Download Polarimeter for an Accelerated Spheromak Patrick Jean-Franc ¸ois Carle

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
c Patrick Jean-François Carle, 2014
Copyright 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 CO2 laser that produces a
Faraday rotation signal of about 0.5◦ for a typical plasma with density and magnetic
field on the order of 1021 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
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
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
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
Contents
Abstract
i
Acknowledgments
iii
Statement of Originality
iv
Contents
v
List of Tables
ix
List of Figures
x
Acronyms
xxiii
Chapter 1:
Introduction
1.1 Fusion Power . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The General Fusion Inc. Project . . . . . . . . . . . . . . . . . .
1.3 Problem: Accelerated Spheromak Magnetic Field Measurement
1.3.1 Immersed Probes . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Particle Spectroscopy . . . . . . . . . . . . . . . . . . . .
1.3.3 Non-Perturbing Polarimetry . . . . . . . . . . . . . . . .
1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 2:
Plasma Waves
2.1 Fundamental Plasma Parameters . . . . . . . . . . . . . . . . . . . .
2.2 Appleton-Hartree Dispersion Relation . . . . . . . . . . . . . . . . . .
2.3 Non-magnetized Plasma . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Magnetic Field Perpendicular to Wave Propagation Direction and the
Cotton-Mouton Effect . . . . . . . . . . . . . . . . . . . . . . . . . .
v
1
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2.5
2.6
Magnetic Field Parallel to Wave Propagation Direction and Faraday
Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Arbitrary Magnetic Field Direction . . . . . . . . . . . . . . . . . . .
Chapter 3:
Polarimetry
3.1 Amplitude-Based Polarimeters . . . . . . . .
3.2 Phase-Based Polarimeters . . . . . . . . . .
3.2.1 Two-Beam Heterodyne Polarimeter .
3.2.2 Three-Beam Heterodyne Polarimeter
3.3 Signal Demodulation . . . . . . . . . . . . .
3.4 Beam Ellipticity . . . . . . . . . . . . . . . .
3.5 Beam Collinearity . . . . . . . . . . . . . . .
3.5.1 Collinearity Error Theory . . . . . .
3.5.2 Rotating Wedge Calibration Device .
3.5.3 Iris for Overlapping Beams . . . . . .
3.6 Gaussian Beam Propagation . . . . . . . . .
3.7 Beam Refraction . . . . . . . . . . . . . . .
3.8 Interference Phase Contrast . . . . . . . . .
3.8.1 Beam Alignment . . . . . . . . . . .
3.8.2 Beam Coherence Length . . . . . . .
3.9 Acousto-Optic Modulator . . . . . . . . . .
26
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32
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55
57
58
60
63
Chapter 4:
Spheromaks and Plasma Injectors
4.1 Helicity and Minimum Energy States . . . . .
4.2 Spheromak Magnetic Geometry . . . . . . . .
4.3 Spheromak Formation and Acceleration . . . .
4.4 λ Profile . . . . . . . . . . . . . . . . . . . . .
4.5 β Limit . . . . . . . . . . . . . . . . . . . . .
4.6 Safety Factor . . . . . . . . . . . . . . . . . .
4.7 Related Spheromak Experiments . . . . . . .
4.8 PI-1 Geometry . . . . . . . . . . . . . . . . .
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66
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80
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93
Chapter 5:
Polarimeter Design
5.1 Plasma Injector and Chord Geometry
5.2 Complementary Injector Diagnostics
5.2.1 Interferometers . . . . . . . .
5.2.2 Magnetic Probes . . . . . . .
5.2.3 Thomson Scattering . . . . .
5.3 Polarimeter Laser . . . . . . . . . . .
5.3.1 Laser stability . . . . . . . . .
5.3.2 Laser Alignment . . . . . . .
vi
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5.4
5.5
Acousto-Optic Modulators (AOMs) . . . . . . . .
Optics . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Lens . . . . . . . . . . . . . . . . . . . . .
5.5.2 Waveplate . . . . . . . . . . . . . . . . . .
5.5.3 Thin Film Polarizer . . . . . . . . . . . . .
5.5.4 Reflectors and Beamsplitters . . . . . . . .
5.6 Polarimeter Component Layout . . . . . . . . . .
5.6.1 Bottom Level . . . . . . . . . . . . . . . .
5.6.2 Top Level . . . . . . . . . . . . . . . . . .
5.7 Signal Measurement . . . . . . . . . . . . . . . .
5.7.1 Hardware and Software Reference Signals .
5.7.2 Raw Signal Magnitude and Noise . . . . .
5.7.3 Bandwidth . . . . . . . . . . . . . . . . . .
5.8 Calibration . . . . . . . . . . . . . . . . . . . . .
5.8.1 Ellipticity Calibration . . . . . . . . . . .
5.8.2 Collinearity Calibration . . . . . . . . . .
5.9 Triggering . . . . . . . . . . . . . . . . . . . . . .
5.10 Electrical and Vibrational Noise . . . . . . . . . .
5.11 Related Work . . . . . . . . . . . . . . . . . . . .
Chapter 6:
Polarimeter Simulations
6.1 Chord Profiles and Simulated Polarimetry
6.2 Characteristic Plasma Frequencies . . . . .
6.3 Cotton-Mouton Effect . . . . . . . . . . .
6.4 Refraction . . . . . . . . . . . . . . . . . .
6.5 WKBJ Approximation . . . . . . . . . . .
6.6 Beam Collinearity . . . . . . . . . . . . . .
6.7 Optically Active Windows . . . . . . . . .
6.8 Plasma Light . . . . . . . . . . . . . . . .
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95
98
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123
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125
126
129
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135
135
137
138
Chapter 7:
Measurements of an Accelerated Spheromak
7.1 Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Noise Floor for a Null Signal . . . . . . . . . . . . . . . . . . . . . . .
7.3 Density Measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Missed Fringes on the Polarimeter and Interferometer Density
Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Polarimeter Dual Densities . . . . . . . . . . . . . . . . . . . .
7.3.3 Correlation between Interferometer and Polarimeter Densities
7.4 Faraday Rotation Measurements . . . . . . . . . . . . . . . . . . . . .
7.5 Axial Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
140
140
141
143
143
146
148
150
159
Chapter 8:
Future Work
8.1 Translation Alignment . . . . . . .
8.2 Beam Profiler . . . . . . . . . . . .
8.3 Shortened Beam Path . . . . . . .
8.4 Multiple Chords and Abel Inversion
Chapter 9:
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Summary
161
161
162
163
164
166
Appendix A:
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
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178
178
181
185
186
187
List of Tables
5.1
Temperatue-wavelength regimes of the polarimeter laser. The 18.0 −
21.6◦ C range is chosen since it is wide and gives a wavelength of
10.591µm, which is close to 10.6µm.
5.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Frequency of a 10.6µm beam compared to characteristic frequencies of
the plasma assuming n = 1022 m−3 , B = 2T, and T = 10eV.
7.1
94
Review of plasma polarimeters in other experiments. RFP stands for
reversed-field pinch.
6.1
. . . . . . . . . . . . . . . . . .
. . . . . 131
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. . . . . . . . . . . . . . . . 152
ix
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]. . . . . . . . . . . . . . . . . . .
1.2
Wendelstein 7-x stellarator [89] is an alternative MCF reactor design
that uses precisely shaped magnetic fields to confine the plasma. . . .
1.3
4
5
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]. . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
6
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.) . . . . . . . . . . . . . . . . . . . . .
1.5
Magnetic probe array with ceramic shielding in formation region. Plasma
damage visible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
8
11
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
17
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
29
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.
3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
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. . . .
3.3
35
A third, linearly polarized beam that does not pass through the plasma
can be used to obtain density information. The additional beam has
frequency ωV , and is combined to the circularly polarized beams with
mirror, M, and beamsplitter, BS. . . . . . . . . . . . . . . . . . . . .
3.4
37
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 DP and DR .
xi
. . . . . . . . . . . . . . . . . . .
41
3.5
Demodulation example for the ωLR 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 ωLR 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 ωLR . (e) The residual 2ωLR spike is filtered out.
3.6
. . . .
42
Simulated phase shift response generated by a rotating λ/2-waveplate
for three test ellipticities where R = 1 and L varies. When the ellipticity is greater or less than 1, the response is non-linear.
3.7
. . . . . .
47
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. . . . . . . . . . . . . .
3.8
47
Side and top views of wedge used to calibrate out beam collinearity
error. The highest and lowest point on the wedge are respectively
labelled Phigh and Plow .
3.9
. . . . . . . . . . . . . . . . . . . . . . . . .
51
Iris can make beams purely overlap, but not collinear since beam intensity peaks are still offset. . . . . . . . . . . . . . . . . . . . . . . .
52
3.10 Gaussian beam profiles. (a) Radial profile shows 99% of beam power is
within circle of radius 1.5W0 . (b) Axial profile shows that beam radius
√
expands to 2W0 at an axial distance of z0 from the beam waist, and
diverges at an angle θ0 when z z0 . . . . . . . . . . . . . . . . . . .
xii
54
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, θR , is
exaggerated for clarity.
. . . . . . . . . . . . . . . . . . . . . . . . .
57
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.
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
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.
. . . . . . . . . . . .
61
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. . . . . . . . . .
62
3.15 Acousto-optic modulator Doppler-shifts an incident laser beam with
frequency ωB by the acoustic wave frequency ωA .
. . . . . . . . . . .
64
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. 65
xiii
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]. . . . . . . . . . . . . . . . . . . . .
4.2
69
PI forms and accelerates spheromaks. As it is accelerated down the
injector, the spheromak is compressed and increases in density and
temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
71
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.
. . . . . . . . . .
72
4.4
Flux conserver placed in the target chamber at the end of the injector.
73
4.5
Select CHI experiments drawn to scale [42]. . . . . . . . . . . . . . .
78
4.6
Approximate dimensions of PI-1 [43]. . . . . . . . . . . . . . . . . . .
78
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 Bφ and poloidal field Bθ are provided. (b) Cross
section of the injector at the PI-1 352 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
82
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.
5.3
84
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. . . . . . . . . . . . . .
5.4
85
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. . . . . . . . . . . . . . . . . . . . . . . . .
5.5
87
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
5.6
about 270µs and 310µs. . . . . . . . . . . . . . . . . . . . . . . . . .
89
Thomson scattering geometry [86]. . . . . . . . . . . . . . . . . . . .
90
xv
5.7
Polarimeter CO2 laser with vertically polarized beam path overlaid.
Actuator that controls the laser shutter is shown. . . . . . . . . . . .
5.8
92
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. . . . .
5.9
95
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. . . . . . . . . . . .
97
5.10 A λ/4 waveplate can be used to convert between linearly and circularly
polarized light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.11 Thin film polarizer reflects s-polarized light and transmits p-polarized
light when the angle of incidence is at the Brewster angle. . . . . . . 101
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 ωL . . . . . . . . . . . . . . . . . . 103
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
xvi
