This is known as the Cotton-Mouton effect [30, 74]. 2.5 Magnetic Field Parallel to Wave Propagation Direction and Faraday Rotation If the magnetic field is parallel to the wave propagation direction, θ = 0, then the Appleton-Hartree dispersion relation simplifies to N 2 = 1 X 1 ± Y. (2.43) In terms of the wavenumber, the two solutions are k L = ω 1 c k R = ω c 1 ω 2 p ω (ω + ω c ). (2.44) ω 2 p ω (ω ω c ), (2.45) The wavenumbers are named L and R since plugging these wavenumbers into Equation 2.29 gives the electric field phase relationship for a left-circularly polarized wave (L-wave) and a right-circularly polarized wave (R-wave): E 1x = ±ie 1y. (2.46)

52 2.5. MAGNETIC FIELD PARALLEL TO WAVE PROPAGATION DIRECTION AND FARADAY ROTATION 27 The L and R-waves can propagate when their respective wavenumbers are real. For the R-wave, this is the case when ω < ω c (2.47) or when ω > 1 2 ( ω 2c + 4ω 2p + ω c ). (2.48) Similarly, the L-wave can propagate when ω > 1 2 ( ω 2c + 4ω 2p ω c ). (2.49) The wavenumbers for the L and R-wave are not equal. This indicates that the L and R-waves experience different phase shifts while traversing the plasma, and it is of interest to calculate the total difference in phase accumulated by the beams after traversing the plasma. Up until now, it has been assumed that the plasma is homogeneous. However, this is not the case in reality and it is not immediately clear, for example, if reflections should be considered due to changes in the plasma s index of refraction. However, if the plasma is approximately homogeneous over a distance of the beam s wavelength, then the total phase shift accumulated through the plasma is simply the integral of the wavenumber with respect to the beam s path length. This is known as the Wentzel Kramers Brillouin Jeffreys (WKBJ) approximation [35, 45]. The condition required for this simplification is that the change in the plasma s index of refraction (ie the wavenumber of the beam) is small over a wavelength of the beam. More precisely, the requirement is 1 k 2 (z) k z 1. (2.50)

53 2.5. MAGNETIC FIELD PARALLEL TO WAVE PROPAGATION DIRECTION AND FARADAY ROTATION 28 This simplification allows the L and R-wave phase shifts to be written as φ L = φ R = k L dz, (2.51) k R dz. (2.52) Faraday rotation is defined as half the phase difference between the left and rightcircularly polarized beams: φ F = 1 2 (φ L φ R ). (2.53) Since the superposition of a left and a right-circularly polarized beam is equivalent to a linearly polarized beam, Faraday rotation can also be interpreted as the angle through which a linearly polarized beam s polarization plane rotates (see Figure 2.2). In the Jones representation [29], the superposition of L and R-waves is E = 1 2 = 1 2 E Lx 1 i E Ly E Rx E Ry e i(φl ωt) (2.54) 1 i e i(φ R ωt) (2.55) where E Lx, E Ly are the x, y-components of the L-wave s electric field, and E Rx, E Ry are the x, y-components of the R-wave s electric field. After some manipulation, it can be shown that (see Appendix A.3 for details) E = 2 e i φ L +φ R 2 2 cos φ F sin φ F e iωt. (2.56)

54 2.5. MAGNETIC FIELD PARALLEL TO WAVE PROPAGATION DIRECTION AND FARADAY ROTATION 29 Φ F E Plasma Figure 2.2: Faraday rotation interpreted as the rotation of the polarization plane of a linearly polarized beam of light after passing through a magnetized plasma. Therefore, a change in the phase difference between the L and R-waves, φ L φ R = 2φ F, is equivalent to a rotation of a linearly polarized wave s polarization plane by an angle φ F. It is now of interest to obtain a formula for the cumulative Faraday rotation incurred by a beam passing through a magnetized plasma. The amount of Faraday rotation is φ F = 1 2 (k L k R ) dz. (2.57) Substituting in Equations 2.44 and 2.45 gives φ F = ω 2c ( 1 ω 2 p ω (ω + ω c ) 1 ω 2 p ω (ω ω c ) ) dz. (2.58) It is possible to simplify the integrand with a Maclaurin series 1 + x = x 1 8 x x3... (2.59)

55 2.6. ARBITRARY MAGNETIC FIELD DIRECTION 30 which converges for x 1. It is assumed that ω c ω, which gives ω 2 p ω (ω ± ω c ) ω2 p 1. (2.60) ω2 Therefore, to first order, the integral simplifies to φ F = 1 ( ω 2 ) p ω2 p dz. (2.61) 4c ω ω c ω + ω c Combining the two fractions with a common denominator gives φ F = 1 2c ( ω 2 p ω c ω 2 ω 2 c ) dz. (2.62) The assumption that ω c ω is used again, and the equation is written in its conventional form with n n 0 and B B 0 : e 3 λ 2 0 φ F [rad] = 8π 2 c 3 m 2 ɛ λ 2 0 nbdz nbdz. (2.63) where λ 0 is the wavelength in free space and the units are SI. If this rotation can be measured, then information on the inner plasma magnetic field parallel to the beam can be obtained. This result is the basis of plasma polarimetry. 2.6 Arbitrary Magnetic Field Direction The Faraday rotation result of Equation 2.63 might not be accurate if the magnetic field is not parallel to the beam. For an arbitrary magnetic field direction, the

56 2.6. ARBITRARY MAGNETIC FIELD DIRECTION 31 Cotton-Mouton effect, which affects beam ellipticity, might be significant and must be considered [14, 26]. The change in ellipticity is defined as the difference in phase between the ordinary and extraordinary waves accumulated over the path through the plasma. Recalling Equations 2.35 and 2.36, the change in ellipticity is ɛ = ω c [ 1 X 1 X(1 X) 1 X Y 2 ] dz. (2.64) where Y = ω c /ω, ω c = eb /m and B = B sin θ. It is again assumed that ω ω p, so X 1. Therefore, the square roots can be Taylor expanded and the equation approximated to ɛ ω 2c [ ] X(1 X) X + dz. (2.65) 1 X Y 2 Rearranging over a common denominator gives ɛ = ω 2c ( XY 2 1 X Y 2 ) dz. (2.66) It is also assumed that ω ω c so Y 1. Applying these assumptions gives e 4 λ 3 0 ɛ [rad] nb 16π 3 ɛ 0 m 3 c dz λ 3 0 nb dz 2 (2.67) The units are SI. If this quantity is small, then the Cotton-Mouton effect can be ignored. Otherwise, further analysis must be done to find a more complete description of the polarization state s evolution in a plasma [21].

57 32 Chapter 3 Polarimetry A polarimeter can give information on a plasma s internal magnetic field by measuring the Faraday rotation phase shift incurred by a beam of light passing through the plasma. This chapter details the theoretical considerations of the PI polarimeter, a three-beam heterodyning configuration that is capable of measuring line-averaged density and Faraday rotation along the polarimeter s chord. Polarimeter design details can be found in Chapter Amplitude-Based Polarimeters A simple amplitude-based polarimeter is shown in Figure 3.1. Here, vertically polarized light is sent through a magnetized plasma. The light undergoes Faraday rotation in the plasma, causing the polarization plane of the incident light to rotate. Therefore, upon exiting the plasma, the light s electric field component in the vertical direction is reduced, having been transferred to the horizontal component. The light then passes through a polarizer, which blocks light polarized in the horizontal direction. The intensity of the remaining light is measured with a square-law detector, which produces a signal proportional to the square of the electric field s magnitude E 2 = EE. As

58 3.2. PHASE-BASED POLARIMETERS 33 E k Plasma P D Figure 3.1: Amplitude-based polarimeter where a vertically polarized beam enters a magnetized plasma, which Faraday rotates the polarization plane. The amount of rotation can be determined by monitoring the change in detected intensity after the vertical polarizer (P). The reduction in the vertical component of the electric field is exaggerated in this example. a result, the detector observes a change in the light s intensity as the plasma rotates its polarization plane. This change in intensity is related to the amount of Faraday rotation. Amplitude-based measurements typically must contend with large sources of noise, arising from fluctuations in the laser power and refraction of the beam in dense plasmas, for example. In addition, the Faraday rotation signal is usually chosen to be small to avoid complications with the Cotton-Mouton effect (see Section 2.6). This can make the signal to noise ratio unacceptably low. 3.2 Phase-Based Polarimeters An alternate approach that is more robust to noise is phase-based polarimetry. In this case, the Faraday rotation signal is embedded in the phase of the light beam and not the amplitude. This can be accomplished by modulating the polarization state of the incident light. A two-beam system that can measure Faraday rotation only is first investigated. This is followed by a three-beam system that can measure both Faraday rotation and line-averaged density.

59 3.2. PHASE-BASED POLARIMETERS Two-Beam Heterodyne Polarimeter Consider two circularly polarized beams of opposite handedness that are superimposed and are travelling in the same direction as shown in Figure 3.2. The left and right-circularly polarized beams have respective electric fields E L and E R that are oscillating at respective frequencies ω L and ω R, with amplitudes A L and A R, and initial phases φ L0 and φ R0. In general, the amplitudes can be complex but in this case they are assumed to be real for simplicity. The Jones representation [29] of the electric fields before entering the plasma is E L = 1 2 E R = 1 2 E Lx E Ly E Rx E Ry = A L 2 = A R 2 1 i 1 i e i(φ L0 ω L t) e i(φ R0 ω R t) (3.1) (3.2) where E Lx, E Ly are the x, y-components of E L, and E Rx, E Ry are the x, y-components of E R. Letting A L = A R = 1 and φ L0 = φ R0 = 0, the combined field is (derivation is similar to the one in Appendix A.3) E = 2 2 cos ( ω L ω R t ) 2 sin ( ω L ω R t ) 2 e i( ω L +ω R 2 t). (3.3) The result is a circularly polarized beam oscillating at lower frequency ω L ω R /2 with an amplitude changing sinusoidally at a higher frequency (ω L + ω R )/2. This modulation of the polarization state allows for an effective measurement of the phase difference between the two beams.

60 3.2. PHASE-BASED POLARIMETERS 35 Figure 3.2: A left and right-circularly polarized beam of different frequencies are sent through a magnetized plasma. Each beam is phase shifted by a different amount due to the Faraday effect. After passing through a polarizer (P), the detector (D) records a signal from which a Faraday rotation measurement can be extracted from the signal s phase. When passing through the plasma, the left and right-circularly polarized beams encounter a different index of refraction. Therefore, the two beams have different wavenumbers, and accumulate different phase shifts. After exiting the plasma, the beams pass through a polarizer where the x-component of their electric fields is blocked. In practice, the polarizer is not perfect and some of the x-component is allowed to pass through. Consider the phase shifted beams after exiting the plasma and after passing through the imperfect polarizer. The electric field of the superimposed beams is E = A L 2 δ L i e i(φ L0 ω L t) + A R 2 δ R i e i(φ R0 ω R t) (3.4) where the fraction of the x-component blocked for the L and R-wave is δ L and δ R, which both range from 0 to 1. The signal S observed on the square-law detector is proportional to the magnitude squared of the electric field incident on the detector. The signal is defined as S = E 2 = (E ) T E (3.5)

61 3.2. PHASE-BASED POLARIMETERS 36 where (E ) T is the conjugate transpose of E. This gives S = A2 L 2 (δ2 L + 1) + A2 R 2 (δ2 L + 1) A LA R (δ L δ R 1) ( e i(φ LR(t)+φ LR0 ω LR t) + e ) i(φ LR(t)+φ LR0 ω LR t). (3.6) 2 where ω LR = ω L ω R. Using an identity for the sum of complex exponentials, the signal simplifies to S = A2 L 2 (δ2 L + 1) + A2 R 2 (δ2 L + 1) A LA R (δ L δ R 1) cos(φ LR (t) + φ LR0 ω LR t) (3.7) 2 where φ LR (t) = φ L (t) φ R (t) and φ LR0 = φ L0 φ R0. The observed beat signal oscillates at the difference frequency between the two superimposed beams and contains the desired phase, φ LR (t). This process is known as heterodyning. The beat term has the polarization coefficient (δ L δ R 1), which affects the signal strength. With no polarizer blocking a component of the circularly polarized beams, the polarization coefficient goes to zero and no beat signal is observed since δ L = δ R = 1. The maximum beat signal is obtained when the polarizer works perfectly, that is when δ L = δ R = 0. Therefore, the efficiency of the polarizer affects the signal strength, but does not distort the desired phase. Assuming a perfect polarizer, and using the definition of Faraday rotation (Equation 2.53), the signal is S = 1 2 (A2 L + A 2 R) A L A R cos (2φ F (t) + φ LR0 ω LR t) (3.8) where the definition for Faraday rotation is φ F (t) = φ LR (t)/2. The beat has a phase of

62 3.2. PHASE-BASED POLARIMETERS 37 Figure 3.3: A third, linearly polarized beam that does not pass through the plasma can be used to obtain density information. The additional beam has frequency ω V, and is combined to the circularly polarized beams with mirror, M, and beamsplitter, BS. φ LR0 plus twice the Faraday rotation, which varies in time as the plasma parameters change. The Faraday rotation signal is embedded in the phase of the waveform. The constant phase φ LR0 adds an offset to the desired signal, which can easily be removed. Therefore, for simplicity, initial phases such as φ L0 and φ R0 are set to zero in the sections to follow Three-Beam Heterodyne Polarimeter The two beam, phase-based polarimeter system described in the last section produces a waveform that contains the Faraday rotation signal in its phase. However, in order to obtain information on the inner magnetic field, it is helpful to also have a measurement of density along the same chord since Faraday rotation is the line integral of nb. A three-beam system, as shown in Figure 3.3, can provide this additional information. As in the two beam system, a left and right-circularly polarized beam are combined, sent through the plasma, and passed through a vertical polarizer before reaching the detector. Consider a third beam that is added, which passes around the plasma. This new beam has a vertically polarized electric field E V with amplitude

63 3.2. PHASE-BASED POLARIMETERS 38 A V and frequency ω V. Its Jones representation is E V = E Vx E Vy = A V 0 i e iω Vt (3.9) where it has been defined with a constant phase i = exp(iπ/2) for convenience. Combining the three beams, the signal at the detector is S = 1 2 (A2 L + A 2 R) + A 2 V 1 2 A LA R ( e i(2φ F ω LR t) + e i(2φ F ω LR t) ) A L A V ( e i(φ L ω LV t) + e i(φ L ω LV t) ) (3.10) 1 2 A R A V ( e i(φ R ω RV t) + e i(φ R ω RV t) ). where ω LV = ω L ω V, and ω RV = ω R ω V. Writing the complex exponentials in terms of sines and cosines gives S = 1 2 (A2 L + A 2 R) + A 2 V A L A R cos (2φ F (t) ω LR t) A L A V cos (φ L (t) ω LV t) (3.11) 2 2 A R A V cos (φ R (t) ω RV t). The overall signal contains three beats, each at one of the three possible difference frequencies. The two new beats contain the phase shifts φ L and φ R, which are given by the integral of the wavenumbers (Equation 2.44 and Equation 2.45) over the path

64 3.2. PHASE-BASED POLARIMETERS 39 length through the plasma: φ L = ω c φ R = ω c 1 1 ωp 2 dz, (3.12) ω (ω + ω c ) ωp 2 dz. (3.13) ω (ω ω c ) If it can be assumed that ω c ω then φ n = φ L φ R ω c 1 ω 2 p/ω 2 dz. (3.14) where φ n can be considered as the phase shift due to the plasma density. Expanding as a Taylor series, dropping higher order terms and rewriting with fundamental parameters gives φ n [rad] φ 0 e 2 4πɛ 0 mc λ 2 0 ndz (3.15) where φ 0 is a constant phase, which can be ignored. Substituting in values for the fundamental constants gives φ n [rad] λ 0 ndz. (3.16) The units are SI. Therefore, the phase shifts of the ω LV and ω LR beats provide information on the plasma electron density. The next section describes how to extract these phase shifts from the detected signal with a demodulation algorithm.

65 3.3. SIGNAL DEMODULATION Signal Demodulation The goal is to extract the phase shifts φ LR (t), φ LV (t) and φ RV (t) from the beats of the three-beam polarimeter detector signal. This is done with a demodulation algorithm [50] that requires a plasma signal S pla and a reference signal S ref as shown in Figure 3.4. The plasma signal is the same as for the three-beam polarimeter previously discussed (Equation 3.11). The reference signal is created with the same initial beams as the plasma signal, but the left and right-circularly polarized beams are not sent through the plasma. Therefore, the reference signal does not experience the time-dependent phase shifts due to the plasma-beam interactions. In order to extract the three phases of interest, the algorithm is repeated three times; once for each phase signal. An example of the demodulation procedure for the ω LR beat follows. Both the reference and plasma signals are the sum of three beat terms. Each beat is a cosine oscillating at a different frequency. As shown in Figure 3.5(a), the Fourier transform of either the plasma or reference signal produces a positive and a negative frequency for each of the three beats. A bandpass filter with cut-off frequencies ω LR 2πf c and ω LR + 2πf c is applied to the initial reference and plasma signals to isolate the ω LR beat. The bandwidth chosen, affects the bandwidth of the final extracted phase shift measurement. Ignoring initial constant phases, the filtered signals S ref and S pla are S ref =A ref cos ( ω LR t), (3.17) S pla =A pla cos (φ LR (t) ω LR t) (3.18) where A ref and A pla are respectively the amplitudes of the filtered reference and

66 3.3. SIGNAL DEMODULATION 41 D R Figure 3.4: An additional reference signal is needed to extract the time-dependent phase shifts from the plasma signal. The plasma signal is the usual signal from the three-beam system that interacts with the plasma. The reference signal uses the same initial beams, but they do not interact with the plasma. The plasma and reference signals are respectively recorded by detectors D P and D R. plasma signals. Rewriting in terms of complex exponentials gives S ref =A ref e iω LRt + A ref e iω LRt, (3.19) S pla =A pla e i(φ LR(t) ω LR t) + A pla e i(φ LR(t) ω LR t). (3.20) Each of these complex exponentials corresponds to a spike in frequency space, as shown in Figure 3.5(b). The negative frequencies are eliminated from the filtered reference signal, which amounts to dropping the second term on the right side of Equation 3.19 if ω LR > 0 (see Figure 3.5(c)). This modified reference signal, S ref, is known as the analytic

67 3.3. SIGNAL DEMODULATION 42 -ω RV -ω LV -ω LR ω LR ω LV ω RV (a) ω -ω LR ω LR ω (b) ω LR ω ω=0 2ω LR ω (c) (d) ω=0 ω (e) Figure 3.5: Demodulation example for the ω LR beat. Horizontal and vertical axes are respectively frequency and magnitude. (a) The Fourier transform of the initial plasma and reference signals. (b) A bandpass filter is applied to the plasma and reference signals to extract the ω LR beat. (c) The Hilbert transform is applied to the filtered reference signal. The negative frequencies are eliminated. (d) The plasma beat signal is multiplied by the modified reference signal, which shifts both frequency spikes up by ω LR. (e) The residual 2ω LR spike is filtered out.

68 3.3. SIGNAL DEMODULATION 43 signal and can be obtained by taking the Hilbert transform of S ref. The analytic reference signal is S ref = A ref e i sgn(ω LR) ω LR t. (3.21) where sgn(ω LR ) is the signum function, which returns the sign of its argument. This allows for proper handling of the case where ω LR < 0. The filtered plasma signal, is then multiplied by the analytic reference signal. As shown in Figure 3.5(d), this shifts the plasma signal s beat frequency spikes by ω LR. The new signal S d has frequency spikes at ω = 0 and 2ω LR : S d = A pla A ref e i sgn(ω LR) φ LR (t) + A pla A ref e i sgn(ω LR) (φ LR (t) 2ω LR t). (3.22) A low-pass filter with cut-off frequency f c is applied to remove the 2ω LR spike (see Figure 3.5(e)) giving S d =A pla A ref e i sgn(ω LR) φ LR (t) (3.23) =A pla A ref cos [sgn(ω LR ) φ LR (t)] + ia pla A ref sin [sgn(ω LR ) φ LR (t)]. The ω LR beat s phase shift is then ( I{S d φ LR (t) = sgn(ω LR ) arctan } ) R{S d } (3.24) where the signal has a bandwidth f c, and R and I respectively represent the real and imaginary parts of their arguments. The φ LR phase shift gives the Faraday rotation measurement since φ F = 1φ 2 LR. The same procedure can be carried out to determine the φ L and φ R phase shifts to obtain density measurements.

