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Development and Applications of Synthetic Microwave Imaging Reflectometry (MIR) Diagnostics

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Development and Applications of Synthetic Microwave Imaging
Reflectometry (MIR) Diagnostics
By
XIAOXIN REN
B.S. (University of Science and Technology of China) 2010
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in
Applied Science
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Approved:
_____________________________________
(Neville C. Luhmann, Jr.) Chair
_____________________________________
(Jonathan P. Heritage)
_____________________________________
(Xiaoguang “Leo” Liu)
Committee in Charge
2016
i
ProQuest Number: 10165767
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To my family for their constant love
ii
Acknowledgements
First of all, I would like to express my sincere appreciation to my advisor, Professor
Neville C. Luhmann, Jr., for his guidance, encouragement, and abundant support during
my study at UC Davis, without whom this dissertation would not be even possible. As an
advisor of a large group, he taught me how to become a better researcher, how to plan
ahead, and how to meet every goal. I feel so grateful and lucky to be one of his students. I
also appreciate kind help and valuable comments from my doctoral committee, Professor
Jonathan P. Heritage, and Professor Xiaoguang “Leo” Liu. Special thanks go to Dr.
Benjamin Tobias for his patience and valuable guidance towards practical aspects of my
research projects. He is always enthusiastic and ready to help, and gives me detailed
explanation on every question I had. I also want to thank Lynette Lombardo, our program
manager. She is a very nice lady, and she paid me every month on time while I was in
Davis. In the fulfilment of my various projects, I would like to thank Dr. Calvin Domier
and Dr. Christopher M. Muscatello, without whom no projects would have been completed.
I also would like to thank Dr. Xiangyu Kong, Dr. Liubing Yu, Dr. Chen Luo, Shao Che,
Dr. Yilun Zhu, Xing Hu and Ming Chen for their special help with my projects. It is hard
to mention everyone, but I would like to acknowledge my gratitude to all the members of
the Davis Millimeter-wave Research Center (DMRC) for the strong support and kind
assistance, and I wish everyone good luck.
Last but not least, I would like to thank my parents in China. Their selfless love
helps me overcome every obstacle over and over again, so that I can reach the point I have
got today. I love you all.
iii
Abstract
The Tokamak is one of the most-researched candidates for producing controlled
thermonuclear fusion power. Its confinement quality is highly limited by micro-scale as
well as MHD turbulence in plasma. The UC Davis Millimeter-Wave group is dedicated to
the development of tools, mainly Electron Cyclotron Emission Imaging (ECEI) and
Microwave Imaging Reflectometry (MIR) to diagnose turbulence in plasmas. Here, ECEI
and MIR are used to diagnose temperature and density fluctuations in the cross section of
a Tokamak. In this dissertation, the focus is on the development of the MIR system, and
more specifically, on the development and applications of a synthetic MIR diagnostic.
The synthetic MIR helps in the design stage by optimizing and validating optical
lens parameters of the imaging system. It also helps to determine the possible operating
space of a MIR system. With the help of synthetic MIR, an MIR system was successfully
installed on the DIII-D Tokamak. MIR then achieved great performance in measuring a
wide range of MHD instabilities, for instance, strong inter-ELM modes in H-mode
operation plasmas. Synthetic MIR then helps to interpret experimental data, and was used
for edge harmonic oscillation studies and provides a proxy between the plasma simulation
codes and real plasma.
This dissertation first emphasizes the importance of fusion and fusion turbulence
diagnostics; it then covers the principles of the MIR system, the hardware of the DIII-D
MIR system, and then goes deeply into the process of synthetic MIR. Afterwards, the
dissertation focuses on the analytical results from synthetic MIR simulations for defining
the MIR instrumental function, and also how it helped to calibrate the DIII-D MIR system
iv
and interpret experimental data. Then, the dissertation concludes with future software and
hardware upgrades for both synthetic and real MIR systems.
v
Content
Acknowledgements....................................................................................................iii
Abstract.....................................................................................................................iv
Content......................................................................................................................vi
Chapter 1 Introduction.........................................................................................1
1.1 Motivation....................................................................................................................1
1.1.1Energy crisis...................................................................................................................1
1.1.2 Nuclear energy...............................................................................................................4
1.2 Fusion and Magnetic confinement fusion.....................................................................8
1.3 Tokamak MHD instabilities........................................................................................17
1.4 Turbulence..................................................................................................................20
1.4.1 Turbulence in nature....................................................................................................20
1.4.2 Plasma turbulence and measurements.........................................................................21
1.5 Layout of the dissertation............................................................................................23
Chapter 2. MIR Principles.......................................................................................25
2.1
Wave Propagation in Inhomogeneous Media.......................................................25
2.2
Microwave Imaging Reflectometry......................................................................28
2.2.1 Reflectometry..........................................................................................................28
2.2.2 2-D Phase screen model...........................................................................................31
2.2.3 2D reflectometry......................................................................................................35
Chapter 3
MIR System on the DIII-D Tokamak..............................................40
3.1 Optics..........................................................................................................................40
3.1.1 Optics Overview..........................................................................................................40
3.1.2 Transmitter configuration and testing......................................................................42
3.1.3 Receiver configuration and testing..........................................................................49
3.2 Electronics modules testing.........................................................................................55
3.2.1 MIR system electronics overview................................................................................55
3.2.2 Module overview.........................................................................................................58
3.2.3 Module testing.............................................................................................................60
3.3
Finalized system on DIII-D...................................................................................65
Chapter 4
Development of Synthetic MIR Diagnostic......................................67
4.1
The need for a synthetic MIR diagnostic..............................................................67
4.2
FWR2/3D code......................................................................................................69
4.2.1 FWR Computation model............................................................................................70
4.2.2 Turbulence model........................................................................................................74
4.3 Synthetic MIR diagnostic model.................................................................................77
4.4 Analysis methods.........................................................................................................82
4.4.1 IQ plot analysis............................................................................................................82
4.5 Correlation analysis....................................................................................................86
Chapter 5: Analytical Simulation Results with Synthetic MIR...............................93
5.2 Curvature matching....................................................................................................94
5.2.1 Transmitter..................................................................................................................94
vi
5.2.2 Receiver.....................................................................................................................100
5.3 Sensitivity measurements..........................................................................................105
5.3.1 Plasma profile............................................................................................................106
5.3.2 Out-of-focus conditions.............................................................................................110
5.3.3 Different resolution size........................................................................................112
5.3.4 Oblique incident angle...........................................................................................114
5.4
MIR instrument function....................................................................................117
5.4.1 L-mode discharge instrument function..............................................................117
5.4.2 H-mode discharge instrument function......................................................................126
5.5
Spectral Analysis................................................................................................131
5.5.1 Spatial Correlation.................................................................................................131
5.5.2 Power spectral density (PSD) analysis..................................................................133
5.5.3 Frequency versus wavenumber relationship..............................................................135
5.6
Summary............................................................................................................136
Chapter 6: MIR Characterization with Synthetic Results and Initial Experimental
Campaign...............................................................................................................139
6.1
Comparison with synthetic simulation results....................................................139
6.1.1 Overview...............................................................................................................139
6.1.2 Comparison from a discharge with a density ramp................................................140
6.1.3 Comparison for a vertically jogging discharge......................................................145
6.2
ELM-inter modes observation............................................................................150
6.4
Comparison of ECEI and MIR...........................................................................154
6.4.1 Alfven eigenmodes................................................................................................154
6.4.2 Simultaneous local imaging...................................................................................158
Chapter 7: Edge Harmonic Oscillation Studies using MIR and Synthetic MIR....161
7.1
QH mode.............................................................................................................161
7.1.1 QH mode...............................................................................................................161
7.1.2 Edge Harmonic Oscillations..................................................................................163
7.2
Experimental measurements of counter-propagation........................................165
7.3
Forward modeling of MIR for EHO studies.......................................................166
7.3.1 Forward modeling of MIR.....................................................................................166
7.3.2 Imaging..................................................................................................................168
7.4
Analysis...............................................................................................................172
7.4.1 Instrumental artifacts investigation........................................................................172
7.4.2 Non-linear rotation investigation...........................................................................173
7.4.3 Possible explanation of counter-propagation.........................................................175
7.5 Conclusion..........................................................................................................176
Chapter 8: Continuing Work and Conclusions......................................................178
8.1
Future Synthetic MIR work...............................................................................178
8.1.1 FWR3D.................................................................................................................178
8.1.2 QH-mode studies...................................................................................................181
8.1.3 Alfven wave studies...............................................................................................184
8.1.4 FWR interface with simulation codes....................................................................187
8.2
Hardware upgrades............................................................................................189
8.2.1 CMOS-based transmitter.......................................................................................189
8.2.2 Phased antenna array.............................................................................................191
vii
8.2.3 Digital Beam-Forming and On-Chip Module............................................................193
8.2.4 EAST MIR.............................................................................................................195
8.3 Conclusion..........................................................................................................197
Appendix I
Introduction to electron cyclotron emission imaging (ECEI)........200
Appendix II Previous MIR devices..................................................................205
Appendix III
Manual on output of FWR2D....................................................211
Appendix IV
A different approach of synthetic MIR........................................213
Appendix V
Phase Jumps and DC Drifts.......................................................217
V.1
Phase Jumps.......................................................................................................217
V.2
DC Drifts............................................................................................................219
References..............................................................................................................222
viii
Chapter 1 Introduction
1.1 Motivation
1.1.1Energy crisis
In recent years, the energy crisis together with global warming [1] has attracted
worldwide attention. The predicted increase of world population from 6.7 billion in 2011
to 8.7 billion in 2035 has caused energy demands to increase substantially. Over 70 % of
the increased energy demand is from developing countries, led by China and India – China
overtook the USA as the top CO2 emitter in 2007. Superimposed on this, the UN Population
Division projects an ongoing trend of urbanization, from 52% in 2011 to 62 % in 2035 and
reaching 70 % worldwide in 2050. Electricity demand is increasing twice as fast as overall
energy use and is likely to rise by more than two-thirds from 2011 to 2035 as addressed in
the World Energy Outlook 2013 [2]. Currently, energy is derived mainly from fossil fuel
resources: coal, oil, and natural gas. Even though fracking technology [3] enabling access
to gas in shale beds is extending the supplies for natural gas in several countries, the
impending exhaustion of those fossil fuel resources is still a major concern of all
governments. Moreover, fossil fuels are major contributors to global warming, which is
resulting in severe environmental and climate problems. Fracking, although it has reduced
carbon emission, has the possibility of contaminating ground water reserves. Figure 1.1
shows the increase of average surface temperature from 1880 to 2009 due to the emissions
1
of CO2. Consequently, there is increasing interest in identifying alternative energy
resources to satisfy the increasing demands for energy, especially cleanly generated energy.
Figure 1.1: Global average temperature increased by more than 1 °C from 1880 to 2009.
The zero point is set at the average temperature between 1961 and 1990. [Image from Fifth
Assessment Report (AR5) from International Panel on Climate Change (IPCC)]
Popular alternative renewable sources that are widely utilized around the world are
solar energy [4], wind energy [5], geothermal energy [6], hydroelectric energy [7], and
biomass energy [8]. Figure 1.2 shows 5 popular renewable energies that are widely used
for electricity generation. Solar energy, which mainly comes from two approaches:
photovoltaics (PV) [9] and concentrated solar power (CSP) [10], is an inexhaustible source
of energy as long as the sun is there generating solar power. It has the benefits of being
totally free, widely available, low pollution, no emission, no noise, and no carbon footprint.
However, issues such as not being able to be function overnight, highly affected by weather
2
conditions, toxicity of photovoltaic panels, and requirements of large space for installation
with associated deleterious environmental consequences, restrict the application of solar
energy as a global solution for the energy crisis. Other renewable energies also suffer from
similar drawbacks: location restrictions, costly unit output, and potential environmental
problems. In addition, renewable resources can only provide limited energy supplies, and
cannot satisfy the current high demand for energy. Taking Japan for example [11]: it
currently has an electricity consumption of 85 kWh/person/day, which is mostly provided
by oil and coal, and is increasing year by year; wind energy combined with solar energy,
for example, has the potential to reach 23 kWh/person/day by 2050; other renewable
energies, including hydro-electricity and geothermal energy can potentially reach a total of
10 kWh/person/day by 2050. These numbers mean that renewable energies, due to their
limitations and shortcomings can only be helpful marginally at providing clean power [11].
In addition to other renewable resources, nuclear power, which will be discussed
below, provides about 11% of the world’s electricity, and is the most environmentally
benign way of producing electricity on a large scale. US nuclear power provides about 20%
up to 2004. Due to its high efficiency for generating energy and being relatively
environmentally friendly, nuclear energy, especially fusion energy, is potentially the next
generation of main source for generating power.
3
Solar energy
Wind energy
Hydropower energy
Electricity
Biomass energy
Geothermal energy
Figure 1.2: Popular clean and renewable energy resources widely applied around the world.
1.1.2 Nuclear energy
Nuclear energy, the generation of energy through the process of mass reduction,
with released energy !E = Δmc 2 , is thought to be the most promising approach for solving
the energy crisis, due to its high efficiency for generating energy. The existing ways for
generating nuclear energy on Earth are fusion and fission. Figure 1.3 shows the process of
fission and fusion. Fission is the process of splitting a massive nucleus into smaller nuclei
4
and thus releasing energy, with the released energy carried mostly by by-produced neutrons
and gamma rays. The most widely used nucleus for fission is Uranium, and the reaction
process is given by:
235
!U + n → !fission! + 2!or!3!n + 200Mev
(1.1)
In contrast to the fission process, fusion is the process of fusing low mass nuclei
into a heavier one, during which, energy is released. The process that appears most
promising for future fusion power plants is:
2
!1
H + 13H → 24 He + 01n+17.6Mev
(1.2)
The released energy is carried by both neutrons and alpha particles.
neutron
(a) Fission
Fission
product
neutron
neutron
Target
nucleus
Fission
product
neutron
(b) Fusion
Tritium
neutron
Fusion
Energy
Deuterium
17.6 Mev
Helium
5
Figure 1.3: The process of fission (a), and fusion (b). Fission is the process of splitting the
heavier mass nucleus into several small nuclei when impacted by a neutron, and during
which, energy is released and mostly carried by neutrons. Fusion is the process of lighr
nuclei fusing together and releasing energy in the process. [Image components licensed by
cc-by-sa-3.0]
Though fission can easily produce huge amounts of energy with a small amount of
Uranium, it is not very environmentally benign, and has the potential for causing largescale disasters such as the Fukushima accident [12] and Chernobyl accident [13]. The
Fukushima accident essentially removed nuclear power from Japan, which originally
consisted of 11% of the total energy consumption in Japan, and slowed the increasing usage
of nuclear power around the world. The Chernobyl accident also caused the abandonment
of nuclear power in Italy. Moreover, the fission products are highly radioactive, and the
issue of disposing of these mixtures of short and long half-live radioactive products is a
severe problem. As one underground disposal example, Yucca Mountain in Nevada is
proposed to be a permanent geologic repository used to dispose the high-level radioactive
byproducts from fission [14], as shown in Fig. 1.4. An alternative approach is to use an
accelerator-driven system (ADS) to transmute long-lived radioisotopes in used nuclear fuel
into shorter-lived fission products, and is being investigated as an alternative to the
geologic storage solution [15].
6
Figure 1.4: Geologic disposal repository for used fission fuel waste at Yucca Mountain.
In contrast to fission, fusion, being relatively environmentally benign, energetically
efficient, and potentially able to supply sufficient energy for thousands of years, is an
extremely attractive approach for solving the energy and global warming crisis, and thus
has drawn the attention of many scientists and nations to achieve. A single gram of DT fuel
used for fusion can produce the energy equivalent of 80 tons of TNT, and fusion is also the
process that fuels every star in the universe [16]. People have been working on controlled
fusion starting in the early 1950s and the technology is still under development to harness
the awesome power of the atom for peaceful purposes [17]-[20]. The early optimism for a
rapid solution of fusion was quickly tempered by the fact that producing and sustaining a
temperature of 100 million Celsius for fusion to happen under a laboratory environment is
difficult to achieve. Over the past 50 years, due to the development of plasma physics
understanding, technologies and materials, fusion confinement devices currently have
found their way to demonstrate fusion energy producing plasmas. A more detailed
introduction to existing fusion devices and technologies will be discussed in the following
section. Fusion is becoming more and more popular around the world, and even finds its
7
position in recent scientific fantasy movies; as shown in Fig. 1.5, the movie captures of
fusion energy devices from “Iron Man” (a) and “X-Men: Days of Future Past” (b).
(a)
(b)
Figure 1.5: (a) Ironman’s chest core, to power his super steel suits [Imaged sourced from
movie “Iron Man”]. (b) Fusion device used to power airship in X-Men released 2014
[Image sourced from movie “X-Men: Days of Future Past”]
1.2 Fusion and Magnetic confinement fusion
The Deuterium-Tritium fuel produces energetic alphas and neutrons. The neutrons,
which carry 80 % of the fusion energy, escape and are harnessed and converted to
electricity; while the alphas, which carry the remaining 20 % fusion energy, need to be
confined in order to heat the plasma. To make the plasma self-heating, the energy carried
by the alpha particles should be sufficient to heat the plasma, which is defined as the
ignition condition. The fusion power confined with He atoms for a mixture of 50/50 %
Deuterium and Tritium could be expressed as [21]:
8
n2
Pα =
συ Eα V
4
!
(1.3)
where σ is the fusion cross section [22], υ is the relative velocity [23], and
συ
denotes the average over an Maxwell-Boltzmann velocity distribution at temperature T,
with a value !1.1×10−24 T 2m3s −1 in a relevant temperature range of 6 – 20 keV; in addition,
E is the alpha particle energy, which is 3.5 MeV in a DT reaction, and V is the reaction
! α
volume. To calculate the power lost in heat, the confinement time τ E is introduced, and
!
is given by:
!
Here,
∂W
W
= − + Ploss (1.4)
∂t
τE
!W is the stored energy in the plasma, and !W = 3nTV . Assuming a stationary
plasma, meaning
∂W
= 0 , we thus have:
! ∂t
3nT
Ploss =
V
τE
!
(1.5)
Ignition requires Ploss = Pα ,
!
12T 2
⇒ nτ ET =
συ Eα
!
(1.6)
⇒ nτ ET = 3×1021 m−3skeV
!
(1.7)
The ignition condition was originally derived by Lawson [21], and is called
Lawson’s Criterion.
Naively, one can use basic electrostatics to show that temperatures of hundreds of
keV (109 – 1010 K) are needed to overcome the electrical repulsion between the nuclei; such
9
a requirement would rule out any practical fusion reactor on Earth. However, quantum
tunneling allows the reaction to proceed even if the barrier is not fully overcome, and this
reduces the required temperature to a "mere" 20 keV. At this optimum temperature, ignition
will occur according to Lawson’s criterion if:
nτ > 1.5×1020 m−3s (1.8)
! E
Much of the past half-century's research, be it in magnetic confinement devices like
JET [24] and its antecedents, or in inertial confinement devices, has focused on attempting
to achieve conditions that satisfy the Lawson Criterion.
For inertial confinement fusion (ICF) [24] [25], a pellet of DT fuel is compressed
by a laser to temperature and density that is sufficiently high to satisfy the Lawson’s
criterion, and it requires:
2
!ρ ⋅r > 3g /cm (1.9)
with ρ the compressed pellet mass density and !r the pellet radius. This criterion
indicates that very high-energy lasers are required. Recent simulation indicates that around
2 MJ and 500 TW are needed to ignite the fuel. The NIF device [27] is the world’s first
laser to satisfy these conditions. However, even though ignition is possible, no ICF facility
thus far has achieved a scientific breakeven (fusion power = laser power). Since lasers are
very inefficient machines, gains of at least 100 are needed for a power plant to produce net
power output. Another drawback of ICF for useful energy production is the pulsed nature
of the process, which requires 10 “shots” per second for a realistic reactor. Figure 1.6 shows
the principles of inertial confinement fusion. Initially, the high-energy laser beam rapidly
heats the surface of the fuel pellet, thus forming a surrounding plasma envelope. Then, fuel
is compressed due to the blow-off of the surrounding plasma. Afterwards, the compressed
10
fuel reaches 20 times higher density in the core and ignites at 100 million Celsius. Finally,
thermonuclear burn spreads rapidly through the compressed fuel, and generates energy
many times the level of the input energy.
Laser energy
Blowoff
Inwardly transported
thermal energy
1. Atmosphere formation
2. Compression
3. Ignition
4. Burn
Figure 1.6: The concept and stages of inertial confinement fusion. The first stage is when
laser beams or laser-produced X-rays rapidly heat the surface of the fusion target forming
a surrounding plasma envelope; the second stage is when fuel is compressed by the rocketlike blow-off of the hot surface material; the third stage is when the fuel core reaches 20
times the density of lead and ignites at 100,000,000 ˚C during the final part of the laser
pulse; the last and fourth stage is when thermonuclear burn spreads rapidly through the
compressed fuel, yielding many times the input energy.
Magnetic fusion applies a strong magnetic field to constrain plasmas within a
limited space. Compared to ICF, magnetic confinement fusion has better confinement,
longer durations, and higher nτ ET value. Particles with extremely high kinetic energy
!
would immediately disperse once generated if they are free to move in any direction as
11
shown in Fig. 1.7(a). To constrain the motion of ionized particles, a strong magnetic field
is applied as shown in Fig 1.7 (b). The problem of the magnetic field geometry shown in
Fig 1.7 (b) is that the plasma is not constrained at the ends and thus there are significant
losses. One solution is to bend the strong magnetic field into a circular shape to guide
electrons and ions moving along the field lines, and thus eliminate the end losses and
prevent contact with walls or other materials, similar to that shown in Fig. 1.7 (c).
(a)
Ion
B
(b)
Electron
(c)
12
Figure 1.7: Shown is the constrained motion of ionized particles with strong magnetic field.
(a) Charged particles with high kinetic energy move in random directions; (b) the particles
move in gyro orbits along the added strong magnetic field; (c) To eliminate end losses, the
magnetic field is bent into a circular shape, and thus also eliminates particles contacting
with wall materials.
However, a simple toroidal magnetic field has a radial gradient in the magnetic field.
The gradient, as well as the curvature of magnetic field, would result in a vertical drift and
separation of ions and electrons. This would then cause a build-up of an electric field in
the vertical direction, which causes an E x B drift, rapidly transporting particles out of the
magnetic field along the horizontal direction, and enhancing the process of losing particles.
Such a process of E x B drift is shown in Fig. 1.8 (a) with a pure toroidal magnetic field.
To compensate for such loss, the vertical electric field needs to be eliminated. An extra
“poloidal” magnetic field is introduced to “average” out the separation of ions and electrons
(over regions with strong and weak field), and thus ameliorate the horizontal drift of
particles.
This process is shown in Fig. 1.8(b) with the helical magnetic field compared
to the pure toroidal field lines in Fig. 1.8 (a). Fusion devices working with a helical
magnetic field generated by a toroidal current to constrain particles are called Tokamaks,
which is a transliteration of the Russian word токамак (тороидальная камера
с магнитными катушками) [28], [29]. A typical example of a Tokamak machine is shown
in Fig. 1.9. In Tokamaks, there are two sets of magnets, an external set surrounding the
vacuum chamber (creating a toroidal magnetic field) and an internal transformer that drives
current in the plasma to produce a poloidal magnetic field. The Tokamak is regarded as the
13
most successful fusion device around the world, due to its long confinement time and
simple mechanical structure.
(a)
(b)
B-field
VEXB
+
-
+
-
B-field
+
-
+
.
+
+
jP.S. +
- - -
E-field
B increasing direction
B increasing direction
Toroidal and poloidal field
Only toroidal field
Figure 1.81: The build-up of the vertical electric field with only a toroidal magnetic field,
which causes a horizontal drift of particles. The drift can be largely eliminated with the
helical magnetic field which would bring back the drifted particles and minimize the buildup of the vertical electric field.
1
The jP.S. here is the Pfirsch-Schluter current, which compensates for the ExB drift.
14
Fig 1.9: Example of a Tokamak device with coils to generate the toroidal and poloidal
magnetic field [30]. [Copyright 2016 by the American Nuclear Society, La Grange Park,
Illinois]
In addition to the tokamak, a different magnetic confinement device called
stellarator [31], [32] is also under investigation. In stellarators, external magnetic coils
generate twisted field lines around the inside of the vacuum chamber to confine the plasma.
Figure 1.10 shows the shape of the magnetic field as well as the confined plasma within it.
Compared to the Tokamak, which suffers considerably from current driven instabilities,
the stellarator can easily operate under more stationary operation, without current driven
instabilities. However, the stellarator has some serious construction problems that limits its
wide application: it has very complex magnetic field coils, while curved coils also lead to
large forces and thus require a strong supporting structure, and it is also difficult to be made
into a compact device. LHD [33] in Japan is the second largest operating stellarator around
the world. Based from the rapid development of intensive 3-D simulations, which helped
to determine the optimal shape of the magnetic field and the shapes of the helical and
15
toroidal coils that creates it, Wendelstein 7-X
(W7X) [34], [35], with its first plasma
produced in December 2015, was constructed and is the largest operating stellarator all
over the world. The current LHD confinement time, due to its small plasma volume, is
close to the former TFTR device [36], but less than that of JET [24], [37]. The Tokamak is
arguably superior to the stellarator due to its easier construction, and comparable or longer
confinement time.
Figure 1.10: Shape of the magnetic field of a stellarator, and the constrained plasma inside
[Free license].
There are numerous tokamaks around the world with different size and discharge
parameters; some of them are JT-60 [38], KSTAR [39], DIII-D [40], ASDEX-Upgrade
[41], EAST [42], HL-2A [43], and JET [24], [37]. Among the tokamaks, JET is the largest
existing tokamak, and is the only tokamak device that can operate with D-T plasma
currently. Both EAST in China and KSTAR in Korea are exploring very long pulse high
performance operation (steady-state [44] [45]), which are essential to the success of the
International Thermonuclear Experimental Reactor (ITER) [46], [47]. One main issue for
16
current tokamak devices is to estimate and determine the operation conditions for ITER.
An empirical scaling law [48] that fits data from existing tokamaks provides confidence
that ITER/power plants will achieve the desired performance. According to the scaling law,
confinement time scales strongly with size; thus, ITER is designed to be the world largest
tokamak with a major radius of around 7.5 m. The next generation after ITER will be
DEMO [49], which is to demonstrate the first electricity production from fusion, and after
DEMO, a commercial fusion power plant is expected. Figure 1.11 shows the evolution of
tokamak devices from Tore Supra to DEMO, which is closest to a fusion power plant.
Figure 1.11: Evolution of tokamak devices from small scale to large scale. [Courtesy of
European Union, 1995-2016]
1.3 Tokamak MHD instabilities
While there remain numerous issues before the success of ITER’s demonstration,
instabilities are the main reason blocking the way for realizing sustainable fusion energy.
17
Large-scale instabilities, mainly magnetohydrodynamic (MHD) [50], [51] instabilities,
which possess wavelengths comparable to the minor radius of a tokamak, are the most
dangerous modes for confinement. As an example, the neoclassical tearing mode (NTM)
[52], [53], which are resistive tearing mode islands destabilized and maintained by helical
perturbations to the pressure-gradient driven “bootstrap” current [54], [55], impose a
limitation to the maximum ratio of plasma pressure and magnetic pressure (also called the
beta value) of a tokamak and thus limit the performance of tokamaks. The suppression of
NTMs, for example by means of electron cyclotron current drive, is under investigation for
the operation of ITER [56], [57]. In addition, the extremely high-energy transfer rate of
edge localized modes (ELMs) for a typical high-confinement mode, which is the standard
“baseline” operation scenario for ITER, need to be eliminated or reduced by a factor of 20
for component lifetime [58]. Even though a superconducting wall would be ideal for ITER
operation by stabilizing all edge modes, available walls have finite resistivity, which with
the effects of error fields [59], [60] could cause sudden disruption [61] of the discharges,
and thus destroy plasma-facing components including the wall.
In addition to the global large-scale MHD instabilities, micro-scale instabilities,
which have wavelength comparable to the ion (or electron) Larmor radius [62], are also
common in plasmas and are driven unstable by different mechanisms [63]. Three examples
of them are the ion temperature gradient driven (ITG) [64], electron temperature gradient
driven (ETG) [65] and trapped electron modes (TEM) [66]. These modes could interact
and transform into each other. Even though they are less dangerous than the MHD
instabilities, and don’t lead to catastrophic results, they do degrade the confinement and
are thought to be the origin of plasma turbulence. Turbulence in the plasma is the main
18
source for anomalous transport of particle and energy, making it 10 times larger than what
neoclassical theory predicts [67], causing much faster loss of particle and energy than
theory predicts. Figure 1.12 shows the schematic of the distribution of plasma instabilities
in both wavenumber and frequency domain, and they do overlap with each other in the
spectral domain.
Figure 1.12: Schematic system of multi-scale turbulence in magnetic confined fusion
devices. Here, !a is the device size, ρ i and ρ e are the ion and electron gyroradius,
!
!
ω and ω *e are the diamagnetic drift frequency of ions and electrons, and ω A is the
! *i
!
!
Alfven frequency. Other terms are: MHD/RMHD - magnetohydrodynamics (resistive
MHD), ITG/ETG-ion/electon temperature gradient, CTIM/DTIM (CTEM/DTEM)
collisionless/dissiptive trapped ion/electron mode, CDBM - current diffusion ballooning
mode. [Reprint permission granted by Atomic Energy Society of Japan, Ref. [68]
19
1.4 Turbulence
1.4.1 Turbulence in nature
Turbulence is the most important unsolved problem of classical physics and has
attracted the attention of physicists, mathematicians, and engineers for over 100 years.
Heisenberg is reported to have said that he'd ask God two questions: why relativity and
why turbulence? Turbulence driven diffusion draws scientific interest since it is usually
many orders of magnitude larger than the classical diffusion and is called anomalous
transport, as one of the main methods for relaxing gradients [69]. Experiments by Reynolds
on flow in a pipe showed that the transition between smooth and turbulent flow is
controlled by a dimensionless number, which is referred to as Reynolds (Re) number [70].
The Reynolds number is proportional to velocity and inversely proportional to viscosity,
and when Re increases, the fluid tends to be in turbulent flow.