5.14 Sketch of the top level of the polarimeter as viewed from above. Beams
ωL and ωR 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 ωLR and ωV are interfered
at the detector. The z-direction is out of the page. . . . . . . . . . . . 107
5.15 Combined ωL and ωR beam conversion to circular polarization with a
phase retarding reflector. . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.16 Simulated radius of the combined ωLR 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. . . . . . . . . . . . . . 109
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 fLR = 15MHz. (b) Software
reference uses the corrected frequency 15MHz + δfLR . The Faraday
rotation signal for the hardware reference is offset for convenience. . . 113
5.18 Apparatus used for calibration of beam ellipticity and collinearity. . . 117
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 ωL and ωR first
converted to circular polarization and then combined.
. . . . . . . . 119
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. . 120
xvii
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. 122
6.1
Simulated density and magnetic field radial profiles at the 352 axial
position for n0 = 1022 m−3 and B0 = 2T. Plasma has an axisymmetric
profile modelled after a coaxial Taylor state. . . . . . . . . . . . . . . 127
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 n0 = 1022 m−3 and B0 = 2T. The poloidal
field Bθ is taken to be purely in the axial direction. The components
of the magnetic field parallel and perpendicular to the chord are respectively Bk and B⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
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. . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.4
Geometry used in refraction simulation to approximate the local transverse density gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . 133
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 n0 . . . . . . . . . . . . . . . . . . . . 134
xviii
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 n0 = 1022 m−3 . . . . . . 136
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 B0 = 2T and n0 =
1022 m−3 .
7.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
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. . . . . . . . . . . . . . . . . 142
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
xix
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4
Interferometer also occasionally missed fringes. Note the large lowfrequency disturbance beginning at 500µs. . . . . . . . . . . . . . . . 147
7.5
Polarimeter gives two density measurements since two beams pass
through the plasma. The density signals are typically very similar. . . 147
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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 = 0.03. Polarimeter misses large density gradients. (c)
Measurements for shot with the best correlation, r = 0.81. . . . . . . 151
xx
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.
7.9
. . . . . . . . . . . 153
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. . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.10 Poor agreement between modelled and measured Faraday rotation before improved alignment with the rotating wedge. This suggest large
collinearity noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.11 Polarimeter measurements when the ωL and ωR 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. . . 156
7.12 Improved agreement between modelled and measured Faraday rotation
after the rotating wedge was used to refine the alignment. . . . . . . . 157
7.13 Faraday rotation filtered down to 300kHz makes the signal clearer at
the expense of detail. . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.14 Inner toroidal field estimated from polarimeter Faraday rotation and
density measurements. Faraday rotation measurement bandwidth is
300kHz. Chord length is 60cm. . . . . . . . . . . . . . . . . . . . . . 159
xxi
7.15 Spline-fit to surface magnetic probe axial field measurements at time
of peak polarimeter Faraday rotation (488µs). . . . . . . . . . . . . . 160
8.1
Proposed beam combination design with two translation stages for improved beam collinearity. . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.2
New lens arrangement with shortened path could reduce undesirable
beam collinearity and refraction effects. . . . . . . . . . . . . . . . . . 164
8.3
Multiple probing chords give spatial resolution with Abel inversion. . 165
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. . . . . . . . . . . . . . . . . . . . . . . 179
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.
. . . . . . . . . . 180
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. . . . . . . . . . . . . . . . . . . . . . . . . . . 182
A.5 Deflected alpha particles can be used to reconstruct magnetic field. [93] 184
xxii
Acronyms
AC
Alternating current.
AOM
Acousto-optic modulator.
CHI
Coaxial helicity injection.
CTF
Compact Toroid Fueler.
CTIX
Compact Toroid Injection Experiment.
CTX
Compact Toroid Experiment.
DC
Direct current.
DT
Deuterium-tritium.
FWHM
Full-width half maximum.
GF
General Fusion Inc..
ICF
Inertial confinement fusion.
ITER
International Thermonuclear Experimental
Reactor.
JET
Joint European Torus.
MARAUDER
Magnetically Accelerated Ring to Achieve
Ultra-high Directed Energy and Radiation.
MCF
Magnetic confinement fusion.
MTF
Magnetized target fusion.
xxiii
NEP
Noise equivalent power.
NIF
National Ignition Facility.
PI
Plasma Injector.
PRR
Phase retarding reflector.
RACE
Ring Accelerator Experiment.
RF
Radio frequency.
SI
International System of Units.
SNR
Signal to noise ratio.
SSPX
Sustained Spheromak Physics Experiment.
TEM
Transverse electromagnetic.
TFP
Thin film polarizer.
UPS
Uninterruptible power supply.
USB
Universal Serial Bus.
WKBJ
Wentzel—Kramers—Brillouin–Jeffreys.
xxiv
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
+ 31 H → 42 He (3.54MeV) + 10 n (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]:
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 = Pfusion /Pheating , are plasma density, n, temperature, T , and confinement
time, τE . 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
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 τE = 5 × 1021 m−3 keVs.
(1.2)
where n is in m−3 , T is in keV, and τE 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:
nG = 1020
Ip
πa2
(1.3)
where Ip 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, τE . 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).
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 1020 m−3 . The goal for ITER is to maintain Q=10 for several seconds with
a density of about 1020 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
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
1020 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 1032 m−3 for a matter of nanoseconds [60]. NIF was expected
to achieve ignition by 2013. 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
1.1. FUSION POWER
(a)
6
(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
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
www.generalfusion.com
1.2. THE GENERAL FUSION INC. PROJECT
(a)
8
(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 1020 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
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 1023 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 1021 m−3
and temperature of 100eV (106 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 sec-
tions to follow give brief examples of some of these diagnostics, and their respective
1.3. PROBLEM: ACCELERATED SPHEROMAK MAGNETIC FIELD
MEASUREMENT
10
strengths and weaknesses. More detailed information can be found in Appendices A.1
and A.2.
1.3.1
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.
1.3.2
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
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.
1.3. PROBLEM: ACCELERATED SPHEROMAK MAGNETIC FIELD
MEASUREMENT
12
1.3.3
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]
φF [rad] = 2.63 ×
10−13 λ20
Z
nB · dl
(1.4)
where λ0 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, Bk .
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
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
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.
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
ωp =
ne2
0 m
1/2
(2.1)
2.2. APPLETON-HARTREE DISPERSION RELATION
16
where n is the electron density, e the elementary charge, 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
λD =
0 T
ne2
1/2
(2.2)
where T is the electron temperature. This equation has been expressed using the
shorthand notation T in place of the implied kB T , where kB 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
4
Λ = πλ3D n.
3
(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/ωp , the length
scale of interest is much greater than λD , and the plasma parameter has Λ 1.
2.2
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
2.2. APPLETON-HARTREE DISPERSION RELATION
17
z
k
θ
B
y
x
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 = ŷB0 sin θ + ẑB0 cos θ
where B0 is the magnetic field magnitude.
(2.5)
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 = n0 + n1
(2.6)
where n0 is the equilibrium part and n1 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:
n1 = < n̂1 ei(kz−ωt)
(2.7)
where n̂1 is the complex amplitude of the oscillation, z is position, and t is time.
The symbol < 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, E1 :
E = E1 = E1x x̂ + E1y ŷ + E1z 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
v1 must considered:
v = v1 = v1x x̂ + v1y ŷ + v1z 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
2.2. APPLETON-HARTREE DISPERSION RELATION
19
from collisions is small if
ων
(2.10)
where ν is the electron-ion collision frequency and is defined using the Spitzer resistivity [17]
ν=
ne4 ln Λ
.
16π20 m1/2 T 3/2
(2.11)
where ln Λ ≈ 10 is the Coulomb logarithm. The equation of motion for the electron
fluid is then
mn
∂v
+ (v · ∇) v
∂t
= −en (E + v × B)
(2.12)
Cancelling the density parameters, and writing in terms of perturbed variables gives
m
∂v1
+ (v1 · ∇) v1
∂t
= −e (E1 + v1 × 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 (v1 · ∇) v1
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
v1 = −i
e
(E1 + v1 × B) .
ωm
(2.14)
2.2. APPLETON-HARTREE DISPERSION RELATION
20
Expanding the cross product, the spatial components of Equation 2.14 are
e
ωc
ωc
E1x + v1y cos θ − v1z sin θ ,
ωm
ω
ω
e
ωc
E1y − v1x cos θ ,
= −i
ωm
ω
e
ωc
= −i
E1z + v1x sin θ
ωm
ω
v1x = −i
v1y
v1z
(2.15)
where the electron gyrofrequency is defined as
ωc =
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
1
∇×B= 2
c
∂E
1
j+
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
1
∂B
∇×
= 2
∂t
c
1 ∂j ∂ 2 E
+ 2
0 ∂t
∂t
.
(2.18)
Faraday’s law of induction is
∂B
= −∇ × E.
∂t
(2.19)
Substituting Equation 2.19 into Equation 2.18 gives the wave equation for electric
2.2. APPLETON-HARTREE DISPERSION RELATION
21
fields in the plasma:
1
∇×∇×E=− 2
c
1 ∂j ∂ 2 E
+ 2
0 ∂t
∂t
.
(2.20)
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 (n0 v1 + n1 v1 ) .
(2.22)
The term containing the factor of two perturbations is small and can be dropped.
This approximates the current density to
j ≈ −en0 v1 .
(2.23)
Substituting Equation 2.23 into Equation 2.20 and expressing the electric field as a
perturbation gives
1
∇ × ∇ × E1 = 2
c
en0 ∂v1 ∂ 2 E1
−
0 ∂t
∂t2
.
(2.24)
Applying the partial derivatives on the right side gives
1
∇ × ∇ × E1 = 2
c
ωen0
2
−i
v 1 + ω E1 .
0
(2.25)
2.2. APPLETON-HARTREE DISPERSION RELATION
22
The left side of Equation 2.25 operates on the spatial part of E1 :
∇ × ∇ × E1 = ∇ (∇ · E1 ) − ∇2 E1
∂E1x ∂E1y ∂E1z
∂ 2 E1
=∇
+
+
−
∂x
∂y
∂z
∂z 2
= ∇ (0 + 0 + ikE1z ) + k 2 (E1x x̂ + E1y ŷ + E1z ẑ)
= k 2 (E1x x̂ + E1y ŷ) .
(2.26)
Substituting this result into Equation 2.25, isolating the velocity terms, and separating into components gives
eω
E1x ,
v1x = i N 2 − 1
mωp2
eω
v1y = i N 2 − 1
E1y ,
mωp2
eω
E1z
v1z = −i
mωp2
(2.27)
where the definition for the plasma frequency, ωp , 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)
2.2. APPLETON-HARTREE DISPERSION RELATION
23
into the equation of motion (Equations 2.15). In matrix form, this is