69 3.4. BEAM ELLIPTICITY Beam Ellipticity Ideally, the plasma beams are perfectly circularly polarized before entering the plasma. However, limitations of the optics introduce ellipticity into the two plasma beams, which can generate errors in the phase measurements unless a more careful analysis is taken. To investigate this in further detail, the usual three-beam electric field superposition is slightly modified: E = A L 2 ɛ L i e iω Lt + A R 2 ɛ R i e iω Rt + A V 0 i e iω Vt (3.25) where the ellipticities of the L and R-wave are respectively ɛ L and ɛ R, which are complex numbers in general but are assumed to be real in this simplified exercise. It is not immediately clear how the plasma affects these initially elliptically polarized beams, since the derived plasma phase shifts, φ L and φ R, only apply for circularly polarized waves. Rewriting in terms of left and right-circularly polarized components gives E = A L 2 + A R 2 +A V 1 i 1 i 0 i + ɛ L ɛ R 1 2 e iω Vt. 1 i 1 i i 1 i e iω Lt e iω Rt (3.26)

70 3.4. BEAM ELLIPTICITY 45 Now, left and right-circularly polarized terms obtain phase shifts of φ L (t) and φ R (t) due to interactions with the plasma. Upon exiting the plasma and passing through the vertical polarizer, the resulting electric field is E =i A L 2 [ (ɛl + 1)e iφ L (ɛ L 1)e R] iφ e iω L t 2 +i A R 2 [ (ɛr 1)e iφ L (ɛ R + 1)e R] iφ e iω R t 2 +ia V e iω Vt. (3.27) The algebra to compute the signal S = E E is straightforward but lengthy. The detailed expression is in Appendix A.4. Using the demodulation algorithm of Section 3.3, the relationship between the filtered, downshifted beat signal and the desired phase shift for the ω LR beat is for the ω LV beat is ( I{S d } ) R{S d } ( I{S d } ) R{S d } and for the ω RV beat is ( I{S d } ) R{S d } LV RV LR (ɛ L + ɛ R ) sin φ LR = sgn(ω LR ), (3.28) 1 ɛ L ɛ R + (ɛ L ɛ R + 1) cos φ LR = sgn(ω LV ) (ɛ L + 1) sin φ L (ɛ L 1) sin φ R (ɛ L + 1) cos φ L (ɛ L 1) cos φ R, (3.29) = sgn(ω RV ) (ɛ R 1) sin φ L (ɛ R + 1) sin φ R (ɛ R 1) cos φ L (ɛ R + 1) cos φ R. (3.30) The demodulated signals are now more complicated functions of the phase shifts and the ellipticities. To properly extract the phase shifts, the ellipticities must be known.

71 3.5. BEAM COLLINEARITY 46 A calibration curve could be generated by passing the elliptically polarized beams through a spinning λ/2-waveplate [23]. The spinning λ/2-waveplate mimics the plasma and generates the response of the phase shift for the given beam ellipticities. The response is linear for circularly-polarized beams. To see this, consider the Jones matrix for a λ/2-waveplate with fast axis at an angle θ λ/2 with respect to the horizontal axis. Applied to the two-beam case for simplicity, this gives E = cos(2θ λ/2) sin(2θ λ/2 ) A L sin(2θ λ/2 ) cos(2θ λ/2 ) 2 = A L 2 1 i e i(2θ λ/2 ω L t) + A R 2 1 i 1 i e iω Lt + A R 2 e i( 2θ λ/2 ω R t) 1 i e iω Rt (3.31) (3.32) The phase difference between the two beams is 4θ λ/2. Therefore, applying the demodulation algorithm, the observed phase shift for the two circularly polarized beams passing through a λ/2-waveplate tracks the rotation angle of the waveplate. The relationship is linear and goes through a full cycle for every quarter rotation of the waveplate. If a non-linear response is observed, at least one beam has some ellipticity as shown in Figure 3.6. If the non-linear response is known, it can be used to generate calibration factors to correct measured phase shifts. 3.5 Beam Collinearity Another source of error is related to the collinearity of the left and right-circularly polarized beams (see Figure 3.7). Consider two non-collinear beams that travel through slightly different paths in a plasma. The effect of the plasma density gradient causes

72 3.5. BEAM COLLINEARITY Phase Difference (degrees) ɛ L =1 ɛ L =0.1 ɛ L = Waveplate Angle, θ λ/2 (degrees) Figure 3.6: Simulated phase shift response generated by a rotating λ/2-waveplate for three test ellipticities where ɛ R = 1 and ɛ L varies. When the ellipticity is greater or less than 1, the response is non-linear. ω L n 0 ω L n 0 ω R Plasma ω R Plasma (a) (b) Figure 3.7: The collinearity of the left and right-circularly polarized polarimeter beams has an effect on the Faraday rotation measurement. (a) Near collinear beams give low error. (b) Non-collinear (exaggerated in this figure) beams can introduce a significant error. one beam to accumulate significantly more phase shift than the other. This additional phase shift can alter the Faraday rotation phase shift measurement. In this way, beam collinearity can be a source of error.

73 3.5. BEAM COLLINEARITY Collinearity Error Theory To better understand the conditions that make these effects significant, consider once again the definition of the Faraday rotation phase shift φ F = 1 (kl k 2 R ) dz. As previously shown, this can be written in terms of ω p and ω c, which respectively depend on density and magnetic field. If the beams are not collinear, then the L and R-waves respectively encounter a slightly different density n L and n R, and parallel magnetic field B L and B R. Therefore, Equation 2.61 transforms into the modified Faraday rotation φ F = 1 ( ω 2 ) pl ω2 pr dz (3.33) 4c ω ω cl ω + ω cr where the L and R-waves respectively encounter plasma frequencies ω pl and ω pr, and gyrofrequencies ω cl and ω cr. It is still assumed that ω = ω L ω R and that the path lengths of the two beams are approximately equal. Combining the two fractions gives φ F = 1 ( ω ( ωpl 2 ) ω2 pr + ω 2 pl ω cr + ωpr 2 ω cl 4c (ω ω cl ) (ω + ω cr ) ) dz. (3.34) The denominator can be simplified by assuming ω ω cr, ω cl. Substituting in fundamental parameters gives φ F e2 λ 0 8πɛ 0 mc 2 (n L n R ) dz + e 3 λ 2 (nl ) 0 B 16π 2 ɛ 0 m 2 c 3 R + n R B L dz. (3.35) The first integral is the contribution of the collinearity error. The second integral is approximately equal to the usual φ F since the differences between the densities and

74 3.5. BEAM COLLINEARITY 49 fields are small along the two paths: ( φ F = e2 λ 0 n + eλ ) 0 8πɛ 0 mc 2 πmc nb dz. (3.36) where n = n L n R. Therefore, the collinearity error is small at a particular point on the beam path when C = λ 0 nb n 1 (3.37) where B is the magnitude of the parallel field, and all units are in SI Rotating Wedge Calibration Device To calibrate out collinearity error, the beams can be more precisely aligned by passing them through a rotating wedge [59]. Consider a wedge with index of refraction n w, radius R, base thickness d 0, and wedge angle α as shown in Figure 3.8. A beam enters perpendicular to the bottom (non-wedged) surface at a point (ρ, φ), where ρ is the radial distance and φ the azimuth angle. The distance traversed by the beam through the wedge is d w = d 0 + (R + ρ cos θ) tan α. (3.38) This leads to a phase shift due to the wedge of φ w = 2πn w λ 0 d w. (3.39)

75 3.5. BEAM COLLINEARITY 50 The difference in the phase shift between two parallel, but non-collinear beams with respective entrance coordinates (ρ L, θ L ) and (ρ R, θ R ) is then φ LR = 2πn w tan α λ 0 (ρ L cos θ L ρ R cos θ R ). (3.40) If the wedge rotates at a angular speed ω w, then the phase difference as a function of time can be written as (see Appendix A.5 for details) φ LR (t) = 2πn w tan α λ 0 ρ cos(ω w t + θ ) (3.41) where ρ = [ ρ 2 L + ρ 2 R 2ρ L ρ R cos(θ L θ R ) ] 1/2 (3.42) and ( ) θ ρl sin θ L ρ R sin θ R = arctan. (3.43) ρ L cos θ L ρ R cos θ R As the wedge rotates, the difference in phase between the two non-collinear beams changes and can be measured using the usual demodulation algorithm. To improve the collinearity, the beams must be aligned to reduce the measured phase difference to zero at all times Iris for Overlapping Beams In a previous attempt to eliminate collinearity noise, the combined ω L and ω R beams were sent through an iris (ie adjustable aperture). The iris aperture diameter was set to about half the beam diameter. The purpose was to clip the area of the beams that did not overlap. Therefore, beam power was lost but the beams emerging from

76 3.5. BEAM COLLINEARITY 51 SIDE VIEW P high P low R tanα α d 0 R TOP VIEW θ P high P low R ρ Figure 3.8: Side and top views of wedge used to calibrate out beam collinearity error. The highest and lowest point on the wedge are respectively labelled P high and P low. the iris would be overlapping. However, the beams are not collinear in the sense that would eliminate collinearity noise. As shown in Figure 3.9. The intensity peaks of their radial Gaussian distributions do not coincide, and the phase shifts obtained by each beam are weighted towards different regions in space. Therefore, passing the ω L and ω R beams through an iris to create purely overlapping beams would not prevent collinearity noise.

77 3.6. GAUSSIAN BEAM PROPAGATION 52 Iris Beam 2 peak Beam 2 Beam 1 Beam 1 peak Figure 3.9: Iris can make beams purely overlap, but not collinear since beam intensity peaks are still offset. 3.6 Gaussian Beam Propagation The plasma beams must travel from the laser, through several conditioning optics, through the plasma and then to the detector. to detector is on the order of several metres. The total path length from laser The beam expands as it leaves the laser due to diffraction. It must remain small enough to pass through optical pieces without clipping the beam and wasting beam power. The beam diameter is controlled with lenses, which either focus or expand the beam by refracting the incident light by different lengths in the lens medium. It is helpful to predict the approximate diameter of the beam at various points along the path. This can be done by modelling the laser beam as a Gaussian beam [72]. Immediately upon exiting the laser, the beam is at its narrowest radius also known

78 3.6. GAUSSIAN BEAM PROPAGATION 53 as the beam s waist, W 0. The radial intensity profile of the beam is Gaussian, as shown in Figure 3.10(a), with 99% of the power located within a circle of radius 1.5W 0. The normalized intensity is I = W ( ) 0 W (z) exp ρ 2 2 W 2 (z) (3.44) where ρ is the radial distance from the beam axis, and W (z) is the radius of the beam a distance z from the waist. As the Gaussian beam propagates away, its radius expands according to W (z) = W (z/z0 ) 2 (3.45) where z 0 = πw 2 0 λ (3.46) is the Rayleigh range, which is the distance after which the radius expands to 2W 0. Far from the waist, the beam diverges approximately at an angle θ 0 = λ πw 0 (3.47) as shown in Figure 3.10(b). In order to stay collimated over long distances, it may be necessary to focus the beam with lenses. The effect of a lens on a Gaussian beam is to focus it to a new waist W 0 = W (z0 /f) 2 (3.48) at a distance away from the lens of z = f 1 + (f/z 0 ) 2. (3.49)

79 3.6. GAUSSIAN BEAM PROPAGATION Normalised Intensity, I Normalised Radial Distance, ρ/w 0 3 (a) Normalised Beam Radius, W/W θ Normalised Axial Distance From Waist, z/z 0 θ 0 (b) Figure 3.10: Gaussian beam profiles. (a) Radial profile shows 99% of beam power is within circle of radius 1.5W 0. (b) Axial profile shows that beam radius expands to 2W 0 at an axial distance of z 0 from the beam waist, and diverges at an angle θ 0 when z z 0.

80 3.7. BEAM REFRACTION 55 The new Rayleigh range, z 0, and divergence, θ 0, follow from these equations. In practice, laser beams are not perfectly Gaussian. A measure of beam quality is the M 2 parameter defined for a specific wavelength as M 2 = θ 0 W0 θ 0 W 0 (3.50) where the actual divergence and waist are respectively θ 0 and W 0. Therefore, using the fact that θ 0 W 0 = λ/π for a Gaussian beam (see Equation 3.47), the relationsihp between the actual divergence and waist is θ 0 = M 2 λ. (3.51) π W 0 The M 2 value of a laser provides a correction to the Gaussian beam approximation. Computer programs based on these equations can be used to quickly estimate the beam size and divergence angle at any given location on the beam s path [87]. 3.7 Beam Refraction Density gradients in the plasma can cause the incident beams to refract [11, 45]. Consider two rays A and B initially travelling in the z-direction and offset in the y-direction (ie the transverse direction) by y as shown in Figure Both rays pass through a small plasma slab of width dz. Rays A and B respectively encounter a refractive index of N(z, y + y) and N(z, y) where it is assumed that N(z, y + y) > N(z, y). Examining points P 0 and Q 0 immediately after the slab, the difference in

81 3.7. BEAM REFRACTION 56 the phase of the two rays is dφ = k 0 N(z, y + y)dz k 0 N(z, y)dz. (3.52) A difference in phase indicates that the wavefronts have been refracted with angle of refraction θ R. The points P 0 and Q 1 are on the same wavefront. Therefore, their phases must be equal. The phase difference between points Q 1 and Q 0 is dφ = N(z)k 0 y tan θ R k 0 yθ R (3.53) where it has been assumed that θ R is small, and N(z) 1 since ω ω p. Substituting Equation 3.53 into Equation 3.52 gives θ R = N(z, y + y)dz N(z, y)dz. (3.54) y This is in the form of the definition of a partial derivative. Letting y 0, the result is θ R = y N(z, y)dz. (3.55) Substituting in fundamental parameters and assuming ω ω p gives θ R e2 λ 2 0 n(z, y) dz 8π 2 ɛ 0 mc 2 y n(z, y) λ 2 0 dz. (3.56) y

82 3.8. INTERFERENCE PHASE CONTRAST 57 Plasma slab P 0 Ray A N(z,y+ y) θ R y θ R Q 1 Ray B Wavefronts N(z,y) dz Q 0 ysinθ Figure 3.11: Ray A and B each encounter a different refractive index due to the plasma s density gradient. Therefore, each ray incurs a different phase shift and the wavefront is refracted. The angle of refraction, θ R, is exaggerated for clarity. The units are SI. Therefore, plasma density gradients transverse to the propagation direction can lead to deflection of the beam. This may be a concern for beam-probing diagnostics such as a polarimeter. 3.8 Interference Phase Contrast A laser beam can be pictured as being made up of infinitely thin rays that are distributed across the beam spot like a Gaussian with the highest concentration at the centre of the beam. When two beams interfere at a detector, the total signal is the sum of the small signals produced from individual rays interfering with one another across the detection plane. In the ideal scenario of perfect phase contrast, each interference between rays produces the same small signal across the entire detector. When this is not the case, the phase contrast is not perfect and the overall signal strength decreases [45].

83 3.8. INTERFERENCE PHASE CONTRAST 58 Detection Plane P1 y P2 0π 2π 4π 0π 2π 4π B1 Figure 3.12: Phase contrast diminished due to angle between two interfering beams. Rays at points P1 and P2 do not produce the same interference due to phase differences. θ B Beam Alignment Consider the interference of two beams, B1 and B2, at a detector. The difference in frequency between the two beams is ω. As shown in Figure 3.12, beam B1 s wavefronts are parallel to the detector but B2 s wavefronts are at a slight angle θ. Therefore, the interference of rays at different points on the detector produces a different signal. The two rays at point P1 are in phase and constructively interfere. However, the two rays at point P2 differ in phase and so produce a different interference. The result of imperfect phase contrast is that the alternating part of the overall detector signal is reduced. Consider once again the example of the interference of two beams at slight angle θ to one another. The beams are initially assumed to have a uniform power distribution (ie as opposed to Gaussian). Beam B1 s wavefront is parallel to the detector. Therefore its phase is constant across the detector plane and is taken to be zero. A ray from beam B2 at the detector plane and at a position y

84 3.8. INTERFERENCE PHASE CONTRAST 59 from the beam centre has phase φ(y) = 2πyθ/λ 0 (3.57) when θ is small and the phase of B2 s central ray at the detector is set to zero. In general, as long as the beams trajectories remain the same, the phase difference between a B1 and a B2 ray at position y is φ(y). The small detector signal produced by the interference of these two rays is (see Section 3.2.1) s(y, t) = cos( ωt + φ(y)) (3.58) where the amplitude of the AC component is set to 1 and the constant offset is ignored. The total signal generated from all rays at the detector is S(t) = y s(y, t). (3.59) It can be shown that the alternating signal goes to zero when φ(±w ) = π where W is the beam radius. Figure 3.13 shows this with a simplified simulation. Rays at ten equally-spaced points on a detector are interfered and the resulting signals summed. The overall signal is shown for values of φ(w ). Perfect phase contrast for a beam angle of θ = 0 gives the largest signal since the ray interference signals, s(y, t), are all in phase and add constructively. As the angle is increased, the ray signals are increasingly phase shifted and spread out. Once the φ(±w ) = π condition is met, the ray signals are spread out over an entire period and their sum produces no

85 3.8. INTERFERENCE PHASE CONTRAST 60 alternating component. Therefore, it is required that φ(w ) < π (3.60) Substituting in Equation 3.57 gives the minimum requirement for phase contrast: θ < λ 0 2W. (3.61) For a beam with a Gaussian profile, the phase contrast condition of Equation 3.60 is relaxed somewhat since ray signals further from the beam centre are weighted less heavily. Simulation results comparing the uniform and Gaussian intensity cases are shown in Figure The minimum angular alignment requirement to have any phase contrast for Gaussian beams is φ(w ) < 5 rad (3.62) θ < 5λ 0 2πW. (3.63) Beam Coherence Length Ideally, a laser generates a beam with a single frequency. In practice, the beam s linewidth ν is finite although typically small. In this sense, the beam can be pictured as a distribution of rays initially in phase but with slightly different frequencies. Therefore, as the beam propagates, its wavefronts become increasingly distorted since the phases of the various rays increase at different rates. If two beams with a large

86 3.8. INTERFERENCE PHASE CONTRAST 61 Intensity (arbitrary units) Overall signal Ray signals Intensity (arbitrary units) Overall signal Ray signals Time (arbitrary units) Time (arbitrary units) (a) (b) Intensity (arbitrary units) Overall signal Ray signals Intensity (arbitrary units) Overall signal Ray signals Time (arbitrary units) Time (arbitrary units) (c) (d) Figure 3.13: Simulation showing phase contrast effects of two beams interfering. Beams are assumed to have uniform power distributions. Ten equallyspaced rays are used. Results are presented for four values of φ(w ), the phase difference between the central and outer rays: (a) φ(w ) = 0, (b) φ(w ) = π/3, (c) φ(w ) = 2π/3, (d) φ(w ) = π. The overall ray signal for the case of φ(w ) = π is not exactly zero due to the small number of rays used in this simplified example.

87 3.8. INTERFERENCE PHASE CONTRAST 62 Normalized Signal Magnitude (arb.) Uniform intensity Gaussian intensity Inner-Outer Ray Phase Difference, φ(w) (rad) Figure 3.14: Simulation showing relationship between inner-outer ray phase difference, φ(w ), for uniform and Gaussian beam intensity profiles. A hundred equally-spaced rays are used along the detector. path difference are interfered, this wavefront distortion can lead to a low interference signal. After a distance L, the phase difference between a ray at the central frequency ν 0 and a ray at ν 0 + ν is φ = 2π νl c /c. (3.64) The distance over which this effect can lead to considerable phase contrast loss is called the coherence length L c. It is approximated as the distance when φ = π, giving L c = c 2 ν. (3.65) In order for this undesirable effect to be small, the path difference between two interfering beams must be much less than the coherence length.

88 3.9. ACOUSTO-OPTIC MODULATOR Acousto-Optic Modulator A crucial aspect of a heterodyne plasma polarimeter is the frequency shifting of the initial laser light. This allows for the interference of beams and the creation of beat signals that carry the phase shifts of interest. The magnitude of the frequency shift is important, since this spreads out the beats in frequency space (Figure 3.5(a)). The further apart the spikes, the greater the maximum possible bandwidth of the phase measurement. One method of frequency shifting a beam of light is to reflect the light off a rotating diffraction grating [24, 76]. The motion of the grating Doppler shifts the incident light. The amount of frequency shift is limited by the rotation speed of the grating [24]. Another option is to use multiple lasers that are de-tuned compared to one another by a few MHz [23]. Alternatively, an acousto-optic modulator (AOM) can be used, which Doppler shifts light by reflecting it off acoustic waves [1]. The frequency shift can be in the tens of MHz. AOMs are used in this experiment. The theory behind an AOM is now briefly described. A simplified picture of the AOM is shown in Figure Inside the AOM, a transducer is coupled to a crystal with index of refraction N. The transducer generates acoustic waves inside the crystal with speed v A = λ A ω A /2π where λ A and ω A are respectively the wavelength and angular frequency of the acoustic waves in the crystal. An absorber is placed on the opposite end of the crystal to minimize reflection of the acoustic waves inside the AOM. The acoustic waves create density variations within the crystal, which locally alters the crystal s index of refraction. Consider a simple example where two rays from an incident light beam enter the AOM at an angle θ as in Figure The light beam has free-space wavelength

89 3.9. ACOUSTO-OPTIC MODULATOR 64 Incident Beam Absorber ω B +ω A ω B Sound waves θ θ 1 st order 0 th order Transducer ω B Figure 3.15: Acousto-optic modulator Doppler-shifts an incident laser beam with frequency ω B by the acoustic wave frequency ω A. λ B and angular frequency ω B. The rays of light are reflected off the moving acoustic wave fronts as a result of the variations in the refractive index, acting like a sinusoidal diffraction grating [20]. In Figure 3.16, ray 2 has travelled a distance 2d = 2λ A sin θ further than ray 1. This generates a phase difference of φ = 2d 2π λ B /N = 4πλ A sin θ λ B /N (3.66) where λ B /N is the wavelength of the light in the crystal. The two rays constructively interfere when the phase difference is equal to 2πm, where here m = 0, 1, 2, 3,... represents the order number of the constructive interference spot. This gives 2λ A sin θ = mλ B /N. (3.67) The reflected rays of light are Doppler shifted by the moving acoustic waves. The

90 3.9. ACOUSTO-OPTIC MODULATOR 65 1 θ θ 2 λ A θ d Figure 3.16: Wave fronts of light, initially in phase, enter an acousto-optic modulator and reflect off acoustic waves producing a diffraction pattern. The moving acoustic waves Doppler shift the frequency of the diffracted light. d θ v A change in frequency due to the Doppler effect is ω B = 2ω B v A sin θ c/n = 2λ Aω A sin θ λ B /N (3.68) where v A sin θ is the velocity component of the acoustic wave in the direction of the light ray, and the velocity of the light beam in the crystal is c/n = ω B λ B /2πN. The factor of two is needed since the moving acoustic waves observe the light at a frequency ω B + ω B, and then re-emit the light at a frequency ω B + 2 ω B according to an observer in the lab frame. Combining Equation 3.67 and Equation 3.68 gives ω B = mω A. (3.69) AOMs can be designed with a high diffraction efficiency for the first order spot (ie m = 1). The first order spot would appear to be scattered at an angle θ, known as the Bragg angle, with a frequency shift equal to the acoustic wave frequency. In the example presented, the light is frequency up-shifted. However, it is also possible to down-shift the frequency by reflecting off the acoustic waves in the opposite direction.