Flows can still be smooth
at high Re number, but become increasingly sensitive to perturbations. Low viscosity fluids,
like air, tend to be more turbulent than high viscosity fluids, say water and oil.
Turbulence is difficult to understand due to its essential nonlinearity resulting from
viscosity and the large range of scales. It is everywhere in nature, from the scale of a coffee
cup to the universe, characterized by unpredictability and strong mixing effects, etc.
Turbulence features are not readily predictable by analytical or computational models,
since the smallest eddies of a turbulent flow can be 1000x smaller than the large-scale
features, thus, requiring around 1 billion points for realistic 3D simulations. Since
turbulence is hard to predict or simulate, the best method for observing turbulence is
20
imaging. Figure 1.14 shows the turbulence drawing from Leonardo da Vinci, which is the
earliest attempt to realistically capture the details of turbulent flows. Some turbulence
phenomena can easily been observed by the naked eye, say water and clouds, while some
need sophisticated observation tools to be seen, say turbulence in the air and the activities
of sunspots. Some tools, which extend human’s observation for turbulence, are, for
example, microscope, telescope, and X-ray photography.
Figure 1.14: The turbulence drawing from Leonardo da Vinci, as the first attempt to capture
the flow details of turbulence.
1.4.2 Plasma turbulence and measurements
Due to the toroidal magnetic topology, turbulence in a tokamak plasma can often
be treated as a quasi-2D problem. Turbulence in plasma has limited smallest scales, which
21
is determined by the ion or electron gyro-orbits, and tends to be driven unstable by smallscale instabilities instead of large-scale stirring. Research results have shown that
turbulence, dominating particle and energy transport inside the plasma, is a serious
impediment to the success of fusion energy that compromises the confinement, since it
transports the energy from the core region to the wall, leading to fast energy and particle
leakage [71]. Detecting the variations of fluctuations both temporally and spatially, thus
revealing the transport properties, is of critical importance for improvement of the
confinement of magnetic fusion.
Experimental results have shown that the variation of turbulence mainly happens
on the poloidal cross section of the plasma (the radial and poloidal directions), while in the
toroidal direction, the correlation distance of the turbulence is much larger. Thus, most
diagnostic approaches are focused on the poloidal cross section. Some of the important
parameters measured for turbulence include: correlation length, correlation time, density
and temperature fluctuation amplitudes, and fluctuation velocities. Two-dimensional
imaging, as an advanced tool for turbulence measurement, has the advantage of
determining the de-correlation time and spatial distance of turbulence easily over the cross
section in the plasma.
The Plasma Diagnostic Group (PDG) at the University of California at Davis has
pioneered in the development of millimeter-wave imaging technologies, such as the
Electron Cyclotron Emission Imaging (ECEI) - measuring plasma temperature turbulence,
and Microwave Imaging Reflectometry (MIR) - measuring density fluctuations [72], [74].
Exploiting large aperture optics and multi-frequency technology, these diagnostic tools
measure the local fluctuations over a 2D plane, and help to provide both the spatial and
22
temporal spectra as well as wave propagation properties (phase and group velocities,
fluctuation amplitude, and frequency) over the measured domain. The ECEI technique
collects passive microwave radiation from the plasma, providing a 2-D image of electron
temperature fluctuations with both good temporal (on µs timescales) and spatial resolution
(on cm spatial scales), and thereby represents the evolution of the electron temperature
fluctuations in space and time [72].
The MIR system is an active radar diagnostic tool, which launches microwaves into
plasma, and measures the phase fluctuations imparted on the reflected wave due to density
perturbations [74]. Even though ECEI and MIR both measure fluctuations with
wavenumbers up to approximately the range of ITG waves (including the whole span of
MHD instabilities), they have different requirements for optimized imaging quality, and
cover slightly different regions of the plasma. This dissertation mainly focuses on the
development and analysis of synthetic MIR diagnostic as well as MIR on DIII-D [75].
More details about the principles and capabilities of both MIR and the synthetic MIR
diagnostic will be discussed in the following chapters. While principles and capabilities
related to ECEI are presented in Appendix I.
1.5 Layout of the dissertation
Work the author mainly conducted is the development and application of the
synthetic MIR diagnostic, and she also participated in the design and configuration of the
current MIR system on DIII-D. The synthetic MIR diagnostic, by combining the plasma
23
profile, density perturbation and pre-designed MIR optical system, simulates the
performance of MIR. The synthetic MIR platform provides a numerical proxy for the actual
MIR system where we can study its sensitivity to optical variations, determine accurately
its local sampling volume, and perform forward-modeling of density fluctuations, which
enabling direct comparison of plasma simulations to experimental measurements. Guided
by synthetic MIR simulations, the MIR system was fabricated and tested under laboratory
conditions, and then installed on DIII-D for commissioning and operation, and has
demonstrated great capability for measuring MHD phenomena on the pedestal of an Hmode discharge [76].
The dissertation content described above is separated into 8 chapters. Chapter 2
focuses on the explanation of the principles of MIR. Chapter 3 introduces the optical design
as well as electronic modules of the MIR system for DIII-D. Chapter 4 introduces the
development of the synthetic MIR diagnostic. Chapter 5 focuses on analytical results from
synthetic MIR, including sensitivity measurements and providing the instrument function
of the MIR system. Chapter 6 shows the initial experimental campaign of the MIR system
on DIII-D and initial comparison between synthetic MIR results with experimental results.
In Chapter 7, synthetic MIR is used for the study of edge harmonic oscillations (EHOs)
and helped to interpret MIR measurements of EHOs [77]. Chapter 8 includes future
upgrades for both synthetic MIR and MIR hardware, and also summarizes the whole
dissertation.
24
Chapter 2. MIR Principles
2.1 Wave Propagation in Inhomogeneous Media
In a cold, homogeneous plasma medium, for waves propagating perpendicular to
the magnetic field lines (in the z-direction), the dispersion relationship as a function of
electron density ( ne ) and magnetic field intensity ( Br ) obtained from the Appleton-Hartree
!
!
dispersion equation [78] is:
2
⎛ ck ⎞
2
⎜ ⎟ = µ = 1−
⎝ω⎠
1−
with
X
2
Y
Y2
±
2(1 − X ) 2(1 − X )
2
nee2
ωce
ω pe
2
ω
=
Y
=
. Here, pe
X = 2 ,
ε 0me
ω
ω
!
(2.1)
is the plasma frequency, and
e Br
is the electron cyclotron
ω pe = 56423 ne with the density ne in cm-3, and ω ce =
!
!
me
!
frequency.
This equation is also valid locally for an inhomogeneous medium when the density
gradient changes slowly compared to the wavelength, and most tokamak discharges satisfy
this condition. We assume that the wave frequency ( ω ) is a constant; then, at one position
inside the plasma, we may have k = 0, meaning that the refractive index µ goes to 0, and
thus the incident wave will be reflected. Such a surface with constant k=0 is called a
cutoff/reflection surface. While at some other positions inside the plasma, we may have
k → ∞ , and thus µ → ∞ , the incident wave is absorbed and such a surface is called a
resonant layer. The polarization of the incident wave can be either parallel (O mode) or
25
perpendicular (X mode) to the magnetic field. In the case of the O mode, the dispersion
relationship equation (2.1) assumes the plus sign, and can be rewritten as:
µ 2 = 1 − X (2.2)
It then gives an O mode cutoff frequency, when !µ = 0 , with cutoff frequency
ω = ω pe . When the polarization is perpendicular (X mode) to the magnetic field, the
dispersion relationship equation (2.1) takes the minus sign, and it changes into:
µ 2 = 1−
X (1 − X )
(2.3)
1− X − Y 2
Examination of this formula indicates there are two cutoff frequencies:
2 1/2
ω = (ωce2 / 4 + ω pe
) ± ωce / 2 (2.4)
Taking the plus sign in this equation, the frequency is defined as the Right-Hand cutoff
for the X mode ( ω rx ), while the frequency with the minus sign is called the Left-hand
cutoff of the X mode ( ω lx ). There is also a resonance layer ( µ → ∞ ) between ωce and the
RX cutoff with the characteristic frequency called the upper hybrid frequency, and the
2
frequency is ωh = ω pe
+ ωce2 , slightly lower than the RX cutoff frequency. Figure 2.1
shows the characteristic frequencies, including ω ce , 2ω ce , ω rx , ω lx , and
ω and
h
ω , corresponding to a DIII-D H-mode discharge with shot number # 157102 at time 2420
o
ms.
26
2ωce
ωce
ωrx
ωlx
ωh
ωo
Freq = ω/2π (GHz)
150
100
50
0
1.0
1.2
1.4
1.6
1.8
2.0
Major Radius (m)
2.2
2.4
Fig 2.1: Shown are the different cutoff layers versus major radius along the plasma
midplane for DIII-D, Shot 157102, 2420, a high confinement mode discharge.
Both O-mode and X-mode cutoff frequencies can be used to measure the density
fluctuations. Since the left-hand cutoff layers are enclosed by O-mode cutoff layers, and
are not accessible from outside of the plasma, the X-mode cutoff frequencies discussed
here are the righted-hand cutoff frequencies. There are several advantages of choosing Xmode over O-mode. The RX cutoff frequencies are much higher than the O-mode cutoff
frequencies, which is 30+ GHz higher for any radial position in the case of Fig. 2.1; thus,
the scattering of the RX wave due to materials on the way of propagation would be smaller,
and more power could reach the plasma cutoff surface and be reflected. In addition, since
there would also exist density perturbations in the plasma before the probing frequency
reaches its cutoff layer, a higher frequency would be less sensitive to these low density
perturbations before the cutoff region, and provide better localization for density
turbulence measurements. Moreover, as can be seen in Fig. 2.1, the O-mode wave can only
27
propagate up to a density given by ω pe , while the X-mode wave can propagate further into
the plasma as the magnetic field increases, thus providing the potential to measure density
perturbations even near the core.
Consequently, for the DIII-D tokamak, the RX cutoff frequency ( ω rx ) is chosen for
detecting density perturbations. Practically, since waves could tunnel from the RX cutoff
layer to the upper hybrid resonant layer, which would then result in a strong absorption of
the wave energy, a targeted reflection layer is best to be chosen away from the resonance
layer (at least one free space wavelength of the probing wave). Due to the existence of the
upper hybrid resonant layer, the incident X-mode wave targeted on the RX cutoff surfaces
can only be launched from the low magnetic field side to reach the cutoff layer and then be
reflected. Compared to the RX cutoff layer, the O-mode cutoff can be reached from both
the low field and high field sides.
2.2 Microwave Imaging Reflectometry
2.2.1
Reflectometry
Reflectometry is a radar-like process where reflected waves from cutoff surfaces,
say from ω rx or ω o , are detected to reveal plasma density information. Reflectometry
has been extensively employed for plasma density profile measurements [79], [80]. This
method makes use of the phase delay induced by the propagation of a probing wave
reflected at a cutoff layer in the plasma. Since plasmas are dispersive media whose
refractive index is a function of plasma density, higher frequency radiation reflects from
28
higher density plasma cutoff layers. After propagation through the plasma, to the cutoff
layer and back, the reflected wave is phase shifted by:
4π f
ϕ( f ) =
c
r1
∫
µ (r )dr −
rc ( f )
π
(2.5)
2
Here, r1 is the position of the plasma edge in radial direction and rc ( f ) is the
cutoff layer position in the radial direction for frequency f . The π/2 factor is introduced
by the reflection on the cutoff layer. Another technique of reflectometry is time-of-flight
measurements, in which the time required for the probing wave to propagate to the cutoff
layer, reflect, and then propagate back out of the plasma is measured. The time delay is
obtained from the phase delay:
τ=
1 dϕ
(2.6)
2π df
The measurement of time-of-flight, even though less commonly used than the phase
measurements, has more advantages when applied to high density, high field fusion
plasmas. The phase delay data (for the case of phase measurements) as well as the group
delay data (in the case of time-of-flight measurements) could then be used to reconstruct
the electron density profile [81], [82].
In addition to measuring density profiles, reflectometry is also widely used for
measuring plasma density turbulence [83][85], due to its high sensitivity to density
fluctuations. Conventional fluctuation reflectometry for measuring density turbulence is 1D reflectometry, without focusing facilities between the beam launching horn and detector
horn, and has been used to simulate Doppler-shift measurements of poloidal rotation [86],
and to investigate the effects of two-dimensional fluctuations on reflectometry
measurements [87]. The conventional reflectometry approach is valid for detecting 2D
29
fluctuations only when the fluctuation wavelength is much larger than the probing
wavelength. As shown in Fig. 2.2 (a), when the poloidal turbulence has much larger
wavelength compared to the probing wavelength, scattering is smaller and 1D
reflectometry would be sufficient. While when the wavelength of the poloidal turbulence
decreases and scattering begins to dominate, the reflected field would have non-negligible
power scattered into different directions, as shown in Fig. 2.2 (b), making1D reflectometry
for measuring density turbulence inappropriate. The 1D reflectometry would not provide
perturbation information from any higher (> 0) order scattering lobes except the zeroth
order scattering component. Sometimes when the scattering is strong, the zeroth order even
disappears, and 1D reflectometry would lose signal [88]. To reconstruct the density
perturbation from the reflected wave propagating in different directions, a focusing or
imaging system is needed to gather as much reflected power as possible, and then focus
the gathered power to one position, which is on the image plane most of the time, to reveal
the original density perturbation. Such a reflectometer with the usage of an optical or
imaging system and with probing wave in the microwave range is called microwave
imaging reflectometry (MIR) [73].
Consequently, a 2-D lens based imaging system, the microwave imaging
reflectometer (MIR), with its physical explanation presented in the following 2D phase
screen model, was suggested by Mazzacatto in 2001 [74] to compensate for the inaccuracy
of 1-D measurements by being able to gather as many as possible of the diffracted
components from the measured position and at the same time rejecting the spurious
diffracted components from other positions. The 2-D millimeter-wave imaging
reflectometer with its validity and limitations widely discussed in [85], [74], [89], [73] has
30
the advantage of permitting the observation of the density fluctuations directly in a 2-D
plane.
Figure 2.2: Reflection with negligible and non-negligible scatterings. When turbulence has
much longer poloidal wavelength than the probing wave wavelength, 1-D imaging is
sufficient to accurately measure density fluctuations, while when the turbulence has
decreased wavelength and scattering cannot be ignored, a 2-D imaging system are needed
to accurately measure density fluctuations.
2.2.2
2-D Phase screen model
The accuracy of MIR as a density fluctuation measurement tool and the reason why
a focusing lens system is needed have been demonstrated using the theory of a 2-D phase
screen model approximation, which was introduced by R. Nazikian, et al. [73] in the
context of the basic principles of reflectometry. To understand this, consider a thin screen,
upon which a sinusoidal perturbation is imposed in the x direction of the form
31
a exp i( K x x − Ωt ) , where K x is the fluctuation wavenumber and Ω is the fluctuation
frequency. A plane wave of the form E0 = A exp i(k0 z − ω0t ) is traveling in the z direction,
where k0 is the wavenumber and ω0 is the frequency, is incident upon the screen. The
modulated wave from the thin screen can be written as E = E0 a exp i ( K x x − Ωt ) ; expanding
this equation into a Bessel series yields
∞
E = A ∑ J m (a) exp i[mK x x + k02 − m 2 K x2 ( z − zc ) + (ω0 + mΩ)t ] (2.7)
m =−∞
In the above, m represents the order of a diffracted wave into a different direction,
and J m (a) represents the mth order Bessel function value at a . The larger a is, the
more energy in each diffracted component there will be. Figure 2.3 shows two
complementary components diffracted into symmetric directions. When the perturbation
of the screen is small, the low order small m components dominate. If only the m= -1, 0,
and 1 terms are retained, the field is reduced to
E = A[1 + 2iδφ sin( K x x − Ωt )exp(iΔK z ( z − zc )]exp i(k0 x − ω0t ) (2.8)
When the amplitude of the perturbation a = δφ is sufficiently small, the above
formula is a valid approximation of the total E field. Here, zc is the position of the cutoff
!
layer in the major radius direction. Consequently, from the approximation, it can be seen
that when z = zc , the field has only a pure phase modulation due to the fluctuation on the
thin screen, and possesses no amplitude modulation. However, when the wave propagates
away from the cutoff surface, the modulation will gradually turn into an indistinguishable
mixture of phase and amplitude modulation. Figure 2.4 (a)-(d) shows the distribution of
the field on the complex plane varying as the propagation distance from the cutoff surface
32
increased. The complexity of the modulation away from the cutoff is due to the different
propagation velocities in the direction of z for each Bessel component. As a result, it is
extremely difficult to determine the original fluctuation information from the jumbled
reflected signal away from the reflection surface as shown in Fig 2.4 (d).
Fig 2.3: Shown are two symmetrically diffracted components of the incident wave after the
fluctuation thin screen. [Reprinted figure with permission from Ref. [73]. Copyright 2001,
AIP Publishing LLC]
By having the reflected wave pass through a lens system, which focuses the
collected beam onto the image plane, as the MIR system does, the signal will again have
the smallest amplitude fluctuation [73], thereby resulting in the phase modulation on this
plane closely related to the original phase modulation on the cutoff. The reformed phase
fluctuation with small amplitude fluctuation on the image plane is shown in Fig. 2.8 (e).
Furthermore, higher orders of diffracted waves passing through the window, not just the
±1 orders, can be gathered by the lens system, to form a more accurate image of the
density fluctuation. Based on the basic principles of imaging, higher orders of diffraction
33
represent finer structures. Nevertheless, MIR is only able to reform the image of density
fluctuations when the receiver is focused at the cutoff surface, and the diffracted
components that can be gathered are limited by the window size of a tokamak device, so a
well-designed optical system is crucial to forming an accurate MIR imaging. Consequently,
this motivates the need for computational optimization of MIR before actual design and
installation, for estimation of the baseline performance of MIR.
(e)
Fig 2.4: Phase and amplitude modulation along the propagation path for a MIR system.
The wave is purely phase modulated as it leaves the cutoff surface, and gradually changes
to an indistinguishable mixture of phase and amplitude modulation as it propagates away.
Then a lens system is applied on the propagation path to refocus the reflected wave to the
image plane where the amplitude modulation is minimized and the phase modulation is
highly related to the original density perturbation on the cutoff surface. [Reprinted figure
with permission from Ref. [73]. Copyright 2001, AIP Publishing LLC]
What sets our system apart from others is that we employ large aperture lenses to
illuminate a large cross section of the plasma and correspondingly receive the phase and
amplitude modulated reflected beam from an equally large cross section of the plasma.
34
2.2.3
2D reflectometry
Shown in Fig 2.5 are the laboratory test results from the initial proof-of-principle
MIR instrument developed for the TEXTOR tokamak. Here, the corrugated wheel is
constructed from an inner wheel with 60 cm in diameter and 20 cm in width, with a
sinusoidally corrugated flexible aluminum strip wrapped around the circumference. A
range of poloidal wavenumber kθ and corrugation depth hcorr , as highlighted on this
figure, could be observed by varying the spacing and height of the corrugated structure. A
single point 2D MIR measurement is configured with a transmitter horn, steering and
focusing reflective mirror, and a receiver horn, as shown in Fig 2.5 (a), the above part. A
simple conventional 1D reflectometry with a combination of transmitter horn and detector
horn is also configured in Fig. 2.5 (a), the below part. Distance from the reflector to the
first component of the two type of reflectometry (the first focusing mirror in 2D
reflectometry configuration and the detector horn in 1D reflectometry) is highlighted in
Fig. 2.5(a), and represented by d . When the distance from the target to source is small in
the 1D setup, for example with d = 10 cm, the 1D reflectometer works well and recovers
the fluctuation accurately, as shown in Fig. 2.5 (b), in the first row. However, when the
distance between the source and target is increased to 30 cm, as shown in the second row
in Fig. 2.5 (b), the 1D reflectometer then fails to recover the correct rotating pattern, with
considerable mismatching between the measured vibration (black curve) and original
vibration (light blue curve). This mismatch didn’t happen in the optics-based 2D imaging
where the target wheel is 235 cm apart from the detector horn, which is located at the image
plane. As shown in Fig. 2.5 (b) in the last row, the measured vibration pattern through the
35
optics-based 2D reflectometry follows closely with the original rotation pattern, even better
than the 1D reflectometry when the detector is located within 10 cm to the target wheel.
(a)
36
1D configuration, d = 10 cm
1D configuration, d = 30 cm
2D configuration, d = 235 cm (at focus)
(b)
Figure 2.5: Comparison of imaging from 1D and 2D reflectometry. (a) Configuration of
the optics-based 2D reflectometry and non-optics-based 1D reflectometry. (b) Measured
imaging pattern for the 1D configuration with the detector located at 10 cm and 30 cm to
the rotating wheel, and also the measured pattern from 2D reflectometry with the detector
located at 235 cm from the rotating wheel. [90]
Not just limited to detecting single point fluctuations, an optics-based MIR system
could be extended to simultaneously measure density fluctuations at different radial and
poloidal positions, thus creating a 2-D image of the fluctuations on the radial-poloidal plane
37
over time. The simultaneous radial detection could be realized by launching multifrequency probes into the plasma at the same time, while the simultaneous poloidal
detection could be realized by adding more detectors on the image plane from a welldesigned optical system. Figure 2.6 shows the simple schematic of an MIR
transmitter/illumination and receiver/detection system. The combined illumination and
detection system employs large aperture refractive lenses to illuminate a large cross section
of the plasma and correspondingly receive the phase and amplitude modulated reflected
beam from an equally large cross section of the plasma. This is what sets the MIR system
discussed in this dissertation apart from most previous MIR systems. One more important
merit which sets the current MIR system apart from all the previous MIR systems is that it
takes serious account of the curvature matching, which is essential for reconstructing
accurate 2D images of the plasma density turbulence and will be discussed in Chapter 5,
and also applied to a sophisticated synthetic simulation approach to pre-perform MIR
performance and guide the optimized optics design. A brief discussion about previous MIR
systems and measurement results are presented in Appendix II.
38
Fig 2.6: Schematics of MIR illumination and detection system. Large aperture refractive
optics as well as multi-frequency probing beam is employed to illuminate a large cross
section of the plasma and correspondingly receiver the phase and amplitude modulated
reflected beam from an equally large cross section of the plasma.
39
Chapter 3
MIR System on the DIII-D Tokamak2
3.1 Optics
3.1.1 Optics Overview
There are two intertwined phases of optical design that are of concern for MIR; one
is the illumination system and the other is the receiver system. A transmitter system is
comprised of a source and an antenna horn combined with E-plane and H-plane lenses; it
transmits high RF frequency through a lens system into the plasma. The reflected signal
then goes through the receiver system, which is also combined with E-plane and H-plane
lenses, to be detected. The receiver would have shared optics with the transmitter. A basic
conceptual optical design for MIR is shown below in Fig. 3.1. Both transmitter and receiver
will share the same collection optics, and then they will have individual lenses before each
to adjust the focal length of the beam to meet different specifications. The radial coverage
is determined by the frequency range transmitted by the transmitter, and the poloidal
coverage is determined by both the window size and the cutoff curvature. The window that
is used by MIR on DIII-D has a maximum vertical height of around 60 cm.
2
The optics design and installation of this MIR system is primarily the work of Christopher M Muscatello,
and is added here for continuity and the convenience of the reader. The electronic module design is
primarily the work of fellow Ph.D student Alexander Spear. The author mainly participated in lab testing of
the electronic modules and optics.
40
Vacuum window
Shared zoom lens
Receiver array
r
tte
i
m
s
LO
n
Tra
Figure 3.1: CAD drawing of the MIR system, with plasma vacuum window, shared zoom
lenses, which is also shared with the ECEI system, and LO, transmitter and receiver array
[Reprinted figure with permission from Ref. [91]. Copyright 2014, AIP Publishing LLC].
The actual lens system design would be much more complicated. Considerable
attention should be devoted to boundary conditions and optical aberrations. Diffraction, a
boundary condition that is caused due to limited aperture size, can be significantly reduced
empirically by making the beam size on each surface less than or equal to 80 % of the
aperture size. Petzval field curvature [92], an optical aberration causing a flat object not to
be brought into focus on a flat image plane, can cause the focal positions of the receiver
array not to lie on a desired curvature; this can be reduced or compensated by using a
41
crescent-shaped mini-lens array or more complicated optical lens design [93]. Other optical
aberrations, such as coma [94] and spherical aberrations [95], can also diminish the
imaging quality of MIR. Reflections between each lens surface should also be carefully
considered for MIR, since any kind of reflection can impose extra phase noise on the
modulated phase information from the density fluctuations. Nevertheless, since the MIR
system shares the same zoom lens with the ECEI system on DIII-D, and the dimensions of
the MIR system are limited to a nearly 1.2 m x 1.2 m square floor space, this has caused
the optical design to be complicated due to the usage of several reflectors.
3.1.2
Transmitter configuration and testing
3.1.2.1 Horn antenna
The current transmitter horn is a WR-15 25 dB Gain pyramidal horn antenna, and
it transmits the TE10 waveguide mode. It is capable of transmitting frequencies from 50
GHz to 75 GHz; while among this range, MIR uses a sub-frequency range covering from
56 GHz to 74 GHz. The transmission horn has a height of 1.16 inches, a width of 1.5 inches,
and a length of 2.75 inches. Its far field pattern can be modeled as an equivalent Gaussian
beam, as the approximation could be obtained by using the formulas below [96], [97]:
λ
(3.1)
wo (E) =
π tanθ1/e (E)
!
λ
(3.2)
wo (H) =
π tanθ1/e (H)
!
42
λ
λ
where θ1/e (E) = 40 , and θ1/e (H) = 55
b
a
!
!
Here, λ is the probing wave free space wavelength, !a is the horn mouth width and
is 1.5 inches here, and b is the horn mouth height and is 1.16 inches here. In addition,
θ (E) is the angle when the intensity of the electric field in the vertical direction has
! 1/e
decreased by 8.7 dB (= 20log 10 e ) in the far field approximation, respectively. Likewise,
!
θ (H) is the angle when the intensity of the magnetic field in the horizontal direction has
! 1/e
decreased by 8.7 dB, respectively. Applying these formulae, the calculated equivalent
Gaussian beam model has a beam waist in the vertical direction (E-plane) of around 12.7
mm, and 10.6 mm in the horizontal direction (H-plane) for 65 GHz. Figure 3.2 shows the
Pyramidal horn antenna as the transmitter and also as a local oscillator, which will be mixed
with the reflected signal to bring it into an IF range of 0.5 GHz to 9.0 GHz.
65.37 GHz LO
65 GHz transmitter
43
Figure 3.2: The transmitter horn attached to the Gunn oscillator. One is the 65 GHz
transmitter and the other is the 65.37 GHz local oscillator.
The antenna pattern of the horn is measured within an anechoic chamber where the
walls, ceilings, and floor are lined with special electromagnetic wave absorbing materials.
It is quite commonly employed for measuring indoor antenna ranges, and most suitable for
frequencies above 300 MHz. The power source is an old-styled backward wave oscillator
(BWO) source with stable amplitude output. Figure 3.3 shows the measured antenna
pattern in the range of -20 to 20 degrees in both the E plane and H plane for three different
frequencies, 56, 65, and 74 GHz. A clear main lobe is measured with this range, with no
obvious side lobes observed. Since the horn mouth has a smaller E plane dimension than
for the H plane, there are more jags in the E plane than in the H plane.
0
56 GHz
65 GHz
74 GHz
E plane
Power (dB)
-5
56 GHz
65 GHz
74 GHz
H plane
-10
-15
-20
-25
-30
-20
-10
0
10
Angle (deg)
20
-20
-10
0
10
Angle (deg)
20
Figure 3.3: Horn antenna pattern measured from -20 to 20 degrees. Since the E plane has
narrow width, there are thus more jags in the E-plane pattern.
3.1.2.2 Transmitter pattern measurements
44
The transmitter is designed with sophisticated optical software, CodeV, and the
transmitter Gaussian beam waist is 12.7 mm in the E plane and 10.6 mm in the H plane.
Figure 3.4 shows the 3D view of the transmitter with the beam propagating from the
transmitter to the plasma cutoff surface. Since there are some rendering problems of the
software, which cannot be helped by the support group due to limited access of the student
license, the lenses in this figure are not properly displayed, but instead rendered as
rectangles. Starting from the source of the transmitter, in less than 30 cm distance away,
one cylindrical E-plane lens (effective curvature is only in the vertical direction) and one
cylindrical H-plane lens (effective curvature is only in the horizontal direction) are used to
control the beam size. The beam then travels 2 m through two reflectors, and reaches two
E-plane doublet objective lenses, which are used to control the zooming field. After this,
the beam reaches the vacuum interface, and an H-plane mirror (located inside the vacuum
interface) is used to tighten the beam in the horizontal direction. The lens material is highdensity polyethylene (HDPE) [98], with refractive index around 1.57. In addition, HDPE
is hard solid, and can be easily fabricated (i.e., machined) and mounted, and it could also
be made relatively thick to satisfy sophisticated requirements of lens design.
45
(a)
(b)
Figure 3.4: 3D views of the transmitter. Shown in (a) is the E-plane/side view and in (b) is
the H-plane/top view. The lenses are not properly displayed, due to some bugs of the
software, and this could not be helped due to the limited support for the student license.
The beam field with diffraction-based propagation is simulated with a beamlet-based
synthetic propagation method (BSP). One example of the simulated beam intensity on a
modeled 58 GHz cutoff surface is shown in Fig. 3.5. As can be seen from this figure, the
field distribution in the horizontal direction is relatively narrow, while the field distribution
in the vertical direction is widely spread. The beam radius in the horizontal direction is
around 24 mm as it arrives at the plasma, while the beam radius in the vertical direction is
around 93 mm as it arrives at the plasma, meaning the illuminated region is around 200
mm vertically (poloidally). The cutoff surface curvature is around 700 mm for simulations
in this condition.
46
Figure 3.5: Simulated beam field intensity as it arrives on the plasma cutoff surface (58
GHz) using the beam synthetic propagation (BSP) option in CodeV.