2
2
X +N −1
i(N − 1)Y cos θ iY sin θ


 −i(N 2 − 1)Y cos θ
X + N2 − 1
0


i(N 2 − 1)Y sin θ
0
X −1



  0 
  
 =  0 ,
  
  
0
(2.29)
where the cyclotron frequency ratio is
Y = ωc /ω,
(2.30)
and the square of the plasma frequency ratio is
X = ωp2 /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:
N2 = 1 −
1 − X − 21 Y 2 sin2 θ ±
X(1 − X)
1 4
Y
4
sin4 θ + (1 − X)2 Y 2 cos2 θ
1/2 .
(2.32)
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.
2.3. NON-MAGNETIZED PLASMA
2.3
24
Non-magnetized Plasma
If the magnetic field is zero, then ωc = 0, and Y = 0. This leads to the special
dispersion relation
k=
1q 2
ω − ωp2 .
c
(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 > ωp .
(2.34)
The plasma frequency, ωp , is a proportional to the electron density, n0 . 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)
N2 = 1 −
(2.36)
X(1 − X)
1 − X − Y 2 sin2 θ
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 2.33. Plugging
this result into Equation 2.29 forces the electric field to be purely in the direction of
the magnetic field, B:
E1x = E1z = 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
1
k=
c
s
2 ω2 − ω2
ω
p
p
.
ω2 − 2
ω − ωp2 − ωc2
(2.38)
The wave can propagate when its wavenumber is real. Solving for the roots of Equation 2.38, this occurs when
or when
1 q 2
2
ω>
ωc + 4ωp + ωc
2
(2.39)
q
1 q 2
ωp2 + ωc2 > ω >
ωc + 4ωp2 − ωc .
2
(2.40)
Substituting Equation 2.38 into Equation 2.29, the electric field components have the
relationship
E1x
ω ωp2 + ωc2 − ω 2
= iE1z
ωp2 ωc
E1y = 0
(2.41)
(2.42)
Therefore, the electric field is perpendicular to the magnetic field. This is known as
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
N2 = 1 −
X
.
1±Y
(2.43)
In terms of the wavenumber, the two solutions are
kL =
kR =
ω
c
ω
c
s
1−
ωp2
.
ω (ω + ωc )
(2.44)
1−
ωp2
,
ω (ω − ωc )
(2.45)
s
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):
E1x = ±iE1y .
(2.46)
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
or when
ω < ωc
(2.47)
1 q 2
2
ω>
ωc + 4ωp + ωc .
2
(2.48)
Similarly, the L-wave can propagate when
ω>
1 q 2
ωc + 4ωp2 − ωc .
2
(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 1.
k 2 (z) ∂z (2.50)
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
φL =
φR =
Z
Z
kL dz,
(2.51)
kR dz.
(2.52)
Faraday rotation is defined as half the phase difference between the left and rightcircularly polarized beams:
φF =
1
(φL − φR ) .
2
(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




1  ERx 
1  ELx 
E =√ 
+ √ 

2 E
2 E
Ly
Ry
 


1  1 
1  1  i(φR −ωt)
= √   ei(φL −ωt) + √ 
e
2 i
2 −i
(2.54)
(2.55)
where ELx , ELy are the x, y-components of the L-wave’s electric field, and ERx , ERy
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)


2 φL +φR  cos φF  −iωt
E = √ ei 2 
.
e
2
− sin φF
(2.56)
2.5. MAGNETIC FIELD PARALLEL TO WAVE PROPAGATION
DIRECTION AND FARADAY ROTATION
29
ΦF
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, φL −φR = 2φF ,
is equivalent to a rotation of a linearly polarized wave’s polarization plane by an angle
φF .
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
1
φF =
2
Z
(kL − kR ) dz.
(2.57)
Substituting in Equations 2.44 and 2.45 gives
ω
φF =
2c
Z
s
ωp2
1−
−
ω (ω + ωc )
s
ωp2
1−
ω (ω − ωc )
!
dz.
(2.58)
It is possible to simplify the integrand with a Maclaurin series
√
1
1
1
1 + x = 1 + x − x2 + x3 − ...
2
8
16
(2.59)
2.6. ARBITRARY MAGNETIC FIELD DIRECTION
30
which converges for |x| ≤ 1. It is assumed that ωc ω, which gives
ωp2
ωp2
≈ 2 1.
ω (ω ± ωc )
ω
(2.60)
Therefore, to first order, the integral simplifies to
1
φF =
4c
Z ωp2
ωp2
−
ω − ωc ω + ωc
dz.
(2.61)
Combining the two fractions with a common denominator gives
1
φF =
2c
Z ωp2 ωc
ω 2 − ωc2
dz.
(2.62)
The assumption that ωc ω is used again, and the equation is written in its conventional form with n ≡ n0 and B ≡ B0 :
e3 λ2
φF [rad] = 2 3 0 2
8π c m 0
≈ 2.63 × 10−13 λ20
Z
Z
nBdz
nBdz.
(2.63)
where λ0 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
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
ω
∆ =
c
Z "
√
1−X −
s
X(1 − X)
1−
1 − X − Y⊥2
#
dz.
(2.64)
where Y⊥ = ωc⊥ /ω, ωc⊥ = eB⊥ /m and B⊥ = B sin θ. It is again assumed that
ω ωp , so X 1. Therefore, the square roots can be Taylor expanded and the
equation approximated to
ω
∆ ≈
2c
Z X(1 − X)
−X +
dz.
1 − X − Y⊥2
(2.65)
Rearranging over a common denominator gives
ω
∆ =
2c
Z XY⊥2
1 − X − Y⊥2
dz.
(2.66)
It is also assumed that ω ωc⊥ so Y⊥ 1. Applying these assumptions gives
e4 λ30
∆ [rad] ≈
16π 3 0 m3 c4
Z
2
nB⊥
dz
Z
−11 3
2
dz
≈ 2.46 × 10 λ0 nB⊥
(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].
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 5.
3.1
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
3.2. PHASE-BASED POLARIMETERS
E
33
P
k
Plasma
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.
3.2. PHASE-BASED POLARIMETERS
3.2.1
34
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 EL and ER that are oscillating at respective frequencies ωL and ωR , with amplitudes AL and AR , and initial
phases φL0 and φR0 . 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

1  ELx
EL = √ 
2 E
Ly

1  ERx
ER = √ 
2 E
Ry



 AL  1  i(φL0 −ωL t)
 = √  e
2 i



 AR  1  i(φR0 −ωR t)
e
= √ 
2 −i
(3.1)
(3.2)
where ELx , ELy are the x, y-components of EL , and ERx , ERy are the x, y-components
of ER . Letting AL = AR = 1 and φL0 = φR0 = 0, the combined field is (derivation is
similar to the one in Appendix A.3)

2  cos
E= √ 
2 sin
ωL −ωR
t
2

Rt
 −i( ωL +ω
).
2
e
ωL −ωR
t
2
(3.3)
The result is a circularly polarized beam oscillating at lower frequency |ωL − ωR | /2
with an amplitude changing sinusoidally at a higher frequency (ωL + ωR )/2. This
modulation of the polarization state allows for an effective measurement of the phase
difference between the two beams.
3.2. PHASE-BASED POLARIMETERS
35
D
Plasma
ωL ωR
P
ωL ωR
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




AL  δL  i(φL0 −ωL t) AR  δR  i(φR0 −ωR t)
+√ 
E= √ 
e
e
2
2 −i
i
(3.4)
where the fraction of the x-component blocked for the L and R-wave is δL and δR ,
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)
3.2. PHASE-BASED POLARIMETERS
36
where (E∗ )T is the conjugate transpose of E. This gives
A2
A2L 2
(δL + 1) + R (δL2 + 1)
2
2
AL AR
−
(δL δR − 1) ei(φLR (t)+φLR0 −ωLR t) + e−i(φLR (t)+φLR0 −ωLR t) .
2
S=
(3.6)
where ωLR = ωL − ωR . Using an identity for the sum of complex exponentials, the
signal simplifies to
S=
A2L 2
A2
AL AR
(δL + 1) + R (δL2 + 1) −
(δL δR − 1) cos(φLR (t) + φLR0 − ωLR t) (3.7)
2
2
2
where φLR (t) = φL (t)−φR (t) and φLR0 = φL0 −φR0 . The observed beat signal oscillates
at the difference frequency between the two superimposed beams and contains the
desired phase, φLR (t). This process is known as heterodyning. The beat term has the
polarization coefficient (δL δR −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 δL = δR = 1. The maximum beat
signal is obtained when the polarizer works perfectly, that is when δL = δR = 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
1
S = (A2L + A2R ) − AL AR cos (2φF (t) + φLR0 − ωLR t)
2
(3.8)
where the definition for Faraday rotation is φF (t) = φLR (t)/2. The beat has a phase of
3.2. PHASE-BASED POLARIMETERS
37
ωV
BS
D
ωL ωR
Plasma
P
ωL ωR
ωV
ωL ωR
M
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 ωV , and is combined to the circularly polarized beams with
mirror, M, and beamsplitter, BS.
φLR0 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 φLR0 adds an offset to the desired signal, which can easily be
removed. Therefore, for simplicity, initial phases such as φL0 and φR0 are set to zero
in the sections to follow.
3.2.2
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 nBk . 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 EV with amplitude
3.2. PHASE-BASED POLARIMETERS
38
AV and frequency ωV . Its Jones representation is




 EVx 
 0  −iω t
EV = 
 = AV   e V
EVy
i
(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
1
1
S = (A2L + A2R ) + A2V − AL AR ei(2φF −ωLR t) + e−i(2φF −ωLR t)
2
2
1
+ √ AL AV ei(φL −ωLV t) + e−i(φL −ωLV t)
2
1
− √ AR AV ei(φR −ωRV t) + e−i(φR −ωRV t) .
2
(3.10)
where ωLV = ωL − ωV , and ωRV = ωR − ωV . Writing the complex exponentials in
terms of sines and cosines gives
1
S = (A2L + A2R ) + A2V − AL AR cos (2φF (t) − ωLR t)
2
2
+ √ AL AV cos (φL (t) − ωLV t)
2
2
− √ AR AV cos (φR (t) − ωRV t) .
2
(3.11)
The overall signal contains three beats, each at one of the three possible difference
frequencies. The two new beats contain the phase shifts φL and φR , which are given
by the integral of the wavenumbers (Equation 2.44 and Equation 2.45) over the path
3.2. PHASE-BASED POLARIMETERS
39
length through the plasma:
Z s
ωp2
ω
φL =
1−
dz,
c
ω (ω + ωc )
Z s
ωp2
ω
φR =
dz.
1−
c
ω (ω − ωc )
(3.12)
(3.13)
If it can be assumed that ωc ω then
ω
φn = φL ≈ φR ≈
c
Z q
1 − ωp2 /ω 2 dz.
(3.14)
where φn 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
e2
λ0
φn [rad] ≈ φ0 −
4π0 mc2
Z
ndz
(3.15)
where φ0 is a constant phase, which can be ignored. Substituting in values for the
fundamental constants gives
−15
φn [rad] ≈ −2.82 × 10
λ0
Z
ndz.
(3.16)
The units are SI. Therefore, the phase shifts of the ωLV and ωLR 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.
3.3. SIGNAL DEMODULATION
3.3
40
Signal Demodulation
The goal is to extract the phase shifts φLR (t), φLV (t) and φRV (t) from the beats of the
three-beam polarimeter detector signal. This is done with a demodulation algorithm
[50] that requires a plasma signal Spla and a reference signal Sref 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 ωLR 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 ωLR − 2πfc and ωLR + 2πfc is applied
to the initial reference and plasma signals to isolate the ωLR beat. The bandwidth
chosen, affects the bandwidth of the final extracted phase shift measurement. Ignoring
0
0
initial constant phases, the filtered signals Sref and Spla are
0
Sref =Aref cos (−ωLR t) ,
0
Spla =Apla cos (φLR (t) − ωLR t)
(3.17)
(3.18)
where Aref and Apla are respectively the amplitudes of the filtered reference and
3.3. SIGNAL DEMODULATION
41
ωV
BS
DP
Plasma
ωL ωR
P
ωL ωR
ωL ωR
ωV
M
ωV
BS
DR
ωL ωR
P
ωL ωR
ωL ωR
ωV
M
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 DP and DR .
plasma signals. Rewriting in terms of complex exponentials gives
0
Sref =Aref eiωLR t + Aref e−iωLR t ,
(3.19)
0
Spla =Apla ei(φLR (t)−ωLR t) + Apla e−i(φLR (t)−ωLR 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 ωLR > 0
00
(see Figure 3.5(c)). This modified reference signal, Sref , is known as the analytic
3.3. SIGNAL DEMODULATION
42
ω
-ωRV -ωLV -ωLR
ωLR ωLV ωRV
ω
-ωLR
(a)
ωLR
(b)
ω
ωLR
ω
ω=0
(c)
2ωLR
(d)
ω
ω=0
(e)
Figure 3.5: Demodulation example for the ωLR 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 ωLR 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
ωLR . (e) The residual 2ωLR spike is filtered out.
3.3. SIGNAL DEMODULATION
43
0
signal and can be obtained by taking the Hilbert transform of Sref . The analytic
reference signal is
00
Sref = Aref ei·sgn(ωLR )·ωLR t .
(3.21)
where sgn(ωLR ) is the signum function, which returns the sign of its argument. This
allows for proper handling of the case where ωLR < 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 ωLR .
The new signal Sd has frequency spikes at ω = 0 and 2ωLR :
Sd = Apla Aref ei·sgn(ωLR )·φLR (t) + Apla Aref e−i·sgn(ωLR )·(φLR (t)−2ωLR t) .
(3.22)
A low-pass filter with cut-off frequency fc is applied to remove the 2ωLR spike (see
Figure 3.5(e)) giving
0
Sd =Apla Aref ei·sgn(ωLR )·φLR (t)
(3.23)
=Apla Aref cos [sgn(ωLR ) · φLR (t)] + iApla Aref sin [sgn(ωLR ) · φLR (t)] .
The ωLR beat’s phase shift is then
φLR (t) = sgn(ωLR ) · arctan
0
={Sd }
0
<{Sd }
(3.24)
where the signal has a bandwidth fc , and < and = respectively represent the real and
imaginary parts of their arguments. The φLR phase shift gives the Faraday rotation
measurement since φF = 12 φLR . The same procedure can be carried out to determine
the φL and φR phase shifts to obtain density measurements.
3.4. BEAM ELLIPTICITY
3.4
44
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:






AL  L  −iωL t AR  R  −iωR t
 0 
+√ 
+ AV   e−iωV t
E= √ 
e
e
2
2 −i
i
i
(3.25)
where the ellipticities of the L and R-wave are respectively L and 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, φL and φR , only apply for circularly
polarized waves. Rewriting in terms of left and right-circularly polarized components
gives
 


AL  1  L − 1 
E =√

 +
2
2
 i
 



AR  1  R − 1 
+√
 +

2
2
 i
 
 0 
+AV   e−iωV t .
i


1  
+
i
 
1  
+
i
1
−i
1
−i



 −iωL t
 e





 −iωR t
 e


(3.26)
3.4. BEAM ELLIPTICITY
45
Now, left and right-circularly polarized terms obtain phase shifts of φL (t) and φR (t)
due to interactions with the plasma. Upon exiting the plasma and passing through
the vertical polarizer, the resulting electric field is
AL E =i √ (L + 1)eiφL − (L − 1)eiφR e−iωL t
2 2
AR +i √ (R − 1)eiφL − (R + 1)eiφR e−iωR t
2 2
(3.27)
+iAV e−iωV t .
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 ωLR beat is
0
={Sd }
0
<{Sd }
(L + R ) sin φLR
,
1 − L R + (L R + 1) cos φLR
(3.28)
= sgn(ωLV )
(L + 1) sin φL − (L − 1) sin φR
,
(L + 1) cos φL − (L − 1) cos φR
(3.29)
= sgn(ωRV )
(R − 1) sin φL − (R + 1) sin φR
.
(R − 1) cos φL − (R + 1) cos φR
(3.30)
= sgn(ωLR )
LR
for the ωLV beat is
0
={Sd }
0
<{Sd }
LV
and for the ωRV beat is
0
={Sd }
0
<{Sd }
RV
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.
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 θλ/2 with respect to the
horizontal axis. Applied to the two-beam case for simplicity, this gives







 cos(2θλ/2 ) sin(2θλ/2 )   AL  1  −iωL t AR  1  −iωR t 
+√ 
E =
e
 (3.31)
 √   e
2 i
2 −i
sin(2θλ/2 ) − cos(2θλ/2 )