91 66 Chapter 4 Spheromaks and Plasma Injectors A key requirement of most fusion schemes is for the plasma to be confined for an extended period of time. Without confinement, ions and electrons quickly diffuse outwards where they recombine and conduct heat to the walls. This cools the plasma and lowers the chance of a fusion reaction occurring. In a spheromak, diffusion can be significantly slowed compared to an unmagnetized plasma. The state of a spheromak is strongly affected by its magnetic fields. A polarimeter would be a useful diagnostic to learn more about these important fields. This chapter gives an overview of spheromak theory [6, 48, 49] relevant to the design of the PI machines. 4.1 Helicity and Minimum Energy States Magnetic helicity [8, 7] is a measure of the twistedness and interconnectedness of magnetic flux tubes. A flux tube is a bundle of magnetic field with constant magnetic flux through any cross section of the tube. Helicity is described mathematically by the volume integral K = A Bd 3 r (4.1) V

92 4.2. SPHEROMAK MAGNETIC GEOMETRY 67 where K is the helicity, A is the magnetic vector potential, and B is the magnetic field. Plasma instabilities and magnetic reconnection can cause the rapid loss of magnetic energy. However, helicity dissipates only as a result of resistive effects and so is roughly conserved in strongly conducting plasmas. A magnetized plasma with some initial helicity is driven by plasma flows to relax to a minimum energy state that is characterized by the helicity and the geometry of the conducting container [82]. For conserved helicity and magnetic flux, the magnetic field of the minimum energy state satisfies [49] B = λb (4.2) where λ depends on the geometry of the system. When λ is uniform, this state is known as a Taylor state. 4.2 Spheromak Magnetic Geometry For magnetostatics (ie currents are constant), Ampere s circuital law is B = µ 0 j. (4.3) where j is the current density, and µ 0 the permeability of free space. Substituting this into Equation 4.2 gives j = λb µ 0 (4.4) Therefore, the spheromak s internal currents are parallel to its magnetic field lines. Charged particles on closed field lines are contained within the spheromak, making it

93 4.2. SPHEROMAK MAGNETIC GEOMETRY 68 possible for temperature to be maintained for a longer time. A spheromak is also disconnected from external magnetic sources. Its internal currents generate its internal magnetic fields. This is unlike other magnetic fusion confinement schemes like the tokamak, which depend on external coils for their stability. Spheromaks can even maintain stability during strong acceleration by a magnetic force [34]. For a perfect Taylor state, field lines wrap around infinitely thin, nested, toroidal flux surfaces. It may be possible for fluctuations to cause field lines to wander away from their prescribed flux surfaces. These are known as stochastic field lines. Stochastic field lines might still be closed and not intersect the wall. Nonetheless, they can potentially transport particles from the hot plasma core to the colder edge. The spheromak s magnetic field is commonly referred to in terms of its toroidal, B φ, and poloidal, B θ components. As shown in Figure 4.1(a), the toroidal field is in the direction of the azimuth, while the poloidal field encircles the toroidal field lines and is the sum of the radial and axial components: B θ = B ρ + B z. The maximum B φ and B θ are generally equal but distributed differently across the spheromak to give a total field path that is helical as seen in Figure 4.1(b). The field at the wall is mainly poloidal. The field at the poloidal magnetic axis, which in coaxial geometry is typically near the midpoint between the two electrodes, is mainly toroidal.

94 4.2. SPHEROMAK MAGNETIC GEOMETRY 69 Bθ BΦ (a) (b) Figure 4.1: Spheromak magnetic structure. (a) Poloidal, Bθ, and toroidal, Bφ, field components. [81]. (b) Spheromak magnetic field is mainly toroidal near centre and poloidal at edges [49]. For a coaxial conducting wall and with the assumption of zero axial surface current, the axisymmetric solution [41] to Equation 4.2 gives kz cos [kz (z z0 )] [J1 (kρ ρ) + f Y1 (kρ ρ)] λ kρ Bz = B0 sin [kz (z z0 )] [J0 (kρ ρ) + f Y0 (kρ ρ)] λ Bρ = B0 Bφ = B0 sin [kz (z z0 )] [J1 (kρ ρ) + f Y1 (kρ ρ)] (4.5) (4.6) (4.7) where B0 is a constant magnetic field, Jn and Yn are respectively Bessel functions of the first and second kind of order n, and λ2 = kρ2 + kz2. The parameters kρ, kz and f are constrained by the boundary conditions to enforce Bρ = 0 at the walls and Bz = 0 at the axial edges (ie the front and back) of the spheromak.

95 4.3. SPHEROMAK FORMATION AND ACCELERATION Spheromak Formation and Acceleration A spheromak can be formed in several ways [48]. PI spheromaks are created by coaxial helicity injection (CHI). The PI machine shown in Figure 4.2 is similar to a coaxial rail gun. A simple picture of the formation process is given in Figure 4.3: 1. Gas (e.g. deuterium) is symmetrically puffed into the evacuated formation chamber. A solenoid inside the inner electrode generates the stuffing field, which will later form the spheromak s poloidal field. 2. A large voltage difference is applied between the inner and outer electrodes, and the gas breaks down (ionizes). The newly formed and highly conductive plasma shorts the circuit, allowing a large current to flow between the electrodes. This current loop generates a toroidal magnetic field, which is frozen into the conducting plasma. 3. The toroidal magnetic field and radial current produce a j B force, that pushes the plasma out of the formation section. The stuffing field is frozen into the plasma and is distorted as the plasma pushes it out. The j B force acting on the plasma must overcome the magnetic tension in the increasingly bent stuffing field lines. 4. The stuffing field is distorted to such an extent that it reconnects at the back of the plasma, leaving the plasma with a poloidal field disconnected from external sources. Given a short time (tens of µs) to recover from the instabilities generated in the formation process, the plasma relaxes to a stable spheromak configuration. The spheromak s toroidal current generates the poloidal field,

96 4.3. SPHEROMAK FORMATION AND ACCELERATION 71 Figure 4.2: PI forms and accelerates spheromaks. As it is accelerated down the injector, the spheromak is compressed and increases in density and temperature. and its poloidal current generates the toroidal field. The overall current and magnetic field paths are helical, satisfying Equation 4.4. Once the spheromak has been formed, a second, larger capacitor bank is triggered, which sends more current through the plasma. This increases the j B force and accelerates the plasma down the tube. The conically narrowing electrodes compress the plasma to higher magnetic field, density and temperature. A copper container, known as a flux conserver, can be placed in the target chamber at the end of the acceleration section (see Figure 4.4) to catch the spheromak and maintain it for as long as possible.

97 4.3. SPHEROMAK FORMATION AND ACCELERATION 72 Gas Puff Stuffing Field (a) Plasma Gun Field Current (b) (c) Poloidal Field Toroidal Field (d) Figure 4.3: Spheromak formation by CHI [56]. (a) Gas puffed into chamber. (b) Gas breakdown and toroidal field formed. (c) Lorentz force pushes plasma out. Stuffing field frozen into plasma. (d) Stuffing field reconnects at back of spheromak, forming poloidal field.

98 4.4. λ PROFILE 73 Figure 4.4: Flux conserver placed in the target chamber at the end of the injector. 4.4 λ Profile An important property of the spheromak is the λ profile. For a perfect Taylor state, the λ profile is uniform. In general, spheromaks [7] approach the Taylor state but do not exactly satisfy it if the λ gradient is non-zero. A non-uniform λ profile occurs when instabilities, which drive the plasma towards the Taylor state, substantially diminish near the Taylor state. The spheromak then settles at a local minimum energy state close to the global minimum Taylor state. In this case, the λ profile is a function of the poloidal flux function Ψ(ρ, z), which is the flux of the poloidal field through a surface of radius ρ and at axial location z [7]. There are several ways to interpret λ. As seen in Equation 4.2, λ is an eigenvalue of the curl operation on B. Therefore, λ depends on the spatial properties of the system (ie geometry of the boundary). From Equation 4.4, λ can also be thought of as proportional to the ratio of j/b. Two λ profiles worth noting are the hollow and peaked profiles. In a hollow λ profile, the current is stronger on the edge of the spheromak than at the magnetic

99 4.5. β LIMIT 74 axis. Conversely, a peaked profile has weaker edge currents and stronger currents near the core. A hollow profile tends to occur during formation as the gun drives current around the spheromak. A peaked λ profile is a sign of a decaying spheromak where the edge currents have been reduced (eg edge cools due to interaction with the wall and results in higher resistivity). A strongly hollow or strongly peaked distribution of λ can drive instabilities. For example in a strongly hollow profile, the inner current, which creates the poloidal field at the edge, is not able to stabilize the edge current. 4.5 β Limit The efficiency parameter β is defined as the ratio of plasma pressure to magnetic pressure: β = nt B 2 /2µ 0. (4.8) A spheromak that can operate at a higher β is more desirable since this means a denser, hotter plasma can be confined with a weaker field. A larger magnetic field typically translates into higher machine cost and complexity. The β limit is a measure of the maximum confinement capabilities of a magnetized plasma, and varies depending on the geometry of the container and the λ profile [49]. 4.6 Safety Factor The total magnetic field of the spheromak follows a helical path due to its poloidal and toroidal components. The safety factor q is a measure of the windedness of a magnetic field line, representing the number of times a field line wraps itself around the spheromak toroidally for one poloidal traverse around the spheromak.

100 4.7. RELATED SPHEROMAK EXPERIMENTS 75 The safety factor is important when considering instabilities propagating perpendicular to B. Waves propagating parallel to B bend the field lines away from their equilibrium paths, which acts as a stabilizer since the field lines naturally want to return to their equilibrium. A magnetic field line with safety factor q = m/n, where m and n are integers, returns to itself after m toroidal and n poloidal transits. This is true for other field lines travelling on this magnetic flux surface, called a rational surface. A wave with a wavelength matching this rational surface mode would resonate and cause the system to go unstable. Therefore, it is important that the spheromak s q profile avoid these rational surfaces. A large safety factor gradient, also known as magnetic shear, is desirable. The interchange mode is an example of a pressure-driven instability where a magnetic field line in a region of higher plasma pressure is interchanged with one from an area of lower plasma pressure. A large shear means adjacent field lines are more different in shape, which makes them more difficult to interchange. 4.7 Related Spheromak Experiments This section briefly reviews other coaxial helicity injection experiments. Most of the machines discussed are shown to scale in Figure 4.5. Spheromak formation dates back to 1960 [2] when it was observed that a plasma ring pushed with sufficient force into a radial magnetic field could break through and cause the field lines to reconnect. The plasma would then emerge with the disconnected magnetic field wrapped around the plasma, and with the newly formed spheromak free to drift away. Around the same time, other experiments were investigating the acceleration of

101 4.7. RELATED SPHEROMAK EXPERIMENTS 76 unmagnetized ring-shaped plasmas [61]. It was not until a couple decades later that the Ring Accelerator Experiment (RACE) [34] successfully accelerated a spheromak. As they were accelerated, the spheromaks were also compressed down the machine s conical tube giving an increase in density to n = m 3 and magnetic field amplification to B = 0.4T. It was found that the RACE data matched well with theoretical momentum and energy balance models. The Compact Toroid Experiment (CTX) [91, 55] achieved the formation of spheromaks with n = m 3, B = 0.5T, and lifetimes in the hundreds of µs. An order of magnitude increase in the lifetime was obtained by changing the flux conserver from a mesh-wall to a solid-wall. The mesh-wall caused a significant fraction of the plasma-confining poloidal field lines to intersect with the wall. Therefore, particles following these lines would collide with neutrals, increasing the local resistivity and promoting the loss of magnetic energy and helicity necessary for good confinement. The Magnetically Accelerated Ring to Achieve Ultra-high Directed Energy and Radiation (MARAUDER) [22] was a follow-up to the RACE experiment for spheromaks with a larger mass (1mg vs. 10µg). With a larger machine and power bank, MA- RAUDER accelerated spheromaks down a conical tube to n = m 3 and B = 1T. The Compact Toroid Fueler (CTF) [67] generated m 3, 50µg spheromaks for the purpose of re-fuelling tokamaks. In order to penetrate a tokamak s magnetic field and reach its core, the spheromak must have kinetic energy greater than the displaced tokamak magnetic field energy, requiring the spheromak s speed to be in the hundreds of km/s. The CTF spheromak was able to penetrate a tokamak with a 1T toroidal field without causing severe disruption. The Compact Toroid Injection Experiment (CTIX) [46, 51] created high-density,

102 4.8. PI-1 GEOMETRY 77 high-speed spheromaks with n = m 3 and B = 1T for fusion-related applications. CTIX showed that the spheromak s density could be increased an order of magnitude by injecting fuel into the spheromak as it was being accelerated down the tube. The Sustained Spheromak Physics Experiment (SSPX) [39] sought to achieve a steady state spheromak but concluded that it was not possible to simultaneously sustain the spheromak indefinitely with CHI and maintain good confinement. Nonetheless, sustainment of the spheromak was achieved for milliseconds with n = m 3 and B = 0.3T. During the sustainment, a quiescent period was observed where fluctuations in the magnetic field were substantially lowered. Ohmic heating from the spheromak s internal currents and resistivity was also obtained, causing the spheromak to reach temperatures on the order of 500eV. 4.8 PI-1 Geometry The geometry of PI-1 including dimensions is shown in Figure 4.6. The geometry of other PI-like machines is largely similar to PI-1. In terms of sputtering undesirable impurities off the electrode walls, a larger electrode radius is favourable since the current is distributed over a larger area. The overall size of the machine has been chosen in consideration of this fact. The electrode gap between the formation electrode and the outer wall, along with the input gas pressure, determine the breakdown voltage required to begin ionizing the gas. This breakdown point can be found from a Paschen curve. The two solenoids located on the outside of the machine are used to tune the shape of the plasma, and to help reduce the breadown voltage by concentrating electrons around the generated magnetic field lines. The magnetized plasma is pushed out into the expansion region,

103 4.8. PI-1 GEOMETRY 78 Figure 4.5: Select CHI experiments drawn to scale [42]. Figure 4.6: Approximate dimensions of PI-1 [43].

104 4.8. PI-1 GEOMETRY 79 which has a larger electrode gap. Here, the plasma is allowed to relax to a Taylor state in preparation for acceleration. The gap between the conical acceleration electrodes shrinks in a self-similar manner. A self-similar geometry implies that the fractional change in the electrode gap is equal to the fractional change in the radius of both electrodes. A self-similarly compressed spheromak maintains its aspect ratio such that its length decreases proportionally with its radius [66]. As the volume of the spheromak decreases, its density and magnetic field increase. For a fractional decrease in radius, A = ρ(z=0), the frac- ρ(z) tional increase in the density is A 3, and the fractional increase in the magnetic field is A 2 [65]. The electrode geometry has been changed in the past to modify the behaviour of the spheromak. For example, a section of the accelerator has been tested with a constant gap geometry to reduce the energy required to push the spheromak down the tube. This would make the spheromak deviate somewhat from self-similar behaviour.

105 80 Chapter 5 Polarimeter Design This chapter describes the design of the three-beam, heterodyning polarimeter constructed to diagnose spheromaks formed and accelerated in the General Fusion plasma injector. The polarimeter is composed of a laser, two acousto-optic modulators, detectors, and a variety of optical pieces to steer and modify the beam. A number of supporting diagnostics are also positioned around the injector to measure the various plasma parameters. 5.1 Plasma Injector and Chord Geometry Diagnostic ports providing access to the plasma are located at numerous axial and angular positions on the injector. The axial position of ports is defined according to the distance in centimetres from the backflange of the injector, near the formation region. The majority of the polarimeter data collected was taken using a chord on the PI- 1 injector at an axial position 352cm from the backflange as shown in Figure 5.1(a). The 352 position is preferred due to relatively easy access to ports, the occasional availability of an array of insertable magnetic probes, and mid-range densities and

106 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 81 magnetic fields. As shown in Figure 5.1(b), the polarimeter s left and right-circularly polarized beams pass vertically through a window at the bottom of the injector at the 144 angular position and out another window at the 36 angular position. The length of the chord in the plasma is about L = 60cm. Since the chord is vertical, the Faraday rotation incurred is due to the vertical component of the magnetic field. For a spheromak, this consists of the vertical component of the toroidal field B φ, and the vertical component of the radial field B ρ, itself a component of the poloidal field B θ. A second injector PI-2 exists which is similar in design to PI-1. Measurements were occasionally made on PI-2 near the end of the injector at the 493 position. The chord configurations are oriented similar to the PI-1 injector chord, but have a shorter length on PI-2 of about 40cm. Unless otherwise noted, all presented data pertain to the PI-1 injector. 5.2 Complementary Injector Diagnostics Other diagnostics relevant to the polarimeter are the interferometers, magnetic probes and Thomson scattering diagnostic, which respectively measure line-averaged density, components of magnetic field, and temperature. The polarimeter measures Faraday rotation, which depends on density and magnetic field. The interferometers and magnetic probes provide other measurements to compare against. The plasma temperature must be known in order to verify the assumption that the collision frequency is much less than the frequency of the polarimeter s beam. This section describes the operation of these diagnostics in further detail.

107 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 82 Backflange Polarimeter beam B Φ B θ 352 cm (a) Inner electrode 0 60cm cm 41cm 90 51cm Outer electrode 180 (b) Polarimeter Beam Figure 5.1: PI-1 injector and chord geometry. (a) The most often used polarimeter chord is positioned 352cm from the backflange of the injector. Sketches of the toroidal field B φ and poloidal field B θ are provided. (b) Cross section of the injector at the PI axial position looking from the end of the injector towards the backflange. The labelling convention for the injector angular positions is shown. The interferometer chord is omitted for clarity.

108 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS Interferometers A schematic of the interferometer is shown in Figure 5.2. A laser diode produces a 1.5µm linearly polarized beam that is transported via fibre optic cable. A fibre optic coupler splits the main beam into three separate beams. In the schematic, the centre beam is not needed and is discarded. The reference beam is reflected back through the fibre and split at the coupler joint so the detectors D1 and D2 each receive a reference beam. The last of the three beams, the sample beam, is collimated at the output of the fibre with a lens and sent through a window on the injector, emerging out another window on the opposing side. The sample beam is then reflected back through the same path and is focussed into the same fibre. The sample beam, now phase shifted by the plasma, travels back through the fibre to the coupler joint, where it is split and combined with the two reference beams. For a 3-way fibre coupler, each output is phase shifted 120 with respect to the other outputs. This behaviour is due to conservation of energy [19]. At each detector, the sample and reference beams interfere, producing a signal that depends on the phase difference between the two beams. The signal is sampled at 20MHz, giving a maximum bandwidth of 10MHz. The interference signal at one detector is 120 out of phase compared to the other. With no plasma along the beam s chord, a constant intensity is observed. When a plasma passes through the beam chord, the sample beams are phase shifted due to the plasma density, which changes the interference signal. The signals at the two detectors must be out of phase in order to track the direction of the density change, which can be ambiguous otherwise. Plotting the two signals

109 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 84 Dump Sample arm Reference arm Plasma 3-way fibre-optic coupler D1 D2 Initial beam Figure 5.2: Interferometer schematic. Laser split into reference and samples beams by fibre optic coupler. Reference beam is reflected back through the fibre. Sample beam is phase shifted by the plasma and reflected back. Both sample and reference beams are split by the coupler on their way to detectors D1 and D2. The signal is phase shifted by 120 at one of the detectors. Reference and sample beams interfere, producing a signal from which the plasma line-average density can be determined. against one another produces an ellipse as a plasma passes by, where a change in the density corresponds to movement between two points on the ellipse (see Figure 5.3(a)). An equation is obtained for the ellipse with a fitting function so the points can be shifted and scaled to a circle centred at the origin. The angular distance between two points on the circle can then be related to a change in plasma line-averaged density using Equation Interferometers are spread out over several axial locations on the machine. As seen in Figure 5.3(b), the interferometers show that the plasma is compressed to higher densities as it is accelerated down the injector. The interferometer chord at the 352 position passes vertically through the injector between the 216 and 324 angular

110 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 85 Detector 1 (arbitrary units) Shot Detector 2 (arbitrary units) (a) Density (x10 21 m -3 ) Shot Time (µs) (b) Figure 5.3: Amplitude-based interferometer diagnostic on GF plasma injector. (a) Two out of phase interference signals are produced and can be plotted against one another to track density changes. (b) Computed densities from interfereometers at three axial locations. positions. This is a mirror image of the polarimeter chord. Therefore, if the plasma is perfectly axisymmetric, the polarimeter and interferometer should produce the same density measurement. The 352 interferometer typically measures an average density between m 3 and m 3.