The transmitter optics were assembled and tested within the laboratory. Since the
two-piece doublet zoom lenses, which are shared with ECEI, cannot be removed from the
DIII-D ECEI system for testing, they are also fabricated in duplicate for testing and lens
configurations, as highlighted on Fig. 3.6. In addition, the translation stage for the optical
lenses is also not mounted for testing. Here, the translation stage would enable remote
control of the positioning of the focal lenses to eliminate inaccuracy from manual
adjustment. Figure 3.6 shows the testing configuration of the transmitter optics. The WR15 pyramidal horn antenna is attached with the Gunn oscillator to provide transmitter
47
power. A detector after the zoom lenses, located at the position of the plasma, is applied to
detect the field strength in the poloidal direction. A measured field strength pattern along
the vertical direction is shown in Fig. 3.7. Since the measurement is made on a flat surface,
the vertical coverage is thus slightly smaller than the coverage on a curved surface.
Figure 3.6: Transmitter testing configuration. For testing convenience and better
collimation, reflectors are not placed. The field strength is detected by antenna after the
zoom lenses.
48
Figure 3.7: Measured transmitter pattern along the vertical direction. Since the
measurements are made on a flat surface, the vertical coverage is slightly smaller than the
poloidal coverage on a curved cutoff surface.
3.1.3
Receiver configuration and testing
3.1.3.1 Mini lens array
The receiver array is composed of 12 individual pairs of dual dipole antenna and
mini lens [99], with the layout shown in Fig. 3.8. A dual dipole antenna [100] with
dimensions of around 40x30 mil is attached to the back of each mini lens. The dipole
antenna uses a Schottky diode [101] across the two arms, as highlighted in Fig. 3.8. This
diode has a series resistance of 10 ohms and a total capacitance of 0.008 pF. It allows a
49
maximum dissipated RF power of 15 mW, with a cutoff frequency of 5 THz. The diode
dimensions are 0.1 (w) x 0.5(l) x 0.01(t) mm, and constitute the main factor that limits the
spacing design of the dual dipole antenna. It down converts the reflected high frequency
(centered at 65 GHz) with an LO frequency (65.37 GHz), into an RF range of 0.5 to 9 GHz.
The elliptical substrate mini lens with dual dipole antenna attached to it can provide the
highest directivity compared to other substrate lenses, with minimal off-axis aberration and
side lobe problems [100]. Figure 3.9 shows the measured field pattern for an individual
pair of dual dipole antenna and mini lens, with the antenna tilted in both the E and H
directions.
The RF signal from 0.5 to 9 GHz from the dual dipole antenna then passes through
a low noise amplifier and is transferred through low loss coaxial cables (mostly long) to
the place/room where the electronic RF/IF modules reside.
(a)
(b)
Schottky diode
Figure 3.8: Curved mini-lens array mounted on the array box (a) as well as the dual dipole
50
antenna (b) attached to the back of each mini lens. The signal from the antenna is passed
through a low noise amplifier, and then is transferred to the RF/IF electronics modules.
0
Power (dB)
-5
56 GHz
65 GHz
74 GHz
E plane
56 GHz
65 GHz
74 GHz
H plane
-10
-15
-20
-25
-30
-20
-10
0
10
Angle (deg)
20
-20
-10
0
10
Angle (deg)
20
Figure 3.9: Receiver antenna pattern in both the E-plane and H-plane, with side lobes
appearing at around -15 dB with respect to the main lobe.
3.1.3.2 Receiver optics configuration and testing
The receiver curvature matching is compensated for by the similarly curved
detector min lens array as shown in Fig. 3.8. The mini lens array curvature also
compensates for the difference in optical path length between the center and edge channels
that appears as a spherical aberration. This configuration yields robust imaging across the
entire poloidal array of antennas for a large set of plasmas conditions. A different approach
is to use meniscus lenses [93]; however, this can be prohibitively expensive and is difficult
to fabricate using high-density polyethylene (HDPE) compared to the curved mini lens
array method. The curved array method is implemented in the DIII-D MIR system due to
its cost effectiveness and ease of fabrication [91], [102]. Both meniscus and parabolic
51
lenses, and especially parabolic lenses, are investigated and will be applied for the coming
EAST MIR optics [103], [104]. More sophisticated lens design is also under progress for
the upgrade plan of the DIII-D MIR system.
The receiver optical system has different design purposes concerning the E plane
(vertical plane) and H plane (horizontal plane). On the E plane, where imaging takes place,
the optics is composed of three parts: the curved mini-lens antenna array, a focal lens, and
objective lenses (also known as zoom lenses for the current DIII-D ECEI system [105],
[106]). Figure 3.10 shows the 3D side view (E-plane) and top view (H-plane) of the
receiver system. Starting on the port side of the optical train, the doublet objective lenses,
which are shared with the transmitter as well as the ECEI system and highlighted on Fig.
3.10, are used to control the shape and size of the focal plane on the object plane (the
plasma cutoff). Next, the focal lens is used to control the positioning of the focal plane
radially in the plasma. Finally, the curved antenna array is designed to be an image of the
object, i.e., the curved cutoff surface. On the H-plane of the receiver optical train, the
effective optical components consist of two curved reflectors used to form a narrow
convergent (nearly collimated) beam at the cutoff surface [107].
52
(a)
Port side
Doublet objective lens
Focal lens
(b)
Curved Antenna array
Port side
Figure 3.10: Shown are the side view (a) and top view (b) of the receiver system. The side
view is the E-plane (vertical plane) where imaging mainly takes place. The top view is the
H-plane (horizontal plane) where the beam is mainly focused and narrowed to converge
power, thereby permitting the beam to pass through the narrow H-plane apertures.
The receiver pattern is also calibrated and measured in the laboratory, in the
poloidal/vertical direction. Figure 3.11 shows the configurations for receiver testing. Since
there is no plasma in laboratory, a long rod reflector made of metal, which is used to
perform like a point source in the vertical direction, is used to mimic the reflection from
the plasma. There is a small horn underneath the rod reflector, which is powered by the
53
combination of several multipliers and a synthesizer. Absorbers are placed around the rod
to minimize reflections from other targets, making the receiver see the clear signal from
the rod. The rod will move up and down, and the reflected signal from the rod then passes
through the zoom lenses as well as other receiver focusing lenses, and arrives at the receiver
array box. The detected signal then goes into the pre-calibrated RF/IF electronic modules,
as shown in Fig. 3.11 left lower corner. The field amplitude from different modules is
plotted with respect to rod positions. Figure 3.12 shows the measured receiver field
amplitude from different modules with respect to the rod positions. The field amplitude for
each channel is distributed evenly and with similar maximum voltage, indicating that the
electronics for edge channels and center channels performs similarly. The total
measurement region is around 20 cm.
Figure 3.11: Receiver antenna testing configuration
54
Fig 3.12: Antenna patterns measured with the position of the targeting rod; different
receivers are focusing at different poloidal positions, evenly spaced, and 2 cm resolution.
This value actually changed to 3 cm in the plasma, due primarily to refraction effects in
the plasma.
3.2 Electronics modules testing
3.2.1 MIR system electronics overview
The transmitter electronics configuration is shown in Fig. 3.13. It employs a 65
GHz Gunn oscillator to generate a central carrier frequency which is mixed, using a double
55
side band mixer, with two separate 1–9 GHz tunable frequency synthesizers (Pronghorn
sources, due to their highly stable frequency and power) to simultaneously generate a set
of four frequencies between 56 to 74 GHz. The four frequencies are power combined and
transmitted from the pyramidal horn antenna into the plasma. Each synthesized IF signal
source has a corresponding frequency offset source with 510 MHz offset, and each of them
mix with the IF signal source to generate 510 MHz reference signals which are used in the
down conversion process.
Figure 3.13: Multi-frequency MIR illumination source formed by up-converting
intermediate frequency (IF) sources and amplifying the up-converted signals. The IF
source frequency range is from 0.5 to 9.0 GHz.
A receiver system prototype, containing the plasma, optics, and electronics is shown
in Fig. 3.14. Before entering the receiver array box, the reflected signal first passes through
a notch filter and dichroic plate. The notch filter is a type of frequency selective surface
(FSS), and it is implemented here to protect the mixer array from stray electron cyclotron
resonance heating (ECRH) power. Since enormous heating power from ECRH might leak
out of the vessel and be received by the detector, a notch filter here can provide very high
rejection at the ECRH frequency, but low insertion loss at frequencies away from the
56
ECRH frequency, thus preventing potential burnout of the mixing array or spurious signals.
The dichroic plate is a perforated metal plate, with arrays of holes, as shown in Fig 3.14. It
uses the cutoff frequency property of a circular waveguide to serve as a high pass filter for
single sideband mixing. Signals reflected from the plasma then reach and are focused onto
the receiver array that contains 12 spatially separated antennas. These antennas contain
high frequency mixer diodes being pumped with a 65.37 GHz Gunn oscillator. The two
Gunn oscillators are offset by 370 MHz to prevent the reflected signals from overlapping
after down conversion. The resulting IF signal from each antenna contains the original 1–
9 GHz probe frequencies with a 370 MHz offset and a phase modulation imposed by the
plasma density fluctuations. The IF signal from each antenna is sent to its own RF/IF
receiver electronics module for down conversion to recover the plasma fluctuation
information.
65.37 GHz
Gunn
Oscillator
To down conversion
IF ± 370 MHz
Figure 3.14: Shown is the reflected signal passing through the optics, notch filter, and
57
dichroic plate, and then gathered with the antennas and down converted to the original 1-9
GHz probing frequencies with a 370 MHz offset and a phase modulation imposed by the
plasma density fluctuations.
3.2.2 Module overview
Each RF/IF receiver module contains two identical down converter electronics
chains, with a total of 1 input and 4 pairs of I/Q outputs. The input signals are first amplified
and split into two down converter chains, where they are mixed with RF LO (~1-9 GHz)
signals and down converted to two IF signals. The IF signals consist of 140 MHz and 880
MHz carriers. They are then mixed with 140 MHz and 880 MHz LO signals to be down
converted to < 1.5 MHz I/Q signals, which represent the density fluctuation signals. The
low frequency signal is then routed to the analog-to-digital converters (ADCs), which
sample at a rate of 1 MHz. Figure 3.15 shows the electronic module without frame. The
four-layer board design is implemented using Rogers RO4350B dielectric material for the
RF signals and FR-4 for the digital and power layers. An unbroken ground plane separates
the RF layer from the power and digital layers allowing for RF signal isolation. The RF
layer utilizes grounded coplanar waveguide transmission lines for 50 Ω matching
throughout the module. The IQ mixer on this figure is a mixer with one input but two output
ports, with a 90-degree phase difference between the two output signals, in-phase and
quadrature components.
58
Figure 3.15: Shown is the MIR IF electronic board module without the frame. Top cover
and LO routing feedthroughs are removed. [Photographed by the author]
A backplane, which is like a bus in computer design to all the modules, is
implemented to make the connections as well as the control of digital signals easy. Since
each module contains many digitally controlled variable gain stages and also RF switches,
numerous digital I/O lines are required. To make the control clean and easier, preprogramed microcontrollers are also embedded on each module to control all the variable
gain amplifiers, and then all the microcontrollers are connected to the backplane. The RF
switches, requiring 6 digital lines on each module, are directly routed to the backplane. The
digital signals on the backplane are then controlled by a master controller, which is
programmable, and can be remotely controlled by the computers, using LabVIEW in this
case. Figure 3.16 shows 8 modules connected to one backplane, while the master controller
is not shown. This kind of design allows for easy future upgrading of system capabilities
59
without having to disassemble the entire system.
Figure 3.14: Shown is the rack of 8 MIR modules connected to the backplane.
[Photographed by Benjamin Tobias, AIP reprint permission]
3.2.3 Module testing
The RF/IF modules were assembled, tested, and characterized at UC Davis by the
author and her colleagues. The 12 by 4 channel system requires 12 modules, and 15 are
fabricated to ensure sufficient back-ups. A testing signal generation board configured as in
Fig. 3.15 was built and used to generate all the four LOs and input signal. Here, as seen
from the figure, a 1 MHz Pronghorn signal (in the 371 MHz PHS) is used to model the low
density fluctuation signals. Since each board needs four different LO signals, two RF LOs,
one 140 MHz and 880 MHz, four LO signals are generated separately. Due to the limited
number of Pronghorn sources (only four), the two RF signals used for testing are the same,
which will not be true during actual measurements. All the 15 boards were tested, and some
of them were fixed manually. One board has misplaced component, and cannot be fixed
60
easily, thus is left out for further measurements.
After fixing the boards with problems, the boards were then tuned to output similar
maximum voltages for each channel by observing the IQ plots through an oscilloscope.
Figure 3.16 shows the measured maximum voltage for 4 different output ports and 7
different frequencies. Since the in-phase and quadrature power output from each individual
I/Q mixer is almost the same, as it should be, the maximum voltage is measured for either
in-phase or quadrature components. The measurement is performed along all the 14 boards.
As can be seen from this figure, the board voltage distribution with respect to each pair of
I/Q output channels as well as different frequencies is flat along all the 14 boards, meaning
that each board has similar performance, and will not introduce extra errors along the
poloidal direction measurements. In addition, we successively acquired 2 spare boards for
replacing.
61
PHS (Pronghorn synthesizer source)
Amplifier
Mixer
PHS: 1~9 GHz
PHS: 1~9 GHz + 510 MHz
Coupler
Coupler
PHS: 370 MHz
PHS:
371 MHz
Power Splitter
140 MHz BPF
Power Splitter
880 MHz BPF
RF LO1 RF LO2
INPUT RF
140 MHz LO
880 MHz LO
Fig 3.15: Diagram to generate the testing signals. There is a 1 MHz reference signal that is
used as the analytical signal of density perturbation. Both RF LOs from the testing signal
are the same, while which will not be true when actually used.
62
Fig 3.16: Measured maximum voltage for 4 different output I/Q pairs and 7 different
frequencies. The measurement is performed along all the 14 boards. The board voltage for
each pair of output channels and frequency is flat along all the boards, meaning that the
board wouldn’t introduce extra errors for poloidal direction measurements.
63
Other tests of the background noise level, channel signal coupling, power response,
and remote computer controls were also performed. Since time ran out for testing due to
the tight DIII-D installation schedule, the data for these tests were not well documented
and thus are not presented in detail here. Figure 3.17 shows one part of the testing facilities,
as well as two examples of the I/Q signal on the oscilloscope. As can be seen from this
figure, the I/Q signals have similar voltage levels from the same channel, and the measured
“density fluctuation” frequency is 1 MHz sinusoidal wave.
64
A140
B880
Fig 3.17: Shown on the left are the testing facilities. On the right are two screenshots of the
reading of the 1 MHz I/Q signal on the oscilloscope, for channels A140 and B880.
3.3 Finalized system on DIII-D
After intensive measurements and calibration of the optical system as well as the
RF/IF modules, the MIR system was transported to the DIII-D tokamak at San Diego for
65
installation. After further calibration and testing after installation, the MIR system is now
working intensively and importantly for the density fluctuation measurements.
Figure 3.18 shows the assembled MIR/ECEI system on DIII-D. The right side is
the combined MIR and ECEI optical system. The MIR system is an active radar which
sends a probe wave into the plasma, while ECEI is a passive radiometric diagnostic system.
The MIR system works at a frequency range of 56 to 74 GHz, while the ECEI system
works at a frequency range from 90 to 130 GHz. They share the same device window as
well as zoom lenses, and are employed to simultaneously provide perturbation information
for electron density and temperature. The optics is located around the tokamak device,
while the electronic modules are located in a different room, as shown in the right part of
Fig 3.18. The signals detected from both MIR and ECEI receiver systems are transferred
to the modules through low loss coaxial cables.
MIR
ECEI
To
p
las
Sha
red
zoo
m
ma
len
ses
Figure 3.18: Combined MIR/ECEI optical system on DIII-D, as well as the location of the
electronics modules.
66
Chapter 4
Development of Synthetic MIR Diagnostic
4.1 The need for a synthetic MIR diagnostic
Due to the rapid development of computer technologies and computational
techniques, pre-modeling and simulations are becoming standard approaches prior to
actual real manufacturing and building. Simulations have been widely applied due to their
capability to save enormous amounts of money and time, while at the same time providing
sufficient data to build a reliable product. This process has been widely used in engineering
and product design to investigate the effect of changes without the need of producing a
physical prototype. A simulation can give results that are not experimentally measurable
with our current level of technology and help to easily capture and gain the understanding
of the whole system. Applied to the MIR system, it will also be of great benefit if we can
predict the expected performance and limitations of such a system before its actual
fabrication and operation. Simulations and modeling could help to optimize the
performance of such a system. In the case of MIR: maximizing the poloidal coverage with
restrictions from aperture size and the on-site platform, tuning of various optical
parameters, testing performance for different plasma operations, and knowing specific
influences due to different optics misalignments. Synthetic MIR can also help to determine
the operational domain of such a system considering the perturbation wavenumber and
fluctuation amplitude.
The synthetic MIR method models the optical process of MIR and also simulates
the response of the MIR signal to different plasma fluctuations. It models the large aperture
lenses and simulates the complex field with sophisticated optical software along the
67
propagation path up to the plasma surface. It also takes the real EFIT [108] equilibrium
profile from different tokamaks, including DIII-D and EAST. External analytical density
fluctuation models or more realistic fluctuation models from plasma simulation software
are readily included as perturbations to the equilibrium. The wave interaction between the
turbulent plasma and the probing microwaves is simulated with a full-wave reflectometry
code (FWR) [109], which was mainly designed to simulate correlation reflectometry in
large Tokamak devices. The synthetic MIR platform enables direct comparison of plasma
simulations to experimental measurements. With the synthetic modeling, we have a
numerical proxy for the actual MIR system where we can study its sensitivity to optical
variations, determine accurately its local sampling volume, and perform forward modeling
of density fluctuations. Figure 4.1 shows the flow chart of how synthetic MIR works, with
the first row indicating the input information to the simulation of the plasma-wave
interaction. From the simulation output, we can then conclude about how MIR would
perform as a density diagnostic tool, and also how to optimize the performance.
68
Table 4.1: Shown is a simple flow chart for the synthetic MIR diagnostic. The first row is
the input parameters to the simulation of the plasma wave-interaction, and the last row is
what we can obtain from the synthetic simulations.
Before this work, no forward synthetic simulation of an MIR system had been
performed prior to the development of the MIR system. It had been difficult to determine
how sensitive the diagnostic performance is to the optical setups. In addition, scientists had
no simulation results to compare with when they obtained experimental data. Being
unaware of how critical the design of the optical system needs to be as well as how plasma
turbulence would affect the reflected wave, previous MIR systems are not able to robustly
measure density fluctuations in the plasma, even though their laboratory tests have shown
their capability to recover fluctuation images of the tested target, such as a corrugated metal
surface [110]. For example, the TEXTOR MIR system has been used to accurately measure
the poloidal rotation velocity with and without neutral beam injection (NBI) [111].
4.2 FWR2/3D code
The FWR code, taking thermal relativistic effects into account, is specifically
developed to simulate correlation reflectometry in large-scale fusion devices. This code
has two versions: FWR2D, which iteratively solves wave propagation in a 2D poloidal
plane; and FWR3D, which solves the full-wave equations over a 3D volume of plasma [5].
The FWR2D code is the key component of a synthetic MIR diagnostic for establishing
design parameters, and predicting the baseline of an MIR system performance. This code
has been validated successfully against laboratory experiments [109], [113] and it has been
69
applied to interpret reflectometer data that were taken on the JT-60U tokamak [38]. Since
FWR3D was only recently developed and is still not ready for massive and efficient
simulations, its simulation results will not be included in the main content of this
dissertation from Chapter 4 to 7; instead, some brief results from it will be presented in
Chapter 8 for discussing about future development of synthetic MIR diagnostic.
4.2.1 FWR Computation model
!
The propagation of the electric field amplitude !E( x ,t ) in an inhomogeneous medium
can be described by the wave equation:
2iω
∂E
+ ℓE = 0
∂t
where!ℓ ≡ c 2∇2 + ω 2ε , andc is speed of light,
(4.1)
ω
is free space wave frequency, and
ε
is the dielectric constant.
An explicit solution of the wave equation applying the finite-difference time-domain
(FDTD) method requires a time step which satisfies the Courant stability condition [114],
[115]:
⎡ 1
1
1 ⎤
vmax Δt ≤ ⎢ 2 + 2 + 2 ⎥
⎣ Δx Δy Δz ⎦
Here,
−1/2
(4.2)
vmax is the maximum wave phase velocity within the medium, which can be
taken as the speed of light, while Δx , Δy , and Δz are the mesh grid cell sizes for
simulation, and should be much smaller than a single wavelength for accuracy. Thus, a
single such run takes of the order of 100 CPU hours (1 CPU hour == using CPU 100 % for
1 hour) for estimation. With this requirement, as well as the fact that hundreds of runs are
70
required to achieve statistically significant results for a given set of parameters, an explicit
solution is impractical. Thus, an implicit algorithm, which is stable for arbitrarily large
time step, is applied [109]:
⎛ 2iω 1 ⎞ n+1 n
⎜⎝ Δt + 2 ℓ⎟⎠ E = s (4.3)
!
With !E n+1 = E[(n +1)Δt ]
⎛ 2iω 1 ⎞ n
The source s n = ⎜
+ ℓ⎟ E , involves only quantities at the nth time level. Since,
Δt
2 ⎠
⎝
!
experimentally, the wavevectors are aligned principally along one of the (Cartesian)
coordinate directions (R), an efficient iterative solution, the line Jacobi method [116], can
be employed to solve for wave reflection near the reflection layer.
While for the region away from the reflection layer, the condition c∇ε / ωε ≪ 1 is
satisfied, further efficiency is achieved there through the usage of the paraxial
approximation. The radial wavevector is defined as kR (R) = ε (R,Z 0 ) , the integrated
!
phase along the centroid direction is φ = ∫ R kR dR . The field could be decomposed into
!
incoming (I) and reflected (R) components along the radial direction:
E(x ,t ) = E PI exp(−iφ )+ E PR exp(iφ ) (4.4)
!
Here, the subscript “p” designates “paraxial”. Time variation of the medium is ignored
since it is slow compared to the wave transit time across the simulation domain. The
incoming and reflected components could then be solved independently through the usage
of Gaussian elimination [117] to obtain time efficiency.
71
In summary, the computation is split into three sub domains: the free space domain
(one step solution, most efficient), the paraxial approximation domain (less efficient) and
FDTD full-wave solver domain (least efficient). The total process can be summarized into
the following steps:
(a) Specify the (complex) incident amplitude at the antenna (detector) plane. This can
be done either by using an analytical model of the beam field or external beam field
from other optics simulation software. The second method is used in this
dissertation
(b) Project the incident complex amplitude onto the plasma boundary using the freespace Green’s function
(c) Solve the paraxial equation for E PI up to the boundary of the full wave solver at
!
R = RFW , which is within a few wavelengths of the reflection point.
!
(d) From RFW inward to the reflection layer, solve the full-wave equation, implicitly
!
for the complex field amplitude, with E PI as the incoming wave amplitude. Iterate
!
the solution until steady state is achieved.
(e) Take the outgoing component E R of !E as the initial condition for the reflected
!
paraxial field with E PR (RFW ,Z ) = E R (RFW ,Z )
!
(f) After solving the paraxial equation for E PR up to the plasma boundary, project the
!
reflected field to the receiver (detector) plane again using the free-space Green’s
function.
Figure 4.2 shows the combination of the antenna/receiver, the free space propagation,
the paraxial propagation, and full wave solution of the FWR2D simulation. An illumination
72
beam is introduced into FWR2D. Near the cutoff layer, a full wave solution of the plasma
wave interaction is obtained. After propagating paraxially to the plasma edge, the resulting
field is convolved with the antenna pattern of the receive optics to obtain the imaging data.
The paraxial region is as long as the region between the plasma edge and a few wavelengths
from the cutoff. Each such iteration of obtaining the reflected field would take around 1 to
5 minutes.
Plasma edge
Vaccum Propagation
Antenna
Receiver
Curved Cutoff layer
Figure 4.2: A block diagram of a synthetic diagnostic technique for the evaluation of
candidate MIR designs. An illumination beam is introduced into FWR2D. Near the cutoff
layer, a full wave solution of the plasma wave interaction is obtained. After propagating
paraxially to the plasma edge, the resulting field is convolved with the antenna pattern of
the receive optics to obtain imaging data.
73
The FWR3D computational model is similar to that of 2D, with decomposition of the
computational domain for efficiency. In 3D, the equilibrium B field curvature will be taken
into account, and the B field is no longer symmetric in the Z (toroidal in 3D) direction, but
along the field pitch direction. Due to the introduction of 3D rotating magnetic field
fluctuations in FWR3D, wave polarization may be changed, and decomposed into X mode
and O mode. The X and O modes are then treated separately, and only the mode with the
input specified polarization is returned. While in FWR2D, since the magnetic field is
constant along the Z direction, wave polarization is conserved. A more complicated 3D
dielectric tensor is introduced and used for solving the E-field equations. Wave absorption
due to the nearby upper-hybrid layer is still not considered. Boundary conditions are treated
differently, where a source is added in the full-wave region to act like an incident wave. In
this new approach, another source is placed inside the calculation area to cancel the
outgoing component of the E field imposed at the boundary; thus, the field can then be set
to zero on the calculation boundary.
4.2.2 Turbulence model
The turbulence FWR utilizes could either come from plasma code simulations, or the
following analytical Gaussian model. The 2D Gaussian distributed density fluctuation
spectrum over the poloidal plane could be described as follows:
⎧ ⎡ (k − k )⋅ R̂ ⎤2 ⎡ (k − k )⋅ Ẑ ⎤2 ⎫
⎪
⎪
m
m
Ik = I0 exp− ⎨ ⎢
⎥ +⎢
⎥ ⎬ (4.5)
⎪⎩ ⎢⎣ Δk ⋅ R̂ ⎥⎦ ⎢⎣ Δk ⋅ Ẑ ⎥⎦ ⎪⎭
!
74
Here, !R̂ and !Ẑ are, respectively, the unit vectors in the radial and vertical directions
on the poloidal plane. k m is the mean wave-vector of the distribution, and !Δk is the
standard deviation. The density perturbation amplitude n!ks for each wavevector (k) and
!
simulation (s) is proportional to the square root of Ik . The inverse Fourier transform of
!
n! yields the relative fluctuation amplitude !n! /n , with !n the (assumed) smoothly
!k
varying equilibrium density profile. Time-dependent fluctuation is introduced in the
poloidal plane, and is constructed for each simulated time slice, and the correlation between
the density fluctuations for two adjacent time slices is [113]:
2
⎛ ⎛ Δt ⎞ 2 ⎞
⎛ ⎛ (d + vt) ⋅ Δk ⎞ 2 ⎞
⎛ n! ⎞
1
n! n! = ⎜ ⎟ exp ⎜ − ⎜ ⎟ ⎟ exp ⎜ − ⎜
⎟⎠ ⎟ cos(d ⋅ km ) (4.6)
2
⎝ n⎠
n2 1 2
⎝ ⎝ τ ⎠ ⎠
⎝ ⎝
⎠
In the above, n! / n is the relative fluctuation level, τ is the de-correlation time, Δt
is the time difference between the two time slices, v is the poloidal velocity of the
turbulence, and d = ( x1 − x2 ) xˆ + ( z1 − z2 ) zˆ is the displacement in the radial and poloidal
directions between the turbulences of the two time slices. Since the de-correlation time is
relatively much longer than the simulation time, making Δt / τ generally small, therefore
⎛ ⎛ Δt ⎞2 ⎞
exp ⎜ − ⎜ ⎟ ⎟ ≈ 1, thus leaving the temporal de-correlation negligible over two time slices.
⎜ ⎝τ ⎠ ⎟
⎝
⎠
The radial correlation length
λrc = 2 / Δk ⋅ xˆ , and the vertical correlation length
λzc = 2 / Δk ⋅ zˆ , and those two conventions omitting the usage of π are chosen to make the
calculation of the Fourier transformation of the Gaussian easier [109].
In the following example, simulations applying the analytic model are made to have
the radial deviation of ∆# to be 0.222 cm-1, corresponding to a radial de-correlation
75
length around 9 cm. Since this length is relatively long compared to the Airy length [118]
of the probing wave when it arrives at the cutoff layer, the radial de-correlation is negligible
over the thin cutoff layer. The evanescent region behind the cutoff is many free space
wavelengths long, such that tunneling effects may be ignored. Since the wave-vector
applied in the FWR code has a convention of k = 2/λ, which is 1/π of the standard
wavenumber !k ( !k = 2π / λ ) for calculation convenience, we applied the standard
representation for wavenumber by multiplying the wavenumber from FWR by π. Thus, the
radial fluctuation wavenumber is fixed at 0 cm-1, and its radial deviation is 0.222 π cm-1.
Figure 4.3 shows the field intensity along the radial direction from the FWR2D simulations
where density turbulence is added in the plasma profile. The probing frequency is 74 GHz,
and the equilibrium profile is a typical DIII-D H-mode discharge.
Figure 4.3: Electric field intensity of the 74 GHz probing and reflected beams going into
and out of the simulation space, where (a) shows the paraxial region of the incoming X-
76
mode wave, (b) shows the full-wave region near the cutoff, and (c) is the paraxial solution
of the outgoing X-mode waves. [Reprinted figure with permission from Ref. [119].
Copyright 2014, AIP Publishing LLC]
4.3 Synthetic MIR diagnostic model
With the combination of sophisticated optical simulations, efficient wave and
plasma interaction simulations with FWR codes, and advanced analysis tools to extract and
analyze density turbulence from reflected field, the synthetic imaging process is able to
quantify and optimize MIR performance.
The optical portion is simulated with an optical algorithm that could use a Gaussian
beam source, and which is also capable of diffraction-based propagation between the
lenses. The beam is propagated in the software from the source through all the refractive
lenses, reflectors and windows, to a plane before the plasma edge. Both fast-Fouriertransformation (FFT) based diffraction and beamlet-based diffraction algorithms are
applied.