 
AL  1  i(2θλ/2 −ωL t) AR  1  i(−2θλ/2 −ωR t)
=√ 
+ √  e
(3.32)
e
2 −i
2 i
The phase difference between the two beams is 4θλ/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
3.5. BEAM COLLINEARITY
47
Phase Difference (degrees)
100
50
0
50
²L =1
²L =0.1
²L =10
1000 10 20 30 40 50 60 70 80 90
Waveplate Angle, θλ/2 (degrees)
Figure 3.6: Simulated phase shift response generated by a rotating λ/2-waveplate for
three test ellipticities where R = 1 and L varies. When the ellipticity is
greater or less than 1, the response is non-linear.
ωL
∆n≈0
ωL
∆n≉0
ωR
Plasma
ωR
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.
3.5. BEAM COLLINEARITY
3.5.1
48
Collinearity Error Theory
To better understand the conditions that make these effects significant, consider once
R
again the definition of the Faraday rotation phase shift φF = 12 (kL − kR ) dz. As
previously shown, this can be written in terms of ωp and ωc , 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 nL and nR , and parallel magnetic
field BkL and BkR . Therefore, Equation 2.61 transforms into the modified Faraday
rotation
φ0F
1
=
4c
Z 2
2
ωpL
ωpR
−
ω − ωcL ω + ωcR
dz
(3.33)
where the L and R-waves respectively encounter plasma frequencies ωpL and ωpR , and
gyrofrequencies ωcL and ωcR . It is still assumed that ω = ωL ≈ ωR and that the path
lengths of the two beams are approximately equal. Combining the two fractions gives
1
φ0F =
4c
Z
!
2
2
2
2
ωcL
ωcR + ωpR
+ ωpL
− ωpR
ω ωpL
dz.
(ω − ωcL ) (ω + ωcR )
(3.34)
The denominator can be simplified by assuming ω ωcR , ωcL . Substituting in fundamental parameters gives
φ0F
e 2 λ0
≈
8π0 mc2
Z
e3 λ20
(nL − nR ) dz +
16π 2 0 m2 c3
Z
nL BkR + nR BkL dz.
(3.35)
The first integral is the contribution of the collinearity error. The second integral is
approximately equal to the usual φF since the differences between the densities and
3.5. BEAM COLLINEARITY
49
fields are small along the two paths:
φ0F
e2 λ0
=
8π0 mc2
Z eλ0
nBk dz.
∆n +
πmc
(3.36)
where ∆n = nL − nR . Therefore, the collinearity error is small at a particular point
on the beam path when
C=
5.36 × 10−3
|∆n| 1
λ0 nBk
(3.37)
where Bk is the magnitude of the parallel field, and all units are in SI.
3.5.2
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 nw ,
radius R, base thickness d0 , 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
dw = d0 + (R + ρ cos θ) tan α.
(3.38)
This leads to a phase shift due to the wedge of
φw =
2πnw
dw .
λ0
(3.39)
3.5. BEAM COLLINEARITY
50
The difference in the phase shift between two parallel, but non-collinear beams with
respective entrance coordinates (ρL , θL ) and (ρR , θR ) is then
φLR =
2πnw tan α
(ρL cos θL − ρR cos θR ) .
λ0
(3.40)
If the wedge rotates at a angular speed ωw , then the phase difference as a function of
time can be written as (see Appendix A.5 for details)
φLR (t) =
2πnw tan α 0
ρ cos(ωw t + θ0 )
λ0
(3.41)
where
1/2
ρ0 = ρ2L + ρ2R − 2ρL ρR cos(θL − θR )
and
0
θ = arctan
ρL sin θL − ρR sin θR
ρL cos θL − ρR cos θR
.
(3.42)
(3.43)
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.
3.5.3
Iris for Overlapping Beams
In a previous attempt to eliminate collinearity noise, the combined ωL and ωR 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
3.5. BEAM COLLINEARITY
51
SIDE VIEW
Phigh
Plow
R tanα
α
d0
R
TOP VIEW
θ
Plow
R
Phigh
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 Phigh
and Plow .
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 ωL
and ωR beams through an iris to create purely overlapping beams would not prevent
collinearity noise.
3.6. GAUSSIAN BEAM PROPAGATION
Iris
52
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. The total path length from laser
to detector is on the order of several metres. 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
3.6. GAUSSIAN BEAM PROPAGATION
53
as the beam’s waist, W0 . 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 1.5W0 . The
normalized intensity is
ρ2
W0
exp −2 2
I=
W (z)
W (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) = W0
p
1 + (z/z0 )2
(3.45)
where
z0 =
πW02
λ
is the Rayleigh range, which is the distance after which the radius expands to
(3.46)
√
2W0 .
Far from the waist, the beam diverges approximately at an angle
θ0 =
λ
πW0
(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
W0
W00 = p
1 + (z0 /f )2
(3.48)
at a distance away from the lens of
z0 =
f
.
1 + (f /z0 )2
(3.49)
3.6. GAUSSIAN BEAM PROPAGATION
54
Normalised Intensity, I
1.0
0.8
0.6
0.4
0.2
0.00.0
1.5
0.5
1.0
Normalised Radial Distance, ρ/W0
2.0
(a)
Normalised Beam Radius, W/W0
3
2
1
0
θ0
θ0
1
2
33
1
1
2
0
2
3
Normalised Axial Distance From Waist, z/z0
(b)
Figure 3.10: Gaussian beam profiles. (a) Radial profile shows 99% of beam power is
within circle√of radius 1.5W0 . (b) Axial profile shows that beam radius
expands to 2W0 at an axial distance of z0 from the beam waist, and
diverges at an angle θ0 when z z0 .
3.7. BEAM REFRACTION
55
The new Rayleigh range, z00 , and divergence, θ00 , 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
M2 =
f0
θe0 W
θ0 W0
(3.50)
f0 . Therefore, using
where the actual divergence and waist are respectively θe0 and W
the fact that θ0 W0 = λ/π for a Gaussian beam (see Equation 3.47), the relationsihp
between the actual divergence and waist is
λ
θe0 = M 2
.
f0
πW
(3.51)
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 3.11. 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 P0 and Q0 immediately after the slab, the difference in
3.7. BEAM REFRACTION
56
the phase of the two rays is
dφ =
Z
k0 N (z, y + ∆y)dz −
Z
k0 N (z, y)dz.
(3.52)
A difference in phase indicates that the wavefronts have been refracted with angle
of refraction θR . The points P0 and Q1 are on the same wavefront. Therefore, their
phases must be equal. The phase difference between points Q1 and Q0 is
dφ = N (z)k0 ∆y tan θR
≈ k0 ∆yθR
(3.53)
where it has been assumed that θR is small, and N (z) ≈ 1 since ω ωp . Substituting
Equation 3.53 into Equation 3.52 gives
θR =
R
N (z, y + ∆y)dz −
∆y
R
N (z, y)dz
.
(3.54)
This is in the form of the definition of a partial derivative. Letting ∆y → 0, the result
is
∂
θR =
∂y
Z
N (z, y)dz.
(3.55)
Substituting in fundamental parameters and assuming ω ωp gives
e2 λ2
θR ≈ 2 0 2
8π 0 mc
Z
∂n(z, y)
dz
∂y
Z
∂n(z, y)
−16 2
≈ 4.48 × 10 λ0
dz.
∂y
(3.56)
3.8. INTERFERENCE PHASE CONTRAST
Plasma
slab
57
P0
θR
N(z,y+∆y)
Ray A
θR
∆y
Q1
N(z,y)
Ray B
Wavefronts
dz
Q0
θ
sin
y
∆
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, θR , 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].
3.8. INTERFERENCE PHASE CONTRAST
Detection Plane
0π
2π
4π
P1
y
58
P2
0π
2π
4π
B2
B1
θ
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.
3.8.1
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
3.8. INTERFERENCE PHASE CONTRAST
59
from the beam centre has phase
φ(y) = 2πyθ/λ0
(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) =
X
s(y, t).
(3.59)
y
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
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<
λ0
.
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 3.14. The minimum angular alignment requirement to have any
phase contrast for Gaussian beams is
φ(W ) < 5 rad
θ<
3.8.2
5λ0
.
2πW
(3.62)
(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
3.8. INTERFERENCE PHASE CONTRAST
Intensity (arbitrary units)
2
0
2
4
2
0
2
4
Time (arbitrary units)
Time (arbitrary units)
(a)
2
0
2
4
Time (arbitrary units)
(c)
Overall signal
Ray signals
4
Intensity (arbitrary units)
Intensity (arbitrary units)
(b)
Overall signal
Ray signals
4
Overall signal
Ray signals
4
Intensity (arbitrary units)
Overall signal
Ray signals
4
61
2
0
2
4
Time (arbitrary units)
(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.
3.8. INTERFERENCE PHASE CONTRAST
Normalized Signal Magnitude (arb.)
1.0
62
Uniform intensity
Gaussian intensity
0.8
0.6
0.4
0.2
0.00
1
2
3
4
5
6
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 ν0 and a ray at ν0 + ∆ν is
φ = 2π∆νLc /c.
(3.64)
The distance over which this effect can lead to considerable phase contrast loss is
called the coherence length Lc . It is approximated as the distance when φ = π, giving
Lc =
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.
3.9. ACOUSTO-OPTIC MODULATOR
3.9
63
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 3.15. 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 vA = λA ωA /2π where λA and ωA 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 3.16. The light beam has free-space wavelength
3.9. ACOUSTO-OPTIC MODULATOR
Incident
Beam
64
Absorber
ωB+ωA
ωB
θ
Sound
waves
θ
Transducer
1st order
0th order
ωB
Figure 3.15: Acousto-optic modulator Doppler-shifts an incident laser beam with frequency ωB by the acoustic wave frequency ωA .
λB and angular frequency ωB . 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 = 2λA sin θ
further than ray 1. This generates a phase difference of
2π
λB /N
4πλA sin θ
=
λB /N
∆φ = 2d
(3.66)
where λB /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
2λA sin θ = mλB /N.
(3.67)
The reflected rays of light are Doppler shifted by the moving acoustic waves. The
3.9. ACOUSTO-OPTIC MODULATOR
65
1
2
θ
θ
λA
θ
d
d
θ
vA
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.
change in frequency due to the Doppler effect is
vA sin θ
c/N
2λA ωA sin θ
=
λB /N
∆ωB = 2ωB
(3.68)
where vA 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 = ωB λB /2πN .
The factor of two is needed since the moving acoustic waves observe the light at a
frequency ωB + ∆ωB , and then re-emit the light at a frequency ωB + 2∆ωB according
to an observer in the lab frame. Combining Equation 3.67 and Equation 3.68 gives
∆ωB = mωA .
(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.
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=
Z
V
A · Bd3 r
(4.1)
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 = µ0 j.
(4.3)
where j is the current density, and µ0 the permeability of free space. Substituting
this into Equation 4.2 gives
j=
λB
µ0
(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
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,
Bφ , and poloidal, Bθ 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: Bθ = Bρ + Bz . The maximum
Bφ and Bθ 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.
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.
4.3. SPHEROMAK FORMATION AND ACCELERATION
4.3
70
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,
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.
4.3. SPHEROMAK FORMATION AND ACCELERATION
72
Stuffing
Field
Gas
Puff
(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.
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
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 /2µ0
(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.
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
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 = 1019 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 = 1020 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, MARAUDER accelerated spheromaks down a conical tube to n = 1021 m−3 and B = 1T.
The Compact Toroid Fueler (CTF) [67] generated 5 × 1021 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,
4.8. PI-1 GEOMETRY
77
high-speed spheromaks with n = 1021 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 = 1020 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,
4.8. PI-1 GEOMETRY
Figure 4.5: Select CHI experiments drawn to scale [42].
Figure 4.6: Approximate dimensions of PI-1 [43].
78
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)
,
ρ(z)
the frac-
tional increase in the density is A3 , and the fractional increase in the magnetic field
is A2 [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.
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 PI1 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
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 Bφ , and the
vertical component of the radial field Bρ , itself a component of the poloidal field Bθ .
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.
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
82
Backflange
Polarimeter beam
BΦ
Bθ
352 cm
(a)
0°
Inner electrode
60cm
33cm
270°
41cm
90°
51cm
Outer electrode
180°
Polarimeter
Beam
(b)
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 Bφ and poloidal field Bθ are provided. (b) Cross
section of the injector at the PI-1 352 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.
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
5.2.1
83
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
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 3.16.
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
Detector 1 (arbitrary units)
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
0.2
85
Shot 38278
0.1
0.0
-0.1
-0.2
-0.2
0.0
0.2
Detector 2 (arbitrary units)
(a)
493
21
-3
Density (x10 m )
2.0
Shot 38278
1.5
352
1.0
118
0.5
0.0
300
350
400
450
Time (µs)
500
550
(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 1021 m−3 and 1022 m−3 .
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
5.2.2
86
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
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
87
Axial Field (Tesla)
1.0
493
0.8
0.6
0.4
0.2
352
118
0.0
-0.2
Shot 25932
Toroidal Field (Tesla)
1.0
0.8
493
0.6
0.4
352
0.2
118
0.0
-0.2
150
200
Time (µs)
250
300
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.
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
88
is picked up by the probes as spikes at about 270µs and 310µs.
5.2.3
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]
Pi re2 cLn
√
dΩdλ sin2 (θE ) exp
Ps dΩdλ =
2λ0 ve π sin(θo /2)
−c2 (λ − λ0 )2
4ve2 λ20 sin(θo /2)
(5.1)
where Pi is the incident laser power, re the classical electron radius, L the scattering
p
volume length, λ0 the incident beam wavelength, ve = 2kB T /m the mean electron
thermal speed, θo the angle of observation, and θE the angle between observation
and laser polarization directions. The wavelength spectrum is spread like a Gaussian
centred at λ0 with standard deviation
p
σλ = 2 kB T /me λ0 sin(θo /2)/c.
(5.2)
Therefore, it is possible to estimate the plasma temperature by fitting a Gaussian to
5.2. COMPLEMENTARY INJECTOR DIAGNOSTICS
Axial Field (Tesla)
1.0
Shot 32877
0.5
5
4
3
2
5
3
1
4
2
5
0.0 4
3
2
1
5
4
3
2
1
Toroidal Field (Tesla)
1
1.0
3
5
4
3
2
1
5
4
2
1
3
1
2
5
4
3
1
2
3
4
5
4
5
Radial Field (Tesla)
5
4
3
2
1
0.5
3
4
0.0 1
5
2
280
320
3
2
1
3
2
4
5
1
4
5
360
400
Time (µs)
ρ = 48.9 cm
ρ = 45.5 cm
ρ = 42.3 cm
ρ = 40.7 cm
ρ = 39.0 cm
5
1
4
2
3
1.0
4
3 1
2
5
ρ = 49.7 cm
ρ = 46.2 cm
ρ = 42.9 cm
ρ = 41.4 cm
ρ = 39.7 cm
5
3
4
2
1
1
2
0.5
0.0
5
4
3
2
1
89
440
3
2
ρ = 48.7 cm
ρ = 46.4 cm
ρ = 43.3 cm
ρ = 41.2 cm
ρ = 39.6 cm
4
5
480
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.
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 10−12 . 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
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 CO2 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 CO2 laser
creates the main polarimeter beam. The specifications are listed below:
• Continuous 8W power output,
• Vertical polarization (see Figure 5.7),
• Wavelength near λ0 = 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 Lc = 1km. The maximum path length for a polarimeter beam is about 10m.
Therefore, phase contrast issues from this effect are expected to be small.
5.3. POLARIMETER LASER
92
Shutter
Actuator
Initial Beam
Polarization
Figure 5.7: Polarimeter CO2 laser with vertically polarized beam path overlaid. Actuator that controls the laser shutter is shown.
5.3.1
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
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 10.591µm. 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.
5.3.2
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
5.3. POLARIMETER LASER
Temperature Range (◦ C)
15.4
17.8
18.0
21.6
-
17.8
18.0
21.6
23.3
94
Wavelength (µm)
10.611
10.233
10.591
10.571
Table 5.1: Temperatue-wavelength regimes of the polarimeter laser. The 18.0−21.6◦ C
range is chosen since it is wide and gives a wavelength of 10.591µm, 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.
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-25-4-10.6 and
GEMF-40-4-10.6, 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 1st -order beam (frequency
shifted) and 0th -order beam (not frequency shifted). The ratio of the 1st -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 θBragg .
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
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 1st 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 1st 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 1st -order
beam is deflected downwards and shifted down in frequency.
When the AOM is rotated by the Bragg angle, the separation of the output 1st order and 0th -order beams is 2θBragg . The beams must be allowed a few inches to
separate sufficiently before the 0th -order can be blocked and the 1st -order steered to
the next stage of the polarimeter. The specifications of the AOMs are
• Maximum optical power density of 5W/mm2 ,
• Diffraction efficiency of 60%,
• Bragg angles θBragg 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 0.0015%.
5.4. ACOUSTO-OPTIC MODULATORS (AOMS)
Cooling
terminals
97
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
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.
5.5. OPTICS
5.5.1
99
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.
5.5.2
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).
5.5. OPTICS
100
45o
Fast axis
Figure 5.10: A λ/4 waveplate can be used to convert between linearly and circularly
polarized light.
5.5.3
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 67.4◦ . 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.
5.5.4
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.
5.5. OPTICS
101
s-component
Angle of
incidence
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
5.6. POLARIMETER COMPONENT LAYOUT
are possible:




 Ep   1  −iωt
E=
=
e
Es
±1
102
(5.3)
where Ep and Es 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:


 1  −iωt
E=
.
e
∓i
(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.
5.6. POLARIMETER COMPONENT LAYOUT
103
ωL
ωR
o
45
ωL
ωR
45o
s
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 ωL .
5.6.1
Bottom Level
On the bottom level (see Figure 5.13), the CO2 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 ωR beam. The other half passes through a 33/66 beamsplitter (BS01) that reflects
one-third of the power down the table. This is the ωV beam, which at a later time
5.6. POLARIMETER COMPONENT LAYOUT
104
y
ωLR
x
ωL
ωV
ωR
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.
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 ωL beam. The ωL beam is then sent
through a λ/2-waveplate (WP). This has the effect of rotating the light from vertical
to horizontal polarization.
The ωL and ωR beams must now be combined. After exiting the 40MHz AOM,
the ωR beam must be steered by mirror M02 so that it can intersect beam ωL 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
0.01◦ . 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 ωL
and ωR 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 ωR is
reflected off beamsplitter BS02, while beam ωL is transmitted through. Beamsplitter
BS02 is also held in a Kinematic mount, which allows the direction of the ωR beam
to be matched to the direction of the ωL beam. In this arrangement, it is possible to
combine beams ωL and ωR 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 ωV 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.
5.6. POLARIMETER COMPONENT LAYOUT
5.6.2
106
Top Level
On the top level (see Figure 5.14), the ωV beam is split by three 50/50 beamsplitters
(BS07, BS08, BS09) such that two beams are produced with an eighth of the initial
ωV 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 ωV beams
are respectively steered towards the plasma and reference detectors.
The combined ωL and ωR 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 ωL
and ωR 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 5.15. 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 ωL and ωR 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 ωLR 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 3.47. 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
5.6. POLARIMETER COMPONENT LAYOUT
107
y
x
ωLR
ωV
Figure 5.14: Sketch of the top level of the polarimeter as viewed from above. Beams
ωL and ωR 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 ωLR and ωV are interfered
at the detector. The z-direction is out of the page.
5.6. POLARIMETER COMPONENT LAYOUT
ωR
To plasma
ωL
Beam
combiner
ωL
λ/2
waveplate
108
Phase
retarding
reflector
Mirror
ωR
Figure 5.15: Combined ωL and ωR beam conversion to circular polarization with a
phase retarding reflector.
radius throughout its path as shown in Figure 5.16. 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 ωLR 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 ωLR beam’s power.
The reference ωLR beam is directed towards another 50/50 beamsplitter (BS06) where
it is combined with an ωV 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 ωLR beam is steered out of the box, through the ZnSe windows on
5.7. SIGNAL MEASUREMENT
1.4
L3
Radius (cm)
1.2
1.0
0.8
0.6
109
Beam
Lens
Window
L0 L1
0.4
0.2
0.00
2
4
6
8
Distance from laser (m)
10
Figure 5.16: Simulated radius of the combined ωLR 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 ωV 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
5.7. SIGNAL MEASUREMENT
110
following specifications:
• Bandwidth of 100MHz,
• Active area of 1mm2 ,
√
• Noise equivalent power (NEP) of 5×10−9 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 Zdet = 50Ω
• Maximum power density of 1W/mm2 .
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.
5.7. SIGNAL MEASUREMENT
111
The signal at the output of the amplifier is measured by a Picoscope 4227 digitizer
with
• 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 ωL and ωR beams are respectively shifted
up in frequency by the 25MHz and 40MHz AOMs. Therefore, when the ωL , ωR and
ωV 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.
5.7.1
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 Sref is generated from
hardware, that is the interference of the ωL , ωR and ωV . 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
Sref = cos(2πfLR t)
(5.5)
where fLR = 15MHz is the beat frequency of the interfering beams. Therefore,
this expression can be sampled at the appropriate sampling frequency (125MHz) to
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 fLR = 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, δfLR to the beat frequency can be obtained by fitting a sinusoid to the
plasma signal to extract a more accurate frequency near fLR = 15MHz. The reference
signal with the frequency correction is
Sref = cos(2π(fLR + δfLR )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 0.091◦ . It might be possible to improve the software reference
by electrically monitoring the AOM frequencies.
5.7.2
Raw Signal Magnitude and Noise
The peak to peak voltage of the signal at the detector can be estimated knowing that
Vdet = P RZdet 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
5.7. SIGNAL MEASUREMENT
Faraday rotation (deg)
60
113
Shot 6900 (PI2)
50
40
30
20
10
0
0.0
0.2
0.4
0.6
Time (ms)
0.8
1.0
(a)
Faraday rotation (deg)
1.0
Hardware
Shot 6900 (PI2)
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
Software
0.0
0.2
0.4
0.6
Time (ms)
0.8
1.0
(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 fLR = 15MHz. (b) Software reference uses the corrected frequency 15MHz + δfLR . 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 Vdig = 12 GP RZdet 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).
5.7. SIGNAL MEASUREMENT
114
The power of the ωL , ωR and ωV beams at the detector is respectively estimated
to be PL = 75mW, PR = 50mW and PV = 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 Vdig = 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 ωV beam is not well aligned with the ωLR 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:
P
1
√
NEP · F fBW
P
= 1 × 108 √
fBW
SNR =
(5.7)
(5.8)
where P is in Watts and fBW is the bandwidth in Hz. The system has fBW = 100MHz.
For an incident power of P = 100mW, the SNR of the signal at the digitizer is about
1000. The digitizer has 12-bit resolution, or 212 = 4096 levels, which is greater than
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.
5.7.3
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 fs . 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 fs /2, known as the Nyquist frequency [63,
80]. The signal contains three beat waveforms which oscillate at one of the three
beat frequencies: f1 < f2 < f3 . 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 fc , also called the bandwidth.
The bandwidth for the Faraday rotation measurement φF 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 φn , it is not always sufficient to use fc = 1MHz since it is a much
5.8. CALIBRATION
116
larger and more rapidly changing quantity. For example with Bk = 1T, the ratio of
Equation 3.16 to Equation 2.63 is large:
φn
≈ 1000.
φF
(5.9)
Therefore, the φn 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 φF beat as far in frequency space as possible from the
φn beats since the φF measurement is relatively small and more sensitive to noise.
Having chosen an fc for the highest frequency beat f3 , the requirement to produce a
measurement without aliasing is
fs > 2(f3 + fc ).
5.8
(5.10)
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
5.8. CALIBRATION
117
Rotating
mount
Optic
DC motor
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.
5.8.1
Ellipticity Calibration
To calibrate for ellipticity, the ωLR beam, which is composed of the circularly polarized
ωL and ωR 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
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.
5.8.2
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 ωR 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
Phase shift (deg)
5.8. CALIBRATION
119
50
0
-50
Initial setup
Improved setup
0
10
20
30
Time (ms)
40
50
(a)
Phase
retarding
reflectors
Mirrors
ωL
ωR
(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 ωL and ωR 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
5.9. TRIGGERING
120
Phase shift (deg)
10
5
0
-5
Poor alignment
Improved alignment
-10
0.04
0.08
Time (s)
0.12
0.16
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
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 5.21. 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.
5.10
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
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
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 2224. 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].
5.11
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.
Plasma Type
B
(T)
n
(m−3 )
Wavelength
(µm)
Bandwidth
(kHz)
Noise
(deg)
GF PI [15]
HIT-SI [37, 40]
MST [12, 23]
ZT-40M [27]
NSTX [92]
DII-D [85]
JET [10, 64]
TEXTOR [57]
Alcator C-mod [9]
MTX [69]
LHD [1]
Accelerated spheromak
Steady state spheromak
RFP
RFP
Tokamak
Tokamak
Tokamak
Tokamak
Tokamak
Tokamak
Stellarator
1
1
0.1
0.5
1
2
3
2
8
6
3
1021
1020
1019
1020
1019
1021
1020
1019
1020
1020
1020
10.6
1000
432.5
184.6
1000
10.6
195
337
118
185
10.6
500
1
50
40
100
1
1
1
500
1
1
0.2
0.1
0.1
0.3
.3
0.3
0.02
0.1
0.2
.2
0.01
5.11. RELATED WORK
Experiment
Table 5.2: Review of plasma polarimeters in other experiments. RFP stands for reversed-field pinch.
124
125
Chapter 6
Polarimeter Simulations
An important parameter that must be chosen is the wavelength of the polarimeter
beam. Faraday rotation is proportional to λ20 as can be seen in Equation 2.63. In this
sense, it is favourable to use a beam with large λ0 . However, undesired plasma effects
tend to become more severe with increasing λ0 . This chapter details simulations that
have been made to help understand the effect of the choice of a particular λ0 on the
performance of the polarimeter.
Plasma-beam interactions are simulated along the polarimeter chord at the PI-1
352 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-2 493 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.
6.1. CHORD PROFILES AND SIMULATED POLARIMETRY
6.1
126
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 Bφ 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 Bθ 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 Bθ 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 kρ =
17.6m−1 , and f = −2.61. 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, Bk , which in this case is in the vertical direction. Since it has
been assumed that Bθ 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
6.1. CHORD PROFILES AND SIMULATED POLARIMETRY
Magnetic field (T)
Density, n (1021 m−3 )
10
8
6
4
2
0
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.030
127
Bφ
Bθ
35
40
45
50
Radial distance, ρ (cm)
55
60
Figure 6.1: Simulated density and magnetic field radial profiles at the 352 axial position for n0 = 1022 m−3 and B0 = 2T. Plasma has an axisymmetric profile
modelled after a coaxial Taylor state.
φ, the parallel magnetic field is
Bk (ρ, φ) = Bφ (ρ) cos φ
(6.1)
Knowing the radial profiles of Bφ (ρ) and n(ρ), the chord profiles of Bk (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 n0 = 1021 m−3 to 1022 m−3 ,
and the maximum magnetic field strength from B0 = 0.5T to 2T. The resulting
simulated Faraday rotation is between φF = −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 n0 = 1022 m−3 is φn = 8100◦ , which
is about three orders of magnitude larger than the Faraday rotation phase shift and
6.1. CHORD PROFILES AND SIMULATED POLARIMETRY
128
Beam chord
y
x
Φ
Φ
BΦ(ρ)
BΦ(ρ,Φ)
BΦ(ρ,Φ)cosΦ
ρ
BΦ(ρ,Φ)sinΦ
(a)
Magnetic field (T)
Density (1021 m−3 )
10
8
6
4
2
0
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0 30
Bφ
Bθ
B
B
20
10
0 10 20 30
Position on chord (cm)
40
50
(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 n0 = 1022 m−3 and B0 = 2T. The poloidal field Bθ is
taken to be purely in the axial direction. The components of the magnetic
field parallel and perpendicular to the chord are respectively Bk and B⊥ .
6.2. CHARACTERISTIC PLASMA FREQUENCIES
129
so can be a significant source of noise (see Section 6.6). The measured φn can be
converted to an average density n̄ 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 B̄φ can also be obtained by dividing the measured Faraday rotation
φF by the constant coefficient from Equation 2.63, the chord length, and the average
density n̄. In the current configuration with L = 60cm, this gives n̄ = 8 × 1021 m−3
and B̄φ ≈ −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 n̄ and toroidal field B̄φ 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 ωc , and the plasma frequency ωp . From Equations 2.11, 2.16 and 2.1 it
6.2. CHARACTERISTIC PLASMA FREQUENCIES
130
Weighting (unitless), Field (unitless)
1.0
0.8
0.6
0.4
0.2
0.030
20
Far. rot. weighting
Normalized field magnitude
10
0
10
20
Position on chord (cm)
30
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 ,
ωc ∝ B,
√
ωp ∝ n.
(6.2)
(6.3)
(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 = 1022 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
6.3. COTTON-MOUTON EFFECT
ω (THz)
180
ν (THz)
0.01
131
ωc (THz)
0.4
ωp (THz)
6
Table 6.1: Frequency of a 10.6µm beam compared to characteristic frequencies of the
plasma assuming n = 1022 m−3 , B = 2T, and T = 10eV.
be stated that
ω ν,
(6.5)
ω ωc ,
(6.6)
ω ωp .
(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
2
∆ ∝ λ30 nB⊥
.
(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
6.4. REFRACTION
132
field: Bφ sin φ. The total field perpendicular to the chord is then
B⊥ = (Bφ sin φ)2 + Bθ2
1/2
.
(6.9)
The chord profiles of n, Bφ and Bθ 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 n0 = 1022 m−3 and B0 = 2T, the change in ellipticity
is found to be ∆ = 0.005◦ . 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 12 ∆s on either side of the beam curve in the local transverse direction as
shown in Figure 6.4. The respective densities at these points are n(x− 12 ∆sx , y+ 12 ∆sy )
and n(x + 12 ∆sx , y − 12 ∆sy ) where ∆sx = ∆s cos θR and ∆sy = ∆s sin θR . For small
6.4. REFRACTION
133
n x − 12 ∆sx , y + 12 ∆sy
Beam tangent at (x, y)
Beam
θR
∆sy
∆s
(x, y)
θR
∆sx
n x + 12 ∆sx , y − 12 ∆sy
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(x − 12 ∆sx , y + 12 ∆sy ) − n(x + 21 ∆sx , y − 12 ∆sy )
∆n
=
.
∆s
2∆s
(6.10)
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 n0 = 1022 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
6.4. REFRACTION
134
30
y-position (cm)
20
10
n0 =1021 m−3
n0 =1022 m−3
n0 =1023 m−3
n0 =1024 m−3
0
10
20
30
20
30
40
50
60
x-position (cm)
70
80
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 n0 .
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 1mm2 . In this configuration, for a displacement of less than
1mm, a maximum plasma density n0 < 1021 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.
6.5. WKBJ APPROXIMATION
6.5
135
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 2.50. Restating this equation for the geometry of the
polarimeter chord at the 352 position (see Figure 5.1(b)) gives
1
W = 2
k
∂k 1
∂y (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 ωL and ωR 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.
136
1.0
1.0
0.8
0.8
0.6
W = k12 | ky | ( ×10−7 )
Density, n (1022 m−3 )
6.6. BEAM COLLINEARITY
0.6
n
W
0.4
0.4
0.2
0.2
0.030
20
10
0
10
y-position (cm)