111 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS Magnetic Probes The magnetic probes used on the injector are simply loops of wire oriented in a particular direction. A changing field through the loop produces a changing voltage at the wires terminals. Once calibrated, integrating the voltage can give an estimate of the magnetic field passing through the loop. Most probes are located at the surface of the injector. Surface probe assemblies consist of two separate wire loops respectively oriented so the normal to the loop plane is either in the axial or azimuthal direction. Therefore, it is possible to measure the axial component of the poloidal field as well as the toroidal field at the surface of the injector. As the spheromak passes by, the probes tend to measure a large poloidal field and a small toroidal field as seen in Figure 5.4, which is expected. However, a significant toroidal field often follows immediately after the poloidal signal drops back down to zero as the spheromak accelerates away. This is the toroidal pushing field created from external currents passing through the injector electrodes. As its name suggests, the pushing field pushes the spheromak down the injector. An insertable array of magnetic probes is also available on some shots. The linear array is enclosed in an alumina ceramic tube and contains five radial, five axial and five toroidal probes spanning approximately 14cm. Sample probe array measurements are shown in Figure 5.5. It can be seen that the inner toroidal probe near the midpoint between the electrodes has the strongest signal. This is approximately the magnetic axis. Conversely, the axial and radial signals are strongest near the outer electrode. The radial field changes sign as the spheromak passes by since the poloidal field wraps around the spheromak from front to back. Also noteworthy is that the plasma formation and acceleration discharges create electromagnetic interference that

112 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 87 Axial Field (Tesla) Shot Toroidal Field (Tesla) Time (µs) Figure 5.4: Sample of magnetic field measurements for probes at the 118, 352 and 493 axial positions and several angular positions. The axial field signals show the location of the spheromak in the injector. The toroidal field signals increase once the spheromak has passed, which corresponds to the toroidal pushing field.

113 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 88 is picked up by the probes as spikes at about 270µs and 310µs Thomson Scattering Thomson scattering [45] is a non-perturbing method of measuring electron temperature. Thomson scattering occurs when an electromagnetic wave accelerates an electron and thus causes it to re-emit radiation at the same frequency as the incident electromagnetic wave. However, the scattered wave appears to have a shifted frequency to an observer in the lab frame due to the electron s thermal motions. For a distribution of electron velocities, the result is a frequency broadening of the detected scattered waves, known as Doppler broadening. The geometry of a Thomson scattering experiment is shown in Figure 5.6. The power of scattered photons for infinitesimal solid angle dω and wavelength dλ is [86] P s dωdλ = ( ) P i recln 2 c 2 2λ 0 v e π sin(θo /2) dωdλ (λ λ 0 ) 2 sin2 (θ E ) exp 4veλ sin(θ o /2) (5.1) where P i is the incident laser power, r e the classical electron radius, L the scattering volume length, λ 0 the incident beam wavelength, v e = 2k B T/m the mean electron thermal speed, θ o the angle of observation, and θ E the angle between observation and laser polarization directions. The wavelength spectrum is spread like a Gaussian centred at λ 0 with standard deviation σ λ = 2 k B T/m e λ 0 sin(θ o /2)/c. (5.2) Therefore, it is possible to estimate the plasma temperature by fitting a Gaussian to

114 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 89 Axial Field (Tesla) Toroidal Field (Tesla) 1.0 Shot ρ = 49.7 cm 4 ρ = 46.2 cm 3 ρ = 42.9 cm 5 2 ρ = 41.4 cm 4 1 ρ = 39.7 cm ρ = 48.9 cm 1 4 ρ = 45.5 cm 2 3 ρ = 42.3 cm 2 ρ = 40.7 cm ρ = 39.0 cm Radial Field (Tesla) ρ = 48.7 cm 4 ρ = 46.4 cm 3 ρ = 43.3 cm 2 ρ = 41.2 cm 1 ρ = 39.6 cm Time (µs) Figure 5.5: Probe array magnetic field measurements. The array is inserted into the injector at the 352 axial, 180 angular position from the bottom. The radial position of the coils, ρ, is given. For reference, the inner and outer electrode radii are 32.8cm and 51.0cm respectively. Two of the toroidal field signals appear to have saturated at about 1T. Noise from the plasma formation and acceleration discharges can be seen at about 270µs and 310µs.

115 5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS 90 Figure 5.6: Thomson scattering geometry [86]. the measured wavelength spectrum. In addition, the density can be obtained within the scattering volume by knowing the temperature, integrating the area underneath the fitted Gaussian and comparing it with a Rayleigh scattering calibration shot [75]. The difficulty is that, for typical plasma densities, the chance of a photon scattering is small, on the order of A powerful laser is needed in order to ensure the Thomson scattered light can be distinguished from the plasma light (eg line radiation, bremsstrahlung). General Fusion s Thomson scattering diagnostic uses a 10ns, 2.5J laser. The scattered light is then collected by a lens and focussed into a fibre optic cable. The fibre relays the light to an imaging spectrometer, which disperses the light onto a 2cm long, 1 16 photomultiplier tube array. The photomultiplier converts the light into an electrical signal that can be analysed on a computer. The focal length and distance of the lens from the laser decides how much light is captured and what section of the beam is imaged. In the current setup, the diagnostic examines light from an approximately 1cm-long portion of the beam near the centre

116 5.3. POLARIMETER LASER 91 of the plasma. 5.3 Polarimeter Laser The wavelength of the beam should be chosen in order to maximize the Faraday rotation phase shift signal while avoiding problematic plasma effects. Considering these relationships, a CO 2 laser with a wavelength of 10.6µm was chosen. The consequences of this choice are further discussed in Chapter 6. An Access Laser Merit-S CO 2 laser creates the main polarimeter beam. The specifications are listed below: Continuous 8W power output, Vertical polarization (see Figure 5.7), Wavelength near λ 0 = 10.6µm, Single longitudinal mode and linewidth ν = 100kHz, Transverse electromagnetic (TEM) mode TEM00 (ie Gaussian beam) with beam quality factor M 2 < 1.1, Beam waist radius of 1.2mm, Divergence half-angle of 2.25mrad. From Equation 3.65, the coherence length of the polarimeter s laser is on the order of L c = 1km. The maximum path length for a polarimeter beam is about 10m. Therefore, phase contrast issues from this effect are expected to be small.

117 5.3. POLARIMETER LASER 92 Shutter Actuator Initial Beam Polarization Figure 5.7: Polarimeter CO 2 laser with vertically polarized beam path overlaid. Actuator that controls the laser shutter is shown Laser stability As the laser warms up from a cold start, the beams interference at the detector produces a signal with a peak-to-peak voltage that is observed to fluctuate. The laser s manufacturer has confirmed that the main beam emerging from the laser changes its direction slightly as the laser warms up. This would explain the signal fluctuations since the three beams interfered at the detector would each take a slightly different path and become offset at the detector. After about a half hour, the signal stabilizes suggesting the laser temperature stabilizes as well. When operating during plasma shots, the laser remains on but the shutter is normally closed for safety reasons. Once the shutter has opened for a shot, a similar variation in signal is observed in the first ten seconds. This is believed to be due to the laser having a different equilibrium temperature with the shutter either open or

118 5.3. POLARIMETER LASER 93 closed. Therefore, it is ensured that the shutter is opened once the capacitors begin to charge charging, which gives about one minute for the laser to warm up before the shot. This usually assures that a large signal is obtained when the shot is fired. The water coolant takes a relatively long path of about 50m to go from the chiller to the polarimeter. The water is also shared with other systems in parallel (ie the polarimeter is not downstream from the water of other machines). The temperature inside the building can also vary by about 10 C between the summer and the winter. Therefore, this might add some instability to the temperature of the laser. A chiller dedicated to the polarimeter and placed a shorter distance away could improve control of the laser s temperature. The laser is water-cooled by a Neslab HX-300 chiller that regulates the water temperature to within 0.1. The temperature of the laser has a small effect on the wavelength of the output beam (see Table 5.1). Although the difference in wavelength is small, such variations can significantly change how the beam interacts with the optics, which can considerably degrade the alignment. The chiller water temperature is set to 18.5 C corresponding to a more exact output wavelength of µm. This temperature band is chosen since it is relatively wide. It is also close to 10.6µm, which is important since many of the polarimeter s optical components are designed for this particular wavelength. The laser requires about half an hour to warm up before the temperature and output beam power stabilize Laser Alignment At 10.6µm, the beam is in the infrared and is therefore invisible to the naked eye. This presents a number of challenges for the alignment of the beam and safety. The

119 5.3. POLARIMETER LASER 94 Temperature Range ( C) Wavelength (µm) Table 5.1: Temperatue-wavelength regimes of the polarimeter laser. The C range is chosen since it is wide and gives a wavelength of µm, which is close to 10.6µm. laser has a shutter, to which a linear actuator is attached with triggering circuitry that allows for remote opening and closing of the shutter (see Figure 5.7). The laser remains on in between plasma shots to keep the laser warm, but the shutter is normally closed, opening only when the capacitor banks begin charging for a plasma shot. During this time the area is clear, and it is safe to have the shutter open. Infrared laser safety goggles must be worn when doing maintenance on the polarimeter such as beam alignment. The invisible beam is aligned primarily using Post-It notes as a beam probe. Postit notes temporarily change to a dark colour when exposed to heat. The beam spot can theen be identified by placing a Post-It note in the beam s path and observing a dark spot (see Figure 5.8). It must be ensured that the laser power is turned down sufficiently low such that the beam does not burn the paper and possibly damage nearby optics. In situations where the beam power is very low, a Macken beam probe can be used that is made of a more sensitive material. However, in order for the spot to be seen on the Macken probe, a UV lamp must be shone onto the probe material. The UV lamp can also be used to enhance the visibility of the beam spot on Post-It notes.

120 5.4. ACOUSTO-OPTIC MODULATORS (AOMS) 95 Laser spot Figure 5.8: Post-It notes change colour when exposed to heat and can be used to locate the invisible beam. For very low power beams, the more sensitive Macken probe is used, which is seen on the far right. 5.4 Acousto-Optic Modulators (AOMs) Two AOMs are used to frequency shift two of the three polarimeter beams creating beat waveforms when the beams are interfered. The beats contain important phase information. The theory of operation of the AOM is described in Section 3.9. The two original AOMs (see Figure 5.9(a)) are Brimrose models GEMF and GEMF , which are respectively tuned for an input frequency of 25MHz and 40MHz. The AOMs are powered by radio-frequency (RF) drivers with respective output signal frequencies of 25MHz and 40MHz. The beam diffracts off the acoustic wave generated inside the AOM crystal. The majority of the output beam power is split between the 1 st -order beam (frequency shifted) and 0 th -order beam (not frequency shifted). The ratio of the 1 st -order beam power to the input beam power is called the diffraction efficiency. This efficiency is maximized when the incident beam is at a slight angle to the acoustic wavefronts called the Bragg angle θ Bragg. For vertically polarized light at the input, the AOM is oriented so the acoustic waves propagate in the upward vertical direction. The Bragg angle is set by rotating

121 5.4. ACOUSTO-OPTIC MODULATORS (AOMS) 96 the AOM slightly with respect to the beam. The axis of rotation is parallel to the table and perpendicular to the direction of the beam of light as shown in Figure 5.9(a). Once rotated, the beam s electric field is slightly off-parallel from the acoustic wave propagation direction. The direction of the AOM s rotation decides whether the 1 st - order beam is shifted up or down in frequency. When the AOM is rotated such that the acoustic wave direction is pointed towards the input light beam, the output 1 st - order beam is deflected upwards and shifted up in frequency. Conversely, when the acoustic wave direction is pointed slightly away from the light beam, the 1 st -order beam is deflected downwards and shifted down in frequency. When the AOM is rotated by the Bragg angle, the separation of the output 1 st - order and 0 th -order beams is 2θ Bragg. The beams must be allowed a few inches to separate sufficiently before the 0 th -order can be blocked and the 1 st -order steered to the next stage of the polarimeter. The specifications of the AOMs are Maximum optical power density of 5W/mm 2, Diffraction efficiency of 60%, Bragg angles θ Bragg of 24.1mrad and 38.5mrad respectively, Germanium crystal with anti-reflection coating for high transmission of light at 10.6µm, AOM crystal is water cooled. The RF driver specifications are Output power of 20W, Frequency accuracy of 0.015% and stability of %.

122 5.4. ACOUSTO-OPTIC MODULATORS (AOMS) 97 Cooling terminals Absorber side Beam entrance / exit hole Mounting post / axis of rotation Power Transducer side (a) (b) Figure 5.9: AOMs are used to frequency shift an incident beam to allow for heterodyning. (a) Original Brimrose AOM in the appropriate orientation to frequency-shift vertically polarized light. (b) New IntraAction AOM that replaced the broken 25MHz Brimrose AOM. The AOMs must be water cooled due to the substantial power directed into the crystal from the transducer. Over time, sediment built up in the AOMs possibly from corrosion within the AOM. The accumulation of sediment impeded the flow of water and prevented the AOM from being properly cooled. Eventually the AOM s O-ring seals broke and slowly leaked the water coolant. Around the same time, the 25MHz AOM began to transmit light poorly. Upon further inspection, it was discovered that the germanium crystal of the 25MHz AOM had fractured. The failure was believed to be caused by the improper cooling of the crystal from sediment build-up. The original 25MHz AOM had to be replaced by a new 25MHz AOM from IntraAction, model AGM-253PC1 (see Figure 5.9(b)). The same 25MHz RF generator is used with the new AOM and works properly. Several AOM configurations are possible and are now discussed. The polarimeter

123 5.5. OPTICS 98 requires three main beams at three slightly different frequencies. Two of the beams are frequency-shifted by the AOMs and the third is left unshifted. When interfered, the signal at the detector is the sum of three beat waveforms. Each waveform oscillates at one of three possible beat frequencies. A beat frequency is the difference frequency between two interfering beams. There are two sets of beat frequencies possible when the three beams are interfered. The first set of beats is 15MHz, 25MHz and 40MHz created by either frequency upshifting both beams or frequency downshifting both beams. The second set is 25MHz, 40MHz, and 65MHz created by upshifting one beam and downshifting another. As explained in Section 5.7.3, the frequency separation of the beats decides the maximum possible bandwidth of the signal. The second set of beats gives slightly more bandwidth, but requires a higher digitizer sampling rate (over 130MS/s) to capture all three beats without aliasing. The first set of beats (ie 15MHz, 25MHz and 40MHz) is used to relax the requirements on the digitizer. 5.5 Optics A variety of optics pieces are used to modify the laser s initial beam. Optics are held in place with mounts fixed to a half-inch thick Thorlabs solid aluminum optical breadboard. The optics pieces typically have a clear aperture of 1 in diameter. Transmission optics are made from ZnSe or Ge, which have low absorption of the 10.6µm beam. These materials have a high index of refraction, which requires them to have anti-reflection coatings. The sections to follow describe the main optics components in further detail.

124 5.5. OPTICS Lens The longest beam path is on the order of several metres. The beam diverges along its path due to diffraction. The beam width must be kept sufficiently small across the entire path in order to pass through optics without clipping the edges. Lenses can control the beam divergence by refracting beam rays over different path lengths inside the lens. Planoconvex lenses are used with the curved side of the lens turned towards the collimated beam. This attempts to equalize the distance that beam rays traverse inside the lens. Therefore, spherical aberrations of the focussed beam s wavefronts can be reduced compared to using a biconvex lens in the same situation Waveplate A waveplate is a birefringent crystal with the property that polarized light has a higher phase speed along a particular axis called the fast axis. Therefore, the thickness of the waveplate can be made to introduce a precise phase difference between two orthogonal components of a beam of light. A λ/2-waveplate creates a 180 phase difference between incident components when the waveplate s fast axis is oriented at a 45 angle with respect to the the incident beam s polarization plane. This could be used to rotate a beam s polarization plane from vertical to horizontal polarization, for example. Similarly, a λ/4-waveplate can introduce a 90 phase shift between components, which could convert linearly polarized light to circularly polarized light and vice-versa (see Figure 5.10).

125 5.5. OPTICS o Fast axis Figure 5.10: A λ/4 waveplate can be used to convert between linearly and circularly polarized light Thin Film Polarizer When discussing the interaction between a surface and an incident beam of light, it is helpful to identify the s and p-polarizations, which are respectively the components of the incident light perpendicular and parallel to the plane of incidence. In general, the s and p-components of the incident light do not reflect with the same magnitude or phase when incident on a surface [5]. At an angle of incidence equal to the Brewster angle, the reflected light is entirely s-polarized. For ZnSe, the Brewster angle is Under normal circumstances, the transmitted light is a mix of s and p-polarized light. Thin film polarizers can be designed with a coating that allows for very high transmission of the p-component and very high reflection of the s-component when oriented at the Brewster angle (see Figure 5.11). These are useful to separate the components of a beam of light Reflectors and Beamsplitters Reflectors (mirrors) are used to steer beams around the polarimeter and injector. As previously mentioned, the phase change between reflected s and p-components is not necessarily zero. Therefore, ellipticity could be introduced into a reflected circularlypolarized beam, which is undesirable for the reasons outlined in Section 3.4.

126 5.5. OPTICS 101 Angle of incidence s-component Incident beam p-component Figure 5.11: Thin film polarizer reflects s-polarized light and transmits p-polarized light when the angle of incidence is at the Brewster angle. Reflectors can be designed to minimize this effect about a particular angle of incidence and for a particular wavelength. These are made by layering coatings over a reflective substrate. The thickness of the coatings can be precisely controlled to create desired phase delay between the reflected s and p components [3, 77]. Most reflectors on the polarimeter optical breadboards are zero-phase reflectors that produces 0 ± 1 phase change between s and p components for an angle of incidence of 45. Deviation from the designed angle of incidence can lead to large phase shifts of about 4 in phase per 1 in angle. Therefore, when it is more difficult to obtain a 45 angle of incidence (eg steering light around injector), a regular silver mirror is used that gives a phase distortion of no more than 5 at any angle. An inexpensive phase retarding reflector (PRR) is also used in the polarimeter to create a 90 ± 3 phase delay between s and p-components, which can convert linearly polarized beams to circular polarization. The PRR delays the p-component of the light. As shown in Figure 5.12, either left or right-circularly polarized light can be made depending on the orientation the incident linearly polarized light. Consider light incident on a PRR at a 45 angle. The linearly polarized light s polarization plane is oriented at a 45 angle to the incidence plane. Two electric field orientations

127 5.6. POLARIMETER COMPONENT LAYOUT 102 are possible: E = E p E s = 1 ±1 e iωt (5.3) where E p and E s are respectively the p and s components of the electric field. The positive and negative orientations of the polarization plane are encapsulated by the ± symbol. After the PRR, the s-component is phase advanced by π/2. As per the chosen convention, phase advances in the negative direction with a positive increase time. Therefore, an advance in phase corresponds to a factor of e iπ/2 = i. Applying this delay to the electric field produces circularly polarized light: E = 1 i e iωt. (5.4) Therefore, the initial positive and negative linear polarization orientations respectively give right and left-circularly polarized light. Beamsplitters can be made from the same principle by replacing the reflective substrate with a material that can transmit light at 10.6µm such as ZnSe or Ge. The coating on polarization-insensitive beamsplitters gives the desired reflection-transmission ratio without introducing a phase shift between the s and p components. 5.6 Polarimeter Component Layout The configuration of the polarimeter s components for one reference and one plasma beam is now described. The polarimeter box has a bottom and a top level. The x and y-directions are shown in Figures 5.13 and 5.14, while the z-direction is out of the page.

128 5.6. POLARIMETER COMPONENT LAYOUT 103 ω L 45 o ω R ω L ω R s 45 o p Figure 5.12: A phase retarding reflector used to convert between linearly and circularly polarized light. The s and p components of the initially linearly polarized light are shown for beam ω L Bottom Level On the bottom level (see Figure 5.13), the CO 2 laser outputs a 10.6µm beam with an initial waist radius of 1.2mm and power of 8W. This beam must be split into three and two of these beams must also be focussed through the AOM entrance/exit holes, which are about 2mm in diameter. It has been attempted to equalize the power of the three beams with the materials on hand. The initial beam from the laser is focussed with a 10 focal length lens (L00). After passing through the lens, the beam is split by a 50/50 beamsplitter (BS00). Half the power is sent immediately towards the 40MHz AOM (AOM40) and becomes the ω R beam. The other half passes through a 33/66 beamsplitter (BS01) that reflects one-third of the power down the table. This is the ω V beam, which at a later time

129 5.6. POLARIMETER COMPONENT LAYOUT 104 ω LR y x ω L ω V ω R Figure 5.13: Sketch of the bottom level of the polarimeter as viewed from above. The initial laser beam is split into three beams, two of which are frequency shifted by the AOMs and combined. The z-direction is out of the page.