The FFT based diffraction method applies Fraunhofer diffraction for far field
propagation, and Fresnel diffraction for near field propagation, and is very efficient
considering the computational time. While the beamlet-based algorithm is slower, it is
more accurate and stable than the FFT based approach. During the first stage of optical
design, due to the large aperture of the optical components and no reflectors used, the FFT
approach is sufficient to give accurate field data to be used in FWR, and thus is applied
most of the time. However, afterwards, since several reflectors are used, the FFT based
approach which is highly sensitive to the defined mesh cell size on each plane failed to
provide stable and accurate field information. The slower beamlet based diffraction
77
propagation is used to acquire complex field amplitude, which is to be propagated into
FWR simulations. Figure 4.4 shows a comparison of the beamlet-based (a) and FFT-based
(b) diffraction beam propagation simulation. The two intensity plots are achieved from the
same surface on the path of beam propagation. There is a clear sharp intensity change on
Fig. 4.4 (b). This happens unexpectedly, and is non-predicable. Thus, beamlet-based
simulation is more often used for receiver/transmitter beam field generation, due to its
better stability.
Figure 4.4: Comparison of the beam intensity distribution on the same surface from CodeV
using (a) beamlet-based and (b) FFT-based diffraction analysis. A clear sudden intensity
change is observed on the edge of the field in (b). Beamlet-based simulation results are
much more stable than that of FFT-based simulation results.
FWR2D takes the electric field from the optical simulation as well as plasma
profiles with electron density, temperature, and magnetic strength information over a 2D
poloidal plane as input. It then simulates wave interaction with the plasma with a fine mesh
over the plasma region, and outputs the reflected field. The time-dependent reflected signal,
78
either on the detector plane or on the boundary between paraxial and vacuum solver, is
used for imaging analysis. Actually, the complex electric field on any surface in the radial
direction along the beam propagation could be used for imaging analysis. We choose the
two surfaces specified above because they are the furthest surfaces from the cutoff surface
achievable from the simulation results. Thus, the amplitude modulation, due to wave
interference along the propagation distance, would be most significant for the field on those
two surfaces. If the density perturbation information can be restored from these two fields
with strongest unwanted amplitude modulation, it would then be evident that lens-based
2D imaging could be a good tool for density perturbation diagnostic. Each simulation (time
slice) of the 2D code takes less than 5 minutes, while the 3D version takes up to 10 minutes,
and is still under development for extended functionality.
Propagating the reflected field numerically from the detector plane to the receiver
antennas through the receiver optical system is difficult and prohibitively time consuming,
since each meaningful ensemble requires more than 1000 iterations and very high spatial
resolution is also needed in order to maintain the details of the reflected signal. An
alternative method that exploits optical reciprocity is used here to generate the numerical
diagnostic signals. The unperturbed receiver field pattern is propagated to the detector
plane, and an overlapping integral over time between the receiver field and reflected field
is calculated to represent the perturbation observed at the receiver location. The integration
calculation for thousands of reflected signals is very efficient. One example of the
numerical diagnostic signal generated through this method is shown in Fig. 3.5 in the
bottom, where a synthetic phase perturbation is plotted with respect to time. In addition,
since the receiver antenna patterns are Gaussian, it is easy to propagate those beams from
79
the receivers to the detector plane without losing details, and thus it is preferred due to both
efficiency and numerical accuracy.
Figure 4.5 illustrates the process to generate synthetic imaging signals. The
transmitter horn antenna pattern as shown in the upper right box is the experimentally
measured pattern. Instead of using the experimentally measured pattern, since it is
measured after the design process when FWR simulations are mostly done, a similar
simulated probing field pattern is used in FWR2D for wave and plasma interaction
simulations. The probing field goes through the FWR2D simulations of the paraxial
propagation and full-wave simulations near the cutoff, and then is reflected back. The
reflected field is then convolved with the unperturbed receiver field on the same plane to
generate the phase modulation, which corresponds to the density perturbation on the cutoff
surface. The laboratory measured receiver array pattern is shown in Fig. 4.5 on the right
middle box. It can be seen that the main lobe for each single receiver is distributed evenly
along the poloidal direction. A Gaussian fit to the main lobe of each receiver antenna
pattern is an excellent approximation; the side lobes are down in amplitude by 15 dB and
thus contribute negligibly to the signal.
80
Vaccum
Figure 4.5: Block diagram of the FWR2D simulation method illustrating the generation of
numerical diagnostic signals.
FWR2D simulation is performed in three parts: the vacuum
region without plasma, the paraxial region with plasma but away from the reflection layer,
and the full wave (FDTD) region near the reflection layer. The probing beam utilizing the
“Transmitter Pattern” enters the plasma and is reflected from the cutoff surface. The
receiver antenna array has a “Receiver Pattern,” with each curve representing the antenna
pattern of each individual channel. The overlapping integral over time between the
reflected field from FWR2D and one single unperturbed receiver field is used to generate
the numerical diagnostic signal for that specific channel.
81
4.4 Analysis methods
4.4.1 IQ plot analysis
Using the above described simulation model, on the plasma edge surface or detector
plane, the convolution between the simulated complex E field corresponding to the receiver
antenna pattern ( Erec ) and the reflected E field from FWR2D ( Eref ) for each time slice is
!
!
calculated using the following formula:
(E ) ⋅E
Ei = ref i rec
(Eref )0 ⋅Erec
(4.7)
where the subscript “i” designates the ith time slice, and (Eref )0 represents the
!
reflected field when there is no turbulence on the cutoff surface. The convolution values
represent the normalized E field of the reflected wave seen on the image plane. The
ensemble of phase of Ii = Ai * cosφ = Re{Ei } represents the density turbulence over time
for the measured position. This ensemble could accurately represent the density
perturbation when the density fluctuation amplitude is relatively small; while when the
fluctuation amplitude increases, unexpected phase jumps, which will be explained later,
happen, and the ensemble needs to be recalculated to accurately represent the correct
density turbulence again. The ensemble of the complex E i values over a complex plane
is called an I/Q plot. Here,
I = Ai * cosφi = Re{Ei }
i
and
82
(4.8)
Q = Ai * sin φi = Im{Ei }
i
(4.9)
In the above, A is the normalized reflected field amplitude with respect to the receiver
field amplitude. In addition, φi is the phase of E i , and is imparted on the wave by the
density perturbation; an ensemble of φi represents measured density turbulence as
described above. Hence, how accurate φi is measured determines the quality of an MIR
system. For an I/Q plot, I designates the in-phase component, and Q designates the
quadrature component, and these definitions are widely used in communication
technologies.
An ensemble of I/Q points is a good measure of whether the imaging quality is good
or bad. Density fluctuations at the cutoff layer change only the phase of the waves that are
reflected. Consequently, at the cutoff layer there are only phase fluctuations, which also
represent the density fluctuations. On their way out of the plasma, the reflected waves
interfere with each other which gives rise to amplitude fluctuations and a change of the
phase of the outgoing waves. As we saw earlier, with an optical imaging system or a
numerical imaging algorithm, the interference effects can be reversed and amplitude
variations can also be reduced to recover a pure phase signal as was imprinted by the
density fluctuations on the reflected wave fronts. If all the scattered waves can be collected
through the optics, a pure phase signal back at the imaging plane should be obtained; while
if some of the scattered waves (propagating outside the aperture, or diffracted due to limited
aperture size) are missed or disturbed, amplitude fluctuations would be enlarged on the
imaging plane, causing worsening of the imaging quality. Therefore, a measure of how
good the image is at the imaging plane is the spread in amplitude fluctuations. Therefore,
83
if an I/Q “fuzzy” ball is seen, it means that imaging is poor and the amplitude distribution
becomes a Rayleigh distribution [120]. If the imaging is perfect, the amplitude distribution
becomes a Delta distribution. In practice, however, good imaging give rise to a Rice
distribution, and when it becomes worse it goes from a Rice distribution to a Rayleigh
distribution [121]. Figure 4.6 shows examples of normalized synthetic I/Q plots, where the
amplitude modulation on the signal increases from left to right, indicating that imaging
quality is decreasing in this direction too.
Figure 4.6: Examples of IQ plots, with decreasing imaging quality. The plasma equilibrium
profile is from an approximation of DIII-D H-mode discharge # 154397. In addition, the
turbulence is from an analytical model as described in section 4.2.2.
An IQ plot, with the points distributed mainly in a circle, normally means that the
receiver is well focused on the cutoff. However, under some circumstances, the phase
ensemble from such “good” IQ plots doesn't correctly represent the expected density
fluctuations. As the fluctuation wavenumber increases, higher orders of the Bessel
expansion become increasingly important, and retaining only up to the first order is no
longer valid. The diffracted components that pass through the window and which are also
captured by lenses are no longer sufficient to construct a good image. When the fluctuation
wavenumber becomes even larger, assuming unchanged fluctuation amplitude, points on
84
the IQ plots will again shape into a good crescent curvature. This phenomenon can be
explained by the fact that the structure of the density fluctuation is too fine to be observed
by MIR; therefore, the cutoff surface appears to MIR like a smooth surface moving back
and forth without poloidal rotating structures, causing unresolved diffraction and resulting
in a well-shaped IQ plot. Considering a receiver spot size on the cutoff surface with a fullwidth at half maximum of 1.0 cm, the maximum wavenumber accessible for MIR is around
1.5 cm-1, according to the Nyquist criterion [122]. Figure 4.7 shows IQ plots for three
fluctuations, using the 2D Gaussian fluctuation model described in section 4.2.2, with
different dominant wavenumbers ( km ), but with the same fluctuation amplitude (n! /n ) at
0.1%. The dominant wavenumber increases from 0.5 cm-1 in (a), to 1.0 cm-1 in (b), and to
3.0 cm-1 in (c); while the I/Q plot or imaging quality changes from small amplitude
modulation, increased amplitude modulation, and then small amplitude modulation again.
The results shown in this figure are from an L-mode discharge, and all the poloidal
fluctuation wavenumbers, which are Gaussian distributed, have a deviation (Δk ) of 0.1 cm1
, meaning all the fluctuations are highly coherent, and possess a small spread in the
wavenumber spectrum. Figures 4.7 (a) and (c) have better distributed IQ plots than that of
Fig. 4.7(b). However, Fig. 4.7 (a) has a wavenumber that is within the measurement range
for MIR, and its phase modulation corresponds to the density fluctuation; while Fig. 4.7
(c) has a wavenumber that is twice the maximum wavenumber MIR can measure, which
should not be resolved according to the Nyquist criterion, and its phase modulation doesn’t
represent the detailed structure of density perturbation as will be described below. In the
next section, we can see that the phase fluctuation is poorly correlated to the density
fluctuation for Fig. 4.7 (c), meaning that the density fluctuation is not actually measured
85
although the IQ plot is in a desirable shape.
Fig 4.7: IQ plots for density fluctuations with wavenumber a) 0.2 cm-1, b) 1.0 cm-1 and c)
3.0 cm-1. The fluctuation level is 0.1%. The first one has 1000 samples; the second and
third figures have 300 samples.
4.5 Correlation analysis
One of the great advantages of the MIR synthetic diagnostic is that the phase
fluctuation versus time achieved from the simulated IQ measurements can be compared
directly to the original specified density fluctuation in the plasma. It is the most direct and
effective way to determine if MIR is working properly, and how well it works for different
density fluctuation parameters. Since the FWR2D code does not directly give the position
of the cutoff surface, linear interpolation to obtain the zero points of refractive index along
the radial direction from the simulation is applied to determine the exact cutoff surface
position and fluctuation structure. The pure density fluctuation on the cutoff surface is
obtained by subtracting the equilibrium position of the cutoff from the perturbed ones. For
analysis convenience, the radial amplitude of density perturbation (ΔR ), which is originally
in units of meters, is represented by a value ( Δϕ ) in radian units using the formula below:
86
Δϕ = 2π (ΔR / λ )
(4.10)
where λ is the free space wavelength of the probing wave.
To quantify the similarities between the density fluctuations and measured phase
fluctuations, which is also a measure of the imaging quality of MIR, both Pearson (linear)
[123] and distance correlations [124] between the numerical diagnostic phase signal and
the original density fluctuations are calculated. Pearson correlation is the most
straightforward measure of the linear similarities between two variables. Its value varies
from -1 to 1, where 1 means totally positive correlation, 0 means no correlation, and -1
means totally negative correlation. Its formula is as follows:
corrp =
!
∑
∑
n
i=1
n
i=1
(x i − x )( yi − y)
(x i − x )
2
∑
n
i=1
(4.11)
( yi − y)
2
where x i is one instance of the ensemble of variable !X , and yi is one instance of the
!
!
ensemble of variable !Y ; !x is the mean value of !X , and ! y is the mean value of !Y .
Compared to Pearson correlation, the distance correlation has a more complicated
mathematical model, and represents a general dependence of two variables, not just the
linear correlation. The distance is zero only if the variables are statistically independent.
Given the variable ensembles used above, the distance correlation is calculated from the
following formulas:
Firstly,
a = x i − x j (4.12.1)
! i,j
b = yi − y j (4.12.2)
! i,j
87
!i, j = 1,2,...,n (4.12.3)
Then, take all doubly centered distances to obtain:
Ai , j = ai , j − ai ,. − a., j + a.. (4.13.1)
Bi , j = bi , j − bi ,. − b., j + b.. (4.13.2)
where ai ,. is the mean of the j-th row, and a., j is the mean of i-th column, and
a.. is
the grand mean of the a matrix. The distance covariance is
1 n
dCov ( X , Y ) = 2 ∑ Ai , j Bi , j (4.14)
n i , j =1
2
n
Therefore, the distance correlation between the two variables X and Y is
dCor ( X , Y ) =
dCov( X , Y )
(4.15)
dCov( X , X )* dCov(Y , Y )
Figure 4.8 shows the comparison between the phase fluctuations and density
fluctuations given the fluctuation conditions as shown in Fig. 4.7 (a). Both linear and
distance correlations are calculated between the two curves. Correlation values vary
between 1 and 0, with 1 meaning high dependence between the compared two arrays, and
0 meaning a total independence relationship; therefore, higher correlation values indicates
better agreement between the density fluctuations and measured phase fluctuations. The
linear correlation value of the two curves is 0.93, while the distance correlation is 0.9,
meaning they are highly correlated as shown in Fig. 4.8.
88
Fig 4.8: Phase fluctuation (blue) compared to density fluctuation (red). The distance
correlation is 0.90, which means the two curves are highly correlated.
In Fig. 4.9, three different correlation values corresponding to the I/Q plots shown in
Fig. 4.7, with wavenumber from left to right being 0.5 cm-1 (this is differentfrom the
wavenumber of 0.2 cm-1 on Fig. 4.7 (a), while since the correlation value for 0.5 cm-1 case
is already 0.99+, the correlation value for the 0.2 cm-1 case would also be 0.99+), 1.0 cm-1
and 3 cm-1. As can be seen from this figure, the correlation value is decreasing as the
fluctuation center wavenumber increases. The lowest correlation value of 0.3 on Fig. 4.9
(c) proves that even when the
I/Q plot indicates good imaging quality, the phase
fluctuation can be poorly correlated to the density fluctuation, meaning that the density
fluctuation is not actually measured in this case. The fluctuation amplitude for all the cases
89
shown in Fig. 4.9 still 0.1 %, and the deviation of the Gaussian distributed poloidal density
fluctuation is also 0.1 cm-1.
Fig 4.9: Measured phase compared with density turbulence, as well as correlation value for
the three different I/Q plots on Fig. 4.7 are shown. The fluctuation level is 0.1%. As the
poloidal fluctuation center wavenumber increases, the correlation value decreases. The
poor correlation value of 0.3 in (c) shows that that even when the I/Q plot in Fig. 4.7 (c)
indicates good imaging quality, the actual imaging could be very poor.
In Fig. 4.10, three different correlation values are labeled on the top for each case where
n! /n increases from left to right. As n! /n increases, the reflected waves scatter into a
wider angle, and the aperture misses more of the scattered wave; thus, the amplitude
fluctuation on the I/Q plane tends to increase. Furthermore, with increasing
n! / n ,
phase
jumps3, which indicate a phase change that is unreasonably larger than expected, occur
more often in the numerical diagnostic process; this phenomenon is not fully understood,
but it could impose accumulated errors on the numerical diagnostic results and thus
contaminate the numerical imaging quality. By increasing the sampling frequency, the
3
Phase jumps are also observed with FM demodulators and interferometers.
90
phase jump effects can be reduced, but at the cost of more time and more disc space for
storing FWR2D simulation outputs. At the beginning of the DIII-D design, the proposed
MIR sampling frequency was 2 MHz, but the ultimate sampling frequency for the real
system is 1 MHz. The sampling frequency of 2 MHz is still retained for numerical analysis
for the reason stated above. As the correlation value decreases, meaning that the imaging
quality varies from good to bad, the I/Q plots tend to deform from annular shaped into more
random distributions as shown in Fig. 4.10. Furthermore, the time traces between the
numerical phase signal and the density fluctuation change from matched to unmatched. In
an actual measurement circumstance, we can only obtain I/Q plots of the measured phase
signal and cannot compare the real density fluctuations with measured phase fluctuations.
As suggested from the correlation values and I/Q plots from Fig. 4.10, we choose a
correlation value of 0.6 as the lower limit of meaningful distance correlation between two
data sets for convenience, and this convention is used in the following content of simulation
results analysis.
91
Fig 4.10: Numerical diagnostic data for three imaging qualities with increasing
n/n
from
left to right are shown, illustrating the characteristics of very good and poor imaging
quality. As the correlation between the numerical diagnostic signal and density fluctuations
at the cutoff surface decreases, data in the I/Q plane deform from an annular distribution
into a statistically noisier one. The numerical diagnostic signal (the phase of points in the
I/Q plane) and the real fluctuation prescribed in simulation are also compared.
92
Chapter 5: Analytical Simulation Results with Synthetic MIR
Synthetic diagnostic results are analyzed to provide a general understanding of how
a particular MIR configuration would perform in measuring density fluctuations. The
analysis, as will be described in the following, helped to optimize the optical design of the
DIII-D MIR system [91]. The MIR synthetic diagnostic helps to provide a quantitative
understanding of the capability of MIR on measuring density fluctuation parameters, and
also how various optical variations would affect the performance of MIR. The numerical
imaging results were crucial in designing the optimized, robust 2-D imaging reflectometer,
which was successfully installed and now operating on the DIII-D tokamak. Both L and H
mode discharges are investigated with the synthetic MIR diagnostic. Since H-mode is more
favorable for future fusion devices, and MIR is more suitable for H-mode diagnostics, as
will be explained later, the installed MIR system is currently configured and optimized for
H-mode operation. L-mode poses more challenges for the MIR system, but has also
currently achieved promising experimental diagnostic results. Most of the simulations
discussed in this section, intended to define the functionality metrics of MIR, are conducted
with the analytic fluctuation model, with adjustable radial and poloidal fluctuation spectra
and amplitude. Fluctuations from more sophisticated plasma turbulence codes will be
employed and discussed in a following chapter. In addition to the synthetic MIR diagnostic
method, which uses large aperture optics for imaging, introduced in this dissertation, there
is a different synthetic MIR method which numerically tracks back the reflected signal to
the original cutoff layer to obtain density fluctuation information and is presented in
Appendix IV.
93
In this chapter, firstly, a detailed discussion about wave-front curvature matching
for both transmitter and receiver is given. Afterwards, a discussion about some obstacles
when analyzing the numerical results, such as phase jump and DC drift, will be given. Then,
sensitivity measurements concerning how different optical misalignments (mainly for the
receiver) would affect the measured results are discussed. Finally, the functionality metrics
of MIR with respect to the fluctuation amplitude and wavenumber for both L-mode and Hmode are given as a summary to the general performance of MIR from numerical
simulations.
5.2 Curvature matching
5.2.1 Transmitter
For the transmitter beam, the most important design aspect is the curvature
matching, which ensures accurate phase inversion and better fidelity than a mismatched
curvature setup. Ignoring the complex simulation of wave interaction inside the plasma
from FWR2D, an easily understandable illumination procedure is presented in Fig. 5.1,
with a Gaussian beam source antenna transmitting waves through a lens system into the
plasma, which then impinge upon the cutoff surface. On this figure, the entire transmitter
lens system, consisting of reflectors and refractive lenses, is represented by a single large
aperture convex lens. The curvature of the probing beam upon the target could be adjusted
by varying the distances between lenses, source, and target; the phase front curvature of
the beam incident on the target should be matched to the cutoff surface curvature.
94
Fig 5.1: Schematic of an MIR probing system. Waves (depicted as geometrical optical
rays) from a horn antenna are transmitted into the plasma after a lens system, and the wave
front curvature should be the same as the plasma cutoff curvature. [Plasma image from
www.sciencephoto.com]
The matching is of critical importance, since mismatching can cause severe
distortion of the reflected signal from the target. In the case of bad curvature matching, the
beams incident on the edge of the target take a total longer/shorter propagation path
(ingoing and outgoing) than the beam impinging upon the center would take, and the beam
would not be reflected back in the same orientation as the incident beam, as shown in Fig.
5.2. Furthermore, the mismatched reflected beam can spread widely, even to the outside of
the collecting optics, thereby losing the collection of higher order of diffractions. When the
probing beam and reflected beam have the identical orientations, in the condition of
curvature matching, the collection efficiency would be maximized.
95
Probing beam wavefront
Cutoff Surface
Unmatched 1
Matched
Unmatched 2
Figure 5.2: Schematic drawing of three different curvature matching conditions. Two
unmatched conditions are on the edges, and one matched condition in the middle. The
arrows mean the reflection direction of the probing wave. Only when the beam phase front
curvature is matched to the cutoff surface curvature as in the middle case, does the beam
come back along the incident path.
When the beam is propagated onto the cutoff surface from the low field side,
curvature matching would require the incident beam to be converging after it passes the
last refractive lens or reflector and before it enters the plasma. This configuration also
means that the beam illumination size at the plasma would be smaller than the size when it
passes through the last device aperture; thus, the poloidal illumination region in the plasma
is always smaller than the window height under this condition. However, the poloidal
coverage can be enlarged with either collimating or diverging probing beams; of them the
96
former would cause a coverage size comparable to the aperture size and the latter would
result in a coverage larger than the aperture size. However, both collimating and diverging
probing beams would cause spread of the reflected beam into a diverging angle upon
reflection, reducing the collection efficiency of the reflected beam (less diffraction
components collected), and resulting in poor imaging quality. Considering the imaging
quality and at the same time maximizing the poloidal coverage, the curvature-matching
setup is most favorable.
Originally, the imaging plane is a flat surface, while the cutoff surface, most of the
time, is a curved surface; thus, to match the flat image surface with the curved object
surface, simple spherical or cylindrical lenses are not sufficient. Some approaches can be
employed to mostly ameliorate this problem, for example, the usage of meniscus lenses, or
polynomial lenses. The usage of both techniques is under development, and will be applied
to the upcoming EAST MIR system [103]. The DIII-D MIR system is still using the
simplest cylindrical lenses, but with curved bent image plane, which will be explained later.
More flexibility with either the current curved image plane or flat image plane will
eventually be added by using more advanced lens configurations.
Figure 5.3 shows the right-handed X-mode cutoff surfaces in the poloidal and radial
plane for an L-mode DIII-D discharge with density and temperature decreased compared
to original shot density and temperature. Figure 5.4 (a) shows the right-handed X-mode
cutoff surfaces in the poloidal and radial plane for an H-mode shot on DIII-D with magnetic
field around 2.0 T. Figure 5.4 (b) shows the zoomed in cutoff surface positions on the
pedestal where rapid density profile change occurs. By comparing the two figures, it can
be seen that the cutoff frequency in L-mode changes slower from edge to core, while the
97
cutoff changes much faster on the steep gradient pedestal region for the H-mode discharge,
and more slowly above the pedestal and up to the core. The pedestal in H-mode acts more
like a mirror to the probing frequencies, by having much smaller density compared to the
cutoff density before the cutoff surface and also rapid density increase after the cutoff
surface; thus, the plasma along the propagation path of the probing wave that would have
negligible contribution to the phase modulation, and wave tunneling can also be ignored.
While for the L-mode, on the contrary, the probing wave would pass through a plasma
region with comparable, but smaller, density compared to the density on the cutoff surface;
thus, the turbulence in this region could contribute non-negligible phase modulation to the
probing wave, which is not favorable for MIR locality determination.
Thus, H-mode,
specifically for the pedestal region, has much better imaging quality than the L-mode
discharges. However, in the region before the core and pedestal top of an H-mode discharge,
the density profile has a small gradient; thus, the turbulence of the plasma, where the
probing beam would pass through, would contribute considerably to the phase modulation
and cause non-negligible scattering of the power, so that imaging would be much more
difficult. Currently, MIR has measured clear and significant features of modes existing on
the pedestal region of H-mode, while its measurements for L-mode and region between
core and pedestal top of an H mode are still under improvement.
98
Figure 5.3: Right-hand X-mode cutoff surface of an L-mode discharge from the DIII-D
tokamak, with low-density profile. Frequencies are labeled on each corresponding line.
(a)
99
(b)
Figure 5.4: (a) Right-handX-mode cutoff surface for an H-mode plasma (shot # 150640 @
3005 ms). (b) Zoomed in cutoff surfaces on the pedestal region of the H-mode.
5.2.2 Receiver
Figure 5.5 shows the schematic of an MIR receiver system, with two receiver
channels displayed. The large aperture lens is a simple representation of the receiver optical
system, and each receiver antenna (~ 1 mm) is attached to one single mini lens (~ 1-2 cm
in radius) for improved detection of the reflected signal [99].
100
Fig 5.5: The receiver system with two channels is shown. Each receiver antenna is attached
to a mini lens.
Similar to the transmitter, the receiver also needs to achieve proper curvature
matching. However, the curvature matching between the transmitter and receiver is quite
different in several ways. For the transmitter, there is a single source, and it is essential to
ensure that the wave-front of the probing wave when it arrives at the cutoff surface matches
with the cutoff surface. Under this condition, the shape of the wave-front, though the wavefront radius can be dramatically changed, cannot be tuned at each point to match with the
cutoff surface. The wave-front curvature can be slightly different to the cutoff surface
curvature; critical curvature matching is not possible. Unlike the transmitter, the receiver
is a point to point matching; receiver channels must match with the curvature of the cutoff
surface, and sightlines must have an incident angle of zero. A simple schematic of the
design aspects of the receiver is shown in Fig. 5.6, and is explained in the following
paragraph.
The two challenges make the design of receiver much harder than the design of the
transmitter. The receiver design process takes advantage of reciprocity of wave
propagation. Instead of making a bunch of cutoff surfaces, the design makes a fixed image
101
array and tunable receiver lenses. Then the field is propagated from the receiver array to
the cutoff surface. The lens system is tuned to achieve different cutoff surface positions
and curvatures. In real experiments, due to the reciprocity, the reflected wave would go
through the pre-designed path and be observed by the receivers. Thus, the first challenge
is how to design the lens system to make the receiver channels match with the curvature of
the cutoff surface. Several methods were tried, and the final approach is to make a curved
image/receiver plane, as shown in Fig. 5.6. The curvature of the image plane is 260 mm.
This image plane could be easily matched with a 770 mm curved cutoff surface with the
final design, which is used in the design process. The receiver lens system combined with
the mini lens array could be matched with cutoff surfaces with curvatures ranging from
500 mm to 1000 mm, satisfying most H-mode measurement requirements. In addition,
making the sightline incident angle equal to zero is more challenging, due to limited device
window size as well as a curved image plane. The sightline angle equal to zero condition,
which also means perpendicular observation, is shown in Fig. 5.6. The current system has
most tilted angles on the edge channels, which are around 8 degrees off zero. How the
sightline incident angle, when angles are not equal to zero, would affect the imaging quality
will be discussed in the following Section. 5.3.
102
Simplified lens system
Receiver/Image plane
Cutoff surface
Matched object plane
Receiver channels
Sightlines with angle equals zero
Figure 5.6: Schematic of the receivers. The design process is with a fixed receiver
plane, and tries to match the “image plane” (highlighted as a matched object plane) of the
system to the cutoff surface as much as possible. The two challenges of the receiver design
is to make the “image plane” of the lens system match with the cutoff surface as much as
possible, and at the same time make the sightlines with angle equal to zero (perpendicular
reflection).
Figure 5.7 shows the structure of the receiver array both in optical software and the
actual fabricated mini lens array. The curved image array is to compensate for the curved
cutoff surface. There are 12 poloidal channels in total, and the distance between two
adjacent channels is 26.75 mm, with each mini lens having a radius of 22 mm (44 cm in
diameter > 26.75 mm) included. With this adjacent channel distance and mini lens size, it
is impossible to align all the mini lenses into one vertical array; thus, the 12 mini lenses are
split into two arrays, one even numbered mini lens array (2, 4, 6, …, 12) and one odd
numbered mini lens array (1, 3, 5, …, 11). A 50-50 % beam splitter with 50 % power
103
transmission and 50 % reflection is mounted between the two min lens arrays to evenly
split the reflected power to the two arrays. In addition, the imaging magnification is 1:2 (1
on cutoff surface, and 2 on receiver plane).
A 50% reflector will be added here
260 mm in curvature
Figure 5.7: Receiver array structure, with one from the optics simulation software (left)
and one a photograph of the fabricated mini lens receiver array (right). Since the diameter
of an individual mini lens is 44 mm larger than the distance between two adjacent channels,
which is 27.65 mm, the receiver array is split into two sub-arrays (even-numbered array,
and odd-numbered array). A 50/50 beam splitter is added between the two arrays to evenly
split the incoming power to the two mini lens arrays, as indicated in this figure.
Another different aspect between the transmitter and receiver focusing concerns the
spot size. For the transmitter, the design is intended to illuminate a region on the cutoff
104
surface as large as possible; thus, the design purpose of the transmitter optics is to make
the spot size on the cutoff surface as large as possible while maintaining tolerable curvature
matching. For the receiver, the design purpose for the optics is to minimize the possible
spot size of each detection position, while at the same time sustaining a proper curvature
matching. The smaller the spot size is, meaning higher instrumental resolution, the larger
the measurable wavenumber could be. However, how small the spot size could reach is
restricted by the minimum Gaussian beam waist achievable on the cutoff surface. The
Gaussian beam waist is mostly restricted by the window size, probing wave frequency,
wave propagation media density, and also expected plasma coverage. Theoretically, the
minimum spot size theoretically achievable from synthetic simulation results is 1.0 cm4,
which is determined by the minimum detectable 2.0 cm in wavelength with 0.1 %
fluctuation amplitude, in the format of full width at half maximum (FWHM), while the
actual spot size from instrumental measurements is around 3.0 cm in FWHM. This actual
spot size is increased due to several factors, including diffraction due to the plasma along
the path of propagation.