20
300.0
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 n0 = 1022 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 B0 = 2T and n0 = 1022 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 φ0 = 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
6.7. OPTICALLY ACTIVE WINDOWS
137
Collinearity parameter, C (unitless)
104
103
102
101
100
10-1 30
20
10
0
10
Position on chord (cm)
20
30
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 B0 = 2T and n0 = 1022 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/5 × 10−6 = 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
φF = V dBk
(6.12)
6.8. PLASMA LIGHT
138
where d is the path length through the material, and Bk the magnetic field in the
material parallel to the direction of the beam.
The polarimeter’s ωLR 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 Bk = 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 0.02◦ . 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.
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]:
−30
dPbr = 6 × 10
gn2
12.4
dλ [W/(cm3 · angstrom · sr)]
exp −
T 1/2 λ2
λT
(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 = 1022 m−3
and average temperature of T = 50eV. The average solid angle Ω is conservatively
set as the ratio of the window area (π · 62 mm2 ) to the surface area of a sphere from
the middle of the cylindrical plasma volume (4π · (600/2)2 mm2 ) giving Ω ≈ 10−4 .
The bremsstrahlung power at the detector is about Pbr ≈ 1mW. The combined laser
power is about 100mW. Therefore, the plasma light noise is expected to be small.
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
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 Aref Apla , 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 ωLR 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 1018 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
7.2. NOISE FLOOR FOR A NULL SIGNAL
Density (1021 m−3 ), Magnetic field (T)
0.6
0.5
0.4
142
PI1 - Shot 34701
Polarimeter density
Original poloidal wall field
Shifted poloidal wall field
0.3
0.2 Noise spike
0.1
0.0
0.1
390
400
410
420
Time(µs)
430
440
(a)
PI1 - Shot 34139
4
Interferometer
Polarimeter
Density (1020 m−3 )
3
2
1
0
1
200
300
400
500
Time(µs)
600
700
(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.
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.
7.3.1
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 λ0 = 10.6µm and the chord length of
0.6m, Equation 3.16 indicates that a 360◦ fringe shift corresponds to a density change
of 4 × 1020 m−3 . A typical plasma at the 352 position has a density of about 1021 m−3 .
7.3. DENSITY MEASUREMENTS
144
Null signal std. dev. (1019 m−3 )
1.0
0.8
0.6
0.4
0.2
0.00.0
0.2
0.4
0.6
0.8
Amplitude product, ArefApla (arbitrary)
1.0
(a)
0.45
Null signal std. dev. (deg)
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.050.0
0.2
0.4
0.6
0.8
Amplitude product, ArefApla (arbitrary)
1.0
(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.
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 λ20 . 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
7.3. DENSITY MEASUREMENTS
2.5
PI2 - Shot 6902
1.5
1.0
0.5
Interferometer
Polarimeter
1.2
Density (1021 m−3 )
Density (1021 m−3 )
1.4
Interferometer
Polarimeter
2.0
PI1 - Shot 34698
1.0
0.8
0.6
0.4
0.2
0.0
0.5
200
146
250
300
Time (µs)
350
400
(a)
0.0
410
415
420
425
430
Time (µs)
435
440
(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.
7.3.2
Polarimeter Dual Densities
The polarimeter gives two density measurements since two beams, ωL and ωR , are
sent through the plasma and interfered with the ωV 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.
7.3. DENSITY MEASUREMENTS
147
5
Density (1021 m−3 )
4
3
Interferometer
Polarimeter
2
PI2 - Shot 6888
1
0
200 300 400 500 600 700 800 900 1000
Time (µs)
Figure 7.4: Interferometer also occasionally missed fringes.
frequency disturbance beginning at 500µs.
2.5
Density (1021 m−3 )
2.0
Note the large low-
Density 1
Density 2
PI2 - Shot 6881
1.5
1.0
0.5
0.0
230 240 250 260 270 280 290 300 310
Time (µs)
Figure 7.5: Polarimeter gives two density measurements since two beams pass
through the plasma. The density signals are typically very similar.
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.
7.3.3
Correlation between Interferometer and Polarimeter Densities
The correlation between the interferometer and polarimeter densities is defined by
the Pearson correlation coefficient:
cov (ni , np )
σi σp
r=
(7.1)
where ni and np are respectively the interferometer and polarimeter density measurements, and σi and σp their standard deviations defined as
σX =
"
X
i
Xj − X̄
#1/2
.
(7.2)
The covariance between two variables X and Y is defined by
cov (X, Y ) =
X
j
Xj − X̄
Yj − Ȳ .
(7.3)
7.3. DENSITY MEASUREMENTS
Density (1021 m−3 )
2.0
149
Density 1
Density 2
1.5 PI2 - Shot 6936
1.0
0.5
0.0
250
300
350
Time (µs)
400
450
(a)
8
8
PI2 - Shot 6936 - Reference Signal
Density 1
6
5
4
Far. Rot.
Density 2
3
2
1
00
PI2 - Shot 6936 - Plasma Signal
7
Magnitude (arbitrary)
Magnitude (arbitrary)
7
6
5
Far. Rot.
Density 1
Density 2
4
3
2
1
10
20
30
40
Frequency (MHz)
(b)
50
60
00
10
20
30
40
Frequency (MHz)
50
60
(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.
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 (< 1021 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
7.4. FARADAY ROTATION MEASUREMENTS
151
7
6
Counts
5
4
3
2
1
00.0
0.2
0.4
0.6
0.8
Correlation coefficient, r (unitless)
1.0
(a)
3.0
Interferometer
Polarimeter
1.0
PI1 - Shot 34701
0.8
2.5
Density (1021 m−3 )
Density (1021 m−3 )
1.2
0.6
0.4
2.0
Interferometer
Polarimeter
PI2 - Shot 6884
1.5
1.0
0.5
0.2
0.0
0.0410 415 420 425 430 435 440 445 450 455
Time (µs)
0.5
230 240 250 260 270 280 290 300 310
Time (µs)
(b)
(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 = 0.03. Polarimeter misses large density gradients. (c) Measurements
for shot with the best correlation, r = 0.81.
7.4. FARADAY ROTATION MEASUREMENTS
Shot #
Field Ratio
33092
33093
33094
33096
33097
33099
33100
33101
33102
1.17
1.88
1.57
1.57
2.15
2.06
2.18
2.34
2.68
152
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 Bθ (axis)/Bz (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
7.4. FARADAY ROTATION MEASUREMENTS
153
PI1 - Shot 34704
Pol. probe contribution
Tor. probe contribution
Model tor. at magnetic axis
Magnetic field (T)
0.8
0.6
0.4
0.2
0.0
360
380
400
420 440
Time(µs)
460
480
500
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.
7.4. FARADAY ROTATION MEASUREMENTS
PI1 - Shot 33092
1.0
0.6
0.4
0.2
0.0
0.2
200
Model
Internal Probe
1.0
Toroidal field (T)
Toroidal field (T)
0.8
PI1 - Shot 33099
1.2
Model
Internal Probe
154
0.8
0.6
0.4
0.2
0.0
300
400
500
Time(µs)
600
700
0.2
200
300
(a)
400
500
Time(µs)
600
700
(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 7.10. 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 ωL and ωR 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 ωL and ωR beams become linearly polarized
155
PI1 - Shot 34139
1.0
0.2
0.5
0.1
0.0
0.0
0.5
1.0
400
420
Density (1021 m−3 )
Faraday rotation (deg), magnetic field (T)
7.4. FARADAY ROTATION MEASUREMENTS
0.1
Polari. Far. rot.
Probe Far. rot. model
0.2
Mag. axis tor. field
Polari. density
440
460
480
500
Time (µs)
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 ωV 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 ωL and ωR 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,
7.4. FARADAY ROTATION MEASUREMENTS
Density
1.0
Shot 29512
0.5
0.5
0.0
0.0
21
Density (x10
-0.5
-1.0
-3
Faraday rotation
-0.5
m )
Faraday rotation (deg)
1.0
156
-1.0
150
200
250
300
350
Time (µs)
400
450
500
Figure 7.11: Polarimeter measurements when the ωL and ωR 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.
0.0
0.5
1.0
400
450
Time (µs)
Polari. Far. rot.
Probe Far. rot. model 0.5
Mag. axis tor. field
Polari. density
500
550
0.5
0.5
0.0
0.0
0.5
1.0
360
380
0.4
0.5
0.2
0.0
0.0
0.2
0.5
400
450
Polari. Far. rot.
Probe Far. rot. model
Mag. axis tor. field
0.4
Polari. density
500
550
600
650
Time (µs)
Faraday rotation (deg), magnetic field (T)
PI1 - Shot 34707
Density (1021 m−3 )
Faraday rotation (deg), magnetic field (T)
PI1 - Shot 34705
Polari. Far. rot.
0.5
Probe Far. rot. model
Mag. axis tor. field
Polari. density
1.0
400 420 440 460 480 500
Time (µs)
0.6
0.5
0.4
0.2
0.0
0.0
Density (1021 m−3 )
0.0
1.0
Density (1021 m−3 )
0.5
Faraday rotation (deg), magnetic field (T)
0.5
Density (1021 m−3 )
Faraday rotation (deg), magnetic field (T)
1.0
PI1 - Shot 34704
1.0
7.4. FARADAY ROTATION MEASUREMENTS
PI1 - Shot 34702
0.2
0.5
360
380
Polari. Far. rot.
Probe Far. rot. model 0.4
Mag. axis tor. field
0.6
Polari. density
400 420 440 460 480 500
Time (µs)
157
Figure 7.12: Improved agreement between modelled and measured Faraday rotation after the rotating wedge was
used to refine the alignment.
7.4. FARADAY ROTATION MEASUREMENTS
0.5
1.0
0.5
0.0
0.0
0.5
360
380
Density (1021 m−3 )
Faraday rotation (deg), magnetic field (T)
PI1 - Shot 34704
158
Polari. Far. rot.
0.5
Probe Far. rot. model
Mag. axis tor. field
Polari. density
1.0
400 420 440 460 480 500
Time (µs)
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 2.63. Figure 7.14 gives this result for PI-1 shot 34704 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.
7.5. AXIAL RESOLUTION
159
PI1 - Shot 34704
Magnetic field, Bφ (T)
0.4
1.0
0.5
0.2
0.0
0.0
0.2
Density, n (1021 m−3 )
Bφ
n
0.6
0.5
0.4
0.6
1.0
388 390 392 394 396 398 400 402
Time (µs)
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, Lsph , 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, vsph , is Lsph divided
by the amount of time it is observed at the polarimeter’s axial position, which is
Surface probe axial field (T)
7.5. AXIAL RESOLUTION
160
0.5
0.4
Shot 34704
Raw points
Spline fit
0.3
0.2
0.1
0.0
100
200
300
400
500
600
Position from back flange (cm)
700
800
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 = vsph /fc .
At the 352 position, vsph ranges from 30km/s to 150km/s. Therefore with fc =
1MHz, δz is between 3cm and 15cm. Typically, Lsph is between 0.75m and 2m,
meaning the number of resolvable scale lengths Lsph /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.
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 ωL and ωR 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 ωR 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 ωR beam to be finely translated along both
8.2. BEAM PROFILER
ωLR
162
To phase retarding reflector
Kinematic mount
ωL
ωR
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
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 ωV beam to the exit
port area and combine it with the ωLR 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 1021 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 1022 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
8.4. MULTIPLE CHORDS AND ABEL INVERSION
1.4
L2 L3
1.2
Radius (cm)
1.0
0.8
0.6
164
Beam
Lens
Window
L0
0.4
0.2
0.00.0
0.5
1.0
1.5 2.0 2.5 3.0
Distance from laser (m)
3.5
4.0
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
8.4. MULTIPLE CHORDS AND ABEL INVERSION
165
F(y1)
F(y)
a
y
dy
r
F(y2)
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]
1
f (r) = −
π
Z
r
a
dF
dy
p
dy y 2 − r2
(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.
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
167
a strong interference, the polarimeter measures the Faraday rotation and density null
signal noise floors to be about 0.1◦ and 1018 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 1021 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 1021 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.
BIBLIOGRAPHY
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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,
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.
Vout
eB
+
V
-
B
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
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, φF , is proportional
to the magnetic field strength in the direction parallel to the beam of light, Bk , the
Verdet constant of the magneto-optic material, V , and the length of the path through
the material L:
φF = Bk 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.
A.2. PARTICLE SPECTROSCOPY CONTINUED
A.2
181
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
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.
A.2. PARTICLE SPECTROSCOPY CONTINUED
183
a Lorentz force
F = qv × B
(A.3)
where q is the particle’s charge, which is negative for the electron and positive for
the proton. Therefore, the electron and proton are pulled in opposite directions. In
the rest frame of the atom, there appears to be an electric field E = v × B and line
emissions from the atom are split. This is called the motional Stark effect.
Applied to plasma diagnostics, the motional Stark effect can be used to measure
the local magnetic field in a similar manner as the Zeeman effect [90]. However,
an advantage of the Stark effect is that the separation between Stark-split lines,
for example using a beam of deuterium atoms, tends to be much higher than for
a comparable Zeeman splitting diagnostic [68]. Therefore, the lines are more easily
resolvable. A motional Stark effect diagnostic must be able to distinguish between
plasma electric fields and the apparent electric field created from the beam of atoms
moving through the magnetic field. This is particularly a problem in colder parts of
the plasma where the conductivity is lower.
The motion of particles themselves can serve as a magnetic field diagnostic.
Charged particles gyrate around magnetic fields due to the Lorentz force (Equation
A.3). The radius of the gyration about a field line is the Larmor radius:
rL =
mv⊥
|q|B
(A.4)
where m is the mass of the particle and v⊥ is the velocity component of the particle
perpendicular to the magnetic field. Therefore, if a charged particle is fired into a
A.2. PARTICLE SPECTROSCOPY CONTINUED
184
Figure A.5: Deflected alpha particles can be used to reconstruct magnetic field. [93]
vessel containing a magnetic field, it is deflected by the magnetic field and can eventually collide with the vessel wall. If the point of contact on the wall can be observed,
then this gives information on the plasma magnetic field if the initial velocity, mass
and charge of the particle is known.
A recent experimental design [93] uses an alpha particle (helium) source placed
at the vessel wall in such a way that the initial angle of the particles is known. The
particles are deflected by the plasma magnetic field and impact a 40 × 40cm detector
located on the wall (see Figure A.5). The advantage is that a high speed beam
generator is not required. This significantly lowers the cost and complexity of the
diagnostic. However, for plasmas with high magnetic field, it is more difficult for
alpha particles to penetrate into the plasma core since the Larmor radius is inversely
proportional to the magnetic field.
A.3. RELATIONSHIP BETWEEN LINEARLY AND CIRCULARLY
POLARIZED LIGHT
185
A.3
Relationship between Linearly and Circularly Polarized Light
The goal is to show the relationship between linearly and circularly polarized light.
Consider left (L-wave) and right (R-wave) circularly polarized light, both at a frequency ω. The x and y-components of the L-wave’s electric field are ELx , ELy . The x
and y-components of the R-wave’s electric field are ERx , ERy . The respective phases
of the L and R-waves are φL and φR . The Jones representation of the superposition
is