130 5.6. POLARIMETER COMPONENT LAYOUT 105 allows for density measurements. The other two-thirds of the power is sent to the 25MHz AOM (AOM25) and emerges out as the ω L beam. The ω L beam is then sent through a λ/2-waveplate (WP). This has the effect of rotating the light from vertical to horizontal polarization. The ω L and ω R beams must now be combined. After exiting the 40MHz AOM, the ω R beam must be steered by mirror M02 so that it can intersect beam ω L at the beam combining-optic (BS02). This can be done by holding mirror M02 in a Thorlabs Kinematic mount, which allows for precise rotations with a resolution of better than In addition, mirror M02 is attached to a translation stage that allows fine translation of the beam with better than 10µm resolution in the y-direction. The ω L and ω R beams intersect at a beam-combining optic, which in the present design is a 50/50 beamsplitter (BS02). Half the power of the beams is lost in the process and is absorbed by a beam block (BB). For the half that is of interest, the beam ω R is reflected off beamsplitter BS02, while beam ω L is transmitted through. Beamsplitter BS02 is also held in a Kinematic mount, which allows the direction of the ω R beam to be matched to the direction of the ω L beam. In this arrangement, it is possible to combine beams ω L and ω R such that they meet at the surface of beamsplitter BS02 and afterwards proceed in the same direction. The newly combined beams are sent through another 10 focal length lens (L01) to refocus them. The ω V beam is also sent through a 10 focal length lens (L02). All beams are then steered upwards towards the top level of the polarimeter with mirrors M03 and M04.

131 5.6. POLARIMETER COMPONENT LAYOUT Top Level On the top level (see Figure 5.14), the ω V beam is split by three 50/50 beamsplitters (BS07, BS08, BS09) such that two beams are produced with an eighth of the initial ω V beam power. The other unused beams are absorbed by beam blocks (BB). This beam splitting is a consideration for future work where up to seven plasma-probing beams and one reference beam could be used. After being split, the two ω V beams are respectively steered towards the plasma and reference detectors. The combined ω L and ω R beams must be converted from linear to circular polarization. This could be done with a λ/4-waveplate. However, a different approach is taken using a much less costly phase retarding reflector (PRR). As the combined ω L and ω R beams are travelling up towards the top level, they are reflected off a PRR that directs them horizontally onto the top level as shown in Figure In order for the beams components to be properly delayed and form circularly-polarized beams, the PRR steers the beams in a direction that is parallel to the table, and at a 45 angle from the x-axis. Alternatively, a λ/2-waveplate could be used to first rotate the polarizations of the linearly polarized ω L and ω R beams by 45. The beams would then have to be steered by the PRR into the x direction. The combined circularly polarized beams form the ω LR beam. This beam must travel several metres to pass through the plasma and arrive back at the polarimeter box for detection. At this point, the divergence of the beam is too large and must be reduced to slow the growth of the beam radius. The divergence and beam waist radius are inversely related as seen in Equation Therefore, before it can be collimated, the beam is expanded to a radius of about half a centimetre. It is then collimated by a 1m focal length lens (L04), which allows the beam to maintain a small

132 5.6. POLARIMETER COMPONENT LAYOUT 107 y x ω LR ω V Figure 5.14: Sketch of the top level of the polarimeter as viewed from above. Beams ω L and ω R are converted to circular polarization and their divergence is reduced. All beams are split into multiple parts, giving the possibility to have multiple polarimeter chords. Beams ω LR and ω V are interfered at the detector. The z-direction is out of the page.

133 5.6. POLARIMETER COMPONENT LAYOUT 108 ω R To plasma ω L Phase retarding reflector Beam combiner Mirror ω L λ/2 waveplate ω R Figure 5.15: Combined ω L and ω R beam conversion to circular polarization with a phase retarding reflector. radius throughout its path as shown in Figure In the current setup, the beam is expanded simply by letting it travel a long enough distance until it has reached the desired radius. This could also be done using a short focal length lens to bring the beam to the desired radius over a much shorter distance, then sending the expanded beam into another lens with an optimal focal length that minimizes the beam radius over some path length. The ω LR beam is split three times with 50/50 beamsplitters (BS03, BS04, BS05) to produce one reference and one plasma beam with 1/8 of the initial ω LR beam s power. The reference ω LR beam is directed towards another 50/50 beamsplitter (BS06) where it is combined with an ω V beam. The combined beams are then passed through a 2 focal length lens (L03), which focusses them through a thin film polarizer (TFP). For the particular configuration used in this experiment, the TFP reflects away the beams horizontally polarized components and transmits the vertically polarized components into the reference detector (D01). The plasma ω LR beam is steered out of the box, through the ZnSe windows on

134 5.7. SIGNAL MEASUREMENT 109 Radius (cm) L0 L1 L3 Beam Lens Window Distance from laser (m) Figure 5.16: Simulated radius of the combined ω LR beam along its path through the plasma and into the detector. The radii of the of the lenses and injector windows are also drawn for reference. the injector and back into the polarimeter box. Once inside the box, it is combined with an ω V beam using a 50/50 beamsplitter (BS10). The combined beams then pass through a 2 focal length lens (L05) and TFP (TFP02) before arriving at the plasma detector (D02) with a radius of about 0.1mm. 5.7 Signal Measurement The detector is a non-cooled Vigo PVM-10.6 HgCdZnTe photovoltaic detector. When the detector material absorbs an incident photon, it can eject an electron and produce a current. A voltage proportional to the current can then be measured across some resistance. The detector can be described as a square-law detector since the number of photoelectrons produced is proportional to the incident power, which is proportional to the square of the electric field s magnitude [33]. Therefore, the voltage measured is proportional to the square of the electric field s magnitude. The detector has the

135 5.7. SIGNAL MEASUREMENT 110 following specifications: Bandwidth of 100MHz, Active area of 1mm 2, Noise equivalent power (NEP) of W/ Hz. NEP is a measure of the sensitivity of the detector. It is defined as the power per square root of bandwidth that gives a signal to noise ratio (SNR) of 1, Linear relationship between current and power with a responsivity of R = 4mA/W, Resistance of Z det = 50Ω Maximum power density of 1W/mm 2. The detector typically produces a small signal on the order of a few mv. The signal is amplified with a Boston Electronics 490X amplifier, which is impedance-matched to each detector. The amplifier is powered with a linear ±5V power supply to avoid the low-frequency switching noise associated with some switched-mode supplies. The specifications of the amplifier are Gain of G = 40, Alternating current (AC) coupled with bandwidth from 1kHz to 100MHz, Noise factor F = 1.9, which is the ratio of input SNR to the output SNR, Output impedance of 50Ω.

136 5.7. SIGNAL MEASUREMENT 111 with The signal at the output of the amplifier is measured by a Picoscope 4227 digitizer Sampling frequency 125MS/s giving a Nyquist frequency of 62.5MHz, Bandwidth of 100MHz, 12-bit resolution, Input impedance of 1MΩ. As outlined in the previous section, the ω L and ω R beams are respectively shifted up in frequency by the 25MHz and 40MHz AOMs. Therefore, when the ω L, ω R and ω V beams are all interfered at the detector, three beat frequencies are observed in the signal: 15MHz, 25MHz and 40MHz. The 15MHz beat contains the Faraday rotation information while the 25MHz and 40MHz beats contain the density phase shift Hardware and Software Reference Signals The demodulation algorithm requires a reference signal and a plasma signal. Up until this point, it has been assumed that the reference signal S ref is generated from hardware, that is the interference of the ω L, ω R and ω V. However, the reference signal can also be created in software by generating points at the desired frequency. For example, to obtain a Faraday rotation measurement, the required reference signal is of the form S ref = cos(2πf LR t) (5.5) where f LR = 15MHz is the beat frequency of the interfering beams. Therefore, this expression can be sampled at the appropriate sampling frequency (125MHz) to

137 5.7. SIGNAL MEASUREMENT 112 generate a reference signal in software. The advantage of a software reference signal is that the reference beams can be eliminated, saving alignment time, beam power, and space. The choice of f LR = 15MHz is not sufficiently accurate and produces an artificial phase delay between the two signals. This results in a Faraday rotation measurement that increases linearly with time. The frequency error is typically in the range of a few 100Hz, giving a slope in the tens of degrees per ms (see Figure 5.17(a)). A correction, δf LR to the beat frequency can be obtained by fitting a sinusoid to the plasma signal to extract a more accurate frequency near f LR = 15MHz. The reference signal with the frequency correction is S ref = cos(2π(f LR + δf LR )t). (5.6) This expression can then be sampled as before to create an improved reference signal. After demodulation, the result is much more accurate as shown in Figure 5.17(b). However, small noise spikes can still be seen, which are believed to be due to small changes in the frequency of the AOM that cannot be reproduced in simulation. The software reference produces a Faraday rotation signal with standard deviation 0.16 while the hardware gives It might be possible to improve the software reference by electrically monitoring the AOM frequencies Raw Signal Magnitude and Noise The peak to peak voltage of the signal at the detector can be estimated knowing that V det = P RZ det where P is the beam power incident on the detector. The signal then passes through the AC-coupled amplifier, where the direct current (DC) component

138 5.7. SIGNAL MEASUREMENT 113 Faraday rotation (deg) Shot 6900 (PI2) Time (ms) (a) Faraday rotation (deg) Hardware Software Shot 6900 (PI2) Time (ms) (b) Figure 5.17: Comparison of Faraday rotation measurements using software and hardware reference signals. The Faraday rotation measurement bandwidth is 0.5MHz. (a) Software reference with f LR = 15MHz. (b) Software reference uses the corrected frequency 15MHz + δf LR. The Faraday rotation signal for the hardware reference is offset for convenience. of the signal is removed and the remaining AC signal is increased by a factor G. The amplified signal is transmitted from the amplifier with a 50Ω output impedance through a 50Ω, 50cm long coaxial cable and into a 50Ω terminator. The digitizer measures the voltage across the terminator, which is V dig = 1GP RZ 2 det since the circuit is matched and the voltage is divided equally between the amplifier and the terminator. The terminator is used to avoid signal reflections [18] due to the impedance mismatch between the amplifier (50Ω) and the digitizer (1MΩ).

139 5.7. SIGNAL MEASUREMENT 114 The power of the ω L, ω R and ω V beams at the detector is respectively estimated to be P L = 75mW, P R = 50mW and P V = 83mW. The total total power on the detector is then approximately 200mW. Therefore, the peak to peak signal at the digitizer is predicted to be around V dig = 800mV. The largest observed signal is only about 350mV, which is equivalent to a total power of 100mW at the detector. This suggests that either power is lost along the beam path or the interference between the combined beams is not maximized (eg beams do not overlap perfectly, phase contrast noise, etc.). Absorption is not expected to be the cause, since all components claim absorption of less than 1%. It is possible the AOM has not been aligned optimally to produce the highest diffraction efficiency. A beam profiler would be a helpful tool to understand where the power is lost. Another explanation for the difference in power is that the beam ω V beam is not well aligned with the ω LR beam. This is usually a difficult beam to align, and would benefit greatly from redesign with translation stages. The detector and amplifier add white noise (eg thermal noise) to the raw signal. The SNR of the signal at the digitizer can be estimated for a particular beam power and bandwidth knowing the NEP of the detector and F of the preamp: 1 P SNR = NEP F fbw (5.7) = P fbw (5.8) where P is in Watts and f BW is the bandwidth in Hz. The system has f BW = 100MHz. For an incident power of P = 100mW, the SNR of the signal at the digitizer is about The digitizer has 12-bit resolution, or 2 12 = 4096 levels, which is greater than

140 5.7. SIGNAL MEASUREMENT 115 the SNR. Therefore, the detectors are expected to be the limiting factor in terms of raw signal noise reduction and not the digitizer quantization error. In theory, the SNR could be improved by increasing the beam power on the detector. However, the detector s maximum recommended power density of 1W has already been reached. The beam radius at the detector is focussed down to about W = 0.1mm. This gives a power density at the detector of about 1.4W since 99% of the beam power is within a circle of radius 1.5W. Therefore, no further increase in the beam power is possible Bandwidth The main polarimeter signal is produced at the detector from the interference of three beams of slightly different frequency. The signal is then amplified and sampled at a frequency f s. The detector, amplifier and digitizer low-pass filter the signal as per their respective bandwidths. The highest frequency component of the signal that can be measured without aliasing is f s /2, known as the Nyquist frequency [63, 80]. The signal contains three beat waveforms which oscillate at one of the three beat frequencies: f 1 < f 2 < f 3. Each beat waveform contains a phase that must be measured. As discussed in Section 3.3, the demodulation algorithm isolates a single beat and extracts the phase from it. At the output of the algorithm, the measured phase has a cut-off frequency f c, also called the bandwidth. The bandwidth for the Faraday rotation measurement φ F is generally kept to 1MHz or lower. The Faraday rotation signal is relatively small, so larger bandwidths tend to drown the signal out in noise from the higher frequency components. For the density phase shift φ n, it is not always sufficient to use f c = 1MHz since it is a much

141 5.8. CALIBRATION 116 larger and more rapidly changing quantity. For example with B = 1T, the ratio of Equation 3.16 to Equation 2.63 is large: φ n φ F (5.9) Therefore, the φ n measurement bandwidth is often set above 1MHz. The upper limit of the bandwidth is set by noise. There is white noise spread over the entire frequency spectrum for example from thermal motion of charges in the detector. As the bandwidth is increased, more noise is added to the signal. Another factor that determines the upper bandwidth limit is the separation of the beats in frequency space. If the beats are too close and the bandwidth too large, the signals can mix with one another significantly and degrade the final measurement. It is also advisable to keep the small φ F beat as far in frequency space as possible from the φ n beats since the φ F measurement is relatively small and more sensitive to noise. Having chosen an f c for the highest frequency beat f 3, the requirement to produce a measurement without aliasing is f s > 2(f 3 + f c ). (5.10) 5.8 Calibration An apparatus to calibrate beam ellipticity (see Section 3.4) and collinearity (see Section 3.5) is now described. The calibration is done by sending the appropriate beams through a rotating optic. As shown in Figure 5.18, the optic is held in place by a mount that is rotated at about 15Hz with a DC motor. The apparatus is often placed on the polarimeter optics table. For this reason, a Sorobothane sheet is glued

142 5.8. CALIBRATION 117 Rotating mount DC motor Optic Sorbothane base Figure 5.18: Apparatus used for calibration of beam ellipticity and collinearity. to the base of the apparatus to minimize the transfer of vibrations to the optics table. The Sorobothane also helps suck the apparatus down onto the table, keeping it stationary during rotation Ellipticity Calibration To calibrate for ellipticity, the ω LR beam, which is composed of the circularly polarized ω L and ω R beams, is sent through a rotating λ/2-waveplate. The rotating waveplate generates an increasing or decreasing Faraday rotation signal depending on the direction of rotation. The waveplate essentially mimics a magnetized plasma, and pushes the Faraday rotation signal quickly through multiple cycles. In this way, the relationship between the true Faraday rotation (ie beams perfectly circularly polarized) and the measured Faraday rotation can be determined. Figure 5.19(a) shows a non-linear response between true and measured Faraday rotation as a result of beam ellipticity in the initial setup. The undesired ellipticity in

143 5.8. CALIBRATION 118 this particular case could account for Faraday rotation error on the order of a factor of 2 given typical plasma parameters. The source of the problem was determined to be during the conversion of the plasma beams from linear to circular polarization with two phase retarding reflectors. In this flawed arrangement (see Figure 5.19(b)), two phase retarding mirrors separately reflected the two plasma beams before they were combined. This setup avoided the use of an expensive half-waveplate, but proved to be more difficult to align properly and produced unacceptably elliptical beams. A suitable half-waveplate was obtained and placed into the system so that both plasma beams were combined before conversion to circular polarization (see Figure 5.15). The result was a dramatic improvement in the linearity of the response as shown in Figure 5.19(a). The error from ellipticity has been reduced to less than 0.1, which is below the resolution of the polarimeter Collinearity Calibration To minimize noise due to beam offset, the beams can be precisely aligned by passing them through a rotating wedge of ZnSe. The wedge angle is 14 arcmin (0.23 ). Non-collinear beams have slightly different path lengths in the wedge. Therefore, they experience different phase shifts and cause a change in the Faraday rotation signal. To improve the collinearity of the beams, their positions can be adjusted by translating and angling them in order to eliminate the signal created from the rotating wedge. Figure 5.20 gives an example of the signal produced from the beam alignment process. At present, the ω R beam can only be translated along the horizontal axis. To align it along the vertical axis, its position must be steered with mirror M2 and then its

144 5.8. CALIBRATION 119 Phase shift (deg) Initial setup Improved setup Time (ms) (a) Phase retarding reflectors Mirrors ω L ω R (b) Figure 5.19: Elliptically polarized beams. (a) Nonlinear and linear responses from calibration with spinning λ/2 waveplate [15]. (b) Original configuration of polarimeter that produced elliptical beams. Beams ω L and ω R first converted to circular polarization and then combined. angle corrected with beamsplitter BS02. Aligning the beam on this axis is a tedious exercise since an adjustment in the position requires a correction in the angle. A modification to the polarimeter s design could be made to allow for translation along both axes. The mount s rotation rate is limited to about 15Hz. If the signal is sampled at the standard rate of 125MHz, millions of samples are needed to collect a full rotation

145 5.9. TRIGGERING Phase shift (deg) Poor alignment Improved alignment Time (s) Figure 5.20: Apparent Faraday rotation during beam alignment by placing a rotating, 14 arcmin (0.23 ) wedge of ZnSe in the path of the plasma beams. Both signals have bandwidths of 500Hz. Signal sampled at 10.4kHz. of the wedge. This slows computation time and makes it difficult to obtain instant feedback between adjusting the position of a beam and seeing the effect on alignment on a computer screen. However, it is possible to significantly lower the number of samples needed by taking advantage of aliasing. When sampled below the Nyquist frequency, the frequency peak of interest shifts to a new frequency as a result of aliasing. Therefore, it is simply a matter of determining where this aliased peak is expected and changing the demodulation frequency in the algorithm accordingly. In this manner, a much lower sampling rate can be used, lowering the computational burden and easing the alignment process. 5.9 Triggering Three polarimeter functions must be enabled remotely with light sent through fibre optic cable. As soon as the capacitors start to charge for a shot, an optical trigger is sent to open the laser shutter, giving it about one minute before the shot is fired to

146 5.10. ELECTRICAL AND VIBRATIONAL NOISE 121 reach a temperature equilibrium. Once the capacitors finish charging, another optical trigger is sent to disconnect the polarimeter from the its power lines and run it off an uninterruptible power supply (UPS). This is a safeguard against electrical spikes travelling through the power lines. The entire polarimeter system is run off battery for only a few seconds. A third optical signal triggers the digitizers to begin recording data. This trigger is ideally synchronized with the triggers of other plasma injector diagnostics, but differences in the receiving trigger electronics can lead to a delay of about 10µs. Each of these optical triggers must be converted into an electrical signal that can be recognized by the relevant instruments. The optical to electrical trigger conversion circuit is shown in Figure The circuit consists of a photodiode, which generates a small current when it receives an optical signal. This current is then amplified with an operational amplifier. A feedback resistor controls the gain and typically has a value around 500kΩ. The resulting electrical trigger signal is then fed to the appropriate module. For the shutter, the signal is sent to a normally open relay that gives power to a linear actuator when closed, which opens the shutter. The powerdisconnect trigger signal is sent to two normally-closed relays connected to the two power lines. The polarimeter box and instruments remain connected to the ground line at all times for safety. The data collection trigger is sent directly to the digitizer external trigger channel Electrical and Vibrational Noise To facilitate alignment, the polarimeter box is located a few metres from the vacuum vessel at the 352 position. Unfortunately, this places the box near the capacitor

147 5.10. ELECTRICAL AND VIBRATIONAL NOISE 122 Figure 5.21: Optical to electrical trigger circuit. The resistor typically has a value of about 500kΩ to provide a large gain to the small photodiode signal. banks, which generate transient noise pulses during discharge that the equipment can pick up. For this reason, the laser, AOMs, and detection equipment are housed in a box shielded by 1/8 thick aluminum plates bolted to an aluminum construction rail frame. Conductive foam strips are glued to the outside of the frame to improve the contact between the plating and the frame. The sealed metal box acts as a Faraday cage that can reflect much of the electromagnetic noise. The box is connected to ground at a single point to avoid ground loops and power is routed through an isolation transformer. Just before a shot, the box is disconnected from the building power lines using a relay. During this short time, the equipment is powered with an uninterruptible power supply. Within the box, standard methods of shielding from radio frequency (RF) noise are used such as carrying power and signals with shielded BNC cables and avoiding ground loops whenever possible. These steps can help to reduce capacitor discharge noise. Initially, the digitizers were connected to a laptop inside the polarimeter box that processed the data and saved it to the network data drive via an ethernet connection. During high power acceleration shots, the noise from the discharge would often reset the digitizers and prevent the data from being collected. The problem was solved

148 5.11. RELATED WORK 123 by removing the ethernet connection, since this was essentially a copper wire that created a ground loop, picked up noise surges and disrupted the digitizers. The ethernet connection was replaced by an optical connection using an Icron Universal Serial Bus (USB) Ranger The only electrical connection to the box during a shot was the connection to ground. Once this change was made, the digitizers no longer reset themselves during a shot. The polarimeter is located in an area that is acoustically noisy due to the proximity of turbopumps and compressors. The injector itself generates significant vibrations during a shot. The polarimeter optical table is not isolated from mechanical vibrations. However, it is not believed such vibrations can cause significant noise since the measured phase signals are carried on a beat wavforms with frequencies in the MHz whereas mechanical vibrations are on the order of 100Hz [1] Related Work The designed polarimeter is among the fastest in the world. It diagnoses dense, accelerated spheromaks, which is unique. Table 5.2 provides a summary of polarimeters in past and current plasma experiments.