5.3 Sensitivity measurements
There are numerous factors that can influence the general performance of MIR
including: curvature matching of the transmitter and receiver, window size, light path, and
reflections between lens surfaces. Curvature matching is discussed in the above section for
4
The minimum spot size should be acquired under the simulation conditions of 2.0 cm in wavelength and
0.1 % in fluctuation amplitude, which still give correlation value above 0.6, as indicated on Fig. 5.15.
105
both receiver and transmitter. The window aperture is a major limitation for MIR operation;
however, since the aperture size (60 cm in height, and 20 cm in width) is not adjustable,
this cannot be improved. The length of the optical light path is restricted by the seriously
space limited platform, and there is little space available for variation. In this section, we
mainly focus on three optical misalignment factors considering the receiver design that are
adjustable during the operation and also have essential impacts on the imaging quality. The
three factors that happen normally in the MIR system are: out of focusing, spot size
variation, and off-normal incidence angles.
5.3.1 Plasma profile
The fitted plasma profiles used for the simulation in this section are shown in Fig. 5.8.
The applied plasma profile is an approximation of shot #150640, a typical ELMing Hmode for DIII-D. The plasma profile is obtained experimentally through the usage of EFIT
[125]. The density and magnetic field profiles are shown in Fig. 5.8(a). The edge density
gradient is steeper than in the actual discharge, and the density goes to zero at !ρ = 1.0 ( ρ
is the square root of the normalized toroidal flux, and !ρ = 1.0 is the last closed flux
surface [28], following which is the scrape-off-layer [28]). We used this reduced
description since it is simpler for simulation while at the same time it does not significantly
affect the simulation results. The magnetic field is decreased to 1T on axis to move the
cutoff layer further inside the plasma; in this case, the paraxial approximation region can
be longer while leaving sufficient space for the reflected wave to propagate to this region’s
edge and to generate a wave field with high level of amplitude fluctuations, and thus
making the imaged IQ plots more reliable if they can eliminate or reduce the amplitude
fluctuations and reveal clear phase fluctuations. Figure 5.8(b) displays the relevant
106
characteristic frequencies generated from the profile in Fig. 5.8(a). The probing frequency
is 74 GHz, and at this frequency the separation between the right-hand X-mode (RX) cutoff
and the upper hybrid layer is 2 cm. The transmission power tunneled to the upper hybrid
layer from the calculation of the last Airy fringe [126] is thus only ~10-6 and is safely
ignored.
4e19
Density (m-3)
1
Density
Magnetic
field
0.9
3e19
0.8
2e19
0.7
1e19
0.6
0
2.1
90
Frequency (GHz)
(b)
2.2
2.3
75
70
65
60
2.1
0.5
2.4
RX cutoff
85
80
Magnetic field (T)
5e19
(a)
Probe frequency: 74 GHz
Upper Hybrid
LX cutoff
O cutoff
2.15 2.2 2.25
major radius (m)
2.3
Figure 5.8: Fitted plasma profiles and related characteristic frequencies used for H-mode
diagnostic sensitivity studies. (a) Density and magnetic field profiles for an approximation
107
of shot #150640, a typical ELMing H-mode on DIII-D. (b) the relevant characteristic
frequencies generated from the profiles in (a). The probing frequency is 74 GHz.
Figure 5.9(a) shows the incident and reflected wave amplitudes on the same surface
of the plasma boundary, which is located at a major radius of R = 2.54 m. The reflected
field is reflected from a cutoff surface without fluctuations, the so-called equilibrium
condition. For the H mode plasma investigated herein, the curvature of the 74 GHz cutoff
surface is 680 mm. The designed optical wave front curvature on this surface is around 780
mm. Since the poloidal coverage is around 20 cm, the maximum mismatching due to
unmatched curvatures for two circles with 680 and 780 mm radius within a height of 20
cm is less than 1 cm theoretically, which is relatively small and tolerable. Figure 4.12 (b)
shows the phase of the incident electric field on the cutoff layer, and it is nearly uniform in
both the vertical (poloidal) and horizontal (toroidal) directions over the region viewed by
the diagnostics receiver which covers around 20 cm in the poloidal direction. The toroidal
phase is achieved by assuming the cutoff radius in the toroidal direction on the mid-plane
is 2.25 m in major radius, which is nearly the location of the 74 GHz cutoff surface.
108
Wave amplitude
(a)
0.14
Incident
Reflected
0.12
0.06
0.02
0
−30 −20 −10 0
Phase (rad)
(b)
10 20 30
3
Poloidal
2
Toroidal
1
0
−1
−2
−10
5
−5
0
10
Poloidal position (cm)
Fig 5.9: Shown is the nominal optical arrangement used for H-mode diagnostic sensitivity
studies. Figure (a) compares the electric field amplitude distributions of the incident and
reflected beams without density fluctuations. Their similarity indicates a ‘matched’
reflection. Figure (b) is the phase of the incident electric field on the cutoff layer in both
poloidal and toroidal directions with the latter taken on the mid-plane. It is nearly constant
in both the vertical (poloidal) and horizontal (toroidal) directions over the range viewed by
the diagnostics receiver antennas (from -10 cm to 10 cm). The toroidal phase is achieved
109
using a toroidal radius of 2.25 m, which is the major radius position of the 74 GHz cutoff
layer.
5.3.2 Out-of-focus conditions
In our design scenario, the image plane (location of the receiver antennas) is fixed,
while the object plane (cutoff layer) is variable; thus, following the moving cutoff surface,
the lens system has to be adjustable to accommodate this. It actually takes empirical
experience to predict the right in-focus position. One can tell whether the cutoff surface is
coming into focus by looking at the variation of the measured I/Q plots. Once the I/Q plots
are distributed primarily on a circle with small amplitude modulation, it is clear that the
cutoff surface is located at the right focal position of the imaging system, and valuable
measurements could be obtained.
Considering the out-of-focus situation, a sensitivity investigation of how much the
cutoff surface could be out of focus while still yielding a valuable measurement was
conducted. Here, off-focal-position (out of focus) means the reflection layer is not located
on the correct object plane in the radial direction, as shown in Fig. 5.10 (a). On this figure,
shown are two situations of focusing: the solid curves indicating the in-focus condition,
and the dotted curves indicating an out-of-focus condition. Figure 5.10 (b) shows the
correlation values changing with the displacements between the receiver focal point and
the cutoff position. The x-axis is the ratio of the off-focal displacement to the Rayleigh
range of the Gaussian beam. The Rayleigh range is the distance along the propagation
direction of a beam from the waist to the location where the area of the cross section is
doubled [127]. Observed from Fig. 5.10 (b), it can be seen that when the focal point is
110
located between the cutoff surface and the optical system (in the + direction), MIR is more
prone to interference from reflections off the cutoff surface than when the focal point is
located more inside the plasma than the cutoff surface (in the – direction). Importantly and
obviously, only when the receiver is at the focused position will the correlation be highest.
The fluctuation amplitude used for this simulation is held constant at 3%, while the
wavenumber distribution is Gaussian with a width of 0.2 π cm-1 and centered at 0.75 π cm1
. The trend of the curvature doesn’t change much when the center wavenumber and
amplitude changes.
Different
focal
position Focused
-
(b)
0.8
Correlation
(a)
0.6
0.4
0.2
+
Focal
position decreasing direction
-0.83
0
0.83
Off focal length
Rayleigh range
Figure 5.10: Shown are the schematics of in-focus and out-of-focus configurations (a) and
how the correlation values change with respect to the displacements between the receiver
focal point and the cutoff position (b). As seen on (b), when the focal point is located
between the cutoff surface and the optical system, MIR is more prone to interference from
reflections off the cutoff surface than when the focal point is located at smaller major radius
than the cutoff surface, and only when the receiver is at the focused position will the
correlation be highest. The fluctuation amplitude used for this simulation is held constant
111
at 3%, while the wavenumber distribution is Gaussian with a width of 0.2 π cm-1 and
centered at 0.75 π cm-1.
5.3.3
Different resolution size
Similarly, the effects of different spot sizes are also examined when the spot size is
the only variation. The spot size means the size of the Gaussian beam waist at the cutoff
surface, and it has been assumed that the receiver is in-focus on the cutoff surface. Spot
size is represented in the form of the full width at half maximum (FWHM), where the
strength of the antenna response is equal to half of its maximum value. A schematic of
different spot sizes on the cutoff surface is shown on Fig 5.11 (a), where the two
configurations (solid and doted curves) are both with an in-focus condition, but their beam
waists on the cutoff surface are different.
Figure 5.11 (b) shows how correlation values change with the various spot sizes.
The x-axis is the ratio of FWHM to probing wave free space wavelength (~ 4.05 mm).
From this figure, we can see that under the same fluctuation conditions, when the spot size
is smaller, meaning higher resolution, the correlation value would be higher than when the
spot size is large. However, this case is not always true; as the targeted range of fluctuation
wavelength or correlation length increases, larger spot size would be a better choice for
actual imaging. This is because a Gaussian beam with larger beam waist will diverge
slower than a beam with smaller beam waist; thus, when the beam with larger beam waist
arrives at the window it would have smaller beam width, leaving the aperture diffraction
effects being smaller. Occasionally, when the fluctuation wavelength is very small
compared to the designed spot size, as discussed in Chap. 4, the receiver can also yield
112
nicely shaped I/Q plots (plots distributed on a circle). However, in this type of scenario,
the phase fluctuation does not match with the prescribed density fluctuation in the plasma
profile. This case can be mitigated by ensuring that the spot size is smaller or of similar
size to the reflection layer fluctuation wavelengths, as also indicated by the Nyquist
sampling criteria. The theoretically averaged spot size ( wmin ) for the current DIII-D MIR
system is 1.4 cm; thus, the smallest accessible wavelength ( λθ ≥ 2* wmin ) would be 2.8 cm,
and the highest wavenumber ( kθ = 2* π / λθ ) is around k = 0.71π cm-1. The fluctuation
amplitude used for this simulation is held constant at 3%, while the wavenumber
distribution is Gaussian with a width of 0.2 π cm-1 and centered at 0.75 π cm-1.
(b)
Different spot
size (FWHM)
Correlation
(a)
FWHM2
1.0
0.8
0.6
0.4
0.2
2.5
5
7.5
10
FWHM
Wavelength
FWHM1
Figure 5.11: Shown is the configuration of different spot size focusing on the cutoff surface
(a), and how the correlation value or imaging quality changes with different spot size (b).
As seen in (b), as the spot size becomes smaller, the correlation value would increase,
where the fluctuation parameters are kept the same for all the data. The fluctuation
amplitude used for this simulation is held constant at 3%, while the wavenumber
distribution is Gaussian with a width of 0.2 π cm-1 and centered at 0.75 π cm-1.
113
5.3.4
Oblique incident angle
In our synthetic diagnostic simulations, the plasma cutoff surface is usually treated
as the source of the reflected signal, and the receiver array is the image plane. Although
the Gaussian beam waist from each individual receiver could be designed to be located on
the cutoff surface in this manner, the incident direction may not be at normal incidence to
the cutoff surface; how oblique the incident direction could be is investigated here. The
receiver is designed taking advantage of optical reciprocity by taking the receiver as the
source, and the plasma cutoff surface as the image plane. There are a number of advantages
of designing a receiver in this manner; one outstanding one of them is that it can make the
design of the receiver easier by making the receiver array fixed, but with the plasma cutoff
surface varying.
Figure 5.12 shows the receiver wave intensity on the detector plane versus the
reflected wave intensity at three different incident angles, with the detector plane located
around 40 cm from the cutoff surface. The green curves are the intensity distribution of the
receiver beam on the detector plane, and the blue curve is the intensity distribution of
reflected beam from the FWR2D simulation on the same detector surface, the central local
minimum of which may due to the scattering of the fluctuation on the cutoff surface. The
fluctuation conditions of this figure are a wavenumber of 0.5 cm-1 and a fluctuation
amplitude of 0.1%. The calculated distance correlation for the 0 degree case is 0.93, for the
5 degree case it is 0.93, and for the 10 degree case it is 0.83. When the wavenumber
increases, the highest oblique viewing angle will decrease, while on the contrary, for small
wavenumbers, the limit of the oblique viewing angle can be up to 20 degrees for
114
measurements with the middle channel. Correlations are even sensitive to angles in the
case of edge channels.
Figure 5.12: Shown are the field intensity comparisons between the reflected field (blue)
and receiver incident field (green) on the detector plane (a) for different incident angles at
the mid-plane (b). In (a), the green curves are for the receiver field and the blue curves are
the intensity of the reflected field. Since the reflected wave (blue) needs to cover all the 12
poloidal receiver channels, one single receiver field (green) is much narrower than the
reflected field. The x-axis is the poloidal position on the detector plane. Correlations for
each case from left to right are 0.93, 0.93, and 0.83.
Figure 5.13 shows how the imaging quality is affected by the incidence angle of
the sightlines of the receiver on the cutoff surface for an edge channel. Figure 5.13 (a)
shows the configuration of different oblique incident angle measurements on an edge
channel which is located above the mid-plane by 7.5 cm, and (b) shows how the measured
correlation value varies with different incident angles. The x-axis of Fig. 5.13 (b) is
115
normalized to the beam divergence angle [128]. The oblique angle investigation is
especially important for edge channels, since it is particularly difficult to achieve normal
incidence for off-axis sightlines. Correlations are above 0.6 for angles in both directions
within 5 degrees, while this expands to more than 15 degrees as the incidence angle moves
towards the mid-plane. These particular acceptable ranges of incident angles may change
with different fluctuation conditions; consequently, normal incidence is important to
resolve high wavenumber or large amplitude fluctuations, while the requirement is relaxed
for low wavenumber and low amplitude fluctuations. In DIII-D H-mode plasmas, where
the cutoff is typically located somewhere along the pedestal, the current MIR system has
an angle of incidence of less than +10 degrees for the edge-most channels.
Different Incidence
angle
Incidence
angle
+
-
(b)
Correlation
(a)
0.9
0.8
0.7
0.6
0.5
−3.6
Mid-plane
−1.2
1.2
3.6
Incidence angle
Beam divergence angle
Figure 5.13: Shown is the schematic of oblique incidence for one edge channel which is
above the mid-plane channel by around 7.5 cm (a), and how the correlation values vary
with different incident angle for this edge channel (b).
116
5.4 MIR instrument function
This section focuses on a discussion concerning the range of fluctuation parameters
that MIR is capable of measuring through the usage of the synthetic MIR method. The
investigation is done for both H-mode and L-mode, and for both coherent and broadband
perturbations. Equi-correlation contours versus the variation of different fluctuation
wavenumbers, amplitude, and focusing positions are given for each case.
5.4.1
L-mode discharge instrument function
The L-mode discharge used for simulation in this section is presented in Figure
5.14. It is changed from shot # 142111 from DIII-D, with density reduced compared to the
original shot information. Figure 5.14 (a) shows the density profile and magnetic profile
along the radial direction in the mid-plane. Magnetic field information is preserved. Figure
5.14 (b) shows the characteristic frequencies of the plasma near the reflection region,
including three resonant layers: upper hybrid resonant layer, the fundamental electron
cyclotron, and the second harmonic electron cyclotron resonant layers; two cutoff layer:
the right-handed cutoff layer and O-mode cutoff layer are also shown. The probing
frequency is 67.5 GHz as highlighted in Fig. 5.14 (b), and it is below the second electron
cyclotron resonant frequency.
117
Density (m-3)
Density
Magnetic
field
1.35e19
1.4
0.45e19
1.2
1.0
0
ωce
90
Frequency (GHz)
80
Electron cyclotron frequency
Probe frequency: 67.5 GHz
60
40
2ωce
RX cutoff
70
50
1.8
1.6
0.9e19
(b)
2.0
Magnetic field (T)
1.8e19
(a)
Upper Hybrid
O cutoff
30
1.8
ωce
1.94 2.08 2.22 2.36
major radius (m)
Figure 5.14: Discharge profile (a) and characteristic frequency of L-mode discharge #
142111. The density is reduced compared to the original discharge information.
The poloidal fluctuation wavelength simulated here is still Gaussian distributed, but
with varied center wavenumber, and fixed deviation of 0.1 cm-1, meaning the fluctuations
are highly coherent cases. The related receiver is focused at the plasma mid-plane (central
channel). The x-axis on the figure is the center wavenumber ( km ) of each Gaussian
distribution, and the y-axis is the fluctuation amplitude in percentage. We chose the
118
minimum acceptable correlation as 0.6, and which is highlighted in red on this figure. The
maximum acceptable coherent wavenumber for L-mode from this figure is around 1.1 cm1
, while the maximum acceptable fluctuation level is greater than 0.6%. Figure 4.18
contains similar equi-correlation contours, but with the receiver focused on a position
below the mid-plane at 19.3 cm. This edge channel has a much smaller wavenumber
limitation, around 0.6 cm-1, and the fluctuation level is up to 0.47%. The poor performance
of the edge channels is due to the diminished curvature matching as well as the reduction
in collected diffracted components to form an image.
-1
Wavenumber Gaussian distribution center (cm )
Figure 5.15: The equi-correlation contours with the receiver focused on the plasma midplane. The solid 0.6 line surrounded region indicates the good correlation region.
[Reprinted figure with permission from Ref. [129]. Copyright 2012, AIP Publishing LLC]
119
Fig 5.16: The equi-correlation contours with the receiver focused below the plasma midplane around 19.3 cm. The edge receiver has poorer performance than the middle channel.
[Reprinted figure with permission from Ref. [129]. Copyright 2012, AIP Publishing LLC]
Figure 5.17 shows two examples of coherent fluctuation spectra for both center and
edge channels used above. The fluctuation used in this figure has a fluctuation center
wavenumber of 0.7 cm-1, a deviation of 0.1 cm-1, and a fluctuation amplitude of 0.1 %.
Figure 5.17 (a) shows the spectrum of the measured signal (blue), matched well with the
spectrum of the density fluctuations on the cutoff surface (red), and the receiver for this is
focused at the mid-plane. Figure 5.17 (b) displays a similar comparison of spectra for the
edge channel 19.3 cm below the mid-plane. Although the correlation of this channel is only
0.02, the spectrum looks true, this may be explained by the fact that even though the phase
120
modulation doesn’t closely following the density fluctuation structure, it is still vibrating
in the same frequency as the original density turbulence. Observing spectra at different
time or positions can help to determine whether the correlations are temporal or spatial.
(a)
Amplitude
(b)
Figure 5.17: Coherent spectra of fluctuation with fluctuation wavenumber of 0.7 cm-1, and
deviation of 0.1 cm-1 and fluctuation level 0.1 %. (a) The spectrum of the measured signal
(blue), matches well with the spectrum of the density fluctuations on the cutoff surface
(red), with a correlation value of 0.9. (b) Shows the same spectral comparison for a channel
measured at 19.3 cm below the mid-plane, with a correlation value of 0.02. The sidelobes
in these two figures come mainly comes from phase jumps (sudden large jump of phase)
of numerical analysis. [Reprinted figure with permission from Ref. [129]. Copyright 2012,
AIP Publishing LLC]
Broadband fluctuations are of importance in plasma turbulence, and are relatively
common
in
both
L-mode
and
H-mode
121
discharges.
Correlation
values
for
broadband/incoherent fluctuations are discussed in this paragraph, with the investigated
fluctuation wavenumbers having a Gaussian distribution centered at 0.0 cm-1. Figure 5.18
shows the broadband spectra comparison between density fluctuations spectra at the cutoff
surface (red) and the measured synthetic diagnostic signal spectra (blue). The analytical
fluctuation model has a Gaussian distribution deviation at 0.5 cm-1 for Fig. 5.18 (a), and
4.0 cm-1 for Fig 5.18 (b). The fluctuation amplitude for both of them is 0.1 %. Each curve
consists of 5,000 time points. A Gaussian fit to the fluctuation data is shown as the
highlighted white curve. It can be seen that when the fluctuation deviation is small, the
measured spectra fits better to the original density fluctuation; while when the density
fluctuation deviation increases to a large number, meaning more fine structured
fluctuations are present, the measured spectra would be narrower than the original spectra,
indicating that MIR failed to measure large wavenumbers.
(b)
Amplitude
(a)
Figure 5.18: Spectra comparison between fluctuations at the cutoff surface (red) and the
synthetic diagnostic signal (blue) is shown for (a), fluctuations with Gaussian distribution
variation of 0.5 cm-1, and (b), Gaussian distribution variation of 4.0 cm-1. The fluctuation
level is 0.1 %. Each plot consists of 5,000 time points. A Gaussian fit to the fluctuation
122
data is shown as the white curve. [Reprinted figure with permission from Ref. [129].
Copyright 2012, AIP Publishing LLC]
Figure 5.19 shows the equi-correlation contours for broadband fluctuations of Lmode; the x-axis is the Gaussian distribution deviation of wavenumber, and the y-axis is
the fluctuation amplitude. The good correlation region has wavenumber up to 2.8 cm-1, and
fluctuation level up to 0.3 %. The receiver applied in this figure is focused at the plasma
mid-plane (center channel). The measurable wavenumber is considerably above the upper
limitation of MIR of 1.5 cm-1, as summarized from the above discussion about spot size.
This is because the fluctuations in all of the cases are a mixture of both low wavenumbers
and high wavenumbers; thus, the good correlation values above the MIR upper limitation
come from the measurement of the low wavenumber components. Figure 4.22 shows the
equi-correlation contours with the receiver focused above the mid-plane around 18.7 cm.
The available maximum wavenumber is around 1.0 cm-1, much smaller than the maximum
resolvable wavenumber of 2.8 cm-1 for the central channel. The tolerable fluctuation
amplitude is around 0.22 %.
123
Figure 5.19: The equi-correlation contours plotted for broadband fluctuations for the center
channel. The x-axis is the Gaussian distribution variation, and the y-axis is the fluctuation
amplitude. The maximum variation and fluctuation amplitude for correlation values above
0.6 are 2.8 cm-1 and 0.3 %, respectively. [Reprinted figure with permission from Ref.
[129]. Copyright 2012, AIP Publishing LLC]
124
0.1
0.3
0. 4
0.6
0.5
0.7
0.5
0.3
0.2
0.6
0.4
0.2
0.3
0.25
0.2
Fluctuation level (%)
0.15
0.3
0.3
0.4
0.2
0.35
0.2
0.5
1
1.5
2
2.5
3
3.5
-1
Wavenumber Gaussian distribution width (cm )
4
Figure 5.20: The equi-correlation contours plotted for broadband fluctuations for one edge
channel above the mid-plane by 18.7 cm. The x-axis is the Gaussian distribution variation,
and the y-axis is the fluctuation amplitude. The maximum variation and fluctuation
amplitude for valuable correlation values (encompassed by the 0.6 contour) are 1.0 cm-1
and 0.22 %, respectively.
Only applicable to the broadband fluctuations, the imaging quality or correlation value
could be improved by the usage of a low pass filter. By directly removing the high
wavenumber components in the spectra from both the measured and original density
fluctuation signals, and then making an Inverse Fast Fourier Transformation to the
modified spectra, the newly constructed density fluctuation curves would have improved
correlation value. Figure 5.21 shows fluctuation comparison curves before and after using
125
a low pass filter. The filter eliminates wavenumber components higher than 0.8 cm-1.
Figure 5.21 (a) has complicated curves with indistinguishable large wavenumber
components, and the distance correlation is just 0.43. In Fig. 5.21 (b), the large
wavenumber components are filtered out, leaving only small wavenumber components
forming the two smoother curves, the distance correlation of which is increased to 0.93,
meaning that the two curves are highly correlated with each other. This property proves
that MIR can still accurately measure fluctuations within its capability range in the
presence of high un-accessible wavenumber components.
Figure 5.21: Shown in (a) is the fluctuation comparison for broadband density fluctuation
with high wavenumber components; shown in (b) is the reconstructed fluctuation curves
after applying a low pass filter to the spectra of (a).
5.4.2 H-mode discharge instrument function
MIR is most suitable for measuring turbulence at the pedestal of an H-mode discharge,
where the density changes dramatically, and the reflection acts more like a wall reflection.
126
The applied H-mode discharge in this section is the same one as the one used in the
sensitivity measurements, which is an approximation of shot #150640, a typical ELMing
H-mode for DIII-D, with magnetic field smaller than the original shot information. Figure
5.22 shows an example of how the correlation values vary as the density fluctuation
wavenumber increases for this H-mode discharge. In Fig. 5.22 (a)-(c), three different phase
signal traces (blue) versus density fluctuations (red) are shown with increasing wavenumbers with fluctuation level ñ/n fixed at 1%. Figure 5.22 (d) gives an example showing
distance correlation values changing with turbulence wavenumber, for ñ/n of 1% and 2%.
As indicated by this figure, if the wavenumber and/or ñ /n increases, the correlation
between the measured signal5 and density fluctuation will decrease [119].
5
This measured signal is obtained from MIR simulations.
127
Figure 5.22: Three measured turbulence and density turbulence are compared under
different fluctuation conditions. As the wavenumber increases, clearly the dominant
fluctuation frequency increases, and as it increases to 2.5 cm-1, the measured phase signal
differs substantially from the density turbulence. (d) shows that the correlation value
decreases with the increasing of wavenumber for ñ/n equal to 1% and 2%.
In Fig. 5.23, correlation contours versus fluctuation amplitude and wavenumbers
are given for the center channel, which has a 1.1 cm vertical offset to the mid-plane, of the
receiver. The poloidal fluctuation wavenumber is distributed with a fixed deviation of 0.2
cm-1. The vertical rotation velocity of the simulated plasma discharge is 2.5 km/s while the
sampling frequency is 2 MHz, twice the actual sampling frequency implemented for the
DIII-D MIR system. Here, Fig. 5.24 and Fig. 5.25 show the correlation contours for two
edge channels, with an offset to the mid-plane by 7.1 cm and -4.8 cm. Simulated
performance of the middle channel is somewhat better than that of the edge channels;
however, results show MIR to be a robust diagnostic for low-k fluctuations for H-mode as
seen in Figure 5.23. The fluctuation amplitude could be up to 5 % while the resolvable
poloidal wavenumber can be up to 2 cm-1 for the middle channel. The fluctuation level is
around 10 times that of the L-mode plasma. This is due to the steeper density gradient at
the pedestal region for H-mode making it better a target for reflectometry imaging. The
accessible center wavenumber is also higher in H-mode than L-mode. For the edge
channels, the fluctuation amplitude can be up to 3 %, and the maximum wavenumber is
around 1.2 cm-1. There is some jaggedness in the contours at the low correlation regions;
the reason for this is that as correlations between the measured signal and reflected signal
decrease with increasing fluctuation amplitudes, there will be a threshold where the
128
correlation values would be meaningless, thus resulting in some jaggedness at the low
correlation values region.
0.6 04.
3.5
0.7
2.5
0.9
0.4
1.5
kθ/π (cm-1)
0.7
1
09.
1
0.5
0.6
0.6
1.5
0.7
0.4
2
04.
0.6
02.
3
0.2
4
Z= 1.1 cm
0.7
Fluctuation level (%)
4.5
2
0.
5
2
2.5
Fig 5.23: Contours of the correlation coefficient (calculated for the numerical diagnostic
signal and the actual density fluctuation imposed at the cutoff surface) are shown versus
fluctuation amplitude and wavenumber, for center channels, which is 1.1 cm vertically
away from the mid-plane.
129
4.5
0.2
Z= 7.1 cm
4
3.5 0.4
3
2.5
2
0.2
0.0
7 .6 0.4
1.5
9
0.
1
0.5
0.2
Fluctuation level (%)
5
0. 0.6 04.
7
1
0.2
1.5
kθ/π (cm-1)
2
2.5
Figure 5.24: Contours of the correlation coefficient for one upper edge channel, which is
focused at above the mid-plane vertically by 7.1 cm.
130
0.2
4.5 .4
Z= -4.8 cm
4
3.5
0.4
3
2.5
2
2
0.
Fluctuation level (%)
5
0.2
.70.6
0.4
1.5
0.9
1
0.5
0
0. .6
7
1
0.4
0.2
1.5
kθ/π (cm-1)
2
2.5
Figure 5.25: Contours of the correlation coefficient for one below edge channel, which is
focused at below the mid-plane vertically by 4.8 cm.
5.5 Spectral Analysis
5.5.1
Spatial Correlation
The above correlation values are mostly temporal correlation values, meaning the
correlation values are calculated depending on the time-dependent signals. To determine
the correlation distance along the poloidal direction, spatial correlations along the poloidal
direction are calculated for different fluctuation configurations, and the simulation results
are shown in Fig. 5.26. The correlation values are calculated from the measured signal at
different poloidal position versus the density fluctuation at the mid-plane. Each fluctuation
has a reasonably high correlation around the mid-plane channel. While, as the poloidal
131
distance increases, the correlation curves decrease sharply and then tend to be noncorrelated. In addition, the correlation values are more sensitive to the density fluctuation
level than the fluctuation wavenumber as seen from the blue and green curves on this
figure. As seen in the figure, when the fluctuation level increases, the poloidal correlation
distance would decrease to around 10 cm. While when the fluctuation level is considerably
small (0.1 %), the correlation distance could be more than 40 cm. The pre-given decorrelation time is 1 ms, and the vertical rotation velocity is 2.5 km/s, meaning that the
given de-correlation distance would be around 2.5 m. Since the largest illumination region
in the poloidal direction is 40 cm for L-mode, and within this range the fluctuations are
highly correlated for small fluctuation amplitudes, as seen in this figure. The longest 2.5 m
correlation distance is not accessible to the current MIR system. When the fluctuation level
increases, the correlation distance would decrease dramatically.
Fig 5.26: The correlation values versus position away from the mid-plane are shown for
four different fluctuation situations. When the fluctuation level is around 0.1%, the
132
correlation distance would exceed the observation range, which is 40 cm. While when the
fluctuation level increases to around 0.3 %, the correlation distance decreases to around 10
cm.