1  ERx 
1  ELx 
E =√ 
+ √ 

2 E
2 E
Ly
Ry
 


1  1 
1  1  i(φR −ωt)
= √   ei(φL −ωt) + √ 
.
e
2 i
2 −i
(A.5)
(A.6)
Factoring out the time dependence, and writing the resulting complex exponentials
in terms of sines and cosines gives


1  cos φL + cos φR + i(sin φL + sin φR )  −iωt
.
E= √ 
e
2 i(cos φ − cos φ ) − (sin φ − sin φ )
L
R
L
R
(A.7)
A.4. POLARIMETER SIGNAL FOR INITIALLY ELLIPTICAL
BEAMS
186
The φL and φR terms can be combined using the sum to product trigonometric identities
cos u + cos v
sin u + sin v
cos u − cos v
sin u − sin v
u−v
u+v
cos
= 2 cos
2
2
u+v
u−v
= 2 sin
cos
2
2
u+v
u−v
= −2 sin
sin
2
2
u+v
u−v
= 2 cos
sin
.
2
2
(A.8)
(A.9)
(A.10)
(A.11)
Using these in Equation A.7 and factoring out like terms gives

2  cos
E= √ 
2 − sin
φL −φR
2
φL −φR
2
cos
φL +φR
+ i sin
R
cos φL +φ
+ i sin
2
2

φL +φR
2
φL +φR
2
 −iωt
.
 e
(A.12)
Simplifying with a complex exponential identity gives

2 φL +φR  cos
E = √ ei 2 
2
− sin
φL −φR
2

 −iωt
.
e
φL −φR
2
(A.13)
The result is a linearly polarized wave. The orientation of the polarization plane
depends on the difference between the L and R-wave phases,
A.4
φL −φR
.
2
Polarimeter Signal for Initially Elliptical Beams
For the three-beam polarimeter scenario including the effects of ellipticity, the total
electric field is given by Equation 3.27. The signal on the square-law detector is
S = E ∗ E = S0 + SLR + SLV + SRV , where the term depending on no beat frequency,
A.5. COMBINATION OF TWO SINUSOIDS OSCILLATING AT THE
SAME FREQUENCY
187
S0 , is
1
1
1
S0 = A2L (2L + 1) + A2R (2R + 1) + A2V − A2L (2L − 1) + A2R (2R − 1) eiφLR + e−iφLR ,
4
4
8
(A.14)
the term depending on the ωLR beat, SLR , is
1
SLR = AL AR 2L R − 2 − (L + 1)(R + 1)eiφLR − (L − 1)(R − 1)e−iφLR e−iωLR t
8
1
+ AL AR 2L R − 2 − (L + 1)(R + 1)e−iφLR − (L − 1)(R − 1)eiφLR eiωLR t ,
8
(A.15)
the term depending on the ωLV beat, SLV , is
AL AV √
(L + 1)eiφL − (L − 1)eiφR e−iωLV t
2 2
AL AV (L + 1)e−iφL − (L − 1)e−iφR e+iωLV t ,
+ √
2 2
SLV =
(A.16)
and the term depending on the ωRV beat, SRV , is
AR AV √
(R − 1)eiφL − (R + 1)eiφR e−iωRV t
2 2
AR AV + √
(R − 1)e−iφL − (R + 1)e−iφR e+iωRV t .
2 2
SRV =
A.5
(A.17)
Combination of Two Sinusoids Oscillating at the Same Frequency
Consider the difference between two sinusoids oscillating at angular frequency ω with
amplitudes ρL and ρR , constant phases θL and θR :
x = ρL cos(θL + ωt) − ρR cos(θR + ωt).
(A.18)
A.5. COMBINATION OF TWO SINUSOIDS OSCILLATING AT THE
SAME FREQUENCY
188
Expanding both cosines and rearranging gives
x = ρ0 cos θ0 cos ωt − ρ0 sin θ0 sin ωt.
(A.19)
ρ0 cos θ0 = ρL cos θL − ρR cos θR ,
(A.20)
ρ0 sin θ0 = ρL sin θL − ρR sin θR .
(A.21)
with the definitions
Therefore, using a trigonometric identity,
x = ρ0 cos(ωt + θ0 )
(A.22)
where ρ0 is obtained from the sum of the squares of Equations A.20 and A.21, and θ0
is obtained from the division of Equation A.21 by Equation A.20 to give
1/2
ρ0 = ρ2L + ρ2R − 2ρL ρR cos(θL − θR )
ρL sin θL − ρR sin θR
0
.
θ = arctan
ρL cos θL − ρR cos θR
(A.23)
(A.24)