149 Experiment Plasma Type B n Wavelength Bandwidth Noise (T) (m 3 ) (µm) (khz) (deg) GF PI [15] Accelerated spheromak HIT-SI [37, 40] Steady state spheromak MST [12, 23] RFP ZT-40M [27] RFP NSTX [92] Tokamak DII-D [85] Tokamak JET [10, 64] Tokamak TEXTOR [57] Tokamak Alcator C-mod [9] Tokamak MTX [69] Tokamak LHD [1] Stellarator Table 5.2: Review of plasma polarimeters in other experiments. RFP stands for reversed-field pinch RELATED WORK 124

150 125 Chapter 6 Polarimeter Simulations An important parameter that must be chosen is the wavelength of the polarimeter beam. Faraday rotation is proportional to λ 2 0 as can be seen in Equation In this sense, it is favourable to use a beam with large λ 0. However, undesired plasma effects tend to become more severe with increasing λ 0. This chapter details simulations that have been made to help understand the effect of the choice of a particular λ 0 on the performance of the polarimeter. Plasma-beam interactions are simulated along the polarimeter chord at the PI position shown in Figure 5.1(b). The 352 chord has a length of 60cm. Most of the usable data has been collected for the 352 position. Therefore, simulation results are not explicitly given for the PI position, but can be inferred. The peak densities and magnetic fields at the 493 position can be expected to be greater by about a factor of 2. The chord length at 493 is 40cm. Simulations are done in two-dimensions (radial-azimuthal plane), since it is assumed that the plasma is much longer in the axial direction than in the radial direction. Therefore, the plasma is relatively uniform in the axial direction.

151 6.1. CHORD PROFILES AND SIMULATED POLARIMETRY Chord Profiles and Simulated Polarimetry The toroidal and poloidal magnetic fields are modelled from the Taylor state for an axisymmetric, coaxial geometry as presented in Section 4.2. The spheromak s toroidal field B φ peaks near the poloidal magnetic axis, which is near the midpoint between the two electrodes. The toroidal field goes to zero at the walls. Conversely, the poloidal field B θ peaks at the edges and is zero at the magnetic axis. In this configuration, the contribution of the radial field is small. For simplicity, it is assumed that B θ is purely in the axial direction and is therefore perpendicular to the beam s chord. This is equivalent to the spheromak being centred along its axial length at the polarimeter chord. This axial position is of greatest interest since it is where the spheromak exposes its highest toroidal field, axial field, and density to the polarimeter chord. The plasma density is assumed to have the same shape as the toroidal field, giving high density at the core and low density at the walls. This is representative of a plasma undergoing diffusion from the core to the wall where charged particles are neutralized. Solving for the required parameters with the 352 electrode geometry gives k ρ = 17.6m 1, and f = The resulting radial profiles are plotted in Figure 6.1. Faraday rotation depends on the component of the magnetic field parallel to the polarimeter s chord, B, which in this case is in the vertical direction. Since it has been assumed that B θ is purely axial and therefore perpendicular to the chord, only the vertical component of toroidal field contributes to the Faraday rotation. As seen in Figure 6.2(a), at a point on the chord with radial distance ρ and azimuthal angle

152 6.1. CHORD PROFILES AND SIMULATED POLARIMETRY 127 Magnetic field (T) Density, n (10 21 m 3 ) B 1.0 φ 1.5 B θ Radial distance, ρ (cm) Figure 6.1: Simulated density and magnetic field radial profiles at the 352 axial position for n 0 = m 3 and B 0 = 2T. Plasma has an axisymmetric profile modelled after a coaxial Taylor state. φ, the parallel magnetic field is B (ρ, φ) = B φ (ρ) cos φ (6.1) Knowing the radial profiles of B φ (ρ) and n(ρ), the chord profiles of B (y) and n(y) can be interpolated. These chord profiles are plotted in Figure 6.2(b). At the 352 position, the maximum density ranges from n 0 = m 3 to m 3, and the maximum magnetic field strength from B 0 = 0.5T to 2T. The resulting simulated Faraday rotation is between φ F = 0.3 and 14. The Faraday rotation is negative since the directions of the beam and parallel toroidal field oppose one another. The unwrapped phase shift due to density for n 0 = m 3 is φ n = 8100, which is about three orders of magnitude larger than the Faraday rotation phase shift and

153 6.1. CHORD PROFILES AND SIMULATED POLARIMETRY 128 y x Beam chord ρ Φ B Φ (ρ) B Φ (ρ,φ) Φ B Φ (ρ,φ)cosφ B Φ (ρ,φ)sinφ Magnetic field (T) Density (10 21 m 3 ) (a) Position on chord (cm) (b) Figure 6.2: Simulated density and magnetic field chord profiles at the 352 axial position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state. (a) Toroidal magnetic field components along the beam chord. (b) Chord profiles for n 0 = m 3 and B 0 = 2T. The poloidal field B θ is taken to be purely in the axial direction. The components of the magnetic field parallel and perpendicular to the chord are respectively B and B. B φ B θ B B

154 6.2. CHARACTERISTIC PLASMA FREQUENCIES 129 so can be a significant source of noise (see Section 6.6). The measured φ n can be converted to an average density n when it is divided by the constant coefficient from Equation 3.16 and the chord length L. A rough estimate of the average toroidal magnetic field B φ can also be obtained by dividing the measured Faraday rotation φ F by the constant coefficient from Equation 2.63, the chord length, and the average density n. In the current configuration with L = 60cm, this gives n = m 3 and B φ 1.7T. The assumed profiles for toroidal magnetic field and density suggest that the central chord points (ie near y = 0) should contribute more to the total Faraday rotation observed than chord points near the wall. Also, the component of the toroidal field parallel to the chord is smaller closer to the wall, adding to this effect. The weighting of points along the chord is shown in Figure 6.3. The result shows that the observed Faraday rotation should be representative of the peak density and toroidal magnetic field near the core of the plasma. At the 352 position, the measured average density n and toroidal field B φ respectively correspond to 78% of the peak density and 85% of the peak toroidal field. 6.2 Characteristic Plasma Frequencies As described in Section 2, the beam frequency ω must be much greater than three characteristic frequencies of the plasma: the collision frequency ν, the electron gyrofrequency ω c, and the plasma frequency ω p. From Equations 2.11, 2.16 and 2.1 it

155 6.2. CHARACTERISTIC PLASMA FREQUENCIES Weighting (unitless), Field (unitless) Far. rot. weighting Normalized field magnitude Position on chord (cm) Figure 6.3: Weighting of Faraday rotation contributions from points along the polarimeter chord at the 352 position. Also shown is the toroidal field magnitude at points along the chord normalized to the peak toroidal field in the plasma. Plasma has an axisymmetric profile modelled after a coaxial Taylor state. is seen that these frequencies hold the following proportionality relationships: ν nt 3/2, (6.2) ω c B, (6.3) ω p n. (6.4) Therefore, the most challenging situation is a plasma with high n and B, and low T. The most difficult scenario for the polarimeter at 352 is taken to be with n = m 3, B = 2T, and T = 10eV. The resulting characteristic frequencies are summarized in Table 6.1 and are compared to the beam frequency for a 10.6µm beam. It can then

156 6.3. COTTON-MOUTON EFFECT 131 ω (THz) ν (THz) ω c (THz) ω p (THz) Table 6.1: Frequency of a 10.6µm beam compared to characteristic frequencies of the plasma assuming n = m 3, B = 2T, and T = 10eV. be stated that ω ν, (6.5) ω ω c, (6.6) ω ω p. (6.7) Therefore, the corresponding assumptions made in previous derivations are expected to be valid. 6.3 Cotton-Mouton Effect It must be verified that the Cotton-Mouton effect does not significantly change the ellipticity of the beam, which can distort the Faraday rotation measurement. As shown in Equation 2.67, the change in ellipticity due to the Cotton-Mouton effect has the relationship ɛ λ 3 0nB 2. (6.8) To estimate the potential significance of this effect, an axisymmetric plasma is simulated with density and magnetic field profiles as previously discussed. The poloidal field is again assumed to be in the axial direction, and therefore contributes to B. As can be seen in Figure 6.2(a), the other contribution to B comes from the toroidal

157 6.4. REFRACTION 132 field: B φ sin φ. The total field perpendicular to the chord is then B = ( (B φ sin φ) 2 + B 2 θ) 1/2. (6.9) The chord profiles of n, B φ and B θ are obtained by interpolating their respective radial profiles at positions (ρ, φ) along the beam s chord. The resulting chord profiles are given in Figure 6.2(b). Running the simulation with 100 points along the chord, and again taking the peak density and field to be n 0 = m 3 and B 0 = 2T, the change in ellipticity is found to be ɛ = Therefore, the Cotton-Mouton effect is not expected to contribute significantly to Faraday rotation measurement error. 6.4 Refraction Refraction may cause the beam to significantly deflect inside the plasma. The angle of refraction is given by Equation 3.56 and depends on the density gradient perpendicular to the beam s direction of travel. To investigate this effect in simulation, a simple 2-dimensional problem is considered as in Figure 5.1(b). The trajectory of the beam is calculated by taking small steps of length dl in the direction of the beam. At each step, the transverse density gradient is approximated and the new direction of the beam is computed. The iteration is repeated until the beam intersects the outer electrode. The local density gradient is computed by considering two points that are spaced equally by 1 s on either side of the beam curve in the local transverse direction as 2 shown in Figure 6.4. The respective densities at these points are n(x 1 2 s x, y+ 1 2 s y) and n(x s x, y 1 2 s y) where s x = s cos θ R and s y = s sin θ R. For small

158 6.4. REFRACTION 133 n ( x 1 2 s x, y s ) y Beam tangent at (x, y) Beam θ R s y (x, y) s s x θ R n ( x s x, y 1 2 s ) y Figure 6.4: Geometry used in refraction simulation to approximate the local transverse density gradient. s, the local transverse density gradient is well approximated by n s = n(x 1 s 2 x, y + 1 s 2 y) n(x + 1 s 2 x, y 1 s 2 y). (6.10) 2 s The refracted beam path is simulated using dl = s = 1mm and for a number of densities as shown in Figure 6.5. The same density profile considered in the previous sections is used. When n 0 = m 3, the total displacement of the beam at the injector exit port is about 0.5mm. The clear aperture diameter of the ZnSe windows is about 13mm. The beam diameter at the windows is about 6mm (see Figure 5.16). Therefore, refraction is expected to have no effect on the beam s passage through the ZnSe vacuum vessel exit window. The total change in the angle of the beam is about 0.1. For beams with radius

159 6.4. REFRACTION y-position (cm) n 0 =10 21 m 3 n 0 =10 22 m 3 n 0 =10 23 m 3 n 0 =10 24 m x-position (cm) Figure 6.5: Simulated refraction of the 10.6µm polarimeter beam passing through the plasma at the 352 axial position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state. Simulation results are given for several peak densities n 0. W = 0.1mm the minimum requirement for phase contrast is that the angular alignment between the beams is less than about 5 (see Equation 3.57). Therefore, in terms of the effect of angular alignment on phase contrast, refraction is negligible. However, the displacement of the beam at the detector due to refraction can be important. For example, for a 5m path from the injector exit port to the detector, the displacement due to the change in angle is approximately 1cm. The active area of a detector is only about 1mm 2. In this configuration, for a displacement of less than 1mm, a maximum plasma density n 0 < m 3 is required. Therefore, it is expected that the signal could be lost when the plasma density exceeds this threshold. The polarimeter s raw interference signal has been observed to briefly disappear when especially high density plasmas pass by.

160 6.5. WKBJ APPROXIMATION WKBJ Approximation As described in Section 2.5, the WKBJ approximation allows density gradient reflections to be ignored. The phase shift incurred by a beam travelling through a plasma can then be represented by a simple integral. The requirement for the approximation to be valid is given by Equation Restating this equation for the geometry of the polarimeter chord at the 352 position (see Figure 5.1(b)) gives W = 1 k k 2 y 1 (6.11) To verify that the WKBJ approximation applies, a two-dimensional problem is again considered. The density chord profile is the same as before (see Figure 6.2(b)). It is necessary to estimate the spatial gradient of the wavenumber in the direction of the beam path. This is done by approximating the gradient over steps of 1mm. The result shown in Figure 6.6 indicates that the WKBJ approximation is valid since W 1 across the entire beam path. 6.6 Beam Collinearity As discussed in Section 3.5, the measured Faraday rotation can be significantly distorted due to density phase noise if the beams ω L and ω R sent through the plasma are offset in space such that each encounters a slightly different density profile. The parameter C from Equation 3.37 describes the magnitude of the difference between density phase noise and the true Faraday rotation at a particular point in space. For collinearity noise to be small, it must be that C 1 over a significant portion of the beam path.

161 6.6. BEAM COLLINEARITY Density, n (10 22 m 3 ) n W W = 1 k 2 k y ( 10 7 ) y-position (cm) Figure 6.6: WKBJ approximation simulation results for 10.6µm beam for the polarimeter chord at the 352 position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state with n 0 = m 3. A simulation is run to estimate C for points along the 352 polarimeter chord, as well as the total magnitude of the phase error introduced due to the collinearity effect. The simulated plasma is again assumed to have an axisymmetric profile modelled after a coaxial Taylor state with B 0 = 2T and n 0 = m 3. The beams are taken to be offset by 1mm, which is expected for an alignment done by eye. The simulation finds that the beam collinearity parameter is greater than 1 for much of the beam path (see Figure 6.7). The magnitude of the maximum density phase noise is φ = 29, while the magnitude of the true Faraday rotation phase shift (ie beams perfectly collinear) is 14. Under these conditions, the density phase noise dominates the Faraday rotation measurement. A beam offset of about 50µm is required to bring collinearity noise down to 10% of the desired signal. This also places a requirement on the angular deviation of the two beams. For an approximately 10m long path, the beams must have an angular deviation of no more than 50µm/10m = 5µrad. The angular deviation can

162 6.7. OPTICALLY ACTIVE WINDOWS Collinearity parameter, C (unitless) Position on chord (cm) Figure 6.7: Simulation of the collinearity parameter, C, at various positions along the polarimeter chord at the 352 position. Plasma has an axisymmetric profile modelled after a coaxial Taylor state with B 0 = 2T and n 0 = m 3. be reduced by aligning the beams at two points separated by a large distance. If the beams positions can only be resolved by eye to within 1mm, the two points must be separated by more than 1mm/ = 200m. 6.7 Optically Active Windows When light is passed through optically active materials, a Faraday effect is observed in the presence of a magnetic field, analogous to Faraday rotation in a magnetized plasma. The strength of this effect is reflected in the Verdet constant, V, of the material. The Faraday rotation of a beam of light in an optically active material is φ F = V db (6.12)

163 6.8. PLASMA LIGHT 138 where d is the path length through the material, and B the magnetic field in the material parallel to the direction of the beam. The polarimeter s ω LR beam passes through 3mm thick ZnSe windows on the vacuum vessel. ZnSe has a relatively high Verdet constant and can therefore cause a significant Faraday rotation at high magnetic fields. The Verdet constant [53] for ZnSe at 10.5µm is V = 20±6 T 1 m 1. The magnetic field at the windows is strongly attenuated since the windows are recessed in a 10cm deep well. Therefore, an upper limit of B = 0.1T at the windows is chosen, though the true field at the windows is likely much less. For transmission through two windows with V = 26 T 1 m 1, the Faraday rotation due to the windows is This is below the current resolution of the Faraday rotation measurement of about 0.1. Therefore, the windows have only a small effect on measurements. 6.8 Plasma Light It was hypothesized that 10.6µm light emitted from the plasma from a particular direction could make its way back to the polarimeter detector and generate noise. However, the light would have to be extremely bright and collimated to begin to match the intensity of the laser light. For this reason, the plasma light effect is believed to be an unlikely source of noise. A series of experiments were done to check this by blocking the laser light from the detector, but allowing a path for the plasma light. The position of the plasma could be monitored with surface probe axial field and interferometer density measurements. No change in the signal was observed when the plasma passed by, suggesting plasma light was not a major source of noise.

164 6.8. PLASMA LIGHT 139 For a pure hydrogen plasma, the bremsstrahlung power emitted per unit volume, per unit solid angle, per unit wavelength in the range λ to dλ is [31]: dp br = gn 2 ( T 1/2 λ exp λt ) dλ [W/(cm 3 angstrom sr)] (6.13) where g is the Gaunt factor (quantum correction), n in cm 3, T is in kev and λ is in angstroms. For this rough calculation, the Gaunt factor is approximated as g 1. The detector is sensitive from about 1µm to 12µm. The volume of plasma considered is a cylinder with radius of 6mm, length of 600mm, average density of n = m 3 and average temperature of T = 50eV. The average solid angle Ω is conservatively set as the ratio of the window area (π 6 2 mm 2 ) to the surface area of a sphere from the middle of the cylindrical plasma volume (4π (600/2) 2 mm 2 ) giving Ω The bremsstrahlung power at the detector is about P br 1mW. The combined laser power is about 100mW. Therefore, the plasma light noise is expected to be small.

165 140 Chapter 7 Measurements of an Accelerated Spheromak Density and Faraday rotation measurements from the polarimeter and interferometer are presented for 18 PI-1 shots and 10 PI-2 shots. The criterion for choosing these shots is that both polarimeter and interferometer density measurements integrate out to zero once the plasma has passed by. The shots are selected from a pool of several hundred where density measurements do not integrate to zero or other important information is missing (eg, interferometer malfunction). Unless otherwise noted, interferometer and polarimeter density measurements have a bandwidth of 5MHz, polarimeter Faraday rotation measurements have a bandwidth of 1MHz, and magnetic probes bandwidths are typically around 0.5MHz. 7.1 Time Delays A time delay typically on the order of tens of microseconds exists between the measurements of the polarimeter and other plasma injector diagnostics such as the interferometer and magnetic probes. This is likely due to differences in the triggering circuitry between the diagnostics. The time delay is measured manually by comparing the timing of the accelerator discharge electrical noise spikes between the polarimeter

166 7.2. NOISE FLOOR FOR A NULL SIGNAL 141 density and the wall probe measurements (see Figure 7.1(a)). This gives the time delay needed to synchronize the polarimeter and magnetic probes. The interferometer does not have a significant delay compared to the magnetic probes. Therefore, the same delay can be used to synchronize the interferometer with the polarimeter (see Figure 7.1(b)). 7.2 Noise Floor for a Null Signal A null signal is defined as the signal measured during a shot when the two beams sent through the injector do not pass through plasma. The null signal gives an estimate of the noise floor for a particular measurement. For example, in Figure 7.1(b), the times representing a null signal range from about 400µs and earlier, and from 500µs and later. The noise floor is calculated as the standard deviation of the first 100µs of polarimeter density and Faraday rotation signals. Figures 7.2(a) and 7.2(b) show the noise floors as a function of the product A ref A pla, where these are respectively the amplitudes of the relevant reference and plasma beat signals. The amplitudes are found by taking the Fourier transform of the raw signal from 0 to 100µs and measuring the height of the relevant beat peak. For example, for Faraday rotation, the amplitude is the height of the ω LR beat in frequency space. As would be expected, an anti-correlation is found between the noise floor and the signal strength. For the selected shots, the smallest noise floor attained is about m 3 for density measurements and about 0.1 for Faraday rotation measurements. The noise spectrum of the null signal is white (ie evenly distributed in frequency), suggesting a likely source is detector thermal noise. To improve the noise floor, the detectors

167 7.2. NOISE FLOOR FOR A NULL SIGNAL 142 Density (10 21 m 3 ), Magnetic field (T) PI1 - Shot Polarimeter density Original poloidal wall field Shifted poloidal wall field Noise spike Time(µs) (a) 4 3 PI1 - Shot Interferometer Polarimeter Density (10 20 m 3 ) Time(µs) (b) Figure 7.1: Polarimeter is typically out of sync with interferometer and magnetic probes. (a) Polarimeter and magnetic probes are synchronized by matching electrical noise spikes from the accelerator discharge. (b) Interferometer is already in sync with magnetic probes, and so can be shifted by the same delay as the probes.