5.5.2
Power spectral density (PSD) analysis
In this section, the power spectral density (PSD) of the measured phase fluctuations
is investigated. The advantage of PSD is that it can always give the correct spectra of the
analyzed data. PSD of mid-plane and edge channels are compared in Fig. 4.27 for a highly
coherent fluctuation, with the Gaussian distributed fluctuation wavenumber with a
deviation of 0.01 cm-1 and a center of 0.4 cm-1, while the fluctuation level is 0.1 %. Strong
sidebands or harmonics of the main frequency are observed on the edge channels, but not
observed on the center channel as shown in Fig. 5.27 (d). Since the diffraction components
are symmetric around the normal sightline of cutoff surface with normal incidence, the
observing sightlines of the edge channels are not normal to the cutoff surface, thus the edge
channels failed to connect symmetric diffraction components that would help to reduce the
sidebands. The dominant frequencies for all the three channels are the same, meaning that
there is little or no observable Doppler drift in the cases. PSD for more incoherent
fluctuations are shown in Fig. 4.28, with wavenumber distribution deviation of 0.1 cm-1
(10 times of that in Fig. 4.27) and no sideband or harmonic modes shows up as in the
coherent case; the distribution of PSD is approximately Gaussian distributed, which is
similar to the given distributions to generate the fluctuations.
133
Figure 5.27: PSD for a highly coherent fluctuation case. For the signal shown in this figure,
the Gaussian beam passing through the receiver system is focused at three different
positions on the cutoff surface, upper (a), lower (b), and middle (c) channels. The Gaussian
distribution of fluctuation wavenumber has a variation of 0.01 cm-1 and a center of 0.4 cm1
, while the fluctuation level is 0.1 %. Shown in (d) is the comparison of the measured
signal for the three different positions.
134
Fig. 5.28: The PSD for 5 different Gaussian distribution of wavenumbers, with each one
having a different center as shown in the legend, but a constant deviation of 0.1 cm-1. The
sampling frequency is 15 kHz, and each case has 1000 sample points.
5.5.3 Frequency versus wavenumber relationship
To tell whether MIR is capable of recovering the rotation velocity of the
fluctuations on the cutoff surface, and also to see if there is any observable Doppler shift
in the measured velocity between the upper and lower channels, the frequency versus
wavenumber dispersion relationship is investigated in this section. Figure 5.29 shows the
dispersion curve for three different channels, with the x-axis corresponding to the
wavenumber and the y-axis corresponding to the frequency. The frequency is the dominant
frequency for each measured spectra. A linear fit for the data obtained from the middle
135
channel is f=1.5061k+0.0572, with f in kHz and k in cm-1, close to a line passing through
the original point. The phase velocity calculated from this equation using the formula is
9.46 km/s, and this value is very close to the given rotation velocity of 10 km/s. In addition,
as seen in this figure, the linear fits of both upper and lower channels are also close to that
of the middle channel; thus, no obvious Doppler shift is observed from the investigation of
the dispersion relation. This might be due to the normal incidence of the illumination beam
on the cutoff surface, whereby the reflected wave will have a small or null Doppler shift.
Fig 4.29: The wavenumber versus frequency for one middle, upper and lower channels.
Dots are the measured points, while the lines are the linear fitting. The linear fitting for all
the channels looks the same; thus, no obvious Doppler shift is observed from MIR signal.
5.6 Summary
136
Based on the above discussion of the synthetic analysis results, MIR is capable of
being a robust diagnostic tool for density fluctuation measurements, especially for H-mode
plasmas. It is able to provide a large poloidal coverage depending upon the height of the
tokamak port, while the radial coverage is determined by the incident frequency range,
which is built to be from 56 GHz to 74 GHz. For coherent fluctuation measurements, the
fluctuation parameters with good (> 0.6) correlation values have the maximum
wavenumber up to 1.1 cm- 1, and fluctuation amplitude up to 1.0% for L-mode, which for
H-mode are up to 2.0 cm-1 and 5 %, respectively. Furthermore, for incoherent fluctuation
measurements, the accessible wavenumber range can be higher. By applying a low pass
filter to widely spread wavenumber distributions, MIR is also capable of restoring the
measured phase as a good measurement of density fluctuations from a wide range of mixed
wavenumbers.
The maximum out-of-focus range from the analysis above is around 50 %
of the probing wave Rayleigh range, while the poloidal resolution on the cutoff surface,
determined by the spot size of the Gaussian beam on this surface, could be as small as 2.0
cm. For the receiver oblique incidence, the most tolerable oblique angle is up to 15 degrees
in the direction towards the mid-plane, while the real MIR system has an oblique angle less
than 10 degrees in this direction, and thus is within the acceptable range.
Despite all the positive attributes of MIR, it is understood that real plasma reactions
will be considerably more complicated than the simulations. Turbulence will be more
complicated with high rotation speed; the cutoff surface will move back and forth making
the illumination beam curvature matching extremely difficult. MHD islands on the path of
the beam may induce large refractive index changes, and thus at last have a large impact
on the measured phase change. Relativistic absorption and upper hybrid absorption may
137
also cause unexpected problems [129], [130] More details regarding the real installation
and electronics are discussed in the following sections.
138
Chapter 6: MIR Characterization with Synthetic Results and
Initial Experimental Campaign6
6.1 Comparison with synthetic simulation results
6.1.1
Overview
MIR was installed on DIII-D in May, 2013, and began to take plasma data shortly
thereafter. It takes spatially resolved “pictures” of the fusion plasma density fluctuations,
which allows for analysis of radial and poloidal correlation lengths, wavenumbers, and
both the phase and group velocity of propagating modes. The MIR system does not rely on
auxiliary tokamak systems, such as neural beam injection, and the probing radiation does
not perturb the plasma while the diagnostic monitors changes in density throughout the
discharge.
One thing which should be noted is that, since the RX cutoff depends on the local
values of the electron density and magnetic field, variations in these profiles lead to
modifications to the radial accessibility, especially the effects from the upper hybrid (UH)
resonance layer which lies close to the RX cutoff layer. Even though the UH resonance
layer usually lies behind the RX cutoff, mode conversion can still happen if the upper
hybrid layer is sufficiently close to the cutoff surface, thereby allowing the evanescent
wave to tunnel in and out of this region and complicating the interpretation of the reflected
signal. To make the reflected signal interpretation easier, the probing region is mostly
6
The term “campaign” is how DIII-D and other Tokamak devices refer to their experiment shots and
goals.
139
chosen near the pedestal top region where the UH layer and the RX cutoff layer would be
sufficiently separated.
The numerical diagnostic results from Chap. 4 indicate that the MIR optical system
installed on DIII-D can accurately resolve poloidal fluctuations with mean wavenumber
kθ up to 2π cm-1, and fluctuation amplitude up to !n! /n = 5%, while this information
!
would be proven to be more trustworthy with experimental results directly compared to
synthetic diagnostic results. In this section, the initial measurements from MIR
characterization discharges are compared to synthetic diagnostic results, to verify the
synthetic simulation results as well as to help interpret the experimental data.
6.1.2
Comparison from a discharge with a density ramp
Since the position of the optical focal plane along the plasma major radius is
determined by the position of the optical system, and this position remains relatively
constant; even though changes in plasma refraction do have some effects on the position
of the focal plane, this dependence tends to be weak. Thus, as the density and magnetic
field evolve throughout the discharge, cutoff surfaces may move in and out of this
particular focal plane. Without real-time focusing, which is under development at UC
Davis [131], MIR image quality thus varies with variation of the plasma profiles. How long
the field depth is actually is experimentally investigated using the density ramping period
of an H-mode discharge. Shown in Fig. 6.1 is one DIII-D typical H-mode discharge
#154397. At the first stage of this discharge, the density is ramping up from 200 ms to 2000
ms. During this time, the magnetic field is kept constant at 1.7 T, meaning that the cutoff
surface position is only dependent on the density profile. In addition, since during this time
140
ELMs are obvious as indicated by the spikes in the last panel of Fig. 6.1, MIR data are best
to be taken during the inter-ELMs periods, with minimum influence from ELMs.
Figure 6.1: DIII-D discharge # 154397, a typical H-mode discharge. Density ramps up from
200 ms to 2000 ms. The magnetic field is constant during this period.
Measurements from the density-ramp period is presented in this section, and
compared to corresponding simulation results using the same discharge profile. As seen
from Fig. 6.2 (a), except for a sudden decrease around 800 ms due to MHD behavior, the
density rises steadily between 500 and 2000 ms. The stationary optical configuration fixes
the focal plane spatially while the density ramp moves the cutoff from smaller to larger
major radius. During this discharge, the optics are configured such that the focal plane is
situated in the pedestal region. The experimental data represented in raw I/Q plots and
histograms of the amplitude distribution are shown in Figs. 6.2 (b) and (c). Data are
presented from a poloidal channel whose focus is located approximately 1 cm above the
141
midplane, and the probing frequency is 62 GHz. At approximately t = 728 ms, an annular
I/Q plot begins forming and continues to improve until approximately 2000 ms when the
amplitude scatter is minimum. The amplitude distribution in the initial state represents that
of Rayleigh fading [194], which occurs when there is no dominant mode of propagation
along a line of sight. In this case, random interference completely dominates and the
channel is out of focus. At t = 728 ms, when the annulus appears, a second distribution
emerges associated with Rician fading, which occurs when the signal propagates along a
dominant line of sight. This point corresponds to the cutoff first entering the effective
depth-of-field. The cutoff continues approaching the focal plane, and at t = 1988 ms, the
cutoff and focal plane are co-located, and the amplitude distribution transforms into a
narrow Gaussian distribution. At t = 728 ms, the location of the 62 GHz cutoff layer at the
mid-plane is located at R = 2.215 m, and at t = 1988 ms, the location of the cutoff has
moved to R = 2.254 m, yielding a coupling distance of approximately 8 (= 2*4) cm. The
coupling distance represents the radial extent within the plasma over which an image can
be formed. This coupling distance is a function of the depth-of-field of the receiver linesof-sight, the phase matching between the illumination beam and the cutoff surface, and the
alignment of the MIR optics.
142
143
Fig 6.2: MIR diagnostic response to ramping density, i.e. radial displacement of the cutoff
surface. Fig (a) shows the density ramp of shot #154397. Rows (b) and (c) show the
experimental data represented in raw I/Q plots and histograms of the amplitude
distribution. Simulations are performed for nearby times, with similar I/Q plots and
amplitude distributions shown in rows (d) and (e).
Similarly, corresponding raw I/Q plots and histograms from the synthetic
simulation, using an identical optical configuration and plasma profile to shot # 154397,
are presented in Figs. 6.2 (d) and (e). For each sampling time, a relevant equilibrium profile
could be obtained and used to generate the synthetic results corresponding to that sampling
time. The fluctuation model used is still the Gaussian distribution analytic model, with
fluctuation amplitude fixed at 3 %. The poloidal wavenumber has a broadband Gaussian
distribution centered at 0 cm-1 with a deviation of 0.3π cm-1. Four windows (each
containing 0.5 ms of data) are chosen to match with the experimental time in Fig. 6.2 (b),
and the correlation value for each time period is 0.65, 0.83, 0.97, and 0.97, respectively, all
of which are above the acceptable correlation value of 0.6. Since simulation conditions are
much simpler compared to the actual experiments, it is reasonable to obtain higher
correlation values from the simulation results. The simulated depth-of-field of the receiver
is approximately 12 cm, longer than expected from experimental results. However, despite
the absolute correlation values, we can observe a similar transition of the numerical I/Q
data versus the real I/Q data. Numerically, amplitude fluctuations on the I/Q plot are large
at t = 505 ms, while they eventually form into a clear annular shape with much reduced
amplitude fluctuations, implying a co-location of the cutoff and focal plane. The histogram
144
in Fig. 6.2 (e) for the I/Q data at t = 505 ms represents a Rayleigh-like distribution
contrasting with the Rician-like distributions with less random amplitude spread seen in
the following time windows. From this figure, we can see that numerical diagnostics can
be a suitable gauge to characterize the performance of MIR, particularly for locating the
optimal conditions for focusing.
6.1.3
Comparison for a vertically jogging discharge
Another comparison is performed for a vertically jogging plasma discharge, which
has nearly constant density and thus fixed radial position of the cutoff over the time period
of interest. Vertical jogs of the plasma make it possible to determine the vertical sensitivity
between the MIR system and plasma discharge. The discharge used for this comparison is
a DIII-D quiescent H-mode discharge # 155117, with some core variables shown in Fig.
6.3. This discharge has reversed plasma current and magnetic field. Similarly to the above
discharge, the magnetic field is constant during the time period of interest at -2.0 T. The
density profile has small variations during the time of interest, while the vertical axis is
moving up and down, as shown by the last panel of “zmaxis”.
145
Figure 6.3: DIII-D discharge # 155117, a quiescent H-mode discharge. The density has
small variations during the period of interest, and not obvious ELMs. The discharge center
is moving up and down as seen from the “zmaxis” signal.
Both experimental results as well as relevant synthetic simulation results are
presented in Fig. 6.4. The density jogs versus time are shown at the top panel of Fig. 6.4.
In this figure, three poloidal channels are shown, each with four illustrative I/Q plots for
different time windows at different vertical positions. Figures 6.4 (b), (c) and (d) show
experimental results of channel 2, channel 6, and channel 11, where small number channels
are at the top in the plasma. Channel 6 is 1.1 cm above the mid-plane, while channel 2 and
channel 11 are 12 cm above and below mid-plane, respectively. At t = 1750 ms, where the
discharge is aligned at the poloidal mid-plane, the real I/Q plots are not a clear annulus for
all the three channels, and this condition holds as the plasma column moves up and then
back to the center. As the discharge moves below the original center by approximately 5
cm at t = 4425 ms, channels 6 and 11 both generate annular I/Q distributions. This result
146
indicates that the optical system at this position is best aligned with the plasma, suggesting
the possibility that the actual optical system is mis-aligned with the tokamak mid-plane by
the same vertical displacement as the plasma.
147
148
Fig 6.4: MIR diagnostic response to vertical displacement of the plasma. Figure (a) shows
the vertical jogging of the discharge versus time for shot #155117. Figures (b), (c), and (d)
show the experimental I/Q plots for four time cases, and they correspond to channel 2
(above the mid-plane), channel 6 (near the mid-plane), and channel 11 (below the midplane), respectively. Figures (e), (g), and (h) show the numerical diagnostic results for the
same four time slices with the same density profile, and they also each correspond to
channels 2, 6, and 11, respectively. The simulation results have similar trends compared to
the experimental results when the numerical receiver has an offset to the tokamak midplane by 5.5 cm.
A simulation is performed for shot 155117 over the same time duration as discussed
above. The fluctuations chosen for the simulations are incoherent, having a Gaussian
distribution with 0 cm-1 center, and 0.3π cm-1 width. This choice is made based on actual
measurements made with MIR which indicate that broadband turbulence centered at 0 cm1
dominates the spectrum for shot #155117 during these times. In addition, the density
fluctuation level is set to be 1%, which emulates the degree of phase modulation observed
in the experimental measurements. Correspondingly, a similar trend in the I\Q plot
variation from the numerical results is achieved with the simulated optical system aligned
below the mid-plane by 5.5 cm; nevertheless, if the optical system is aligned as usual
without a vertical shift of 5.5 cm, the annular I/Q plots emerge at t = 1740 ms and 3435 ms,
suggesting that the MIR optical system has a displacement with the plasma discharge midplane by 5.5 cm. The relevant I/Q plots with the numerical optics set 5.5 cm below the midplane are shown in Figs. 6.2 (e), (f) and (g), corresponding to channels above the mid-plane
to below the mid-plane. As seen in these figures, at t = 4435 ms, for all the three channels,
149
I/Q plots are annularly shaped, meaning the best alignment of the optical system with the
plasma. At t = 1740 ms and 3435 ms, where the misalignment between the optical midplane and plasma mid-plane is around 5 cm, channel 2 gives an annular I/Q distribution,
while channels 6 and 11 have poorer I/Q plots. At t = 2535 ms, with the worst misalignment
around 10 cm, none of the three channels yields a good I/Q plot. The simulated time
windows are several milliseconds different from their relevant experimental time windows.
However, within this difference, the vertical positions of the discharge change negligibly,
so the influence on the comparison of the real and numerical simulation data is negligible.
The simulation results and its comparison with experimental results show that the real
optical system has a displacement with the plasma mid-plane by approximately 5 cm for
this specific shot, indicating that simulations are not only an indispensable guide for
designing the actual MIR system, but also serve as validation for the optimal operating
conditions during real measurements.
6.2 ELM-inter modes observation
The period between ELMs of an H-mode is interesting due to the presence of a
variety of MHD instabilities. MIR showed its capabilities to measure density fluctuations
by measuring a set of up-sweeping modes during the periods of ELMs, and which are also
observed with other diagnostic tools. Discharge # 154412 is a typical ELMing H-mode
discharge on DIII-D. It has reversed magnetic field, and is a fully non-inductive discharge.
The inter ELMs periods of such discharges have a rich spectrum of MHD activities that
are detected by multiple diagnostic tools, including MIR, and remain an active area of
investigation. The measurements presented for this discharge were not directly obtained by
150
this author, but instead obtained by her colleagues [102] and are provided here for better
continuity of the dissertation and for the convenience of the readers.
The MIR initial diagnostic results about these inter-ELM modes are compared to
measurement results from CO2 interferometry [133], ECE-radiometry [134], and Mirnov
coils [135], as shown in Fig. 6.5. The CO2 interferometer on DIII-D consists of three
vertical chords and one horizontal chord. It measures the phase shift of a laser beam
traversing the plasma caused by the electron density along a certain chord. The localized
density fluctuations measured from the interferometer near the plasma edge are calculated
from the spectral cross-power of the horizontal chord and one vertical chord located at R
= 2.1 m. The edge localized spectra measured from the CO2 interferometer are shown in
Fig. 6.5 (a). The ECE radiometer on DIII-D is a 1D temperature fluctuation diagnostic tool
as mentioned in Chap. 2. It has 40 channels in the radial direction, with an IF bandwidth
of 1 GHz, and the spatial resolution is about 2 cm at the half radius and 4 cm at the center
and low-field side edge. The ECE spectrum, represented in Fig. 6.5 (b), is the cross-power
between two adjacent channels near the mid-plane, just inside the last-closed flux surface
at R = 2.156 m and 2.182 m. The Mirnov coil set is an external fluctuation diagnostic tool,
which measures the Mirnov oscillations around the plasma [136]. The magnetic spectrum
from Mirnov coils on Fig. 6.5 (c) is the cross-power between two high-frequency Mirnov
probes located 50 degrees above the mid-plane, and separated toroidally by 2.2 degrees,
representing magnetic fluctuations near the outboard plasma edge. The MIR spectrogram
is obtained from auto-power of a single channel, which is located 5 cm above the midplane and less than 1 cm inside the last closed flux surface, which is at R = 2.185 m, as
151
shown in Fig. 6.5 (d). The region between the vertical dashed lines on this figure indicates
the period when the best focusing is achieved with MIR.
When compared to other diagnostic systems, MIR performed well as a density
fluctuation diagnostic tool. For example, the up-sweeping modes, which are clearly seen
on the interferometer spectrogram but not so obvious from either ECE radiometer or
Mirnov coils results, are also observed clearly on the spectrogram of MIR results. Every
diagnostic tool sees a clear coherent fluctuation at 100 kHz, but MIR also sees a clear 110
kHz fluctuation. This could be due to the fact that MIR has highly sensitivity to small
density perturbations, and it has better localization detection ability than other lineintegrated diagnostic tools, say interferometry, and also than other external measurement
tools, say Mirnov coils. The ECE radiometer is not designed for density fluctuation
measurements, and thus it may detect different perturbations between the density
fluctuations and temperature fluctuations. Another feature, which is uniquely seen on the
Mirnov magnetic coils is a strong coherent 70 kHz fluctuation, is not obvious on the other
three diagnostic tools. This 70 kHz fluctuation may reside in the core and cannot be seen
by the others, since other spectra are primarily measuring at the edge. Since the Mirnov
coils detect the magnetic field perturbation around the coils, which arise from tearing
modes, its measurement results could also indicate magnetic perturbations caused by strong
core modes.
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Figure 6.5. Spectra observed for DIII-D shot #154412. (a) shows the cross-power between
the horizontal and one vertical chords of the CO2 interferometer; (b) shows the cross-power
between two ECE channels located at R = 2.156 m and 2.182 m; (c) shows the cross-power
between two high frequency magnetic coils located outside the plasma and near the midplane; (d) shows the auto-power of one MIR channel above the mid-plane by 5 cm, and
less than 1 cm inside the last closed flux surface. [Courtesy of Dr. Christopher M.
Muscatello]
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6.4 Comparison of ECEI and MIR
6.4.1
Alfven eigenmodes
The investigation of Alfven eigenmodes [137], and how to eliminate or mitigate
their influence on the transport of energetic particles is one of the main topics of current
plasma physics. Thinking of the magnetic field lines as strings of a guitar, plucking of the
string could cause the perturbation to propagate7, which is just the mechanism of Alfven
waves in plasma. Due to the toroidal and closed magnetic field lines in tokamak devices,
there exists many Alfven eigenmodes, where waves cannot be damped normally and can
grow to non-negligible scale. Alfven eigenmodes could lose energy to resonant particles,
and thus cause the particle to escape from the plasma and be lost. In addition, and more
importantly, Alfven eigenmodes could cause resonance islands and eventually produce a
flattened profile of the ion pressure, which then lowers the temperature and density at the
core, thereby reducing the fusion yield [138]. The investigation of wave-particle
interactions is needed to better predict and produce the operating parameters for ITER.
Moreover, since if ITER is going to operate at a self-heating steady state and self-heated
by energetic alpha particles, the condition of Alfven eigenmodes transporting alpha
particles out of their confined orbit should be avoided or mitigated [139].
ECE-Imaging has been proven to be a strong diagnostic tool by providing a wide
2D image of the Alfven mode structures, and enables the direct comparison between
experimental and theoretically expected wave structures [140], [141]. In contrast, prior
7
This is a simple mechanical analogy to explain the Alfven wave dispersion, the wave on a guitar string is
a standing wave, doesn’t actually propagate, while Alfven wave on magnetic lines does propagate.
154
fluctuation measurements only provide either 1D profiles or single-point time-series
measurements. Figure 6.6 shows one example of ECEI measured mode structures
compared to the ideal MHD code NOVA [142], [143], which is used to simulate 2D
structures of the possible shear Alfven eigenmodes [144] in the plasma. The agreement
between the ECEI measured mode structure and the simulated mode structure proves that
ECEI could be a powerful tool for the validation of eigenmode solving codes. Here, in Fig.
6.6, (a) and (c) shows two fundamental radial sheared Alfven mode structures, with n = 3
and n = 2; while, in (b), shown is the first identified radial harmonic of the n = 3 mode in
(a).
155
Figure 6.6: Shown are the eigenmode structures from ECEI and NOVA simulations from
[140]. The excitation frequencies and simulation times from NOVA are in good agreement
with the experimental results from ECEI. Here, (a) and (c) shows two fundamental radial
sheared Alfven modes (n=3, 2); while (b) shows a radial null in the fluctuation amplitude
and is identified as the first radial harmonic of the n = 3 mode shown in (a). [Reprinted
figure with permission from Chapter 8, Ref 6, Copyright 2011 by the American Physical
Society.]
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However, since ECE-Imaging may be cutoff due to over-dense conditions for high
qmin scenarios, and has poor diagnostic capability for Alfven modes for these conditions,
MIR, which also provides 2D imaging data for mode structures, but does not rely on
radiation from the plasma, could fill the critical gap of measuring the Alfven mode
structures under high qmin H-mode scenarios that are under development on DIII-D for
ITER steady state. While since there has been no chance to explore high qmin conditions
with MIR at its full operation capability thus far, Alfven mode diagnosis and
characterization is performed for an L-mode condition, with shot # 161127. Figure 6.7
shows the detection of MIR measurements of both TAE and upsweeping RSAE modes. At
the beginning, as shown in (a), the TAE and RSAE are two separated modes with different
propagation velocities in the same direction. Later on, as shown in (b), the RSAE mode
starts to up sweep, and gradually reaches the frequency where TAE resides. At last, as seen
in (c), the RSAE and TAE are coupled to each other, and grow to a strong unstable Alfven
mode. Since MIR is more suitable for the detection of fluctuations in H-modes, we would
expect it to detect the same or even better quality of Alfven mode structures or spectra than
in L-mode.
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Figure 6.7: RSAE and TEA modes detected by MIR for an L-mode. In (a), both TAE and
RSAE are two separate modes; in (b), the RSAE begins to upsweep; while in (c), the TAE
and RSAE are coupled to one single mode, which is more unstable than each of them.
[Courtesy of Dr. Benjamin Tobias]
6.4.2
Simultaneous local imaging
The purpose of installing the ECEI and MIR systems on the same port of DIII-D
and also sharing the same zoom lenses is to enable a simultaneous local imaging of
temperature and density fluctuations over an overlapped plasma cross section. In addition,
by making the setup, these diagnostics could complement each other for diagnosing
important quantities such as fluctuation amplitude and wavenumber, turbulence correlation
and de-correlation length and time, propagation velocity, and also cross-phase between
density and temperature fluctuations.
Figure 6.8 shows one example of simultaneously measured mode structures from
ECE-Imaging and MIR for an inter-ELM up-sweeping mode. The plasma discharge is still
shot # 154412; the displayed mode is one branch of the up-sweeping modes between two
ELMs at time 1864 ms with frequency 58 kHz, which is highlighted as the white square in
the top spectrogram of Fig. 6.8. The real part of the complex amplitude of the particular
spectral component within this white square is recorded for each channel and plotted on a
two-dimensional spatial grid as the 2D mode structure. A similar approach is used to
generate the 2D mode structure from the spectra measured by ECEI. The series of panels
in the middle row is the phase sequences representing density fluctuations over the
measured 2D region from MIR, while the bottom row represents the series of temperature
158
fluctuations measured from ECEI. Every adjacent panel represents a π /4 phase shift of the
fluctuation, with the color contours proportional to fluctuation amplitudes. The left panel
of Fig. 6.8 shows a portion of the EFIT equilibrium reconstruction constrained by the
motional Stark effect (MSE) diagnostic [146], [147]. The plasma region measured by MIR
is determined by calculating the RX cutoffs for the four probe frequencies (62, 63, 67, and
68 GHz) using fitted profiles for both density and magnetic field. As observed from the
mode structures in this figure, the 58 kHz mode has a poloidal wavelength λθ around 30
cm, or a poloidal wavenumber kθ of 3.3 m-1. The density fluctuations peak near the last
closed flux surface, and decrease monotonically inward.
Figure 6.8: Simultaneously measured mode structure from MIR (middle panels row)
and ECEI (bottom panels row) for an inter-ELM mode. The mode for generating the
159
mode structures is highlighted on the top panel as the white square. The diagnostic region
is highlighted on the left panel, which is an equilibrium fitted profile for shot # 154412.
The cutoff layers correspond to the highest and lowest MIR probe frequencies used for
this particular discharge. [Courtesy of Christopher M. Muscatello]
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Chapter 7: Edge Harmonic Oscillation Studies using MIR and
Synthetic MIR
7.1 QH mode
7.1.1
QH mode
The Quiescent H-mode (QH-mode) is a type of H-mode without strong bursts of
edge-localized modes (ELMs) [148]-[150]. It was first discovered in DIII-D in 1999 when
the counter-injection of neutral beams was combined with cryo-pumping to lower the
plasma density [151], [152]. It was then subsequently investigated on ASDEX-Upgrade,
JT-60U, and JET. H mode operation is the choice for next step Tokamak devices due to its
better confinement time and favorable density profile. However, since the ELMs existing
in a typical H-mode operation would cause damage to divertors as well as other plasma
facing components (for example, the wall), and thus reduce their lifetime (or even result in
catastrophic failure), they must be eliminated or mitigated for safe operation on ITER. QHmode operation, which works under the kink-ballooning boundary condition, thus has no
strong ELM bursts, but also preserves the superior energy confinement of a typical H mode,
is one candidate operation scenario for ITER. The mechanism of how a QH-mode stabilizes
without ELMs is still not clear and also remains an intensive topic of QH-mode studies.
Figure 7.1 shows two QH mode discharges, with co and counter neutral beam injections
with respect to the plasma current direction. The QH-mode pulse length is limited by the
neutral beam injection durations. EHOs [153]-[156] are believed to be the key element of
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QH-mode, enhancing particle transport at the edge while allowing the edge plasma to reach
equilibrium below the ELM stability boundary.
Figure 7.1: Shown are two QH-mode discharges with neutral beams in the same direction
with respect to the plasma current and in the counter direction with respect to the plasma
current. Both shots have no strong divertor burst signals (fs04).
Generally, QH mode operates at a lower pedestal electron density and a higher pedestal
electron temperature than most of the H-modes in DIII-D. However, with the help of the
theory and predictions of the edge peeling-ballooning (kink) modes, the QH-mode
operating space has been significantly extended in recent years. QH-mode with balanced
NBI or co-injection of NBI has been created demonstrating that counter-NBI and edgecounter rotation are not essential conditions for QH-mode, as shown in Fig. 7.1. The ELM
boundary is significantly higher in more strongly shaped plasmas, which broadens the
operating space available for QH-mode and leads to improved core performance. Changing
162
edge rotations allows controlled change in edge density and pressure by a factor of two
while retaining the ELM-free state: reduced edge rotation causes increased pedestal density
and pressure. When the rotation decreases to some value, the coherent EHO modes will be
replaced by broadband MHD modes. Coherent EHO rotates in the counter direction with
respect to the plasma current, but broadband MHD rotates in the co-direction of plasma
current, suggesting that they are different instabilities. Current research focuses more on
the broadband turbulence, since compared to coherent EHO modes, QH-mode sustained
by broadband turbulence has a wider pedestal range allowing for higher maximum pedestal
height, and thus better operating space.
7.1.2
Edge Harmonic Oscillations
Edge harmonic oscillations (EHOs) were first seen on magnetic probes [157]. The
nature of the oscillations detected by the magnetic pickup loops changes when the ELMs
cease, from a bursting behavior to a much more continuous oscillation. The oscillation is
periodic, but not sinusoidal. Its spectrogram usually shows several clear harmonics, while
the multiple harmonics/multiple n numbers are simply the Fourier harmonics needed to
describe such a non-sinusoidal oscillation. EHOs usually rotate in the same direction as the
injected neutral beams [157]. Figure 7.2 shows an example (DIII-D shot # 157102) of QHmode with EHOs versus a typical H-mode (DIII-D shot # 154412) with strong ELMs and
complex inter-ELM MHD activities.