168 7.3. DENSITY MEASUREMENTS 143 could be cooled to lower the thermal noise. 7.3 Density Measurements Recall that the plasma density affects the phase of a beam passing through. The polarimeter and interferometer measure line-averaged density by comparing the phase of a beam that passes through the plasma to a beam that does not pass through the plasma. This section discusses the density measurements captured by the two diagnostics Missed Fringes on the Polarimeter and Interferometer Density Signals If the change in density between two sampling points is too large, then the correct phase change might be incorrectly measured if a fringe is missed. For example, if the actual phase change is 2π + π/3, the observed phase change would incorrectly be measured as π/3. Although there are small differences between the polarimeter and interferometer density measurements that could be due to asymmetries in the plasma, larger density differences are likely due to one or both diagnostics missing at least one fringe. A solution is to increase the sampling rate and bandwidth of the measurements, which would effectively slow the rate of change in phase. Another possible solution is to reduce the beam wavelength since the phase change is proportional to wavelength. Given the polarimeter beam s wavelength of λ 0 = 10.6µm and the chord length of 0.6m, Equation 3.16 indicates that a 360 fringe shift corresponds to a density change of m 3. A typical plasma at the 352 position has a density of about m 3.

169 7.3. DENSITY MEASUREMENTS Null signal std. dev. (10 19 m 3 ) Amplitude product, A ref A pla (arbitrary) (a) Null signal std. dev. (deg) Amplitude product, A ref A pla (arbitrary) (b) Figure 7.2: Noise floor data from null signals for polarimeter measurements of (a) density and (b) Faraday rotation. The density and Faraday rotation null signals respectively have a bandwidth of 5MHz and 1MHz. The amplitude product variable is normalized so the highest measured value is 1.

170 7.3. DENSITY MEASUREMENTS 145 Therefore, even missing a single fringe can produce a notable effect. An obvious indication of a missed fringe is when the density signal does not integrate out to zero after the plasma has passed by (see Figure 7.3(a)). However, it is also possible that fringes have been missed even when the density measurements do return to zero. For example, a fringe representing an increase in density could be missed and balanced out by a fringe representing a decrease in density at a later time. This is believed to be the reason for occasional large differences observed between the polarimeter and interferometer s respective maximum densities (see Figure 7.3(b)). The polarimeter has incorrectly missed fringes that the interferometer has correctly measured. This would be expected to happen since the polarimeter s wavelength is about 10 times as long as the interferometer s. The long polarimeter wavelength is needed to resolve Faraday rotation measurements, which are small but increase with λ 2 0. In this regard, the polarimeter would experience phase shifts 10 times as large and would be more likely to miss large changes in density. The alternative explanation is that there is a large asymmetry in the plasma, which is unlikely since it is not observed at other axial positions. The interferometer also occasionally missed fringes as shown in Figure 7.4. While the interferometer s beam has a much shorter wavelength, the signal is sampled at 20MHz compared to the polarimeter s 125MHz. A faster sampling rate provides smaller changes in density between sample points. In addition, the interferometer density signal is typically much noisier than the polarimeter s, which can also increase the chance of missing a fringe. Therefore, while the interferometer is believed to be a more accurate representation of the maximum plasma density, it is not orders of magnitude better. A stronger polarimeter signal, and a higher sampling rate could offset

171 7.3. DENSITY MEASUREMENTS Interferometer Polarimeter Interferometer Polarimeter Density (10 21 m 3 ) PI2 - Shot 6902 Density (10 21 m 3 ) PI1 - Shot Time (µs) (a) Time (µs) (b) Figure 7.3: Polarimeter has a higher chance of missing fringes since its wavelength is 10 times longer than the interferometer s. (a) Polarimeter density measurement does not return to zero after the plasma has passed, clearly indicating a fringe has been missed. (b) Polarimeter density measurement returns to zero after plasma passes, but there is a large difference in maximum densities, which is caused by the polarimeter missing fringes and does not represent an actual asymmetry in the plasma. the negative impact of the polarimeter s long wavelength. A large, low-frequency disturbance (see Figure 7.4) occurring at around the time the plasma passes by can also be sometimes seen on interferometer signals. These are likely due to vibrations generated during the shot, which directly affect the amplitude-based interferometer measurements Polarimeter Dual Densities The polarimeter gives two density measurements since two beams, ω L and ω R, are sent through the plasma and interfered with the ω V beam that does not pass through the plasma. From the perspective of a density measurement, the difference between the two polarimeter density signals is negligible as seen in Figure 7.5.

172 7.3. DENSITY MEASUREMENTS Density (10 21 m 3 ) Interferometer Polarimeter PI2 - Shot Time (µs) Note the large low- Figure 7.4: Interferometer also occasionally missed fringes. frequency disturbance beginning at 500µs. Density (10 21 m 3 ) Density 1 Density 2 PI2 - Shot Time (µs) Figure 7.5: Polarimeter gives two density measurements since two beams pass through the plasma. The density signals are typically very similar.

173 7.3. DENSITY MEASUREMENTS 148 Occasionally, one signal might not integrate out to zero once the plasma has passed by, producing a large difference between the two signals (see Figure 7.6(a)). Whether or not the signal integrates out to zero depends on its beat signal strength. As seen in Figures 7.6(b) and 7.6(c), the plasma and reference signals for the 25MHz beat ( Density 1 ) are significantly stronger than for the 40MHz beat ( Density 2 ). This extra signal strength allows the 25MHz beat density to avoid missing a fringe in this case. This is another indication that it is important to have as high a signal strength as possible Correlation between Interferometer and Polarimeter Densities The correlation between the interferometer and polarimeter densities is defined by the Pearson correlation coefficient: r = cov (n i, n p ) σ i σ p (7.1) where n i and n p are respectively the interferometer and polarimeter density measurements, and σ i and σ p their standard deviations defined as [ ( σ X = Xj X )] 1/2. (7.2) The covariance between two variables X and Y is defined by i cov (X, Y ) = j ( Xj X ) ( Y j Ȳ ). (7.3)

174 7.3. DENSITY MEASUREMENTS 149 Density (10 21 m 3 ) Density 1 Density 2 PI2 - Shot Time (µs) (a) PI2 - Shot Reference Signal 7 PI2 - Shot Plasma Signal Magnitude (arbitrary) Far. Rot. Density 1 Density 2 Magnitude (arbitrary) Far. Rot. Density 1 Density Frequency (MHz) (b) Frequency (MHz) (c) Figure 7.6: Large differences between polarimeter density measurements. (a) Occasionally, one density measurement misses a fringe while the other does not. This is due to differing signal strengths between the signals beats. The beat with the weaker signal strength misses the fringe as can be seen from the Fourier transforms of the raw (b) reference and (c) plasma signals.

175 7.4. FARADAY ROTATION MEASUREMENTS 150 The Pearson correlation coefficient gives 1 for a perfect positive correlation, and zero for no correlation. The range of the data over which to calculate the correlation is selected manually for each shot as the region about the peak density with significant density features. The correlation results for all examined shots are shown in Figure 7.7(a). Most shots had a correlation of about 0.5 or better indicating there is some correlation between the two density measurements. As seen in Figure 7.7(b), large density gradients are missed by the polarimeter and give a low correlation, whereas higher correlations are obtained from shots with slower changes in density as in Figure 7.7(c). There is also likely some small asymmetry in the plasma, which could explain the smaller differences between the two measurements. In terms of accuracy at measuring the maximum density in the plasma, the interferometer performs better than the polarimeter since the polarimeter is more likely to miss a fringe when there is a large density gradient. However, the polarimeter s higher sensitivity and robustness to vibrational noise, gives it the advantage when measuring low density plasmas (< m 3 ) or finer plasma density features. 7.4 Faraday Rotation Measurements The polarimeter obtains Faraday rotation by measuring the phase difference between a left and a right-circularly polarized beam sent through the plasma. Faraday rotation depends on the line-integrated product of density and magnetic field component parallel to the beam. As shown in Section 6.6, if the plasma-probing beams are not sufficiently collinear, each beam can acquire significantly different phase shifts due to the density gradient. This is believed to be one of the main sources of noise for

176 7.4. FARADAY ROTATION MEASUREMENTS 151 Counts Correlation coefficient, r (unitless) (a) Density (10 21 m 3 ) Interferometer Polarimeter PI1 - Shot Density (10 21 m 3 ) Interferometer Polarimeter PI2 - Shot Time (µs) (b) Time (µs) (c) Figure 7.7: Correlation between the interferometer and polarimeter density measurements. (a) Distribution of Pearson correlation coefficients for all analysed shots. (b) Density measurements for shot with worst correlation, r = Polarimeter misses large density gradients. (c) Measurements for shot with the best correlation, r = 0.81.

177 7.4. FARADAY ROTATION MEASUREMENTS 152 Shot # Field Ratio Table 7.1: Ratios of the maximum internal probe array toroidal field at the magnetic axis to the maximum axial field from the wall probe. The mean ratio is 2 with a standard deviation of 0.4. polarimeter Faraday rotation measurements. Faraday rotation measurements can be compared to a Faraday rotation model generated with Equation (2.63) using other diagnostics as inputs. The model assumes axisymmetry. The radial density profile is modelled after the toroidal field profile for the coaxial geometry Taylor state (see Section 6.1). The density profile is constrained by the line-averaged polarimeter density measurement. Internal magnetic field probe measurements were not available during most polarimeter shots, so the toroidal field also had to be modelled. From previous shots with an inserted magnetic probe array, the magnetic axis (ie. peak toroidal field) is located at about the midpoint between the two electrodes, ρ = 0.41m. In addition, the ratio of the surface probe peak axial field to the magnetic axis peak toroidal field is approximately constant. The field ratio is measured to be B θ (axis)/b z (wall) = 2±0.4 using data from 9 shots (see Table 7.1). The spheromak s toroidal field at the magnetic axis is modelled by taking the surface probe axial field and multiplying it by the field ratio. This accounts for fields

178 7.4. FARADAY ROTATION MEASUREMENTS 153 Magnetic field (T) PI1 - Shot Pol. probe contribution Tor. probe contribution Model tor. at magnetic axis Time(µs) Figure 7.8: Model toroidal field at the magnetic axis is generated from the sum of the wall probe s modified poloidal and toroidal fields. The poloidal field measurement at the wall is multiplied by 2 to approximate the spheromak toroidal field. The pushing field contribution at the magnetic axis is obtained from the wall probe toroidal field multiplied by a factor of 1.2 to account for its 1/ρ dependence. observed early in the shot created by the spheromak. The profile of the spheromak s toroidal field is estimated using the Taylor state profile for coaxial geometry from Section 6.1. At later times, once the spheromak has passed, the pushing field dominates. The toroidal pushing field is the result of currents flowing through the injector electrodes. The strength of the pushing field at a given radius inside the injector can be determined by scaling the toroidal field measured by a wall probe by 1/ρ. This gives a profile of the toroidal pushing field, which can be added to the profile of the spheromak field to obtain an estimate of the total toroidal field profile in the injector (see Figure 7.8). The resulting modelled field typically matches well with the actual toroidal field measured with internal probes as seen in Figure 7.9.

179 7.4. FARADAY ROTATION MEASUREMENTS PI1 - Shot Model Internal Probe PI1 - Shot Model Internal Probe Toroidal field (T) Toroidal field (T) Time(µs) (a) Time(µs) (b) Figure 7.9: Comparison of modelled toroidal field at the magnetic axis to the field measured by internal probes for the (a) worst and (b) best match for maximum toroidal field. Before the rotating wedge was used to refine the alignment, the measured Faraday rotation typically agreed poorly with the modelled Faraday rotation as shown in Figure Pre-alignment measurements are also sometimes characterized by a sustained positive Faraday rotation. Given the known direction of the plasma toroidal magnetic field, the laser beam s direction of travel through the plasma, and the handedness of the circularly polarized plasma beams, the Faraday rotation signal should be predominantly negative. Therefore, signals with long periods of positive Faraday rotation are unlikely to be good measurements since this would require a sustained toroidal field in the direction opposite the expected one. A simple experiment can be done to evaluate whether or not the ω L and ω R beams are significantly non-collinear by passing the beams through a thin film polarizer before entering the plasma. The polarizer strongly reflects away one component of the field (eg horizontal polarization) and strongly transmits the orthogonal component (eg vertical polarization). In doing so, the ω L and ω R beams become linearly polarized

180 7.4. FARADAY ROTATION MEASUREMENTS 155 Faraday rotation (deg), magnetic field (T) PI1 - Shot Polari. Far. rot. Probe Far. rot. model Mag. axis tor. field Polari. density Time (µs) Density (10 21 m 3 ) Figure 7.10: Poor agreement between modelled and measured Faraday rotation before improved alignment with the rotating wedge. This suggest large collinearity noise. and lose their ability to detect true Faraday rotation. However, they can be used to measure density by the usual interference with the ω V beam at the detector. In this scenario, if the beams are sufficiently collinear, the measured Faraday rotation should be zero. Conversely, if the beams are significantly non-collinear, a Faraday rotation signal should be observed, which is related to the different density phase shifts experienced by the ω L and ω R beams. Figure 7.11 presents the results of this experiment, and indicates that the beams are non-collinear since a significant Faraday rotation signal is observed. Results after the improved beam alignment are in better agreement with the model (see Figure 7.12). The calibrated polarimeter generally produces Faraday rotation measurements in the correct direction. Discrepancies between the model and measurement can primarily be attributed to capacitor discharge noise at early times,

181 7.4. FARADAY ROTATION MEASUREMENTS Density Shot Faraday rotation (deg) Faraday rotation Density (x10 21 m -3 ) Time (µs) Figure 7.11: Polarimeter measurements when the ω L and ω R beams are sent through a polarizer before passing through the plasma. Non-zero Faraday rotation indicates beam collinearity is not negligible. Faraday rotation and density signal bandwidths are respectively 100Hz and 1MHz. limitations of the model, and to a lesser degree collinearity noise. The remaining collinearity noise is believed to be responsible for the positive Faraday rotation occasionally observed in regions of high density with large density gradients. Under these conditions, the error can be on the order of the signal magnitude.

182 Faraday rotation (deg), magnetic field (T) Faraday rotation (deg), magnetic field (T) PI1 - Shot Polari. Far. rot. Probe Far. rot. model Mag. axis tor. field Polari. density Time (µs) PI1 - Shot Polari. Far. rot. Probe Far. rot. model Mag. axis tor. field Polari. density Time (µs) Density (10 21 m 3 ) Density (10 21 m 3 ) Faraday rotation (deg), magnetic field (T) Faraday rotation (deg), magnetic field (T) PI1 - Shot Polari. Far. rot. Probe Far. rot. model Mag. axis tor. field Polari. density Time (µs) PI1 - Shot Polari. Far. rot. Probe Far. rot. model Mag. axis tor. field Polari. density Time (µs) Figure 7.12: Improved agreement between modelled and measured Faraday rotation after the rotating wedge was used to refine the alignment Density (10 21 m 3 ) Density (10 21 m 3 ) 7.4. FARADAY ROTATION MEASUREMENTS 157

183 7.4. FARADAY ROTATION MEASUREMENTS 158 Faraday rotation (deg), magnetic field (T) PI1 - Shot Polari. Far. rot. Probe Far. rot. model Mag. axis tor. field Polari. density Time (µs) Density (10 21 m 3 ) Figure 7.13: Faraday rotation filtered down to 300kHz makes the signal clearer at the expense of detail. As shown in Figure 7.13, further low-pass filtering of the Faraday rotation signal can reduce the white noise as well as the suspected electrical noise. The Faraday rotation signal is then somewhat clearer, but this comes at the expense of detail in the signal. Due to the favourable geometry at the 352 axial position (see Section 6.1), a good estimate of the inner toroidal field can be obtained from polarimeter measurements using Equation Figure 7.14 gives this result for PI-1 shot with the chord length taken to be 60cm. Regions of low density are omitted due to small signal to noise. The high field spikes (>1T) at early times are believed to be due to collinearity and electrical noise. Near the peak in density, the inner toroidal field estimate is about 0.6T. In general, the calibrated polarimeter estimates a magnetic field on the order of 1T, which is consistent with inserted probe array measurements of the toroidal field on other similar shots.

184 7.5. AXIAL RESOLUTION 159 PI1 - Shot Magnetic field, B φ (T) B φ n Time (µs) Density, n (10 21 m 3 ) Figure 7.14: Inner toroidal field estimated from polarimeter Faraday rotation (filtered down to 300 khz as shown in Figure 7.13) and density measurements. Chord length is 60cm. 7.5 Axial Resolution The spheromak is accelerated down the injector at high speeds. As the spheromak passes by the polarimeter chord, it is axially scanned. The spheromak can be resolved axially if the speed of the polarimeter s measurements is sufficiently fast given the spheromak s length and velocity. Given the properties of the spheromak, the axial resolution of a polarimeter measurement is now estimated as a function of the measurement bandwidth. The spheromak s length and speed must be estimated from surface probe axial fields. The spheromak s axial length, L sph, is taken as the full-width half maximum (FWHM) of the spline-fit to surface probe axial fields at the time of the maximum Faraday rotation signal (see Figure 7.15). The spheromak s speed, v sph, is L sph divided by the amount of time it is observed at the polarimeter s axial position, which is

185 7.5. AXIAL RESOLUTION 160 Surface probe axial field (T) Shot Raw points Spline fit Position from back flange (cm) Figure 7.15: Spline-fit to surface magnetic probe axial field measurements at time of peak polarimeter Faraday rotation (488µs). measured as the FWHM of the poloidal surface probe signal (as in Figure 5.4). The polarimeter s axial resolution is defined as δz = v sph /f c. At the 352 position, v sph ranges from 30km/s to 150km/s. Therefore with f c = 1MHz, δz is between 3cm and 15cm. Typically, L sph is between 0.75m and 2m, meaning the number of resolvable scale lengths L sph /δz ranges from 5 to 67. For a 1.0m-long spheromak passing the 352 position at 100km/s, the polarimeter can resolve 10 scale lengths across the spheromak. Therefore, the need for a bandwidth in the MHz is justified in order to axially resolve the spheromak.

186 161 Chapter 8 Future Work This chapter lists recommendations for future work to improve the polarimeter s performance. 8.1 Translation Alignment In the current polarimeter setup, the ω L and ω R beams are combined using a single translation stage (see Section 5.6) and their collinearity is checked with a rotating ZnSe wedge (see Section 5.8.2). Since only one translation stage is used, the beams can only be aligned along one axis with a sufficient degree of sensitivity. The ω R beam is effectively translated on the other axis by steering it with a Kinematic mirror. The issue is that this changes the angular deviation of the beams. Therefore, the angular deviation must be corrected with a separate Kinematic mirror, which introduces once again a translation misalignment. This iterative process makes it very difficult to align the beams to within the required 50µm to avoid significant beam collinearity noise (see Section 6.6). The translation error can be corrected by adding an additional translation stage as shown in Figure 8.1. This would allow the ω R beam to be finely translated along both

187 8.2. BEAM PROFILER 162 ω LR To phase retarding reflector Kinematic mount ω L ω R z y x Translation stages Figure 8.1: Proposed beam combination design with two translation stages for improved beam collinearity. axes with no change to the angular deviation. The translation stages can be translated to an accuracy of within 10µm, which should satisfy the collinearity requirements. 8.2 Beam Profiler For a total beam path length of 10m, the angular deviation criterion (see Section 6.6) requires the beams to be aligned at two points separated by 200m if done by eye. Such a large distance between measurement points is difficult to obtain in practice, since the invisible beam would have to be refocussed and steered at multiple points over a large path. The angular alignment could be greatly simplified using a high resolution

188 8.3. SHORTENED BEAM PATH 163 beam profiler. A profiler can resolve the beam position to better than 100µm. This would relax the needed distance between measurement points down to about 20m, which is much more feasible. 8.3 Shortened Beam Path To reduce the collinearity error from angular misalignment of the plasma beams, the path length from the point of beam combination to the detector should be reduced as much as possible. The largest change towards this goal is to place the polarimeter detection equipment (polarizer, detector) at the exit window of the injector. This could save about 4m of path length. The difficulty would be to ensure that the equipment is electrically isolated, and to adequately steer the ω V beam to the exit port area and combine it with the ω LR beam. Placing the detector close to the exit window would also greatly reduce the chance of signal loss due to beam refraction in the plasma. Currently, a maximum density of only about m 3 can be detected before signal loss becomes likely due to beam displacement from refraction. If the detector could be brought right to the 352 exit window, the maximum possible density could be increased to about m 3. In the current setup, the divergence of the beam is reduced in order to keep the beam width small across the entire beam path. To do this, the beam is expanded to a larger radius and then passed through a long focal length lens. The beam is expanded slowly by propagating it across a length of about 1m. This distance can be reduced with the appropriate combination of lenses. For example, a diverging lens could first expand the beam over a short distance, then a second lens with an optimally selected focal length could collimate the beam to minimize its divergence. Figure 8.2 shows

189 8.4. MULTIPLE CHORDS AND ABEL INVERSION L2 L3 Beam Lens Window Radius (cm) L Distance from laser (m) Figure 8.2: New lens arrangement with shortened path could reduce undesirable beam collinearity and refraction effects. a configuration with a 10 focal length diverging lens (L2) used to expand the beam and a 15 focal length converging lens (L3) used to collimate it. The two lenses are separated by 10. This new lens configuration, could save about 1m in path length. The general arrangement of the optics in the polarimeter could also be simplified using only the bottom level and splitting the beams outside the box. This could save another 1m in path length. With these modification, the beam s path length could be reduced by about half. Combined with a beam profiler, this would bring the required angular alignment distance down to about 10m. 8.4 Multiple Chords and Abel Inversion A density and Faraday rotation radial profile can be estimated with Abel inversion of line-integrated measurements from multiple polarimeter chords. These profiles can

190 8.4. MULTIPLE CHORDS AND ABEL INVERSION 165 a y r F(y) dy F(y ) 1 F(y ) 2 Figure 8.3: Multiple probing chords give spatial resolution with Abel inversion. then potentially give a profile of the magnetic field component parallel to the chord. The Abel inversion formula giving the radial profile of a function f(r) is [45] f(r) = 1 π a r df dy dy y2 r 2 (8.1) where y is the offset of the chord (see Figure 8.3), a is the outer radius of the injector, and F is a line-integrated measurement of f(r) along a chord. The gradient of F with respect to y can be approximated as the difference between adjacent line-integrated chord measurements divided by the chord spacing. As the number of chords increases, the approximation improves. Abel inversion assumes axisymmetry of the sought-after variable.

191 166 Chapter 9 Summary A polarimeter has been designed and constructed in a novel effort to diagnose PI s accelerated spheromaks. The PI machine is an important component of GF s magnetizedtarget fusion reactor design. The polarimeter measures Faraday rotation and lineintegrated density in the plasma using three beams at slightly different frequencies. These frequency offsets are created with acousto-optic modulators. Two of the polarimeter beams are counter-rotating circularly polarized and are sent through the injector. A third, linearly polarized beam passes around the injector. The three beams are interfered to produce three beat signals, each corresponding to the interference between two beams. The desired Faraday rotation and density measurements can then be computed from the phase of the beat signals, offering greater robustness to noise. Faraday rotation is related to the line-integrated product of density and magnetic field parallel to the beams path. In the current configuration, the Faraday rotation measurement depends principally on the toroidal field near the centre of the spheromak, where the toroidal field is largest. In this way, the polarimeter can provide a non-perturbing estimate of inner magnetic field. With careful alignment to produce

192 167 a strong interference, the polarimeter measures the Faraday rotation and density null signal noise floors to be about 0.1 and m 3 for bandwidths of 1MHz and 5MHz respectively. Polarimeter-measured density agrees with measurements from an adjacent interferometer, revealing that the plasma has small asymmetries. Density measurements typically fail for plasmas with densities greater than m 3 when refraction in the plasma causes the beam to be displaced off the detector creating signal loss. A proposed improvement to the system is to reduce the beam path from the injector exit window to the detector, which would minimize plasma refraction effects. Reliable Faraday rotation measurements require a well-calibrated system. The largest source of Faraday rotation error is believed to be due to the collinearity of the two beams sent through the plasma. After an improved beam alignment using a spinning ZnSe wedge, the collinearity effect was greatly reduced and Faraday rotation measurements more often agreed with a model generated from magnetic probe data. The polarimeter currently measures Faraday rotation on the order of 1 for plasmas with density and magnetic field on the order of m 3 and 1T. Beam collinearity is still considered to be a significant source of error, which can be further improved with modifications to the beam alignment process and by shortening the beam path. The polarimeter could be expanded to use multiple beam chords, which would allow for a more refined estimate of a density and magnetic field profile. Improvements to the polarimeter will continue to shape it into a highly valuable diagnostic.

193 BIBLIOGRAPHY 168 Bibliography [1] T. Akiyama et al. CO2 laser polarimeter for electron density profile measurement on the Large Helical Device. Review of Scientific Instruments 74.5 (2003), pp [2] H. Alfven, L. Lindberg, and P. Mitlid. Experiments with plasma rings. Journal of Nuclear Energy. Part C, Plasma Physics, Accelerators, Thermonuclear Research 1.3 (1960), p [3] J. H. Apfel. Graphical method to design multilayer phase retarders. Applied Optics 20.6 (Mar. 1981), pp [4] R. Aymar, P. Barabaschi, and Y. Shimomura. The ITER design. Plasma Physics and Controlled Fusion 44 (2002), p [5] R. Azzam and N. Bashara. Ellipsometry and polarized light [6] P. Bellan. Fundamentals of plasma physics [7] P. Bellan. Spheromaks. Imperial College Press, [8] M. A. Berger. Introduction to magnetic helicity. Plasma Physics and Controlled Fusion 41.12B (1999), B167.

194 BIBLIOGRAPHY 169 [9] W. F. Bergerson et al. Far-infrared polarimetry diagnostic for measurement of internal magnetic field dynamics and fluctuations in the C-MOD Tokamak (invited). Review of Scientific Instruments 83.10, 10E316 (2012), pages. [10] G. Braithwaite et al. JET polari-interferometer. Review of Scientific Instruments 60.9 (1989), pp [11] S. J. E. Brockington et al. Plasma density gradient measurement using laser deflection. Review of Scientific Instruments 76.6, (2005), p [12] D. L. Brower et al. Laser polarimetric measurement of equilibrium and fluctuating magnetic fields in a reversed field pinch (invited). Review of Scientific Instruments 74.3 (2003), pp [13] M. Brown. Experimental studies of magnetic reconnection. Physics of Plasmas 6 (1999), p [14] K. J. Button. Infrared and Millimeter Waves Volume 2: Instruments. Academic Press, [15] P. J. F. Carle, S. Howard, and J. Morelli. High-bandwidth polarimeter for a high density, accelerated spheromak. Review of Scientific Instruments 84.8, (2013), p [16] F. Chen. An Indispensable Truth. Springer, [17] F. Chen. Introduction to plasma physics and controlled fusion. Plenum Press, [18] D. Cheng. Field and Wave Electromagnetics. Addison-Wesley series in electrical engineering. Addison-Wesley Longman, Incorporated, 1996.

195 BIBLIOGRAPHY 170 [19] M. A. Choma, C. Yang, and J. A. Izatt. Instantaneous quadrature low-coherence interferometry with 3 3 fiber-optic couplers. Opt. Lett (Nov. 2003), pp [20] R. Corey, A. Schmidt, and P. Saulnier. Using a moving diffraction grating to simulate the function of an acousto-optic modulator. American Journal of Physics 64.5 (1996), pp [21] F. De Marco and S. E. Segre. The polarization of an e.m. wave propagating in a plasma with magnetic shear. The measurement of poloidal magnetic field in a Tokamak. Plasma Physics 14.3 (1972), p [22] J. Degnan et al. Compact toroid formation, compression, and acceleration. Physics of Fluids B: Plasma Physics 5 (1993), p [23] W. X. Ding et al. Upgrade of far-infrared laser-based Faraday rotation measurement on MST. Review of Scientific Instruments 81.10, 10D508 (2010), p. 10D508. [24] G. Dodel and W. Kunz. A far-infrared polari-interferometer for simultaneous electron density and magnetic field measurements in plasmas. Infrared Physics (1978), pp [25] T. Dolan. Fusion research. Pergamon Press, [26] R. M. Erickson et al. Polarimetric Measurement of Plasma Poloidal Magnetic Field via Heterodyne Phase Shift Methods. Plasma Science, IEEE Transactions on 12.4 (Dec. 1984), pp

196 BIBLIOGRAPHY 171 [27] R. M. Erickson. Far-Infrared Polarimetry/lnterferometry for Poloidal Magnetic Field Measurement on ZT-40M. Tech. rep. LA T. Los Alamos National Laboratory, [28] T. Fowler. The fusion quest. Johns Hopkins University Press Baltimore MD, [29] G. R. Fowles. Introduction to modern optics. Dover, [30] C. Fuchs and H. J. Hartfuss. Cotton-Mouton Effect Measurement in a Plasma at the W7-AS Stellarator. Physical Review Letters 81 (8 Aug. 1998), pp [31] S. Glasstone, R. Lovberg, and A. Ruark. Controlled Thermonuclear Reactions: An Introduction to Theory and Experiment. Literary Licensing, LLC, [32] M. Greenwald. Density limits in toroidal plasmas. Plasma Physics and Controlled Fusion 44.8 (2002), R27. [33] D. Griffiths. Introduction to electrodynamics. Prentice Hall, [34] J. H. Hammer et al. Experimental Demonstration of Acceleration and Focusing of Magnetically Confined Plasma Rings. Physical Review Letters 61 (25 Dec. 1988), pp [35] H.-J. Hartfuß and T. Geist. Millimeter-Waves in Plasmas. Wiley Online Library, 2013, pp [36] R. Hawryluk et al. Principal physics developments evaluated in the ITER design review. Nuclear Fusion 49.6 (2009), p

197 BIBLIOGRAPHY 172 [37] N. K. Hicks. A New Design of a Polarimeter for Non-invasive Internal Magnetic Field Measurements on the HIT-SI Spheromak. Innovative Confinement Concepts, Seattle, Washington, USA [38] W. Hogan et al. The National Ignition Facility. Nuclear Fusion 41 (2001), p [39] E. B. Hooper et al. Sustained Spheromak Physics Experiment (SSPX): design and physics results. Plasma Physics and Controlled Fusion (2012), p [40] J. Howard et al. A single-probe-beam double-heterodyne polarimeter-interferometer for plasma Faraday rotation measurements. Journal of Instrumentation 7.07 (2012), P [41] S. Howard. Interaction Dynamics of High Reynolds Number Magnetized Plasma Flow on the CTIX Plasma Accelerator. PhD thesis. University of California Davis, [42] S. Howard et al. Development of Merged Compact Toroids for Use as a Magnetized Target Fusion Plasma. Journal of Fusion Energy 28.2 (2009), pp [43] S. Howard et al. General Fusion Plasma Injector Project Design Summary and Status Report. Tech. rep. General Fusion Inc., [44] O. A. Hurricane et al. Fuel gain exceeding unity in an inertially confined fusion implosion. Nature (Feb. 2014). [45] I. Hutchinson. Principles of Plasma Diagnostics. Cambridge University Press, 2002.

198 BIBLIOGRAPHY 173 [46] D. Hwang et al. Advances in CTIX Accelerator Study. English. Journal of Fusion Energy (2007), pp [47] T. Intrator et al. High sensitivity Faraday rotation technique for measurements of magnetic fields with immunity to x-ray effects. Review of Scientific Instruments 73.1 (2002), pp [48] T. R. Jarboe. Review of spheromak research. Plasma Physics and Controlled Fusion 36.6 (1994), p [49] T. R. Jarboe. The spheromak confinement device. Physics of Plasmas 12.5 (2005), p [50] Y. Jiang et al. Application of a digital phase comparator technique to interferometer data. Review of Scientific Instruments 68.1 (1997), pp [51] J. N. Johnson et al. Simulated and experimental compression of a compact toroid. Applied Physics Letters 95.1, (2009), pages. [52] M. Keilhacker et al. The scientific success of JET. Nuclear Fusion (2001), p [53] M.-H. Kim et al. Measurement of the infrared complex Faraday angle in semiconductors and insulators. Journal of the Optical Society of America B: Optical Physics 28.2 (Feb. 2011), pp [54] R. Kirkpatrick, I. Lindemuth, and M. Ward. Magnetized target fusion. An overview. Fusion Technology 27.3 (1995). [55] S. O. Knox et al. Observations of spheromak equilibria which differ from the minimum-energy state and have internal kink distortions. Physical Review Letters 56 (8 Feb. 1986), pp

199 BIBLIOGRAPHY 174 [56] T. W. Kornack. Magnetic Reconnection Studies on SSX. Tech. rep. Swarthmore College, [57] H. Koslowski and H. Soltwisch. Electron density and q profile measurements with the far-ir interferometer-polarimeter on the TEXTOR tokamak. Fusion Engineering and Design (1997), pp [58] M. Laberge. Experimental Results for an Acoustic Driver for MTF. English. Journal of Fusion Energy 28.2 (2009), pp [59] L. Lin, W. X. Ding, and D. L. Brower. Collinearity alignment of probe beams in a laser-based Faraday effect diagnostic. Review of Scientific Instruments 83.10, 10E320 (2012), 10E320. [60] J. D. Lindl et al. The physics basis for ignition using indirect-drive targets on the National Ignition Facility. Physics of Plasmas 11.2 (2004), pp [61] J. Marshall. Performance of a Hydromagnetic Plasma Gun. Physics of Fluids 3 (1960), pp [62] E. I. Moses. Ignition on the National Ignition Facility: a path towards inertial fusion energy. Nuclear Fusion (2009), p [63] A. Oppenheim, R. Schafer, and J. Buck. Discrete-Time Signal Processing. Prentice Hall, [64] F. P. Orsitto et al. Analysis of Faraday rotation in JET polarimetric measurements. Plasma Physics and Controlled Fusion 53.3 (2011), p [65] P. Parks. Self-similar adiabatic compression of highly elongated field reversed configurations. Nuclear Fusion 39.6 (1999), p. 747.

200 BIBLIOGRAPHY 175 [66] R. E. Peterkin Jr. Direct Electromagnetic Acceleration of a Compact Toroid to High Density and High Speed. Phys. Rev. Lett. 74 (16 Apr. 1995), pp [67] R. Raman et al. Experimental Demonstration of Nondisruptive, Central Fueling of a Tokamak by Compact Toroid Injection. Physical Review Letters 73 (23 Dec. 1994), pp [68] S. Reyes-Cortes et al. Measurement of the plasma radial electric field by the motional Stark effect diagnostic on JET plasmas. Review of Scientific Instruments 74.3 (2003), pp [69] B. Rice and E. Hooper. Poloidal field measurements during sawteeth and disruptions on MTX. Nuclear Fusion 34.1 (1994), p. 1. [70] H. F. Robey et al. Precision Shock Tuning on the National Ignition Facility. Physical Review Letters 108 (21 May 2012), p [71] J. Rommers and J. Howard. A new scheme for heterodyne polarimetry with high temporal resolution. Plasma Physics and Controlled Fusion 38 (1996), p [72] B. Saleh and M. Teich. Fundamentals of photonics. John Wiley & Sons, [73] S. E. Segre and V. Zanza. Finite electron temperature effects in plasma polarimetry and interferometry. Physics of Plasmas 9.7 (2002), pp [74] S. E. Segre. A review of plasma polarimetry - theory and methods. Plasma Physics and Controlled Fusion 41.2 (1999), R57. [75] J. Sheffield et al. Plasma Scattering of Electromagnetic Radiation: Theory and Measurement Techniques. Academic press, 2010.

201 BIBLIOGRAPHY 176 [76] H. Soltwisch and H. Koslowski. Observation of magnetic field perturbations during sawtooth activity in tokamak plasmas. Plasma Physics and Controlled Fusion 39 (1997), A341. [77] W. H. Southwell. Multilayer coating design achieving a broadband 90 degree phase shift. Applied Optics (Aug. 1980), pp [78] W. M. Stacey, N. E. Hertel, and E. A. Hoffman. Radioactive waste produced by demonstration and commercial fusion reactors extrapolated from {ITER} and advanced databases. Fusion Engineering and Design 29 (1995), pp [79] T. H. Stix. Waves in plasmas. Springer, [80] N. Storey. Electronics: A Systems Approach. Pearson/Prentice Hall, [81] J. Taylor. Relaxation and magnetic reconnection in plasmas. Reviews of Modern Physics 58.3 (1986), pp [82] J. Taylor. Relaxation of toroidal plasma and generation of reverse magnetic fields. Physical Review Letters (1974), pp [83] D. M. Thomas. Poloidal magnetic field measurements and analysis with the DIII-D LIBEAM system (invited). Review of Scientific Instruments 74.3 (2003), pp [84] K. Tokimatsu et al. Evaluation of CO2 emissions in the life cycle of tokamak fusion power reactors. Nuclear Fusion 40 (2000), p [85] M. A. Van Zeeland et al. CO2 laser polarimeter for Faraday rotation measurements in the DIII-D tokamak). Review of Scientific Instruments 79.10, 10E719 (2008).

202 BIBLIOGRAPHY 177 [86] K. Warner and G. M. Hieftje. Thomson scattering from analytical plasmas. Spectrochimica Acta Part B: Atomic Spectroscopy 57.2 (2002), pp [87] G. B. Warr and J. Howard. A three-dimensional Gaussian-beam ray-tracing program for designing interferometer/polarimeter plasma diagnostics. Review of Scientific Instruments 72.5 (2001), pp [88] J. Wesson. Tokamaks. Clarendon Press, Oxford, [89] R. C. Wolf and W.-X. Team. Wendelstein 7-X: An Alternative Route to a Fusion Reactor. Contributions to Plasma Physics 49.9 (2009), pp [90] D. Wróblewski et al. Motional Stark effect polarimetry for a current profile diagnostic in DIII-D. Review of Scientific Instruments (1990), pp [91] F. J. Wysocki et al. Improved energy confinement in spheromaks with reduced field errors. Physical Review Letters 65 (1 July 1990), pp [92] J. Zhang et al. A sensitivity assessment of millimeter-wave polarimetry for measurement of magnetic fluctuations associated with microtearing modes in NSTX-U. Plasma Physics and Controlled Fusion 55.4 (2013), p [93] S. J. Zweben et al. Measurement of the internal magnetic field of plasmas using an alpha particle source. Review of Scientific Instruments (2004), pp

203 178 Appendix A A.1 Immersed Probes Continued The magnetic coil probe is simply a multi-turn loop of wire (see Figure A.1). Faraday s Law states that a loop of wire subjected to a changing magnetic flux generates a voltage across its terminals. For a tightly wound loop of wire with N turns and area, A, this gives V = NA db dt (A.1) where B is the component of the magnetic field perpendicular to the loop plane. Therefore, if the voltage across a calibrated magnetic probe is monitored, the change in magnitude of this component of the magnetic field can be calculated. The voltage may then be integrated over time to give the magnitude of the magnetic field over time. A Hall effect magnetic probe, when immersed in a magnetic field, provides a direct relationship between measured voltage and magnetic field strength. The probe is constructed with a current passing through a semiconductor plate (see Figure A.2). In the presence of magnetic field, the electrons in this current can deflect to one side of the plate. This creates a potential difference between opposite sides of the plate,

204 A.1. IMMERSED PROBES CONTINUED 179 B V Figure A.1: Magnetic coil probes measure the change in magnetic field with time. They are located primarily at the surface to avoid perturbing the plasma. V out B e- B + V - Figure A.2: Hall effect can be used to measure magnetic field without having to integrate the signal over time. which can be measured to estimate the magnetic field strength. The advantage of a Hall probe is that it does not require integration of the voltage to produce a magnetic field measurement. Integration error is a common issue with magnetic coil probes in noisy environments, such as at GF. Another magnetic field diagnostic that does not require signal integration is the Faraday rotation probe. Faraday rotation is the rotation of the polarization plane of

205 A.1. IMMERSED PROBES CONTINUED 180 Figure A.3: The Faraday rotation probe [47] observes the rotation of linearly polarized light through a magneto-optic material to directly measure local magnetic field. The probe is optically based and so is immune to electrical noise, but must be inserted into the plasma. linearly polarized light as it passes through a magneto-optic material immersed in a magnetic field. This can also be thought of as a phase lag between left-handed and right-handed circularly polarized light. The amount of rotation, φ F, is proportional to the magnetic field strength in the direction parallel to the beam of light, B, the Verdet constant of the magneto-optic material, V, and the length of the path through the material L: φ F = B V L (A.2) Recent experiments [47] sent linearly polarized light from a HeNe laser through a ZnSe crystal. The rotation and magnetic field could then be calculated by measuring the intensity of the components of light transmitted through two polarizers at right angles to one another (see Figure A.3). The device was sensitive to magnetic fields in the range of 0.004T to 1T and was essentially immune to electrical noise.

206 A.2. PARTICLE SPECTROSCOPY CONTINUED 181 A.2 Particle Spectroscopy Continued The Zeeman effect can be used to measure magnetic field intensity and direction by observing the splitting of line emissions in an atom. Electrons bound to atoms occupy quantized energy levels. Once excited to a higher energy state, for example by the absorption of a photon of light, electrons can transition back down to a lower energy state and emit a photon in the process. This emitted light, known as line emission, is observed at a very specific frequency corresponding to the difference in energy between the electron s excited state and lower energy state. Observing the fine structure of this emission, the line is split due to the interaction between the electron s spin and its orbit around the atom, which can be thought of as inducing a small magnetic field. In a larger external magnetic field, this splitting is even more pronounced (see Figure A.4) and is known as the Zeeman effect. As a plasma diagnostic, it is common to observe Zeeman-split line emissions from a high energy lithium beam injected into the plasma (see Figure A.4). An advantage of using lithium is its high collisional excitation rate, which increases the number of excited atoms and therefore the intensity of the line emissions. In addition, lithium has a relatively large spectral separation between its Zeeman-split lines, making it easier to resolve the effect. The high energy beam is needed for lithium atoms to penetrate deeper into the plasma, allowing for measurement of fields closer to the plasma core. The Stark effect is the electric field analogue to the Zeeman effect. In the case of the Stark effect, an electric field alters the orbit of the electron around the nucleus, thereby affecting the energy levels and splitting line emissions. If a hydrogen atom with velocity, v, is fired into a magnetic field, B, the electron and proton experience

207 A.2. PARTICLE SPECTROSCOPY CONTINUED 182 (a) (b) Figure A.4: Observation of Zeeman splitting from a high energy lithium beam in a plasma allows for measurement of magnetic field [83]. (a) Lithium Zeeman splitting. (b) High energy lithium beam and observation optics mounted on a tokamak.