EHO is predicted to be a low n peeling mode driven unstable by rotational shear at
edge conditions slightly below the ELM stability limit in the absence of rotation [158].
163
There are two mechanisms which lead to the mode saturation at a finite amplitude. First,
as the mode grows to finite amplitude, its magnetic fields interact with the vacuum vessel
wall, slowing down the plasma rotation, decreasing the rotational shear and, hence,
reducing the drive for the mode. The increased particle transport leads to reduced edge
density and edge density gradient, which in turn reduces the edge bootstrap current. The
reduction in edge pressure gradient and edge current density also reduces the drive for the
mode, thereby providing a second mechanism for the plasma to reach a steady state in the
presence of a finite amplitude mode. A more complete theory which also helps to explain
how the EHO enhances the edge particle transport is still not clear and under development.
Figure 7.2: Examples of quiescent H-mode (left) and conventional H-mode (right). The
quiescent H-mode has strong coherent EHOs measured with MIR (upper left), and no
bursts of Dα radiation at the divertor (lower left). The conventional H-mode is ELMing
with frequent spikes of Dα emission (lower right). MIR measured inter-ELMing modes
have higher frequencies (upper right) for conventional H-mode [77].
164
7.2 Experimental measurements of counter-propagation
MIR on DIII-D [91], [102] provides edge-localized fluctuation data demonstrating a
widely-held view concerning the physics of QH-mode on DIII-D: rotational shear modifies
the spectrum of unstable modes toward low-n and saturated edge-harmonic oscillations,
while sustaining the higher n-numbered ELM-inducing fluctuations. It has been used to
compare the poloidal wavenumber spectra of both the coherent and so-called broadband
EHO types. These measurements could help to validate a model for EHO excitation and
control, since MIR data often exhibits unique features when compared to other diagnostics.
Apparent counter propagation is one unique feature observed in MIR data when imaging
near the top of the pedestal, and this is not observed in Mirnov coil array data.
Figure 7.3 shows MIR measurements from two radial channels for discharge #
157102, a typical QH-mode with strong coherent EHOs and configured with normal Bt,
reversed Ip, and counter-current neutral beam torque. Both the dominant mode (n = 1) and
harmonics (n = 2 and n = 3) diagnosed with the 57 GHz channel propagate in the same
direction with respect to the toroidal torque, which is consistent with Mirnov coil array
data. However, localized density fluctuation data from the 58 GHz channel, which is
located more near the top of pedestal, shows that the dominant mode propagates counter to
that measured from the 57 GHz channel, while the harmonics still co-propagate with the
toroidal torque. This is inconsistent with external measurement results from the Mirnov
coil array data. Possible sources such as instrumental artifacts were investigated using
forwarding modeling of MIR; however, they do not reproduce the observed effect of
165
counter-propagating harmonic behavior, thus leading us to conclude that they are in fact
due to unclear subtleties of the EHO mode structure.
Figure 7.3: Apparent counter propagation in MIR data. The dominant mode and harmonics
measured with the 57 GHz channel are propagating in the same direction, consistent with
Mirnov coil array data. However, data from the 58 GHz channel, located more near the top
of pedestal, measures the dominant mode counter propagating with higher harmonics [77].
7.3 Forward modeling of MIR for EHO studies
7.3.1
Forward modeling of MIR
Unlike the previously discussed synthetic MIR simulations, which were primarily
aimed at helping to optimize the design aspects and determine the operating domain of
MIR, forward modeling of MIR significantly helps in understanding the MIR experimental
results, and determining possible instrumental artifacts. In addition, forward modeling of
MIR also helps to verify the plasma simulation results by directly comparing the synthetic
MIR diagnostic results with the experimental measurement results. Since counterpropagation between the fundamental and harmonic EHO modes is observed in MIR data,
166
but not consistent with the measurement results of Mirnov coils, which is an external
diagnostic tool, synthetic MIR combined with sophisticated EHO simulations is employed
to determine if the measured counter-propagation comes from possible instrumental
artifacts, such as transmitter and receiver misalignments, or device aperture introduced
unreal harmonics. From the simulation results, it is revealed that the counter-propagation
doesn’t arises from instrumental artifacts, but could be due to more complicated subtleties
of EHO instabilities located near the top of the QH-mode pedestal.
Both FWR2D and FWR3D are applied to simulate and determine possible
instrumental artifacts that could cause the counter-propagation. However, in this chapter,
only simulation results from FWR2D are presented due to its better stability, and results
from FWR3D are not very good and not presented since there may exist some unconfirmed
errors. The process of generating synthetic diagnostic signals from time-dependent M3DC1 [159], [160] simulation results is shown in Fig. 7.4. Here, the M3D-C1 code is a linear,
single fluid MHD code. It includes the plasma and open field-line region inside the
computational domain; thus, it eases the difficulty of diverted geometry and unstructured
mesh with high-order elements. Here, single fluid model means ω i = ω e = ω E×B , with
!
ω
obtained from CER [161] measurements. The simulations are performed in real
! E×B
DIII-D toroidal tokamak geometry, with a resistive plasma (Spitzer model [162]), resistive
wall, and also with/without toroidal rotation. For each linear M3D-C1 run, a specified
toroidal mode number (n) is assigned and the mode with the highest growth rate is returned.
In Fig. 7.4, a time-series (or periodic sequence) of perturbed field data from simulation is
combined with the prescribed EFIT equilibrium profile of shot # 157102 on DIII-D. Each
time frame is imaged with a 2D array in the radial and poloidal directions. The radial
167
direction is probed with frequencies from 55 GHz to 61 GHz with 1 GHz steps and the
coverage is from 220 cm to 227 cm in major radius. The poloidal direction has a coverage
of 20 cm with 12 channels. The reflected field from FWR is combined with the MIR
receiver field to produce a synthetic diagnostic signal over the 2D imaging plane. The
synthetically measured 2D images are then compared with the EHO mode fluctuation, and
excellent matching between them is achieved.
Figure 7.4: The process of forward modeling MIR diagnostic output from the 3D MHD
code M3D-C1. Time-dependent output from M3D-C1 has been provided for shot # 157102
with the n = 3, 5 coherent EHO mode., are inserted as perturbed fields into the simulation
domain of FWR. The converged electric field pattern at the outgoing microwave boundary
is taken at each time step of the simulation and compared with the receiver antenna pattern
to produce a quadrature (amplitude and phase) signal timepoint [77].
7.3.2
Imaging
A series of adjacent synthetic images produced by the synthetic diagnostic is shown
in Fig. 7.5, with the upper row representing forward modeling results, and the lower row
168
representing plasma simulation output results. The probing frequency range from 55 to 61
GHz covers the top pedestal region where the peak of the fluctuation amplitude resides.
The 12 poloidally evenly spaced channels in each frequency/density channel have a
resolution of approximately 3.0 cm. Since the poloidal coverage is only 20 cm, and is much
less than the wavelength of the most destabilized coherent EHO modes (n <= 5), it is
challenging to resolve the long wavelengths (kpol ~ 0.1/cm) within this restricted window.
The 3D MHD code M3D-C1 is being used to model the QH-mode on DIII-D with the
presence of a low-n (≤ 5) EHO mode. M3D-C1 does not obtain a saturated n = 1 or n = 2
mode for shot # 157102; thus, the destabilized higher modes n = 3 and n = 5 are applied.
As shown on Fig. 7.5, by scaling the phase of the reflectometer signal (upper row)
appropriately with the density perturbation displacement (lower row), the two sequences
are nearly indistinguishable, indicating that the forward modeling accurately resolved the
input density fluctuations. The peak density fluctuation associated with the coherent EHO
fluctuation from M3D-C1 simulation is located at 2.26 m, while the measured maximum
fluctuation in the synthetic MIR signal is at 2.258 m, with only 2 mm difference. This
difference is due to the fact that the peak of the fluctuation is located between two probed
cutoff layers; thus, no data are collected from the peak fluctuation position.
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Figure 7.5: Synthetic images of phase perturbation produced by synthetic diagnostic (upper
row) and images of density perturbation displacement from M3D-C1 simulation output
(lower row). The n = 3 electron density fluctuation produced by M3D-C1 is shown at left.
The perturbation on the upper row is nearly indistinguishable with the lower row with
scaled phase and displacement perturbtion [77].
Figure 7.6 shows the comparison of the experimentally measured EHO (c), simulated
EHO from M3D-C1 (a) and forward modeling measured EHO mode structure (b). As seen
in this figure, the M3D-C1 simulation output is similar to the synthetic output for the
position of the mode, while the experimentally measured EHO didn’t exhibit a very clear
mode lobe, but the position of the mode is similar to the former two. Since the counter
propagation observed in MIR data measured near the top of pedestal could be due to either
imperfections of the instrument, or subtleties of the eigenmode structure, instrument
imperfections as well as partial non-uniform rotation of coherent EHO are investigated
with three different forward modeling configurations: linear EHO modes with ideal MIR
170
optics, linear EHO modes with mis-aligned MIR optics, and nonlinear EHO modes with
ideal MIR optics, as will be discussed below.
10
(a)
(c)
(b)
z (cm)
5
0
-5
-10
220 222 224 226
220 222 224 226
225
226
227
Major Radius (cm)
Figure 7.6: Comparison of the M3D-C1 output (a), forward modeling imaging (b) and
experimentally measured EHO mode structure (c). The M3D-C1 simulation output is
similar to the synthetic output for the position of the mode, while the experimentally
measured EHO didn’t show a very clear mode lobe, but the position of the mode is similar
to the former two.
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7.4 Analysis
7.4.1
Instrumental artifacts investigation
Linear modes, n = 3 and n = 5 destabilized from M3D-C1 simulation, with maximum
fluctuation amplitude magnified to the experimental level (around 5.2 mm, one free space
wavelength of the probing wave) are measured with MIR forward modeling with and
without optical misalignment. Figure 7.7 shows the wavenumber distribution over each
frequency/radial channel for mode n = 3. The two innermost radial channels, with smallest
fluctuation amplitude, show different wavenumbers than the other edge channels. This
could be due to the fact that the two innermost edge channels are located above the pedestal
region, where EHOs as an edge oscillation are not present; hence, the unclear instabilities
and subtleties above the pedestal could cause changed wavenumber compared to channels
located on or below the pedestal. Figure 7.7 (c) shows the wavenumber distribution when
transmitter and receiver are mis-aligned by 4 cm vertically. This number is chosen since
the real receiver and transmitter have an offset of around 3 cm experimentally. Compared
to the case when receiver and transmitter in Fig. 7.7 (b) are well aligned, the wavenumber
seems distributed more widely, but the peak value of the wavenumber is still measured
correctly. The spreading of wavenumber fails to cause a counter-propagation as observed
with the MIR data experimentally, suggesting that the small vertical displacement between
the receiver and transmitter doesn’t account for the observed counter-propagation.
172
Figure 7.7: Wavenumber distribution along the radial direction obtained from (a) M3D-C1
output, (b) forward modeling output with ideal optical alignment and (c) forward modeling
output with mis-aligned optics. This measurement becomes less certain when modest
optical misalignments are introduced, but the peak of the distribution remains centered on
the correct wavenumber [77].
7.4.2
Non-linear rotation investigation
Both M3D-C1 output and forward modeling yield a central wavenumber k around 6.6
rad/m for mode n = 3. The experimental measurement of the wavenumber for the dominant
n = 1 mode is 2.5 rad/m, which is around 1/3 times of that of n = 3. This means the n = 3
mode has similar phase velocity as that of the n = 1 mode. However, experimental
measurement of the n = 3 EHO component gives a local poloidal wavenumber of ~ 3.14
rad/m, which is less than that predicted by the forward modeling of M3D-C1 data. This
uncertainty, as well as the observed counter-propagation, may be due to the non-sinusoidal
nature of the density fluctuation in experiment, which is not inherent to the M3D-C1 result.
173
This has been partially explored by varying the time dependence of the oscillation of the
linear EHO mode. The fundamental magnetic flux displacement obtained from M3D-C1
output is nearly:
d(x ,t ) = a0 cos(ω t + ΔΦ(x)) with ΔΦ(x) = kx + ϕ 0
!
!
(7.1)
Higher fluctuation harmonics, with their amplitudes less than half of that of the
original n = 3 component, are added to generate a new non-linear mode structure:
d(x ,t ) = a0 cos(ω t + ΔΦ(x))+ a1 cos(2* ω t + 2* ΔΦ(x))
+ a2 cos(3* ω t + 3* ΔΦ(x))+ ⋅⋅⋅+ a6 cos(7* ω t + 7* ΔΦ(x))
with
a ≥ 2* a1 ≥ ⋅⋅⋅ ≥ 26 * a6
!0
(7.2)
Figure 7.8: Wavenumber distributions of linear (a) and nonlinear (b) time-denpent
fluctuations. (c) and (d) shows the corresponding wavenumber distribution from forward
modeling for the dominant and 2nd harmonic of n = 3, while both yield the correct central
wavenumber [77].
174
Figure 7.8 (a) and (b) shows the mode structure for both sinusoidal and
nonlinear/non-sinusoidal perturbations. Forward modeling measurements for the nonsinusoidal case suggest that harmonic wavenumbers are also diagnosed correctly, as
highlighted in Figs. 7.8 (c) and (d). This result suggests that the non-uniform rotation of
the EHO mode is not the source of the counter-propagation. Since the experimental
fluctuation has a dominant mode number of n = 1, which is not destabilized with M3D-C1
simulations for shot 157102, more realistic mode structures are needed to eliminate nonlinearity as a reason for counter-propagation. Further exploration with FWR3D is also
warranted to determine more remaining uncertainties.
7.4.3
Possible explanation of counter-propagation
Since the forward modeling indicates that neither the measured vertical
displacement between the transmitting and receiving antennas nor the non-uniform rotation
of EHO modes account for the measured counter-propagation, we are led to speculate as
to whether the counter propagation is a similar diagnostic artifact as seen in ECEI data
[163], which is known to correspond with a fluctuation phase inversion due to magnetic
islands. This weak tearing parity has been overlooked in the past, but a similar feature has
now been observed in BES and other ECE data, as shown in Fig. 7.9. The MIR artifact
signal may be due to the similar phase inversion in density fluctuations caused by EHO
near the top of pedestal.
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Figure 7.9: Phase inversion observed in both ECE and BES data. As the coherent EHO
mode (n = 1) decreasing towards into plasma, before its disappearance, an obvious phase
inversion is captured for shot # 157102, and the similar reason could cause the counterpropagation seen in MIR data [77].
7.5
Conclusion
MIR on DIII-D measures detailed characteristics concerning EHO, and helps to
explain how EHO modes are destabilized, how they saturate, and by what mechanisms they
limit the evolution of the H-mode pedestal. Both optical displacement and non-uniform
EHO rotation are investigated as possible instrumental artifact sources using the
forwarding modeling of MIR, but they do not reproduce the observed effects of counter
propagating harmonic behavior. Excluding these possible explanations, the apparent
counter-rotation of the mode structure near the top of the pedestal could in fact be due to
176
phase/radius inversion of the EHO at that location that is consistent with observations from
ECE-Imaging and BES. More sophisticated, nonlinear models, applying both FWR2D and
more advanced FWR3D will be applied to explore remaining uncertainties.
177
Chapter 8: Continuing Work and Conclusions
8.1 Future Synthetic MIR work
8.1.1
FWR3D
Currently, most FWR simulations are performed with the 2D version, since it has
better stability and has been developed for around 10 years [109]. The new FWR3D version
[112] is still in the process of debugging and optimization, including the addition of more
features and dealing with more sophisticated physics. Compared to FWR2D, FWR3D has
more difficulties in solving the boundary conditions. Just like FWR2D, FWR3D also first
propagates the beam from the antenna plane to the vicinity of the reflection layer using the
paraxial approximation, as described in Chap. 3. To solve for the boundary condition
between the paraxial region and full-wave region, FWR3D defines two surfaces, which are
a quarter of a free space wavelength apart, close to the paraxial region, but inside the fullwave region. The code then propagates the field still using the paraxial approximation to
the two surfaces. The two surfaces then are treated as two sources inside the full-wave
solver, with their back propagation canceled to ensure that no beam is propagated back to
the paraxial region; thus, the field can then be set to zero on the calculation boundary. The
reflected field on one of the two surfaces is then propagated back through the paraxial
region and up to the detector plane. The computation requires large memory space and is
also processor time consuming. To make the simulation finish within a time comparable to
that of the FWR2D version and also make full usage of available free memory, a message
passing interface (MPI), which is a standard library for defining message passing protocols
178
for shared/distributed memory systems, is adopted; MPI combined with parallel processing
dramatically reduces the overall time for finishing an FWR3D simulation.
The current FWR3D code is benchmarked with FWR2D by simply extending the 2D
profile toroidally into 3D without adding any new features to the third dimension. Since
FWR3D is becoming more and more mature, synthetic simulations in the future will be
launched mostly with FWR3D, which deals with more realistic discharge situations. Figure
8.1 shows an initial comparison between the FWR2D and FWR3D results. The 3D version
takes account of the 3D magnetic field effects and perturbations. Shown in the figure is the
coherent coefficient g:
g≡
!
E!
E!
2
Here, !E! is the reflected signal and !g is a measure of the fluctuation level. If g → 0 ,
!
it means large fluctuations; while if g → 1 , it means small perturbations. From Fig. 8.1,
!
the 3D results have smaller perturbations compared to the 2D case with the same
fluctuation amplitude, as seen in the red and cyan curves in this figure. The fluctuations
are from XGC-1 [164] simulations with NSTX [165] equilibrium profiles. Simulation
results from our collaborators from NSTX indicated that fluctuation levels scales from
FWR3D along the measured frequency are closer to the experimentally measured
fluctuation levels, as compared to FWR2D [166]. More detailed comparison between the
FWR3D and FWR2D synthetic simulations results is under investigation, in order to
determine more detailed differences between them. The blue and green curves are two
179
cases of FWR2D results with different amplitudes, and their positions on Fig. 8.1 show
that the coherent coefficient g is indicating the right scale of fluctuation levels.
Figure 8.1: Comparison of the coherent signal from FWR2D and FWR3D. The 3D results
have smaller perturbations generally compared to the 2D case under the same fluctuation
amplitude assumption, and are more closely matched with experimentally measured results.
In addition, in FWR3D, the magnetic field will not be symmetric along the toroidal
direction, but will rotate like a helix. Due to the introduction of this 3D rotation of the
magnetic field as well as its perturbation in FWR3D, the prescribed wave polarization may
change inside the plasma. To account for this feature, the probing beam field in the plasma
is decomposed into extraordinary (polarized perpendicular to the magnetic field) and
180
ordinary (polarized parallel to the magnetic field) modes. The two modes are then treated
separately inside the plasma and combined afterwards; then, the component that has the
same polarization as specified in the input file is returned. In FWR2D, since the magnetic
field is constant along the Z direction, wave polarization is conserved. A more complicated
3D dielectric tensor is introduced and used for solving the E-field equations. Wave
absorption due to the nearby upper-hybrid layer is still not considered in FWR3D.
8.1.2
QH-mode studies
MIR measures the detailed spectrum of the edge-localized fluctuations for QH-mode,
including EHOs and broadband fluctuations, and can be a valuable tool for interpreting the
edge transport physics of QH-modes. Figure 8.2 shows the measured dispersion
relationship for shot # 157102 with both the presence of coherent EHOs and broadband
MHD fluctuations. Clearly seen in this figure, the coherent EHOs and broadband
fluctuations are propagating in opposite directions, suggesting that they are different
fluctuations, and this has been proven with more evidence from MIR as well as other
diagnostic tools. QH-modes sustained with broadband are more attractive, since they
provide more space for increasing the edge density profile with wider pedestal, and are
currently under wide investigation [167].
181
Figure 8.2: Shown is the measured spectrum from MIR with the presence of both coherent
EHOs and broadband fluctuations. Clearly seen in this figure, the two fluctuations are
propagating in different directions, suggesting they are different instabilities. [Courtesy of
Dr. Benjamin Tobias]
Since MIR is playing an important role for analyzing the physics for both broadband
and coherent EHO fluctuations, synthetic MIR simulations, which could help to interpret
and validate the theoretical hypothesis of edge transport theories of QH-mode are also
gaining in importance. Currently, synthetic MIR is being applied to image simulated
fluctuations from both the M3D-C1 and JOREK [169] codes, and will be, with the
references imaging results, compared with experimentally measured results. We are
collaborating with Dr. Xi Chen from General Atomics to investigate rotational shear effects
on EHOs. Dr. Chen has used M3D-C1 to simulate n = 1 to n = 7 EHO modes for shot
157102 with the inclusion of a resistive wall as well as rotational shear. For the resistive
effects, from her simulation, it is found that increased resistivity will increase the growth
rate, but not affect the mode structure. With the rotational shear effects, it can be found that
182
the modes will be slightly elongated in the poloidal direction. We are also collaborating
with Dr. Feng Liu from the ITER group about using forward modeling to study EHO modes
with JOREK [168], [169] nonlinear simulation results. JOREK is a nonlinear MHD code
developed with the aim of studying the nonlinear evolution of the MHD instabilities in full
toroidal X-point geometry. For the first time, the nonlinear MHD simulation of DIII-D QHmodes (shot 145117 and 153440) was carried out by the ITER team. Both ideal and
resistive wall boundary conditions for low toroidal number modes, the influence of toroidal
rotation as well as the effect of the vacuum vessel in the destabilization and saturation of
edge modes for DIII-D QH-modes have been studied. It also demonstrated that there is
indeed nonlinear coupling of n number.
Compared to JOREK, M3D-C1 is more favorable for the investigation of the
rotational shear on edge fluctuations of QH-modes. In JOREK simulations, the toroidal
rotation is approximated as a simple parallel rotation; thus, the linear growth rate is not
significantly changed with respect to the toroidal rotation shear. However, it has been
experimentally confirmed that the toroidal rotation is of great significance and cannot be
simply approximated as parallel rotation. However, JOREK can generate non-linear results
compared M3D-C1, which currently can only generate meaningful linear results. Figure
8.3 shows one example of the EHO simulation results from both M3D-C1 and JOREK.
183
Density fluctuation
Normalized density fluctuation
M3D-C1 # 157102
JOREK # 145117
1
1
0.5
Z/m
Z/m
0.5
0
0
-0.5
-0.5
-1
-1
1
1.5
R/m
2
1.2 1.4 1.6 1.8 2.0 2.2
R/m
Figure 8.3: Comparison of the linear M3D-C1 EHO mode simulation (left) with non-linear
JOREK EHO simulations (right).
8.1.3
Alfven wave studies
Alfven wave diagnostics still remains an important aspect for future MIR work.
These instabilities can be excited by fusion-generated alpha particles, or other confined
fast-ions during plasma heating, and then enhance particle transport or loss, and modulate
plasma pressure and current distribution. Energetic particles, which are led out of their
original confined orbits by gaining energy from Alfven instabilities, could also damage the
first wall components, which presents a critical issue for sustainable operation of advanced
devices. ECE-Imaging has been proven to be a powerful tool for measuring the 2D poloidal
184
mode structure of low frequency modes in the plasma. For instance, its clearly measured
Fast-ion induced Alfven eigenmode mode structure is the first observation from
experiments of how the fast-ion population would influence the mode structure, and it
agreed closely with predictions from simulation codes [145]. Figure 8.4 shows the
comparison of the simulated Fast-ion influence on mode structure with the experimentally
measured mode structure obtained by ECEI [145]. Figure 8.4 (a) is the mode structure for
n=4 reversed sheared Alfven eigenmode (RSAE) predicted from ideal MHD
approximation, which excludes the influence of fast ions on RSAE. Figure 8.4 (b) is the
simulated RSAE mode structure generated from a reduced MHD mode, including the
influence of fast ions on RSAE. On this figure, there is a clear outward spiraling or radial
shearing in the direction of the ion diamagnetic drift. This predicted mode structure
matched closely with that measured from ECEI data, as shown in Fig. 8.4 (c).
Quantification and understanding of this shearing could help to better understand the
interactions between the fast ion population, say alpha particles, and Alfven eigenmodes,
especially for future advanced devices, such as ITER.
185
Figure 8.4: Shown is the simulated mode structure for n = 4 RSAE, as well as the mode
structure measured by ECEI. (a) Simulated mode structure with ideal MHD
approximations, not taking the fast ion effects into account; (b) simulated mode structure
with reduced MHD and nonperturbative models, including effects on mode structure by
fast ions; (c) measured RSAE mode structure from ECE-Imaging. [Courtesy of Dr.
Benjamin Tobias]
MIR is also capable of providing the detailed mode structure of Alfven eigenmodes.
Since ECE-Imaging has issues near the pedestal and edge where the plasma is optically
thin and signals could be contaminated by reflections from the wall, the development of
MIR to diagnose Alfven modes near this region is consequently of critical importance.
186
MIR has shown its capability to measure Alfven instabilities for L mode conditions, as
shown in Fig. 8.5, which is also shown in Chap. 6. In this figure, it is the first experimental
campaign from MIR measuring up-chirping RSAEs and stable TAEs. The RSAE gradually
merges with TAE to cause stronger instabilities. Combined with synthetic MIR, MIR could
help to validate these instability models from simulations. Since MIR generally performs
better in H-mode diagnostic studies than for L-mode, we would speculate that it would
perform better for diagnosing Alfven modes for H-modes.
Figure 8.5. Demonstration of MIR measured RSAE and TAE instabilities during a L-mode.
(a) is the measured spectra; (b), (c) and (d) shows the upchirping behavior of RSAE, which
at last merges with TAE to form a larger instability. [Courtesy of Benjamin Tobias]
8.1.4
FWR interface with simulation codes
UC Davis is also collaborating with Dr. Lei Shi from PPPL to develop a fully
integrated interface between the reflectometry codes FWR2/3D and a number of other
plasma simulation codes including XGC-1, GTC, GTC, M3D-C1, and JOREK.This would
provide great convenience to provide synthetic imaging data from any diagnostic simulated
plasma turbulence. Figure 8.6 shows how such a synthetic diagnostic helps to validate
plasma simulation codes through the usage of indirect comparison between diagnostic
signals. The turbulent instabilities in real plasmas are not known a priori, while they
187
influence the measured diagnostic signals, say the voltage measured by ECE-Imaging and
the phase measured by MIR. Synthetic diagnostics simulate the diagnostic response from
diagnosing the plasma from simulation codes. The synthetic diagnostic signal can be
directly compared to the real diagnostic signal, thus indicating if the simulated plasma (or
plasma modes) could be a valid representation of the real plasma (or plasma modes). The
difficulty of developing such an interface lies in the diverse nature of each code’s variable
systems as well as programming languages. Different plasma simulation software codes
are developed independently by different institutes, and they would have different
programming preferences, causing the compatibility to be a major problem, since the
interface, mainly developed by PPPL so far, is written in Python. In addition, due to
different purposes of the different codes, the output format and variables from each code
are also different; some of them even do not have a fixed output format and will not
automatically output needed information for synthetic diagnostic purposes. Currently,
PPPL has developed a standard library for interfacing XGC-1 and GTS with reflectometry
codes, and is adding GTC to the plasma code group. The UC Davis group is investigating
the M3D-C1 and JOREK format, and are trying to integrate them into the interface library.
188
Figure 8.6: Shown is the schematic of how the synthetic diagnostic helps to validate plasma
simulation codes through “indirect comparison”. Building up a synthetic diagnostic
interface integrating a number of simulation codes is needed for convenient usage of such
an approach. [Modified from Dr. Lei Shi’s schematic]
8.2 Hardware upgrades
8.2.1
CMOS-based transmitter
The current multi-frequency MIR transmitter used on DIII-D has a significant
limitation on the power that can be generated at each frequency. The waveguide amplifier
approach delivers broad frequency coverage and agile tuning capability, but with low
output power at a large number of simultaneous output frequencies. The maximum output
power for the current power amplifier is 15 dBm to ensure linear operation with four input
frequencies. Consequently, the maximum power generated per frequency (total of 4
frequencies) for the current system is 2 mW. The output power would drop to 0.5 mW with
8 output frequencies. Thus, if the current MIR system were to be upgraded to even more
189
probing frequencies, the power per frequency would be too low to have a tolerable signalto-noise ratio (SNR) from the reflected wave.
A CMOS-based multi-frequency transmitter is under development at UC Davis and
is to be employed for MIR upgrades. The new “system-on-chip” transmitter is being
developed by fellow student Yu-Ting Chang and will work over a wide frequency range
from 56 GHz to 74 GHz [170]. It will employ multiple mixers/amplifiers together with full
band power combiners to boost the output power per frequency. The difficulty for
designing such a system lies in the considerable complexity and large area demands due to
the number of up-converting mixers, power amplifiers, and power combiners required.
Moreover, the power dividers and amplifiers are also required to split the LO signal power
and pump the up-converting mixers. However, the tremendous advances in the
development of CMOS technology makes it possible to integrate circuits on a single chip,
miniaturize circuit designs, reduce manufacturing costs, and enable system-wide
improvements of generating the 8 or 16 output frequencies. Figure 8.7 shows one example
of the on-chip amplifier chain with 3 stages. This would be only one element for the overall
transmitter on-chip system, and the transmitter under development would be much more
complicated with more mixers, amplifiers and power dividers. The new transmitter would
be optimized for the future MIR systems, with the advantages of: high output power, ultra
wideband operation, ultra low-noise, protected and highly stable performance.
190
Figure 8.7: One example of on-chip amplifier chain with several stages. The transmitter
under development would be more complicated. [Courtesy of Yu-ting Chang]
8.2.2
Phased antenna array
The current transmitter utilizes a rectangular horn at V band (50-75 GHz) to
transmit the power into the plasma. Since the Gaussian beam approximation pattern for
such a horn is fixed, there is no flexibility to control the wave-front from the source. This
kind of configuration is not desirable, since it doesn’t provide timely tuning of the wavefront as the plasma cutoff surface moves and changes, and also cannot provide good
curvature matching for all the probing frequencies.
Considering the more sophisticated needs for a transmitter, a more advanced phased
antenna array (PAA) based MIR system is under development, with tunable wave-pattern
from the source. A phased antenna array system typically consists of a large number of
radiating elements, which have been arranged in a rectangular or triangular board, with
each radiating element including a phase shifter or a time delay device. A phased antenna
array is shown in Fig. 8.8 (a). Beams are steered by sequentially shifting the phase of the
signal emitted from each radiating element, to provide constructive/destructive
191
interference, so as to steer the beams in the desired direction or phase front curvature as
shown schematically in Fig. 8.8 (a) [171], [172]. The operation bandwidth of the PAA
system is limited by the bandwidth of all the radiating elements, especially the phase
shifters or delay devices.
Fig 8.8: (a) Schematic of a beam shaping phased antenna array based on curved time delay
arrangement, (b) One designed sample of phased antenna array with 8 elements.
A PAA instead of a simple horn can generate various different Gaussian beams to
satisfy the harsh transmitting requirements at the limitations of lens system design and its
dimensions. Moreover, the delayed/shifted phase can be controlled through voltage, and
thus enables remotely and electronically controlling the transmitter beam shape. This could
then help to make the adjustment of the lenses easier, and can also provide a simultaneous
phase front following the variation of the plasma discharge. The phased antenna array
would be combined with the CMOS-based system-on-chip transmitter to form an overall
transmitter system.
192
8.2.3 Digital Beam-Forming and On-Chip Module
For the current mini-lens based receiver system, motorized translation stages are used
to adjust the position of the lenses to achieve the desired focusing. However, such an
adjustment is time consuming and not efficient to change the focusing during a single shot,
since each shot will last for at most 6 s. To compensate for such a disadvantage, digital
beam-forming which helps to focus electronically is under development. With digitalbeam-forming technology, the beam would be focused digitally by adding compensation
phase to the measured IQ signal. This is a similar approach to the numerical focusing as
discussed in Chap. 3, but is done with electronic components and can be done in real-time
within a plasma discharge. The basic idea for such an approach is to add the in-phase and
quadrature signals from the current/upgraded receiver system, which was originally used
to obtain the phase variation as the density fluctuation signal, to a field-programmable gate
array (FPGA), and program the FPGA to add the needed compensated phase. Thus, the
difficulty of implementing such a system lies in how to determine the needed compensation
phase for the measured signal. The amount of needed phase can be calculated from optical
simulations. Figure 8.9 shows the schematic for the current beam-forming based receiver.
Compared with the current receiver, it adds a pre-amp before the first stage of mixer, which
is not applied in the current system. Each signal from each receiver is divided and evenly
distributed to each FPGA module.
193
Antenna
Array
...
PA
Ref1
...
4-6MHz BP
4-6MHz BP
4-6MHz BP
...
4-6MHz BP
Pre-Amp
Ch1-2
Chn-1
Ch2-2
Chn-2
PA
...
...
...
Ref1
4-6MHz BP
Pre-Amp
Ch2-1
PA
...
...
4-6MHz BP
4-6MHz BP
FPGA Beamformer
Pre-Amp
4-6MHz BP
Ch1-1
FPGA Beamformer
Power
Divider
FPGA Beamformer
Ref1
4-6MHz BP
Ch1-m
Ch2-m
Chn-m
Figure 8.9: Schematic of the digital beam-forming based receiver system. Each FPGA
module corresponds to one mini-lens receiver. [Modified from Fengqi Hu’s schematic]
In addition, the current receiver electronic modules suffer from considerable
environmental noise, including noise from the RF heating system, voltage spikes, radiation
“bursts” from ELMs, and mechanical vibrations. In addition, optical coupling and high
gain after the receiver antenna provides high sensitivity to instability. Thus, the system
needs ‘clean’ power and good electrical isolation to reduce the signal-to-noise ratio.
Moreover, coupling in free space also requires good alignment and mechanical isolation.
A more advanced robust and compact monolithic microwave integrated circuit (MMIC)
based system is under development in a UC Davis/PPPL collaboration. It would be
packaged in multilayer liquid crystal polymer (LCP [173], [174]), with this substrate
chosen for its excellent electrical and superior mechanical properties. This LCP package
194
will replace the heavy and bulky microwave modules. The new circuits would have overall
reduced noise temperature, plentiful LO power, and good electrical isolation due to the
hermetically sealed package. The LO signal can also be stabilized by drive “on board”, and
low frequency noise can also be eliminated.
8.2.4
EAST MIR
A next generation MIR system is to be installed on the EAST tokamak in China. This
new MIR would have more sophisticated optical design and optimized electronic module
design. For the optical part, non-spherical lenses, such as hyperbolic and parabolic lenses,
are used to provide more flexibility for curvature matching and depth-of-field adjustment.
The optical system designed for EAST will have decoupled adjustment capability for three
main optical parameters: image plane curvature, image plane radial location, and size of
image plane (smaller means higher resolution, larger means worse resolution) [103], [104].
This means that the three parameters can be changed independently, and this property is
not true for the installed DIII-D MIR system. The antenna array is back to a planar array
again, with all the field adjustment realized with translation of lenses. Figure 8.10 shows
the simple diagram for the designed EAST optical system designed by Dr. Yilun Zhu from
the University of Science and Technology of China (USTC).
195
Figure 8.10: The EAST MIR optical system designed by Dr. Yilun Zhu from USTC
[Courtesy of Dr. Yilun Zhu].
Compared to the initial DIII-D MIR system [91], instead of using one high frequency
Gunn oscillator, the new system would use low frequency synthesizers combined with
multipliers to fulfill the input frequency range requirement; by doing so, the probing
frequency range would have no gaps (since the initial DIII-D MIR had a range of
frequencies around the LO frequency of 65 GHz which cannot be accessed), and the
extended eight probing frequencies could be chosen with more freedom (they don’t have
to be symmetric around 65 GHz).
In addition, the RF and IF parts will be embedded on
different boards, providing extra spacing for heat sinks, isolation, and noise reduction.
Figure 8.10 shows the RF and IF broad design for the EAST MIR system designed by Dr.
Alex Spear from UC Davis.
The current EAST modules and optical system are under fabrication and testing, and
are to be installed on EAST in Spring, 2016.
196
Figure 8.10: Modules for EAST MIR, with the RF and IF parts separated on two boards,
designed by Dr. Alex Spear from UC Davis [Courtesy of Dr. Alex Spear].
8.3
Conclusion
This dissertation has introduced the needs to diagnose micro-scale and MHD
turbulence in plasma, as well as the two diagnostic tools that are developed by UC Davis
millimeter group for detecting plasma turbulence: the passive ECEI system for detecting
temperature fluctuations over a 2D poloidal plane, and the active MIR system for detecting
density fluctuations in a similar 2D poloidal plane. The two approaches share the same
device port and part of the optical design, and provide simultaneously measuring of density
and temperature turbulence.
The dissertation then focuses on the development of the MIR system, and more
specifically, on the synthetic approach of MIR diagnostics. The synthetic MIR approaches
takes real plasma equilibrium profiles combined with plasma turbulence from either
simulation codes or other turbulence diagnostic tools as the plasma input; it then propagates
the beam field from the real designed optics into plasma by the usage of the code FWR2/3D,
197
a sophisticated reflectometry code for simulating wave interaction with plasma; the
reflected field obtained from the simulation results of FWR2/3D is then combined with the
receiver field from the real designed receiver system to generate the synthetic MIR
diagnostic signal. This approach is firstly used to optimize the design aspects of the MIR
system, as well as determine the possible operating space of a MIR system. With the help
of the synthetic MIR diagnostic, the MIR system was finally fabricated and installed on
the DIII-D tokamak. Measurement results from MIR for characterization discharges are
compared to synthetic MIR diagnostic results, showing great agreement, and such a process
also discovered a 5 cm vertical displacement of the receiver with respect to the plasma midplane. During its initial experimental campaign, MIR achieved great agreement for
measuring detailed inter-ELM modes when compared to measurements from other
diagnostic tools: interferometer, ECE radiation, and Mirnov coils. Simultaneously
measured mode structures from ECEI and MIR for one inter-ELM mode are also compared.
MIR also showed its capacities for measuring Alfven eigenmodes.
In 2015, MIR on DIII-D was extensively used for imaging edge modes for H-mode
(QH-modes). While such MIR data measured interesting features of EHO modes during an
QH-mode with counter-propagation at pedestal top region, this feature needs to be better
understood. The possible sources for observing such a feature could come from
instrumental artifacts of MIR, say, optical misalignments and beam scattering. The
synthetic method of MIR analysis is used to investigate possible instrumental artifacts.
Advanced M3D-C1 simulations generate saturated EHO modes with experimentally
calibrated equilibrium plasma profiles. The synthetic imaging data are then directly
compared to the input plasma profile, where great agreement between their spectra is
198
achieved. This result indicates that the experimentally measured counter-propagation,
which is not observed by other external measurement tools, such as Mirnov Coils, does not
result from MIR instrument artifacts. It might be due to subtleties that are located on the
pedestal top which are not accessible to external measurement tools, and needs further
explanations, say varying the propagation mode structure.
In the future, synthetic MIR is to be implemented more completely with FWR3D,
which takes the toroidal magnetic field pitch into account and introduces toroidal variations
compared to FWR2D. A synthetic MIR interface with a wide range of plasma simulation
codes is also under development in collaboration with PPPL, and this could help to validate
the simulation model by directly comparing experimental measured MIR results with
synthetic MIR diagnostic results. The MIR hardware is also to be upgraded to allow for
better signal-to-noise ratio, and tunable flexibilities. In the short term, the CMOS-based
on-chip amplifier and mixer, as well as a robust and compact monolithic microwave
integrated circuit (MMIC) based system packaged in multilayer liquid crystal polymer
would be applied to provide high-transmitted power and improve the receiver system
signal-to-noise ratio. In the longer term, a phased antenna array would also be implemented
on the transmitter for fast steering of the output beam wave-front. Digital beam-forming
technology will also replace the current mini-lens receiver array, to digitally realize
focusing of the reflected beam for better accuracy.
199
Appendix I
Introduction to electron cyclotron emission
imaging (ECEI)
The ECEI system is a passive radiometric detection system, detecting the electron
temperature fluctuation over a 2D poloidal plane, with its principles briefly described here.
To ensure that emission at the electron cyclotron frequency can be observed by an external
detector, the radiation has to come out from the plasma freely without encountering a
resonance or reflection layer, exemplified in Fig. 2.1, the first harmonic is generally the
most accessible of the harmonics. The first harmonic of
ωce is below the upper hybrid
resonant layer and right-handed cutoff layer, and thus cannot be detected by contemporary
ECE systems.
Due to the existence of resonant and reflection layers above the fundamental
harmonic of
ωce , the second harmonic is chosen as the preferred radiation frequency to
be detected by external diagnostic devices, such as ECE radiometers [175] [176] and ECEI.
Since the second harmonic is within the millimeter wave range, from 90 GHz to 125 GHz
for DIIII-D ECEI system, it suffers less damping when radiated out of plasma than higher
harmonics, and would have more matured industry components ready to use.
The radiation intensity arriving at the observation point is determined by [177]:
I(ω ,T ) = I B (ω ,T )(1− e −τ )
(I.1)
!
200
where ω is the observed frequency,
source, and
!T is the local temperature of the radiation
is the optical thickness [178]. Here, I B (ω ,T ) indicates the radiation
!
τ
intensity from a black-body, and it can be expressed as:
2hω 3
1
I B (ω ,T ) = 2 hω /kT
c e
−1 (I.2)
!
In the above, !h is the Planck’s constant, and is 4.135667662(25) × 10−15 !ev ⋅ s .
The electron cyclotron frequency is on the order of 100 GHz, in a Tokamak with the Bfield on the order of 2 T; thus, hω (~ 10-4 eV) represents an energy much less than the
plasma temperature (!kT ~1 keV), and the observed radiation intensity can be
approximated as:
2ω 2
I(ω ,T ) = 2 kT(1 − e −τ )
c
(I.3)
To retain an expression for the observed radiation intensity that depends only on
the observed frequency and the electron temperature, the plasma must appear as a blackbody source, where radiation emanates from an optically thick region of the plasma.
Here, optical thickness is the natural logarithm of the ratio of the incident to transmitted
radiant power through a material:
⎛ φi ⎞
τ = ln ⎜ t ⎟ = −lnT
⎝φ ⎠
(I.4)
with φ the radiant flux transmitted by the material, and φ is the radiant flux
i
t
received by that material, and T is the transmittance of that material. In other places in
this dissertation or this appendix, T means temperature.
201
Optical thickness of the second harmonic of X-mode ECE and the fundamental
frequency of O-mode is shown in Fig. I.1. With τ ≫1 , equation (I.3) can be simplified
into:
I(ω ,T ) =
2ω 2
kT (I.5)
c2
Since the optical thickness of the third harmonic is thinner than the second
harmonic for X-mode waves, the third harmonic is not chosen as the preferred frequency
range for detection. The measured intensity with fluctuation I! thus corresponds to the local
temperature fluctuation T! within regions where the plasma is optically thick.
Figure I.1: Optical thicknesses, τ, for the ordinary and extraordinary mode at the
fundamental and first harmonic. This plot is generated from the equilibrium profile of
DIII-D shot 157102 at time 2420 ms. This shot is an example of an H-mode without
ELMs.
202
The extraordinary mode evaluated around 110 GHz exhibits excellent optical thickness
through most of the plasma and emission is as from a black-body.
Conventional ECE radiometry has been applied to measure plasma electron
temperature along the radial direction for over thirty years, and it also provides reliable
temperature fluctuation measurements with proper calibration [175] [176]. ECE-Imaging
is most like an extended ECE radiometry, which covers a 2D imaging array on the poloidal
plane. Figure I.2 shows a simple schematic of the ECEI system with a 1-D detector antenna
array. The detected high frequency of the 2nd harmonic electron cyclotron radiation is then
down converted with the use of a local oscillator to reveal the temperature information. Its
individual antenna imaging volume is reduced to around 1 cm-3.
Figure I.2: Schematic of the 2D ECE-Imaging
The UC Davis group has developed a series of ECEI systems for several major tokamaks
around the world, and they are employed on DIII-D [179], KSTAR [180], EAST, HL-2A
[181], TEXTOR [182], ASDEX-Upgrade [183], and HT-7 [184]. ECEI has been widely
203
used for plasma physics studies, including Alfven eigenmodes, tearing modes, sawtooth
instabilities, and edge localized modes. A magnetic reconnection event associated with
sawtooth crash was observed by the ECEI system on TEXTOR [185], and for the first time,
showed that the sawtooth crash may occur at the high field side.
204
Appendix II
Previous MIR devices
There are four MIR systems prior to the DIII-D MIR (this work for DIII-D will be
the fifth MIR, with a sixth for EAST under development) installed on different magnetic
confinement devices around the world. One was installed on TEXTOR in Germany [186]
operating from 2002 to 2008, one is installed on LHD in Japan [187], one is installed on
KSTAR in Korea [188], one is installed on a reversed pinch machine TPE-RX [189], and
another one was developed for HL-2A [189].
In Fig. II.1, shown is the combined proof-of-concept ECEI/MIR system previously
installed on TEXTOR. In this apparatus, the transmitter beam is fixed at 88 GHz
illuminating a poloidal range of 10 cm; translatable combination of lens and mirrors is
applied to adjust the focus to match the location of the cutoff; a wide aperture optical
system is used for collecting the reflected waves and forming an image of the plasma cutoff
layer onto a linear array of 16 detectors. Data were gathered and analyzed to provide
information concerning density fluctuations. Figure II.2 shows some results from the proofof-principle tests. The probing frequency and imaging focal plane are held fixed, while the
density ne is ramped up. The cutoff surface was swept through and beyond the focal plane
of the optics, with phase and amplitude modulation depicted by polar plot with amplitude
and phase as the coordinates plot as shown in Fig. II.2 (a) - (d). The plots of (a) and (d) are
from off-focal positions of the lens system (one is with low density while another is with
over-dense density), while (b) and (c) are in-focus, showing that amplitude modulation on
the IQ plots are minimized when the cutoff is in focus, demonstrating the possibility of
imaging of density perturbations. The spectrum shown in Fig II. 3 (a) is from off-focus and
205
gives no specific fluctuation wavenumber information, even though it could results from
broadband fluctuation with MIR in focus; in contrast, the spectrum of Fig. II. 3 (b), from
an in-focal position, give some obvious peak wavenumbers that correspond to the
fluctuation wavenumbers on the cutoff surface. The results show that MIR is capable of
measuring density fluctuations, and that measuring at the in-focus position is of great
importance.
Figure II.1: Combined proof-of-concept ECEI/MIR system previously installed on
TEXTOR in 2002 and operated through 2008. The MIR system shares two front-end
mirrors and the port window with ECEI. [Reprinted figure with permission from Ref.
184. Copyright 2003, AIP Publishing LLC]
206
Fig II.2: IQ plots of a single MIR channel during a density ramp in an Ohmic discharge.
[Reprinted figure with permission from Ref. 184. Copyright 2003, AIP Publishing LLC]
(a)
(b)
Figure II.3: Phase power spectra from off-focus position and in-focus position.
[Reprinted figure with permission from Ref. 184. Copyright 2003, AIP Publishing LLC]
An improved MIR system compared to the TEXTOR one is installed on the
KSTAR tokamak. Compared to the 1st generation of MIR on TEXTOR, which applied one
fixed probing frequency and reflective lenses, the KSTAR MIR system [187] has two
207
major probing frequencies, and uses refractive lenses to provide more flexibility. Due to
figure permission problem, the KSTAR MIR system is not shown here, but can be referred
in Ref. [Lee, Woochang, et al. "Microwave imaging reflectometry system for
KSTAR."Plasma and Fusion Research 6 (2011): 2402037-2402037]. The design of the
KSTAR MIR system suggested that extreme care is needed to design sophisticated optical
system taking care of the optical aberration problems. Compared to the TEXTOR MIR
system which applied mostly mirrors, the KSTAR MIR system mostly employed refractive
lenses. Large mirrors with precise curvature are expensive and difficult to manufacture. In
addition, the reflective approach of the configuration could introduce aberrations on the
image plane. Refractive lenses could induce more flexibility, however, the standing waves
between two close surfaces need to be treated properly. The KSTAR MIR system is
combined with the low field side ECEI and shared the same front-end optics together with
multi-frequency illumination.
The MIR system installed on the DIII-D tokamak is the main topic of this dissertation.
It has 4 discrete but tunable probing frequencies ranging from 56 – 74 GHz and 20 cm
poloidal coverage. With the assistance of synthetic MIR simulations, the DIII-D MIR
system was designed with optimized optical parameters, and also has an expectation of
how well it would perform under different plasma conditions.
In addition, there are many
improvements in hardware applied since the TEXTOR MIR system. The next generation
of MIR system to be installed on EAST in China is under development, with 8
simultaneous probing frequencies and more advanced optical design and coupling. Figure
208
II.4 shows the evolution of MIR system for density turbulence measurement for tokamak
devices.
Figure II.4: Evolution of MIR system for measuring density turbulence on Tokamak
devices.
In addition to tokamak devices, MIR is also implemented on other magnetic
confinement fusion devices, including the stellarator and reversed field pinch (RFP) [191].
One of them is the combined MIR/ECEI system on LHD [187]. This system measures 3D ne fluctuations, with 7 poloidal, 5 toroidal channels, and four simultaneous probing
frequencies (60.4, 61.8, 64, and 64.6 GHz). The imaging optics have a movable mainmirror to change focal position. Due to the twisted nature of the plasma in a stellarator, the
illumination beam is hard to be reflected in the same direction. The optimum region of the
main mirror angle is very narrow, with a sensitivity of about 1 degree. Another one is the
209
MIR system installed on RFP-RX [189], with one single O-mode illumination frequency
and also mirrors. The imaging detector is comprised of a 4 X 4 planar Yagi antenna array
[194], with a spatial resolution of 3.7 cm in both poloidal and toroidal directions [192].
MIR on TPE-RX is the first demonstration of MIR as a turbulence diagnostic, showing that
MIR is capable of measuring the motion of the cutoff surface with the condition
4k⊥ dL / D < 1 . Here, L , D , k⊥ and d are distance between cutoff and lens, diameter
of the lens, perpendicular wavenumber, and radial displacement of the cutoff surface,
respectively.
210
Appendix III
Manual on output of FWR2D
The complex field amplitude from the full-wave solver, the paraxial solver, and
free space solver are all output into a netCDF file, a format widely used for storing
scientific data in the form of arrays. Figure III.1 shows the list of variables stored in the
output file. “s_Er” and “s_Ei” are the converged complex field solution obtained from the
full wave solver, they have two dimensions in both radial (R), with fine mesh, and poloidal
(Z) direction. Here, “r” designates the real part and “i” represents the imaginary part of the
field solution. “p_Er” and “p_Ei” are the complex field obtained from the paraxial solver.
The radial grid of the paraxial solver is much coarser than that of the full wave solver,
though they have the same dimensions of mesh in the Z direction. Both incident and
reflected field are saved for the paraxial solution; thus, “p_Er” and “p_Ei” have one more
dimension than the full wave data indicating the direction (ingoing or outgoing) of the field.
“a_Er” and “a_Ei” are the incident and reflected field on the antenna/receiver/detector
plane. Since the free-space solver uses the Green’s function for projection, it thus only has
two field solutions on the first plane and last plane on the propagation path. The three
dimensions for “a_Er” and “a_Ei” represent the direction of propagation, the polodial and
toroidal directions. Since there is no information in the toroidal direction for FWR2D
simulations, there will only be valuable values on the axis of Z = Z0 for both “a_Er” and
“a_Ei”. In addition to the field information, other parameters, for example, the dielectric
constant (“s_epsi” and “s_epsr”), and coordinate information (variables with “_x” and “_y”)
are also saved. Some parameters in the list are not used; for example, “ep_epsr” and
“ep_epsi”.
211
Figure III.1: Output variables stored from one FWR2D simulation.
212
Appendix IV
A different approach of synthetic MIR
In addition to the synthetic MIR diagnostic approach discussed in this dissertation,
which simulates the actual optical setup, an alternative synthetic diagnostic approach that
numerically retraces the measured reflected signal on the detector plane up to the image
plane is discussed in [193]. It calculates the electric field on the cutoff surface right after
the reflection by adding a phase term (& − ( ) (* − ,*
to the fast Fourier
transformation (FFT) of the electric field on the detector plane, as the formula shows below:
E(x c ) = F −1 (F(E(x 0 ))e
!
(i( x c −x0 ) k02 −K 2y )
) (IV.1)
Here, E(x c ) is the electric field on the cutoff surface, while E(x0 ) is the electric
!
!
field detected on the detector plane. In addition, k0 is the free space probing wave
!
wavenumber, while K y is the poloidal rotating fluctuation wave wavenumber on the
!
cutoff surface. This approach would be valid under the assumption that the fluctuation
frequency is much smaller than the free space probing wave frequency, and which is
satisfied generally for microwave reflectometry, since plasma turbulence is usually in kHz
~ MHz range, while the probing wave is in the sub-THz range. The added phase term is
intended to remove the phase mixing that occurs as a result of propagation away from the
2
2
cutoff layer, and when this term equals ( x c −x0 ) k0 −K y , the amplitude fluctuation would
!
be minimized, indicating a numerical propagation to the image plane. However, since K y
!
is not known ahead of the measurement results, simple guesses for the added phase term
213
would be tried to reveal the point when the amplitude fluctuation would be minimized, and
this may not be very efficient for fast measurements. Figure IV.1 shows a proof-ofprinciple example of the numerical synthetic imaging diagnostic from [193]. It shows
several distributions of I/Q plots. At the detector plane, the phase (along the red curve) and
amplitude (perpendicular to the red curve) are mixed, and no density perturbation
information from the phase perturbation can be revealed. Depending on the characteristic
time ratio, f, which is the ratio of the edge de-correlation time to turbulence de-correlation
time, an image from a single detector can be formed where the amplitude fluctuations are
greatly reduced. When f is much larger than 1, it means that the eddy has enough time to
move past the region illuminated by the beam, and f should be large enough to have the
synthetic imaging work properly.
214
Figure IV.1: Contour plots of the distribution of the real and imaginary electric field
amplitudes. At the detector plane, the phase (along the red curve) and amplitude
(perpendicular to the red curve) are mixed. Depending on the characteristic time ratio, f,
an image from a single detector can be formed where the amplitude fluctuations are greatly
reduced [193].
Compared to the optical imaging, this technique has the advantage of removing the
limited aperture introduced effects, optical aberrations as well as lens design and
configuration. However, there are some critical drawbacks for practically making usage of
this approach. Firstly, it requires great fidelity of the receiver antennas meaning highresolution antenna arrays, which is hard to be satisfied. In addition, since the fluctuation
wavenumber is difficult to know or estimate a priori and could be frequently varying with
time, this could cause great difficulties in determining the right phase term to be added.
Sometimes, the density fluctuations would have strong subtleties, say non-sinusoidal
structures and other irregular structures; in addition, broadband fluctuations with wide
wavenumber distributions in both co and counter rotation directions would be present.
These could cause great difficulty for the wide application of this numerical approach.
However, for the optical-based reflectometry system, its imaging capability doesn’t depend
on the condition of density perturbation, with both broadband and coherent fluctuations
able to be accurately measured; even though there is an upper limit for the maximum
accessible wavenumber. The imaging quality of MIR and its dependence on choice of
optics will be discussed in the following. In addition, the MIR approach with the usage of
215
optical lens is relatively straightforward and easy for implementation, which could also
been easily designed with great flexibility to fulfill different diagnostic purposes.
216
Appendix V
Phase Jumps and DC Drifts
V.1 Phase Jumps
When the fluctuation amplitude is small compared to the wavelength of the probing
wave in free space, the maximum phase change is from -π to π; consequently, the measured
phase fluctuation can easily match well with the density fluctuation within this range. Here,
the measured phase fluctuation is obtained by calculating the phase of each complex I/Q
value from the I/Q plots:
φ = phase(IQt )
!t
!t >0
(V.1)
The mathematically achievable phase range is from -π to π. As the fluctuation
amplitude increases and becomes larger than the free space wavelength of probing wave,
the density perturbation induced phase perturbation will exceed the range of 2*π; therefore,
phase jumps would occur. One example of the phase jump is shown in Fig. 4.8(a). On this
figure, the blue curve is the measured phase fluctuation, and the red curve is the density
fluctuation represented converted into radian units by using the following formula:
Δx(t )
φ(t ) = 2π
λ
!
(V.2)
Here, !φ(t ) is the converted time dependent phase in radians, !Δx(t ) is the time
dependent density perturbation amplitude, and it is measured at one fixed poloidal location.
λ is the free space wavelength of probing wave. It can be seen that the density fluctuation
has an amplitude much larger than 2* π, while the phase fluctuations are terminated at π or
- π and then restart from – π or π, causing phase jumps. The calculated distance correlation
217
for the two curves is 0.2, and the Pearson correlation is 0.02. This great mismatch between
the phase fluctuation and density fluctuation is mainly due to phase jumps in the measured
signal and could be eliminated by the following method.
As shown in Fig. 4.8 (b), using the same data as in Fig. 4.8 (a), but just by adding
or subtracting 2*π from the jumped position, a much smoother phase curve is obtained,
and the distance correlation is improved from 0.2 to 0.92, meaning that the two curves are
highly correlated. The wavenumber distribution of the fluctuation used in this figure is a
highly coherent fluctuation with a Gaussian distribution centered at 0.2 cm-1, with a
deviation of 0.1 cm-1, and fluctuation amplitude of 0.8 %. However, this method has a
strong limitation when applied to a wider range of fluctuation parameters, especially when
the central wavenumber increases. When the fluctuation wavenumber or fluctuation
amplitude increases, the correlation value between the phase perturbation and density
fluctuation will decrease, and most of the critical phase jumps around 2*π will be replaced
by a number of random phase jumps more than π but less than 2*π. By simply adding or
subtracting 2*π to the positions when phase jump happens, the correlation values will not
increase significantly. Thus, to compensate for this problem, a different approach for
calculating the measured phase perturbation from the I/Q plots is applied, as discussed in
the following section (Sec. 4.3.2).
218
Figure V.1: Shown in (a) is the limitation of the direct comparison of phase fluctuations
(in blue) to density fluctuations (in red), and a low correlation value is obtained between
the two curves. Shown in (b) is the rearranged curve of phase fluctuations (in blue) to
density fluctuations (in red), and an increased correlation value is obtained.
V.2 DC Drifts
Since phase jumps would cause severe problems for analyzing fluctuations with
fluctuation amplitudes greater than λ (less than 1 cm), while for real perturbations, the
fluctuation amplitude could easily exceed this range, and can be several free space
wavelengths of the probing wave, a different approach of calculating the accumulative
phase perturbation is introduced here to eliminate this limitation, and make the synthetic
MIR method applicable to a full range of fluctuation amplitudes and wavenumbers. The
accumulative phase perturbation of the measured signal in a time sequence is calculated
from the following formula:
IQ
φt+1 = phase( t+1 )+ φt
IQt
!
219
(V.3)
Therefore, the phase perturbation at time
!t +1 is the phase difference from time
!t to !t +1 , plus the already known phase perturbation at time !t . Since the phase difference
from
!t to !t +1 can be controlled to be small with properly chosen sampling frequency,
the phase jumps can be eliminated by using this method. This method can only be used for
time dependent fluctuations, meaning the fluctuation is correlated in the time domain. If
the fluctuations for each time slice are independent or random, this method will fail.
However, this method could introduce a new issue with the measured signal, which is the
DC drift. On Fig. 4.9, shown in the blue curve is the fluctuation signal that is drifting away
from the original position. A typical method for dealing with the drift is using a Hamming
window to exclude the drift component. The function of the Hamming window [4, 5 ] with
n points in the window is as follows:
2π n
w(n) = 0.54 − 0.46cos(
) !0 ≤ n ≤ M −1
M
−1
!
(V.4)
The Hamming window calculates the weight of each point in this window, with the
maximum weight being 1.0, and it is then normalized before being applied to the measured
phase signal by the following formula:
wnorm(n) =
!
w(n)
∑ i=1 w(i)
n
(V.5)
Then, the drift components at each time t is:
DC = ∑ i=t−n/2 wnorm(i)φ(i) (V.6
! t
t+n/2
220
On Fig. 4.9, the red curve shows the unperturbed drifting components calculated
from the above formula by choosing the Hamming window size 10 % (n=100) of the total
sampled points (T=1000). The DC component will be abstracted before calculation of
correlation values.
Figure V.2: Example of using a Hamming window for smoothing; the window size n is
10% of the total sampling points used in this case.
221
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